Formalization: Estimating Optimal Treatment Value with Many Covariate Cells and Shrinking Overlap
The complete Lean development behind this paper — every definition, lemma, and theorem of its module, including helpers the paper text never cites. Identifiers link within this page, into the Causalean library, or out to the official Mathlib docs. Note: this index was generated at a different commit than the paper's audited pin (15d86b3) — statements may have moved since the audit.
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Basic 37 declarations The finite PMF abstraction retains arbitrary and null covariate cells.
Vanishing-overlap optimal value: finite observed and causal models
The finite PMF abstraction retains arbitrary and null covariate cells. The measure-theoretic PO and MinimaxATE systems use a different model, so only their generic minimax risk primitives are imported.
For the displayed parameters, Obs is the object specified by this definition.
For the displayed parameters, DiscreteLaw is the object specified by this definition.
Definition (Lean source)
def jointMass
For the displayed parameters, jointMass is the object specified by this definition.
Definition (Lean source)
def cellIdx
For the displayed parameters, cellIdx is the object specified by this definition.
def cellVector
For the displayed parameters, cellVector is the object specified by this definition.
Definition (Lean source)
parameters
def
given by
fun j => jointMass P x (j.val / 2 = 1) (j.val % 2 = 1)
def cellMass
For the displayed parameters, cellMass is the object specified by this definition.
Definition (Lean source)
parameters
def
cellMass P x :
ℝ
def armMass
For the displayed parameters, armMass is the object specified by this definition.
Definition (Lean source)
def propensity
For the displayed parameters, propensity is the object specified by this definition.
Definition (Lean source)
parameters
def
propensity P x :
ℝ
def outcomeMean
For the displayed parameters, outcomeMean is the object specified by this definition.
Definition (Lean source)
parameters
def
outcomeMean P a x :
ℝ
def productLaw
For the displayed parameters, productLaw is the object specified by this definition.
Definition (Lean source)
parameters
given by
Measure.pi (fun _ : Fin n => P.pmf.toMeasure)
def FullObs
For the displayed parameters, FullObs is the object specified by this definition.
For the displayed parameters, PotentialLaw is the object specified by this definition.
def fullMass
For the displayed parameters, fullMass is the object specified by this definition.
Definition (Lean source)
def observedMarginal
For the displayed parameters, observedMarginal is the object specified by this definition.
Definition (Lean source)
parameters
d :
ℕ
Q :
def
given by
pmf := Q.pmf.map (fun z => (z.1, z.2.1, z.2.2.1))
def poCellMass
For the displayed parameters, poCellMass is the object specified by this definition.
def poAtom
For the displayed parameters, poAtom is the object specified by this definition.
Definition (Lean source)
parameters
def
poAtom Q x a y0 y1 :
ℝ
def poArmAtom
For the displayed parameters, poArmAtom is the object specified by this definition.
def poTreatMass
For the displayed parameters, poTreatMass is the object specified by this definition.
def poPotentialMass
For the displayed parameters, poPotentialMass is the object specified by this definition.
def poRegression
For the displayed parameters, poRegression is the object specified by this definition.
Definition (Lean source)
parameters
def
poRegression Q r x :
ℝ
def oracleValue
For the displayed parameters, oracleValue is the object specified by this definition.
Definition (Lean source)
parameters
d :
ℕ
Q :
def
oracleValue Q :
ℝ
given by
def IidSampling
For the displayed parameters, IidSampling is the object specified by this definition.
Definition (Lean source)
parameters
def
IidSampling P μn :
Prop
given by
μn = productLaw P n
def Overlap
For the displayed parameters, Overlap is the object specified by this definition.
Definition (Lean source)
parameters
def
Overlap ε P :
Prop
given by
clause 1
ε ≤ propensity P x
clause 2
propensity P x ≤ 1 - ε
def Consistency
For the displayed parameters, Consistency is the object specified by this definition.
Definition (Lean source)
parameters
def
Consistency Q :
Prop
given by
Q.pmf z = 0
def ConditionalExchangeability
For the displayed parameters, ConditionalExchangeability is the object specified by this definition.
Definition (Lean source)
parameters
def
ConditionalExchangeability Q :
Prop
given by
structure ObservedClass
For the displayed parameters, ObservedClass is the object specified by this definition.
Definition (Lean source)
structure CausalClass
For the displayed parameters, CausalClass is the object specified by this definition.
Definition (Lean source)
parameters
assumes
def observedValue
For the displayed parameters, observedValue is the object specified by this definition.
Definition (Lean source)
parameters
d :
ℕ
P :
def
observedValue P :
ℝ
given by
For the displayed parameters, Estimator is the object specified by this definition.
Definition (Lean source)
For the displayed parameters, ModelLaw is the object specified by this definition.
Definition (Lean source)
def CausalModelLaw
For the displayed parameters, CausalModelLaw is the object specified by this definition.
Definition (Lean source)
parameters
d :
ℕ
ε :
ℝ
def
CausalModelLaw d ε :
Type
given by
{Q : PotentialLaw d // CausalClass ε Q}
For the displayed parameters, observedRisk is the object specified by this definition.
For the displayed parameters, minimaxRisk is the object specified by this definition.
Definition (Lean source)
parameters
n d :
ℕ
ε :
ℝ
def
minimaxRisk n d ε :
ℝ
given by
minimaxValueReal (observedRisk n (d := d) (ε := ε))
def causalRisk
For the displayed parameters, causalRisk is the object specified by this definition.
Definition (Lean source)
parameters
def
causalRisk n est Q :
ℝ
given by
For the displayed parameters, causalMinimaxRisk is the object specified by this definition.
Definition (Lean source)
parameters
n d :
ℕ
ε :
ℝ
def
causalMinimaxRisk n d ε :
ℝ
given by
minimaxValueReal (causalRisk n (d := d) (ε := ε))
For the displayed parameters, logAlphabet is the object specified by this definition.
For the displayed parameters, rateScale is the object specified by this definition.
Definition (Lean source)
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.ArmwiseExtension 20 declarations Armwise anchored four-cell extension
Armwise anchored four-cell extension
def armMassVecFormula
For the displayed parameters, armMassVecFormula is the object specified by this definition. For the displayed parameters, armMassVecFormula is the defined object.
def totalMassVecFormula
For the displayed parameters, totalMassVecFormula is the object specified by this definition. For the displayed parameters, totalMassVecFormula is the defined object.
Definition (Lean source)
def anchoredDenomFormula
For the displayed parameters, anchoredDenomFormula is the object specified by this definition. For the displayed parameters, anchoredDenomFormula is the defined object.
Definition (Lean source)
parameters
def
anchoredDenomFormula ε u a :
ℝ
given by
def armDelta
For the displayed parameters, armDelta is the object specified by this definition. For the displayed parameters, armDelta is the defined object.
def armwiseExtensionFormula
Ambient formula used by the analytic API; its paper domain is pinned below.
Definition (Lean source)
parameters
ε :
ℝ
u :
Fin 4 → ℝ
def
armwiseExtensionFormula ε u :
ℝ
given by
if u
= 0 then 0 else max (totalMassVecFormula u * u (cellIdx false true) / anchoredDenomFormula ε u false) (totalMassVecFormula u * u (cellIdx true true) / anchoredDenomFormula ε u true)
def NonnegativeFourVector
The coordinatewise nonnegative four-vector domain of the armwise construction.
Definition (Lean source)
def armMassVec
Arm mass on the nonnegative four-vector domain.
Definition (Lean source)
parameters
def
armMassVec u _hu a :
ℝ
def totalMassVec
Total mass on the nonnegative four-vector domain.
Definition (Lean source)
def anchoredDenom
Anchored denominator on the nonnegative four-vector domain.
Definition (Lean source)
parameters
def
anchoredDenom ε u hu a :
ℝ
given by
For the displayed parameters, armwiseExtension is the object specified by this definition. For the displayed parameters, armwiseExtension is the defined object.
Definition (Lean source)
parameters
def
armwiseExtension ε u hu :
ℝ
given by
if u
= 0 then 0 else max (totalMassVec u hu * u (cellIdx false true) / anchoredDenom ε u hu false) (totalMassVec u hu * u (cellIdx true true) / anchoredDenom ε u hu true)
theorem cellVector_nonnegative
Observed atom masses supply the construction's domain without an extra assumption. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
Proof (Lean source)
lemma cellVector_nonnegative {d : ℕ} (P : DiscreteLaw d) (x : Fin d) :
NonnegativeFourVector (cellVector P x) := by
intro j
exact toReal_nonneg
In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma sqrt_pair_le_sqrt_twice_sum {x y : ℝ} (hx : 0 ≤ x) (hy : 0 ≤ y) :
sqrt x + sqrt y ≤ sqrt (2 * (x + y)) := by
have hsq : (sqrt x + sqrt y) ^ 2 ≤ 2 * (x + y) := by
nlinarith [sq_nonneg (sqrt x - sqrt y),
Real.sq_sqrt hx, Real.sq_sqrt hy]
nlinarith [Real.sq_sqrt (show 0 ≤ 2 * (x + y) by positivity),
Real.sqrt_nonneg (2 * (x + y)), Real.sqrt_nonneg x, Real.sqrt_nonneg y]
In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma weighted_sqrt_pair_bound {p s q₀ q₁ ε : ℝ}
(hp : 0 < p) (hε : 0 < ε) (hse : ε * p ≤ s)
(hq₀ : 0 ≤ q₀) (hq₁ : 0 ≤ q₁) (hsum : q₀ + q₁ = s) :
p / s * (sqrt q₀ + sqrt q₁) ≤ sqrt (2 * p / ε) := by
have hs : 0 < s := lt_of_lt_of_le (mul_pos hε hp) hse
have hA := sqrt_pair_le_sqrt_twice_sum hq₀ hq₁
rw [hsum] at hA
have hA0 : 0 ≤ sqrt q₀ + sqrt q₁ := by positivity
have hB0 : 0 ≤ sqrt (2 * p / ε) := Real.sqrt_nonneg _
have hAsq : (sqrt q₀ + sqrt q₁) ^ 2 ≤ 2 * s := by
nlinarith [Real.sq_sqrt (show 0 ≤ 2 * s by positivity)]
have hBsq : (sqrt (2 * p / ε)) ^ 2 = 2 * p / ε :=
Real.sq_sqrt (by positivity)
have h1 : 0 ≤ p ^ 2 * (2 * s - (sqrt q₀ + sqrt q₁) ^ 2) :=
mul_nonneg (sq_nonneg _) (sub_nonneg.mpr hAsq)
calc
p / s * (sqrt q₀ + sqrt q₁) =
(p * (sqrt q₀ + sqrt q₁)) / s := by ring
_ ≤ sqrt (2 * p / ε) := by
apply (div_le_iff₀ hs).2
have htarget : (p * (sqrt q₀ + sqrt q₁)) ^ 2 ≤
(sqrt (2 * p / ε) * s) ^ 2 := by
calc
_ ≤ 2 * p ^ 2 * s := by nlinarith [h1]
_ ≤ (2 * p / ε) * s ^ 2 := by
calc
_ = (2 * p * s / ε) * (ε * p) := by field_simp
_ ≤ (2 * p * s / ε) * s :=
mul_le_mul_of_nonneg_left hse (by positivity)
_ = _ := by ring
_ = _ := by rw [mul_pow, hBsq]
nlinarith [mul_nonneg (le_of_lt hp) hA0, mul_nonneg hB0 (le_of_lt hs)]
theorem observed_armwise_sqrt_bound
In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Proof (Lean source)
lemma observed_armwise_sqrt_bound (d : ℕ) (ε : ℝ) (P : DiscreteLaw d)
(hε : 0 < ε) (hP : ObservedClass ε P) (x : Fin d)
(hp : 0 < cellMass P x) :
(∑ a : Bool, cellMass P x / armMass P a x *
(∑ y : Bool, sqrt (jointMass P x a y))) ≤
2 * sqrt 2 * sqrt (cellMass P x / ε) := by
have hq (a y : Bool) : 0 ≤ jointMass P x a y := ENNReal.toReal_nonneg
have hcell : cellMass P x = armMass P false x + armMass P true x := by
simp [cellMass, armMass]
ring
have htrue : cellMass P x * propensity P x = armMass P true x := by
rw [propensity]
field_simp [ne_of_gt hp]
have hfalse : cellMass P x * (1 - propensity P x) = armMass P false x := by
rw [mul_sub, mul_one, htrue, hcell]
ring
obtain ⟨hlo, hhi⟩ := hP.overlap x hp
have hfalsefloor : ε * cellMass P x ≤ armMass P false x := by
have h : ε ≤ 1 - propensity P x := by linarith
calc
ε * cellMass P x = cellMass P x * ε := by ring
_ ≤ cellMass P x * (1 - propensity P x) :=
mul_le_mul_of_nonneg_left h (le_of_lt hp)
_ = armMass P false x := hfalse
have htruefloor : ε * cellMass P x ≤ armMass P true x := by
rw [← htrue]
nlinarith [mul_nonneg (le_of_lt hp) (sub_nonneg.mpr hlo)]
have hb (a : Bool) :
cellMass P x / armMass P a x *
(∑ y : Bool, sqrt (jointMass P x a y)) ≤
sqrt (2 * cellMass P x / ε) := by
cases a with
| false =>
simpa [armMass, Fintype.sum_bool, add_comm] using
(weighted_sqrt_pair_bound hp hε hfalsefloor
(hq false false) (hq false true)
(by simp [armMass, add_comm]))
| true =>
simpa [armMass, Fintype.sum_bool, add_comm] using
(weighted_sqrt_pair_bound hp hε htruefloor
(hq true false) (hq true true)
(by simp [armMass, add_comm]))
have hsum : (∑ a : Bool, cellMass P x / armMass P a x *
(∑ y : Bool, sqrt (jointMass P x a y))) ≤
2 * sqrt (2 * cellMass P x / ε) := by
simpa [Fintype.sum_bool, two_mul] using add_le_add (hb true) (hb false)
calc
_ ≤ 2 * sqrt (2 * cellMass P x / ε) := hsum
_ = 2 * sqrt 2 * sqrt (cellMass P x / ε) := by
rw [show 2 * cellMass P x / ε = 2 * (cellMass P x / ε) by ring,
Real.sqrt_mul (by norm_num)]
ring
In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
r t z c :
ℝ
hr :
0 < r
ht :
0 < t
hz :
0 ≤ z
hzr :
z ≤ r
_hc :
0 ≤ c
_hct :
c ≤ t
conclusion
|z / r - c / t| ≤ (|z - c| + |r - t|) / t
Proof (Lean source)
lemma armwise_ratio_bound {r t z c : ℝ} (hr : 0 < r) (ht : 0 < t)
(hz : 0 ≤ z) (hzr : z ≤ r) (_hc : 0 ≤ c) (_hct : c ≤ t) :
|z / r - c / t| ≤ (|z - c| + |r - t|) / t := by
have hrt : 0 < r * t := mul_pos hr ht
have hmain : |t * z - r * c| ≤ r * (|z - c| + |r - t|) := by
apply abs_le.mpr
rcases le_total r t with h | h
· rw [abs_of_nonpos (sub_nonpos.mpr h)]
constructor <;> nlinarith [le_abs_self (z - c), neg_le_abs (z - c),
mul_nonneg (sub_nonneg.mpr h) hz,
mul_nonneg (sub_nonneg.mpr h) (sub_nonneg.mpr hzr)]
· rw [abs_of_nonneg (sub_nonneg.mpr h)]
constructor <;> nlinarith [le_abs_self (z - c), neg_le_abs (z - c),
mul_nonneg (sub_nonneg.mpr h) _hc,
mul_nonneg (sub_nonneg.mpr h) (sub_nonneg.mpr _hct)]
have halg : z / r - c / t = (t * z - r * c) / (r * t) := by
field_simp
rw [halg, abs_div, abs_of_pos hrt]
apply (div_le_div_iff₀ hrt ht).2
nlinarith [hmain]
In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
su sv zu zv ru rv δ δa :
ℝ
hru :
0 < ru
hrv :
0 < rv
hzu :
0 ≤ zu
hzru :
zu ≤ ru
hzv :
0 ≤ zv
hzrv :
zv ≤ rv
hsv :
0 ≤ sv
hδ :
|su - sv| ≤ δ
hδa :
|zu - zv| ≤ δa
conclusion
|su * zu / ru - sv * zv / rv| ≤ δ + sv / rv * (δa + |ru - rv|)
Proof (Lean source)
lemma armwise_coordinate_ratio_bound {su sv zu zv ru rv δ δa : ℝ}
(hru : 0 < ru) (hrv : 0 < rv) (hzu : 0 ≤ zu) (hzru : zu ≤ ru)
(hzv : 0 ≤ zv) (hzrv : zv ≤ rv) (hsv : 0 ≤ sv)
(hδ : |su - sv| ≤ δ) (hδa : |zu - zv| ≤ δa) :
|su * zu / ru - sv * zv / rv| ≤
δ + sv / rv * (δa + |ru - rv|) := by
have hquot : 0 ≤ zu / ru := div_nonneg hzu (le_of_lt hru)
have hquot1 : zu / ru ≤ 1 := (div_le_one hru).2 hzru
have hr := armwise_ratio_bound hru hrv hzu hzru hzv hzrv
have hid : su * zu / ru - sv * zv / rv =
(su - sv) * (zu / ru) + sv * (zu / ru - zv / rv) := by ring
rw [hid]
calc
_ ≤ |su - sv| * (zu / ru) + sv * |zu / ru - zv / rv| := by
simpa [abs_mul, abs_of_nonneg hquot, abs_of_nonneg hsv] using
(abs_add_le ((su - sv) * (zu / ru)) (sv * (zu / ru - zv / rv)))
_ ≤ δ + sv / rv * (δa + |ru - rv|) := by
have h1 : |su - sv| * (zu / ru) ≤ δ := by
calc
_ ≤ δ * (zu / ru) := mul_le_mul_of_nonneg_right hδ hquot
_ ≤ δ * 1 := mul_le_mul_of_nonneg_left hquot1
(le_trans (abs_nonneg _) hδ)
_ = δ := mul_one _
have h2 : sv * |zu / ru - zv / rv| ≤
sv / rv * (|zu - zv| + |ru - rv|) := by
simpa [div_eq_mul_inv, mul_assoc, mul_left_comm, mul_comm] using
(mul_le_mul_of_nonneg_left hr hsv)
have h3 := mul_le_mul_of_nonneg_left hδa (show 0 ≤ sv / rv by positivity)
nlinarith
In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma totalMassVec_pos_of_nonzero {w : Fin 4 → ℝ}
(hw : ∀ j, 0 ≤ w j) (hne : w ≠ 0) : 0 < totalMassVecFormula w := by
have h0 := hw (cellIdx false false)
have h1 := hw (cellIdx false true)
have h2 := hw (cellIdx true false)
have h3 := hw (cellIdx true true)
have hsum : 0 ≤ totalMassVecFormula w := by
dsimp [totalMassVecFormula, armMassVecFormula]
positivity
rcases eq_or_lt_of_le hsum with hz | hz
· exfalso
apply hne
funext j
fin_cases j <;> simp [totalMassVecFormula, armMassVecFormula, cellIdx] at hz h0 h1 h2 h3 ⊢ <;>
linarith
· exact hz
theorem armwiseExtension_nonneg_le_totalMassVec Lemma armwiseExtension_nonneg_le_totalMassVec in the paper ↗
In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Proof (Lean source)
lemma armwiseExtension_nonneg_le_totalMassVec {ε : ℝ} (hε : 0 < ε)
{w : Fin 4 → ℝ} (hw : ∀ j, 0 ≤ w j) :
0 ≤ armwiseExtensionFormula ε w ∧ armwiseExtensionFormula ε w ≤ totalMassVecFormula w := by
by_cases hzero : w = 0
· subst w
simp [armwiseExtensionFormula, totalMassVecFormula, armMassVecFormula]
· have hs := totalMassVec_pos_of_nonzero hw hzero
have ha (a : Bool) :
0 ≤ totalMassVecFormula w * w (cellIdx a true) / anchoredDenomFormula ε w a ∧
totalMassVecFormula w * w (cellIdx a true) / anchoredDenomFormula ε w a ≤
totalMassVecFormula w := by
have hd : 0 < anchoredDenomFormula ε w a := by
unfold anchoredDenomFormula
exact lt_of_lt_of_le (mul_pos hε hs) (le_max_right _ _)
have hwa := hw (cellIdx a false)
have hwb := hw (cellIdx a true)
have hle : w (cellIdx a true) ≤ anchoredDenomFormula ε w a := by
unfold anchoredDenomFormula armMassVecFormula
exact le_trans (by linarith) (le_max_left _ _)
constructor
· positivity
· apply (div_le_iff₀ hd).2
exact mul_le_mul_of_nonneg_left hle (le_of_lt hs)
simp only [armwiseExtensionFormula, if_neg hzero]
exact ⟨le_max_of_le_left (ha false).1,
max_le (ha false).2 (ha true).2⟩
theorem armwise_modulus_extension
Analytic extension of the modulus; the paper realization below restricts its domain. In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
Proof (Lean source)
lemma armwise_modulus_extension :
(∀ ε : ℝ, 0 < ε → ∀ u v : Fin 4 → ℝ,
(∀ j, 0 ≤ u j) → (∀ j, 0 ≤ v j) → v ≠ 0 →
|armwiseExtensionFormula ε u - armwiseExtensionFormula ε v| ≤
2 * (∑ a : Bool, armDelta u v a) +
2 * (∑ a : Bool, totalMassVecFormula v / anchoredDenomFormula ε v a * armDelta u v a)) ∧
(∀ (d : ℕ) (ε : ℝ) (P : DiscreteLaw d),
0 < ε → ObservedClass ε P → ∀ x : Fin d, 0 < cellMass P x →
(∑ a : Bool, cellMass P x / armMass P a x *
(∑ y : Bool, sqrt (jointMass P x a y))) ≤
2 * sqrt 2 * sqrt (cellMass P x / ε)) := by
constructor
· intro ε hε u v hu hv hv0
let δ : ℝ := ∑ a : Bool, armDelta u v a
have hδa (a : Bool) : 0 ≤ armDelta u v a := by
simp only [armDelta, Fintype.sum_bool]
positivity
have hδ : 0 ≤ δ := Finset.sum_nonneg (fun a _ => hδa a)
have hMass (a : Bool) :
|armMassVecFormula u a - armMassVecFormula v a| ≤ armDelta u v a := by
simpa [armMassVecFormula, armDelta, Fintype.sum_bool, sub_add_sub_comm, add_comm] using
(abs_add_le (u (cellIdx a false) - v (cellIdx a false))
(u (cellIdx a true) - v (cellIdx a true)))
have hTotal : |totalMassVecFormula u - totalMassVecFormula v| ≤ δ := by
have h := abs_add_le
(armMassVecFormula u false - armMassVecFormula v false)
(armMassVecFormula u true - armMassVecFormula v true)
have hf := hMass false
have ht := hMass true
simpa [totalMassVecFormula, δ, Fintype.sum_bool, sub_add_sub_comm, add_comm] using
(le_trans h (add_le_add hf ht))
have hsv : 0 < totalMassVecFormula v := totalMassVec_pos_of_nonzero hv hv0
have hdv (a : Bool) : 0 < anchoredDenomFormula ε v a := by
unfold anchoredDenomFormula
exact lt_of_lt_of_le (mul_pos hε hsv) (le_max_right _ _)
have hw (a : Bool) : 0 ≤ totalMassVecFormula v / anchoredDenomFormula ε v a := by
exact div_nonneg (le_of_lt hsv) (le_of_lt (hdv a))
have hdenom (a : Bool) :
|anchoredDenomFormula ε u a - anchoredDenomFormula ε v a| ≤
armDelta u v a + ε * δ := by
have hmax := abs_max_sub_max_le_max
(armMassVecFormula u a) (ε * totalMassVecFormula u)
(armMassVecFormula v a) (ε * totalMassVecFormula v)
have hscaled : |ε * totalMassVecFormula u - ε * totalMassVecFormula v| ≤ ε * δ := by
rw [← mul_sub, abs_mul, abs_of_pos hε]
exact mul_le_mul_of_nonneg_left hTotal (le_of_lt hε)
have hmaxbound :
max |armMassVecFormula u a - armMassVecFormula v a|
|ε * totalMassVecFormula u - ε * totalMassVecFormula v| ≤
armDelta u v a + ε * δ := by
apply max_le
· linarith [hMass a, mul_nonneg (le_of_lt hε) hδ]
· linarith [hδa a]
exact le_trans (by simpa [anchoredDenomFormula] using hmax) hmaxbound
by_cases hu0 : u = 0
· have hFv := armwiseExtension_nonneg_le_totalMassVec hε hv
have htot : totalMassVecFormula v ≤ δ := by
have h := hTotal
have hzero : totalMassVecFormula (0 : Fin 4 → ℝ) = 0 := by
simp [totalMassVecFormula, armMassVecFormula]
rw [hu0, hzero, zero_sub, abs_neg, abs_of_nonneg (le_of_lt hsv)] at h
exact h
have hweighted : 0 ≤ ∑ a : Bool,
totalMassVecFormula v / anchoredDenomFormula ε v a * armDelta u v a :=
Finset.sum_nonneg (fun a _ => mul_nonneg (hw a) (hδa a))
rw [show armwiseExtensionFormula ε u = 0 by simp [armwiseExtensionFormula, hu0],
zero_sub, abs_neg, abs_of_nonneg hFv.1]
change armwiseExtensionFormula ε v ≤
2 * δ + 2 * (∑ a : Bool,
totalMassVecFormula v / anchoredDenomFormula ε v a * armDelta u v a)
linarith
· have hsu := totalMassVec_pos_of_nonzero hu hu0
have hdu (a : Bool) : 0 < anchoredDenomFormula ε u a := by
unfold anchoredDenomFormula
exact lt_of_lt_of_le (mul_pos hε hsu) (le_max_right _ _)
have harm (a : Bool) :
|totalMassVecFormula u * u (cellIdx a true) / anchoredDenomFormula ε u a -
totalMassVecFormula v * v (cellIdx a true) / anchoredDenomFormula ε v a| ≤
2 * δ + 2 * (totalMassVecFormula v / anchoredDenomFormula ε v a) * armDelta u v a := by
have hzu := hu (cellIdx a true)
have hzv := hv (cellIdx a true)
have hzu_le : u (cellIdx a true) ≤ anchoredDenomFormula ε u a := by
exact le_trans (by dsimp [armMassVecFormula]; linarith [hu (cellIdx a false)])
(le_max_left _ _)
have hzv_le : v (cellIdx a true) ≤ anchoredDenomFormula ε v a := by
exact le_trans (by dsimp [armMassVecFormula]; linarith [hv (cellIdx a false)])
(le_max_left _ _)
have hcoord : |u (cellIdx a true) - v (cellIdx a true)| ≤ armDelta u v a := by
simp only [armDelta, Fintype.sum_bool]
exact le_add_of_nonneg_right (abs_nonneg _)
have hbase := armwise_coordinate_ratio_bound (hdu a) (hdv a)
hzu hzu_le hzv hzv_le (le_of_lt hsv) hTotal hcoord
have he : (totalMassVecFormula v / anchoredDenomFormula ε v a) * ε ≤ 1 := by
have he' : ε * totalMassVecFormula v ≤ anchoredDenomFormula ε v a :=
le_max_right _ _
calc
_ = (ε * totalMassVecFormula v) / anchoredDenomFormula ε v a := by ring
_ ≤ 1 := (div_le_one (hdv a)).2 he'
have hmul := mul_le_mul_of_nonneg_left (hdenom a) (hw a)
nlinarith [mul_nonneg (sub_nonneg.mpr he) hδ,
mul_nonneg (hw a) (hδa a)]
have hmax := abs_max_sub_max_le_max
(totalMassVecFormula u * u (cellIdx false true) / anchoredDenomFormula ε u false)
(totalMassVecFormula u * u (cellIdx true true) / anchoredDenomFormula ε u true)
(totalMassVecFormula v * v (cellIdx false true) / anchoredDenomFormula ε v false)
(totalMassVecFormula v * v (cellIdx true true) / anchoredDenomFormula ε v true)
have hsum :
max (2 * δ + 2 * (totalMassVecFormula v / anchoredDenomFormula ε v false) *
armDelta u v false)
(2 * δ + 2 * (totalMassVecFormula v / anchoredDenomFormula ε v true) *
armDelta u v true) ≤
2 * δ + 2 * (∑ a : Bool,
totalMassVecFormula v / anchoredDenomFormula ε v a * armDelta u v a) := by
simp only [Fintype.sum_bool]
apply max_le <;> nlinarith [mul_nonneg (hw false) (hδa false),
mul_nonneg (hw true) (hδa true)]
simpa [armwiseExtensionFormula, hu0, hv0, δ] using
(le_trans hmax (le_trans (max_le_max (harm false) (harm true)) hsum))
· intro d ε P hε hP x hp
exact observed_armwise_sqrt_bound d ε P hε hP x hp
theorem armwise_modulus
In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
conclusion 1
ε :
ℝ
0 < ε
ε ≤ 1 / 2
u v :
Fin 4 → ℝ
(hu : NonnegativeFourVector u)
(hv : NonnegativeFourVector v)
v ≠ 0
|armwiseExtension ε u hu - armwiseExtension ε v hv|
conclusion 2
Proof (Lean source)
lemma armwise_modulus :
(∀ ε : ℝ, 0 < ε → ε ≤ 1 / 2 → ∀ u v : Fin 4 → ℝ,
(hu : NonnegativeFourVector u) → (hv : NonnegativeFourVector v) → v ≠ 0 →
|armwiseExtension ε u hu - armwiseExtension ε v hv| ≤
2 * (∑ a : Bool, armDelta u v a) +
2 * (∑ a : Bool, totalMassVec v hv / anchoredDenom ε v hv a * armDelta u v a)) ∧
(∀ (d : ℕ) (ε : ℝ) (P : DiscreteLaw d),
2 ≤ d → 0 < ε → ε ≤ 1 / 2 → ObservedClass ε P →
∀ x : Fin d, 0 < cellMass P x →
(∑ a : Bool, cellMass P x / armMass P a x *
(∑ y : Bool, sqrt (jointMass P x a y))) ≤
2 * sqrt 2 * sqrt (cellMass P x / ε)) := by
exact ⟨fun ε hε _ => armwise_modulus_extension.1 ε hε,
fun d ε P _ hε _ => armwise_modulus_extension.2 d ε P hε⟩
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.ConsistencyFrontier 6 declarations The final paragraph of the weighted-separation roadmap reduces uniform consistency to convergence of the uncapped rate.
Uniform consistency from the risk frontier
The final paragraph of the weighted-separation roadmap reduces uniform consistency to convergence of the uncapped rate. These helpers prove that reduction from two-sided risk bounds; they do not assert the still-open bounds themselves.
Capping a nonnegative sequence at one preserves convergence to zero. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma cappedRate_tendsto_zero_iff {ι : Type*} {l : Filter ι}
(r : ι → ℝ) (hr : ∀ i, 0 ≤ r i) :
Tendsto (fun i => min 1 (r i)) l (𝓝 0) ↔ Tendsto r l (𝓝 0) := by
constructor
· intro h
have he : ∀ᶠ i in l, min 1 (r i) < 1 :=
(tendsto_order.mp h).2 1 (by norm_num)
apply h.congr'
filter_upwards [he] with i hi
by_cases hri : r i ≤ 1
· exact min_eq_right hri
· rw [min_eq_left (le_of_not_ge hri)] at hi
exact (lt_irrefl (1 : ℝ) hi).elim
· intro h
exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds h
(fun i => le_min (by norm_num) (hr i)) (fun i => min_le_right _ _)
theorem rateScale_nonneg
The rate is nonnegative, including the zero-sample convention for division. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma rateScale_nonneg {n d : ℕ} {ε : ℝ} (hd : 1 ≤ d) (hε : 0 ≤ ε) :
0 ≤ rateScale n d ε := by
exact le_min (by norm_num)
(div_nonneg (Nat.cast_nonneg d)
(mul_nonneg (mul_nonneg (Nat.cast_nonneg n) hε) (logAlphabet_pos hd).le))
theorem rateScale_tendsto_zero_iff
The capped paper rate vanishes exactly when its uncapped ratio vanishes. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma rateScale_tendsto_zero_iff (N d : ℕ → ℕ) (ε : ℕ → ℝ)
(hd : ∀ k, 1 ≤ d k) (hε : ∀ k, 0 ≤ ε k) :
Tendsto (fun k => rateScale (N k) (d k) (ε k)) atTop (𝓝 0) ↔
Tendsto (fun k => (d k : ℝ) /
((N k : ℝ) * ε k * logAlphabet (d k))) atTop (𝓝 0) := by
apply cappedRate_tendsto_zero_iff
intro k
exact div_nonneg (Nat.cast_nonneg (d k))
(mul_nonneg (mul_nonneg (Nat.cast_nonneg (N k)) (hε k))
(logAlphabet_pos (hd k)).le)
theorem observedWorstRisk_nonneg
Worst-case squared observed risk is nonnegative without a boundedness assumption. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
0 ≤ worstCaseRiskReal (observedRisk n (ε := ε)) est
Proof (Lean source)
lemma observedWorstRisk_nonneg {n d : ℕ} {ε : ℝ} (est : Estimator n d) :
0 ≤ worstCaseRiskReal (observedRisk n (ε := ε)) est := by
exact worstCaseRisk_nonneg (observedRisk_nonneg est)
theorem minimaxRisk_le_observedWorstRisk
Every admissible estimator has worst-case risk at least the minimax value. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma minimaxRisk_le_observedWorstRisk {n d : ℕ} {ε : ℝ} (est : Estimator n d) :
minimaxRisk n d ε ≤
worstCaseRiskReal (observedRisk n (ε := ε)) est := by
exact minimaxValue_le_worstCaseRisk_of_nonneg
(fun e P => observedRisk_nonneg e P) est
theorem uniformConsistency_iff_of_frontier
Once the uniform lower bound and the explicit estimator's upper bound are proved, their risk sandwich gives the roadmap's exact triangular consistency criterion. The bounds are inputs to this assembly lemma, not new model assumptions. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
c C H₀ κ :
ℝ
D₀ :
ℕ
hc :
0 < c
hH₀ :
0 < H₀
hκ :
0 < κ
hD₀ :
2 ≤ D₀
hlower :
∀ n d ε
hupper :
∀ n d ε
if
1 ≤ n
and
2 ≤ d
and
0 < ε
and
ε ≤ 1 / 2
then
armwiseWorstRisk H₀ κ D₀ hH₀ hκ hD₀ n d ε ≤ C * rateScale n d ε
N d :
ℕ → ℕ
ε :
ℕ → ℝ
hlegal :
∀ k, 2 ≤ d k ∧ 0 < ε k ∧ ε k ≤ 1 / 2
conclusion
(∃ est : ∀ k, Estimator (N k) (d k), Tendsto (fun k => worstCaseRiskReal (observedRisk (N k) (d := d k) (ε := ε k)) (est k)) atTop (𝓝 0))
Proof (Lean source)
lemma uniformConsistency_iff_of_frontier
(c C H₀ κ : ℝ) (D₀ : ℕ) (hc : 0 < c)
(hH₀ : 0 < H₀) (hκ : 0 < κ) (hD₀ : 2 ≤ D₀)
(hlower : ∀ n d ε, 1 ≤ n → 2 ≤ d → 0 < ε → ε ≤ 1 / 2 →
c * rateScale n d ε ≤ minimaxRisk n d ε)
(hupper : ∀ n d ε, 1 ≤ n → 2 ≤ d → 0 < ε → ε ≤ 1 / 2 →
armwiseWorstRisk H₀ κ D₀ hH₀ hκ hD₀ n d ε ≤ C * rateScale n d ε)
(N d : ℕ → ℕ) (ε : ℕ → ℝ) (hN : Tendsto N atTop atTop)
(hlegal : ∀ k, 2 ≤ d k ∧ 0 < ε k ∧ ε k ≤ 1 / 2) :
((∃ est : ∀ k, Estimator (N k) (d k),
Tendsto (fun k => worstCaseRiskReal
(observedRisk (N k) (d := d k) (ε := ε k)) (est k)) atTop (𝓝 0)) ↔
Tendsto (fun k => (d k : ℝ) /
((N k : ℝ) * ε k * logAlphabet (d k))) atTop (𝓝 0)) := by
have hn : ∀ᶠ k in atTop, 1 ≤ N k := hN.eventually (eventually_ge_atTop 1)
have hrate := rateScale_tendsto_zero_iff N d ε
(fun k => le_trans (by norm_num) (hlegal k).1) (fun k => (hlegal k).2.1.le)
constructor
· rintro ⟨est, hest⟩
apply hrate.mp
have hdiv : Tendsto (fun k => worstCaseRiskReal
(observedRisk (N k) (d := d k) (ε := ε k)) (est k) / c) atTop (𝓝 0) := by
simpa using hest.div_const c
apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hdiv
· exact Filter.Eventually.of_forall fun k =>
rateScale_nonneg (le_trans (by norm_num) (hlegal k).1) (hlegal k).2.1.le
· filter_upwards [hn] with k hk
apply (le_div_iff₀ hc).2
rw [mul_comm]
exact (hlower _ _ _ hk (hlegal k).1 (hlegal k).2.1 (hlegal k).2.2).trans
(minimaxRisk_le_observedWorstRisk (est k))
· intro hr
refine ⟨fun k => armwiseEstimatorWitness H₀ κ D₀ hH₀ hκ hD₀
(N k) (d k) (ε k), ?_⟩
have hmul : Tendsto (fun k => C * rateScale (N k) (d k) (ε k)) atTop (𝓝 0) := by
simpa using (hrate.mpr hr).const_mul C
apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hmul
· exact Filter.Eventually.of_forall fun k => observedWorstRisk_nonneg _
· filter_upwards [hn] with k hk
exact hupper _ _ _ hk (hlegal k).1 (hlegal k).2.1 (hlegal k).2.2
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.Estimator 42 declarations The final estimator is a finite sum over counts and marks.
Observable capped-Poisson Jackson factorial estimator
The final estimator is a finite sum over counts and marks. The Jackson polynomial is selected from the canonical tensor convolution representation.
def sampleCellCount
For the displayed parameters, sampleCellCount is the object specified by this definition.
def sampleArmCount
For the displayed parameters, sampleArmCount is the object specified by this definition.
def sampleSuccessCount
For the displayed parameters, sampleSuccessCount is the object specified by this definition.
def projectUnit
For the displayed parameters, projectUnit is the object specified by this definition.
def empiricalValue
For the displayed parameters, empiricalValue is the object specified by this definition.
Definition (Lean source)
parameters
n d :
ℕ
o :
Fin n → Obs d
def
empiricalValue o :
ℝ
given by
projectUnit <| ∑ x : Fin d, (sampleCellCount o x : ℝ) / n * max ((sampleSuccessCount o x false : ℝ) / sampleArmCount o x false) ((sampleSuccessCount o x true : ℝ) / sampleArmCount o x true)
def jacksonDegree
For the displayed parameters, jacksonDegree is the object specified by this definition.
Definition (Lean source)
def poissonIntensityFormula
For the displayed parameters, poissonIntensityFormula is the object specified by this definition.
Definition (Lean source)
parameters
n :
ℕ
def
poissonIntensityFormula n :
ℝ
given by
n / 8
def markedCount
For the displayed parameters, markedCount is the object specified by this definition.
Definition (Lean source)
def pilotCenterFormula
For the displayed parameters, pilotCenterFormula is the object specified by this definition.
def pilotHalfWidthFormula
For the displayed parameters, pilotHalfWidthFormula is the object specified by this definition.
Definition (Lean source)
parameters
def
pilotHalfWidthFormula H₀ _hH₀ m L Np j :
ℝ
given by
H₀ * (sqrt (pilotCenterFormula m Np j * L / m) + L / m)
def pilotLower
For the displayed parameters, pilotLower is the object specified by this definition.
Definition (Lean source)
parameters
def
pilotLower H₀ hH₀ m L Np j :
ℝ
given by
def pilotUpperFormula
For the displayed parameters, pilotUpperFormula is the object specified by this definition.
Definition (Lean source)
parameters
def
pilotUpperFormula H₀ hH₀ m L Np j :
ℝ
given by
pilotCenterFormula m Np j + pilotHalfWidthFormula H₀ hH₀ m L Np j
def pilotCenter
Pilot empirical mass on the paper's positive intensity domain. For the displayed parameters, pilotCenter is the object specified by this definition.
def pilotHalfWidth
Pilot half-width with all positive calibration inputs explicit. For the displayed parameters, pilotHalfWidth is the object specified by this definition.
Definition (Lean source)
parameters
def
pilotHalfWidth H₀ _hH₀ m L hm _hL Np j :
ℝ
given by
H₀ * (sqrt (pilotCenter m hm Np j * L / m) + L / m)
def pilotUpper
Pilot upper endpoint on the positive width and intensity domain. For the displayed parameters, pilotUpper is the object specified by this definition.
Definition (Lean source)
parameters
def
pilotUpper H₀ hH₀ m L hm hL Np j :
ℝ
given by
pilotCenter m hm Np j + pilotHalfWidth H₀ hH₀ m L hm hL Np j
def pilotMidpoint
For the displayed parameters, pilotMidpoint is the object specified by this definition.
Definition (Lean source)
parameters
def
given by
fun j => (pilotLower H₀ hH₀ m L Np j + pilotUpperFormula H₀ hH₀ m L Np j) / 2
def pilotRadiusFormula
For the displayed parameters, pilotRadiusFormula is the object specified by this definition.
Definition (Lean source)
parameters
def
given by
fun j => (pilotUpperFormula H₀ hH₀ m L Np j - pilotLower H₀ hH₀ m L Np j) / 2
def pilotRectangle
For the displayed parameters, pilotRectangle is the object specified by this definition.
In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma armwise_continuousOn_rectangle (ε : ℝ) (hε : 0 < ε)
(c r : Fin 4 → ℝ) (hr : ∀ j, 0 < r j)
(hQ : ∀ v ∈ Causalean.Mathlib.Analysis.JacksonApproximation.centeredRectangle c r,
∀ j, 0 ≤ v j) :
ContinuousOn (armwiseExtensionFormula ε)
(Causalean.Mathlib.Analysis.JacksonApproximation.centeredRectangle c r) := by
let Q := Causalean.Mathlib.Analysis.JacksonApproximation.centeredRectangle c r
intro v hv
have hvpos : ∀ j, 0 ≤ v j := hQ v hv
by_cases hv0 : v = 0
· have hmass : ContinuousAt totalMassVecFormula v := by
unfold totalMassVecFormula armMassVecFormula
fun_prop
have hmass0 : totalMassVecFormula v = 0 := by simp [hv0, totalMassVecFormula, armMassVecFormula]
have ht : Tendsto totalMassVecFormula (𝓝[Q] v) (𝓝 (0 : ℝ)) := by
simpa [hmass0] using hmass.tendsto.mono_left nhdsWithin_le_nhds
have hlow : ∀ᶠ u in 𝓝[Q] v, (0 : ℝ) ≤ armwiseExtensionFormula ε u := by
filter_upwards [self_mem_nhdsWithin] with u hu
exact (armwiseExtension_nonneg_le_totalMassVec hε (hQ u hu)).1
have hhigh : ∀ᶠ u in 𝓝[Q] v, armwiseExtensionFormula ε u ≤ totalMassVecFormula u := by
filter_upwards [self_mem_nhdsWithin] with u hu
exact (armwiseExtension_nonneg_le_totalMassVec hε (hQ u hu)).2
simpa [ContinuousWithinAt, hv0, armwiseExtensionFormula] using
(tendsto_of_tendsto_of_tendsto_of_le_of_le'
tendsto_const_nhds ht hlow hhigh)
· have hs : 0 < totalMassVecFormula v := by
have h := totalMassVec_pos_of_nonzero hvpos hv0
exact h
have hden (a : Bool) : anchoredDenomFormula ε v a ≠ 0 := by
have := lt_of_lt_of_le (mul_pos hε hs)
(le_max_right (armMassVecFormula v a) (ε * totalMassVecFormula v))
exact ne_of_gt this
have hform : ContinuousAt (fun u : Fin 4 → ℝ =>
max (totalMassVecFormula u * u (cellIdx false true) / anchoredDenomFormula ε u false)
(totalMassVecFormula u * u (cellIdx true true) / anchoredDenomFormula ε u true)) v := by
have ht : ContinuousAt totalMassVecFormula v := by
unfold totalMassVecFormula armMassVecFormula
fun_prop
have ha (a : Bool) : ContinuousAt (anchoredDenomFormula ε · a) v := by
dsimp [anchoredDenomFormula, armMassVecFormula, totalMassVecFormula]
fun_prop
have hf (a : Bool) : ContinuousAt
(fun u : Fin 4 → ℝ => totalMassVecFormula u * u (cellIdx a true) /
anchoredDenomFormula ε u a) v := by
exact (ht.mul (continuousAt_apply (cellIdx a true) v)).div (ha a) (hden a)
exact (hf false).max (hf true)
have hev : ∀ᶠ u : Fin 4 → ℝ in 𝓝 v, u ≠ 0 :=
eventually_ne_nhds hv0
have heq : (armwiseExtensionFormula ε) =ᶠ[𝓝[Q] v]
(fun u : Fin 4 → ℝ =>
max (totalMassVecFormula u * u (cellIdx false true) / anchoredDenomFormula ε u false)
(totalMassVecFormula u * u (cellIdx true true) / anchoredDenomFormula ε u true)) := by
filter_upwards [hev.filter_mono nhdsWithin_le_nhds] with u hu
simp [armwiseExtensionFormula, hu]
have hgoal := hform.continuousWithinAt.congr_of_eventuallyEq heq
exact hgoal (by simp [armwiseExtensionFormula, hv0])
def jacksonPolynomialPair
For the displayed parameters, jacksonPolynomialPair is the object specified by this definition.
Definition (Lean source)
parameters
def
given by
by classical exact if hK : 0 < K then if hε : 0 < ε then if hr : ∀ j, 0 < r j then if hQ : ∀ v ∈ centeredRectangle c r, ∀ j, 0
≤ v j then let h := affineJackson_exists_mvPolynomial_four hK c r hr (armwiseExtensionFormula ε) (armwise_continuousOn_rectangle ε hε c r hr hQ) (choose h, choose (Classical.choose_spec h)) else (0, 0) else (0, 0) else (0, 0) else (0, 0)
def jacksonCoeff
For the displayed parameters, jacksonCoeff is the object specified by this definition.
Definition (Lean source)
parameters
def
jacksonCoeff ε K c r α :
ℝ
given by
def factorialCellValue
For the displayed parameters, factorialCellValue is the object specified by this definition.
Definition (Lean source)
parameters
def
factorialCellValue ε K m c r Ne :
ℝ
given by
+ ∑ α : Fin 4 → Fin (2 * (K - 1) + 1), jacksonCoeff ε K c r α * ∏ j : Fin 4, factorialLift m (c j) (Ne j) (α j).val / (r j) ^ (α j).val
def armRadiusFormula
For the displayed parameters, armRadiusFormula is the object specified by this definition.
def armWeightFormula
For the displayed parameters, armWeightFormula is the object specified by this definition.
Definition (Lean source)
parameters
def
armWeightFormula ε c a :
ℝ
given by
1 + totalMassVecFormula c / anchoredDenomFormula ε c a
def clippingScaleFormula
For the displayed parameters, clippingScaleFormula is the object specified by this definition.
Definition (Lean source)
def clippedCellValueFormula
For the displayed parameters, clippedCellValueFormula is the object specified by this definition.
Definition (Lean source)
parameters
def
clippedCellValueFormula ε K d m c r Ne :
ℝ
def cappedStatisticFormula
For the displayed parameters, cappedStatisticFormula is the object specified by this definition.
Definition (Lean source)
parameters
def
cappedStatisticFormula hk H₀ hH₀ κ ε o marks :
ℝ
given by
projectUnit <| ∑ x : Fin d, let Np := markedCount hk o marks true x let Ne := markedCount hk o marks false x let c := pilotMidpoint H₀ hH₀ m L Np let r := pilotRadiusFormula H₀ hH₀ m L Np clippedCellValueFormula ε K d m c r Ne
def poissonWeight
For the displayed parameters, poissonWeight is the object specified by this definition.
Definition (Lean source)
parameters
rate :
ℝ
k :
ℕ
def
poissonWeight rate k :
ℝ
given by
exp (-rate) * rate ^ k / (k.factorial : ℝ)
def armwiseEstimatorFormula
For the displayed parameters, armwiseEstimatorFormula is the object specified by this definition.
Definition (Lean source)
parameters
def
armwiseEstimatorFormula H₀ κ D₀ hH₀ _hκ _hD₀ n d ε o :
ℝ
given by
if d < D₀ then empiricalValue o else if (n : ℝ) * ε < (d : ℝ) / logAlphabet d then 1 / 2 else ∑ k ∈ range (n + 1), if hk : k
≤ n then poissonWeight (n / 4) k * ∑ marks : Fin k → Bool, (1 / 2 : ℝ) ^ k * cappedStatisticFormula hk H₀ hH₀ κ ε o marks else 0
def poissonIntensity
The Poisson comparison intensity for a positive sample size. For the displayed parameters, poissonIntensity is the object specified by this definition.
Definition (Lean source)
def pilotRadius
The pilot radius on the positive width, intensity, and logarithm domain. For the displayed parameters, pilotRadius is the object specified by this definition.
Definition (Lean source)
parameters
def
given by
fun j => (pilotUpper H₀ hH₀ m L hm hL Np j - pilotLower H₀ hH₀ m L Np j) / 2
theorem pilotRadius_positive
In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
0 < pilotRadius H hH m L hm hL Np j
Proof (Lean source)
lemma pilotRadius_positive (H : ℝ) (hH : 0 < H) (m L : ℝ)
(hm : 0 < m) (hL : 0 < L) (Np : Fin 4 → ℕ) (j : Fin 4) :
0 < pilotRadius H hH m L hm hL Np j := by
have hc : 0 ≤ pilotCenterFormula m Np j := by unfold pilotCenterFormula; positivity
have hh : 0 < pilotHalfWidthFormula H hH m L Np j := by
unfold pilotHalfWidthFormula
exact mul_pos hH (add_pos_of_nonneg_of_pos (Real.sqrt_nonneg _) (div_pos hL hm))
change 0 < pilotRadiusFormula H hH m L Np j
dsimp [pilotRadiusFormula, pilotLower, pilotUpperFormula]
rw [max_def]
split_ifs <;> linarith
theorem pilotMidpoint_nonnegative
In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
0 ≤ pilotMidpoint H hH m L Np j
Proof (Lean source)
lemma pilotMidpoint_nonnegative (H : ℝ) (hH : 0 < H) (m L : ℝ)
(hm : 0 < m) (hL : 0 < L) (Np : Fin 4 → ℕ) (j : Fin 4) :
0 ≤ pilotMidpoint H hH m L Np j := by
have hc : 0 ≤ pilotCenterFormula m Np j := by unfold pilotCenterFormula; positivity
have hh : 0 ≤ pilotHalfWidthFormula H hH m L Np j := by
unfold pilotHalfWidthFormula
exact mul_nonneg hH.le (add_nonneg (Real.sqrt_nonneg _) (div_nonneg hL.le hm.le))
dsimp [pilotMidpoint, pilotUpperFormula]
exact div_nonneg (add_nonneg (le_max_left _ _) (add_nonneg hc hh)) (by norm_num)
def armRadius
An arm radius is the sum of two strictly positive coordinate radii.
Definition (Lean source)
def armWeight
Sensitivity on nonnegative pilot centers and the public positive overlap domain.
Definition (Lean source)
def clippingScale
The paper's strictly positive clipping scale; the unrestricted analytic extension is clippingScaleFormula.
def clippedCellValue
For the displayed parameters, clippedCellValue is the object specified by this definition.
Definition (Lean source)
parameters
ε :
ℝ
hε :
0 < ε ∧ ε ≤ 1 / 2
K d :
ℕ
m :
ℝ
c r :
Fin 4 → ℝ
hc :
∀ j, 0 ≤ c j
hr :
∀ j, 0 < r j
Ne :
Fin 4 → ℕ
def
clippedCellValue ε hε K d m c r hc hr Ne :
ℝ
def cappedStatistic
For the displayed parameters, cappedStatistic is the object specified by this definition.
Definition (Lean source)
parameters
n d k :
ℕ
hk :
k ≤ n
hn :
0 < n
hd :
2 ≤ d
H₀ :
ℝ
hH₀ :
0 < H₀
κ ε :
ℝ
hε :
0 < ε ∧ ε ≤ 1 / 2
o :
Fin n → Obs d
def
cappedStatistic hk hn hd H₀ hH₀ κ ε hε o marks :
ℝ
given by
m :
poissonIntensity n hn
L :
logAlphabet d
hm :
by dsimp [m, poissonIntensity, poissonIntensityFormula] (: 0 < m)
positivity let hL : 0 < L := by have hdR : (1 : ℝ)
≤ d := by exact_mod_cast (show 1 ≤ d by omega) apply Real.log_pos exact lt_of_lt_of_le (Real.one_lt_exp_iff.mpr (by norm_num)) (le_mul_of_one_le_right (Real.exp_pos 1).le hdR) let K := jacksonDegree κ d projectUnit <| ∑ x : Fin d, let Np := markedCount hk o marks true x let Ne := markedCount hk o marks false x let c := pilotMidpoint H₀ hH₀ m L Np let r := pilotRadius H₀ hH₀ m L hm hL Np clippedCellValue ε hε K d m c r (pilotMidpoint_nonnegative H₀ hH₀ m L hm hL Np) (pilotRadius_positive H₀ hH₀ m L hm hL Np) Ne
def armwiseEstimator
For the displayed parameters, armwiseEstimator is the object specified by this definition.
Definition (Lean source)
parameters
H₀ κ :
ℝ
D₀ :
ℕ
hH₀ :
0 < H₀
_hκ :
0 < κ
_hD₀ :
2 ≤ D₀
n d :
ℕ
ε :
ℝ
hn :
1 ≤ n
hd :
2 ≤ d
hε :
0 < ε ∧ ε ≤ 1 / 2
o :
Fin n → Obs d
def
armwiseEstimator H₀ κ D₀ hH₀ _hκ _hD₀ n d ε hn hd hε o :
ℝ
given by
if d < D₀ then empiricalValue o else if (n : ℝ) * ε < (d : ℝ) / logAlphabet d then 1 / 2 else ∑ k ∈ range (n + 1), if hk : k
≤ n then poissonWeight (n / 4) k * ∑ marks : Fin k → Bool, (1 / 2 : ℝ) ^ k * cappedStatistic hk (lt_of_lt_of_le (by decide : 0 < 1) hn) hd H₀ hH₀ κ ε hε o marks else 0
def armwiseEstimatorWitness
For the displayed parameters, armwiseEstimatorWitness is the object specified by this definition.
Definition (Lean source)
parameters
H₀ κ :
ℝ
D₀ :
ℕ
hH₀ :
0 < H₀
hκ :
0 < κ
hD₀ :
2 ≤ D₀
n d :
ℕ
ε :
ℝ
def
given by
⟨armwiseEstimatorFormula H₀ κ D₀ hH₀ hκ hD₀ n d ε, Measurable.of_discrete⟩
For the displayed parameters, armwiseWorstRisk is the object specified by this definition.
Definition (Lean source)
parameters
H₀ κ :
ℝ
D₀ :
ℕ
hH₀ :
0 < H₀
hκ :
0 < κ
hD₀ :
2 ≤ D₀
n d :
ℕ
ε :
ℝ
def
armwiseWorstRisk H₀ κ D₀ hH₀ hκ hD₀ n d ε :
ℝ
given by
worstCaseRiskReal (observedRisk n (d := d) (ε := ε)) (armwiseEstimatorWitness H₀ κ D₀ hH₀ hκ hD₀ n d ε)
Worst-case causal squared risk of the same observed-sample armwise estimator.
Definition (Lean source)
parameters
H₀ κ :
ℝ
D₀ :
ℕ
hH₀ :
0 < H₀
hκ :
0 < κ
hD₀ :
2 ≤ D₀
n d :
ℕ
ε :
ℝ
def
causalArmwiseWorstRisk H₀ κ D₀ hH₀ hκ hD₀ n d ε :
ℝ
given by
worstCaseRiskReal (causalRisk n (d := d) (ε := ε)) (armwiseEstimatorWitness H₀ κ D₀ hH₀ hκ hD₀ n d ε)
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.FixedSampleFactorial 7 declarations Deterministic sample split and multinomial factorial statistic
Deterministic sample split and multinomial factorial statistic
def pilotSize
For the displayed parameters, pilotSize is the object specified by this definition.
Definition (Lean source)
parameters
n :
ℕ
def
pilotSize n :
ℕ
given by
n / 2
def evalSize
For the displayed parameters, evalSize is the object specified by this definition.
Definition (Lean source)
def pilotIndices
For the displayed parameters, pilotIndices is the object specified by this definition.
def evalIndices
For the displayed parameters, evalIndices is the object specified by this definition.
def pilotCount
For the displayed parameters, pilotCount is the object specified by this definition.
Definition (Lean source)
def evalCount
For the displayed parameters, evalCount is the object specified by this definition.
Definition (Lean source)
def multinomialFactorialLift
For the displayed parameters, multinomialFactorialLift is the object specified by this definition.
Definition (Lean source)
parameters
d :
ℕ
n :
ℕ
hn :
1 ≤ n
o :
Fin n → Obs d
b :
Obs d → ℝ
h :
Obs d → ℕ
_hh :
(∑ j : Obs d, h j) ≤ evalSize n hn
def
multinomialFactorialLift hn o b h _hh :
ℝ
given by
∑ t ∈ piFinset (fun j : Obs d => range (h j + 1)), (∏ j : Obs d, ((h j).choose (t j) : ℝ) * (-b j) ^ (h j - t j)) * (∏ j : Obs d, ((Ne j).descFactorial (t j) : ℝ)) / (ne.descFactorial (∑ j : Obs d, t j) : ℝ)
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.JacksonPacket 9 declarations Localized high-frequency Jackson packet
Localized high-frequency Jackson packet
def packetKernel
For the displayed parameters, packetKernel is the object specified by this definition.
def kernelCoeff
For the displayed parameters, kernelCoeff is the object specified by this definition.
def packetCoeffPlus
For the displayed parameters, packetCoeffPlus is the object specified by this definition.
Definition (Lean source)
parameters
N :
ℕ
j :
ℤ
def
packetCoeffPlus N j :
ℝ
given by
if |j| ≤ (2 * N - 2 : ℕ) then kernelCoeff N j / ((4 * (N : ℤ) + j : ℤ) : ℝ) ^ 2 else 0
def packetCoeffMinus
For the displayed parameters, packetCoeffMinus is the object specified by this definition.
Definition (Lean source)
parameters
N :
ℕ
j :
ℤ
def
packetCoeffMinus N j :
ℝ
given by
if |j| ≤ (2 * N - 2 : ℕ) then kernelCoeff N j / ((4 * (N : ℤ) - j : ℤ) : ℝ) ^ 2 else 0
def packetOscillation
For the displayed parameters, packetOscillation is the object specified by this definition.
Definition (Lean source)
parameters
N :
ℕ
u :
ℝ
def
packetOscillation N u :
ℝ
given by
packetKernel N u * cos ((4 * N : ℝ) * u)
def packetAntideriv
For the displayed parameters, packetAntideriv is the object specified by this definition.
def torusDistance
For the displayed parameters, torusDistance is the object specified by this definition.
In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma packetKernel_normalized_integral (N : ℕ) (hN : 0 < N) :
(∫ u in Icc (-Real.pi) Real.pi,
packetKernel N u / (2 * Real.pi)) = 1 := by
have hden : (2 * Real.pi : ℝ) ≠ 0 := by positivity
convert jackson_integral_eq_one N hN using 1
apply MeasureTheory.integral_congr_ae
filter_upwards with u
simp only [packetKernel]
field_simp
In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ c C : ℝ,
0 < c
c ≤ C
conclusion 1
N :
ℕ
2 ≤ N
Summable fun j : ℤ
=> |packetCoeffPlus N (j + 2) - 2 * packetCoeffPlus N (j + 1) + packetCoeffPlus N j|
Summable fun j : ℤ
=> |packetCoeffMinus N (j + 2) - 2 * packetCoeffMinus N (j + 1) + packetCoeffMinus N j|
(∑' j : ℤ, |packetCoeffPlus N (j + 2) - 2 * packetCoeffPlus N (j + 1) + packetCoeffPlus N j|)
≤ C / N ^ 3
(∑' j : ℤ, |packetCoeffMinus N (j + 2) - 2 * packetCoeffMinus N (j + 1) + packetCoeffMinus N j|)
≤ C / N ^ 3
r :
ℤ
|r| < (2 * N + 2 : ℕ) ∨ (6 * N - 2 : ℕ) < |r|
c / N ≤ |packetAntideriv N 0|
|packetAntideriv N 0| ≤ C / N
u :
ℝ
|packetAntideriv N u| ≤ C / (N * (1 + N * torusDistance u) ^ 2)
Proof (Lean source)
lemma jackson_packet_localization :
∃ c C : ℝ, 0 < c ∧ c ≤ C ∧
∀ N : ℕ, 2 ≤ N →
(∫ u in Icc (-pi) pi,
packetKernel N u / (2 * pi)) = 1 ∧
(Summable fun j : ℤ =>
|packetCoeffPlus N (j + 2) - 2 * packetCoeffPlus N (j + 1) +
packetCoeffPlus N j|) ∧
(Summable fun j : ℤ =>
|packetCoeffMinus N (j + 2) - 2 * packetCoeffMinus N (j + 1) +
packetCoeffMinus N j|) ∧
(∑' j : ℤ,
|packetCoeffPlus N (j + 2) - 2 * packetCoeffPlus N (j + 1) +
packetCoeffPlus N j|) ≤ C / N ^ 3 ∧
(∑' j : ℤ,
|packetCoeffMinus N (j + 2) - 2 * packetCoeffMinus N (j + 1) +
packetCoeffMinus N j|) ≤ C / N ^ 3 ∧
(∀ r : ℤ, |r| < (2 * N + 2 : ℕ) ∨ (6 * N - 2 : ℕ) < |r| →
(∫ u in Icc (-pi) pi,
packetOscillation N u * cos (r * u)) = 0 ∧
(∫ u in Icc (-pi) pi,
packetOscillation N u * sin (r * u)) = 0) ∧
(∀ u : ℝ, deriv (deriv (packetAntideriv N)) u = packetOscillation N u) ∧
(∫ u in Icc (-pi) pi, packetAntideriv N u) = 0 ∧
c / N ≤ |packetAntideriv N 0| ∧
|packetAntideriv N 0| ≤ C / N ∧
(∀ u : ℝ,
|packetAntideriv N u| ≤
C / (N * (1 + N * torusDistance u) ^ 2)) ∧
(∫ u in Icc (-pi) pi,
|packetAntideriv N u| / (2 * pi)) ≤ C / N ^ 2 := by
obtain ⟨c, Ccenter, hc, hcC, hcenter⟩ :=
packet_center_bounds
obtain ⟨Cpoint, hCpoint, hpoint⟩ :=
packet_pointwise_localization
obtain ⟨Cl1, hCl1, hl1⟩ :=
Causalean.Mathlib.Analysis.Approximation.Chebyshev.Jackson.packet_l1_localization
let C : ℝ := max 512 (max Ccenter (max Cpoint Cl1))
have h512C : (512 : ℝ) ≤ C := le_max_left _ _
have hcenterC : Ccenter ≤ C :=
(le_max_left Ccenter (max Cpoint Cl1)).trans (le_max_right 512 _)
have hpointC : Cpoint ≤ C :=
((le_max_left Cpoint Cl1).trans (le_max_right Ccenter _)).trans
(le_max_right 512 _)
have hl1C : Cl1 ≤ C :=
((le_max_right Cpoint Cl1).trans (le_max_right Ccenter _)).trans
(le_max_right 512 _)
refine ⟨c, C, hc, hcC.trans hcenterC, ?_⟩
intro N hN
have hNpos : 0 < N := by omega
have hNR : (0 : ℝ) < N := by exact_mod_cast hNpos
have hplus (j : ℤ) := packetCoeffPlus_eq_staged N hNpos j
have hminus (j : ℤ) := packetCoeffMinus_eq_staged N hNpos j
have hantifun : packetAntideriv N =
packetAntideriv N := by
funext u
exact packetAntideriv_eq_staged N hNpos u
refine ⟨packetKernel_normalized_integral N hNpos,
packetCoeffPlus_secondDiff_summable N,
packetCoeffMinus_secondDiff_summable N, ?_, ?_, ?_, ?_, ?_, ?_, ?_, ?_, ?_⟩
· have h :=
weightedCoeff_delta2_l1
N hN 1 (inl rfl)
simp_rw [hplus]
change (∑' j : ℤ,
|weightedCoeff N 1 (j + 2) -
2 * weightedCoeff N 1 (j + 1) +
weightedCoeff N 1 j|) ≤
C / (N : ℝ) ^ 3
calc
_ ≤ 512 / (N : ℝ) ^ 3 := by
convert h using 1
congr 1
funext j
simp only [delta2,
delta]
congr 1
ring
_ ≤ _ := div_le_div_of_nonneg_right h512C (by positivity)
· have h :=
weightedCoeff_delta2_l1
N hN (-1) (inr rfl)
simp_rw [hminus]
change (∑' j : ℤ,
|weightedCoeff N (-1) (j + 2) -
2 * weightedCoeff N (-1) (j + 1) +
weightedCoeff N (-1) j|) ≤
C / (N : ℝ) ^ 3
calc
_ ≤ 512 / (N : ℝ) ^ 3 := by
convert h using 1
congr 1
funext j
simp only [delta2,
delta]
congr 1
ring
_ ≤ _ := div_le_div_of_nonneg_right h512C (by positivity)
· intro r hr
simpa only [packetOscillation_eq_staged] using
packet_shifted_support
N hN r hr
· intro u
rw [hantifun]
simpa only [packetOscillation_eq_staged] using
packet_twice_deriv N hN u
· rw [hantifun]
exact packet_zero_mean N hN
· rw [packetAntideriv_eq_staged N hNpos]
exact (hcenter N hN).1
· rw [packetAntideriv_eq_staged N hNpos]
exact (hcenter N hN).2.trans
(div_le_div_of_nonneg_right hcenterC hNR.le)
· intro u
rw [packetAntideriv_eq_staged N hNpos u, torusDistance_eq_staged]
exact (hpoint N hN u).trans
(div_le_div_of_nonneg_right hpointC (by positivity))
· simp_rw [packetAntideriv_eq_staged N hNpos]
exact (hl1 N hN).trans
(div_le_div_of_nonneg_right hl1C (by positivity))
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.LowerDense 15 declarations The absolute-value Fejér certificate supplies symmetric, supported, moment-matched priors with an inverse-degree gap, as required by (43)--(44).
Dense boundary-propensity lower experiment
The absolute-value Fejér certificate supplies symmetric, supported, moment-matched priors with an inverse-degree gap, as required by (43)--(44). The two treated Poisson coordinates have the exact Gram identity (47); the untreated coordinates are common to both hypotheses and contribute one.
theorem dense_symmetric_priors_absGap_lower Lemma dense_symmetric_priors_absGap_lower in the paper ↗
The Fejér inverse approximation bound and the symmetric dual-prior construction give the inverse-degree absolute-value gap in (44). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
K :
ℕ
hK :
0 < K
conclusion
∃ P : AbsMomentMatchedPriors K, (1 / 50 : ℝ) / K ≤ (∫ t, |t| ∂P.ν₁) - (∫ t, |t| ∂P.ν₀)
Proof (Lean source)
lemma dense_symmetric_priors_absGap_lower (K : ℕ) (hK : 0 < K) :
∃ P : AbsMomentMatchedPriors K,
(1 / 50 : ℝ) / K ≤ (∫ t, |t| ∂P.ν₁) - (∫ t, |t| ∂P.ν₀) := by
obtain ⟨P⟩ := exists_symmetric_momentMatched_absGap_for_degree K
refine ⟨P, ?_⟩
rw [P.abs_gap]
have h := mul_le_mul_of_nonneg_left (bestUniformApproxErrorAbs_lower K hK)
(by norm_num : (0 : ℝ) ≤ 2)
convert h using 1 <;> first | ring | rfl
Symmetry cancels the signed first moment, so only the absolute-value moment contributes to the centers of the target in (46). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma dense_symmetric_integral_id_zero (ν : Measure ℝ) (hsym : IsSymmetric ν) :
(∫ t, t ∂ν) = 0 := by
have h : (∫ t, t ∂Measure.map (fun t : ℝ => -t) ν) = -(∫ t, t ∂ν) := by
rw [integral_map (by fun_prop) (by fun_prop), integral_neg]
rw [hsym] at h
linarith
For the displayed parameters, denseTreatedMean is the object specified by this definition.
Definition (Lean source)
parameters
r a t :
ℝ
j :
Fin 2
def
denseTreatedMean r a t j :
ℝ
given by
r * (1 / 2 + if j = 0 then -(a * t) else a * t)
For the displayed parameters, denseTreatedLaw is the object specified by this definition.
Definition (Lean source)
parameters
r a t :
ℝ
given by
For the displayed parameters, denseTreatedLikelihood is the object specified by this definition.
Definition (Lean source)
parameters
r a t :
ℝ
N :
Fin 2 → ℕ
def
denseTreatedLikelihood r a t N :
ℝ
given by
∏ j, exp (denseTreatedMean r a 0 j - denseTreatedMean r a t j) * (denseTreatedMean r a t j / denseTreatedMean r a 0 j) ^ N j
theorem denseTreatedMean_nonneg
Legal amplitudes and supported prior parameters yield legal Poisson means. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
0 ≤ denseTreatedMean r a t j
Proof (Lean source)
lemma denseTreatedMean_nonneg {r a t : ℝ} (hr : 0 ≤ r) (ha : 0 ≤ a)
(hahi : a ≤ 1 / 2) (ht : |t| ≤ 1) (j : Fin 2) :
0 ≤ denseTreatedMean r a t j := by
have hat : |a * t| ≤ 1 / 2 := by
rw [abs_mul, abs_of_nonneg ha]
exact (mul_le_mul_of_nonneg_left ht ha).trans (by simpa using hahi)
rw [abs_le] at hat
unfold denseTreatedMean
split_ifs <;> apply mul_nonneg hr <;> linarith [hat.1, hat.2]
theorem denseTreatedLikelihood_gram
Equation (47): both treated outcomes contribute, giving exactly 4 r a². The calculation is valid at support endpoints, including zero alternative means. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
r :
ℝ
hr :
0 < r
a t s :
ℝ
conclusion
(∫ N, denseTreatedLikelihood r a t N * denseTreatedLikelihood r a s N ∂denseTreatedLaw r a 0)
= exp (4 * r * a ^ 2 * t * s)
Proof (Lean source)
lemma denseTreatedLikelihood_gram {r : ℝ} (hr : 0 < r) (a t s : ℝ) :
(∫ N, denseTreatedLikelihood r a t N * denseTreatedLikelihood r a s N
∂denseTreatedLaw r a 0) = exp (4 * r * a ^ 2 * t * s) := by
have hstar (j : Fin 2) : 0 < denseTreatedMean r a 0 j := by
simp [denseTreatedMean]
positivity
have hfun (N : Fin 2 → ℕ) :
denseTreatedLikelihood r a t N * denseTreatedLikelihood r a s N =
∏ j, (exp (denseTreatedMean r a 0 j - denseTreatedMean r a t j) *
(denseTreatedMean r a t j / denseTreatedMean r a 0 j) ^ N j) *
(exp (denseTreatedMean r a 0 j - denseTreatedMean r a s j) *
(denseTreatedMean r a s j / denseTreatedMean r a 0 j) ^ N j) := by
unfold denseTreatedLikelihood
rw [← Finset.prod_mul_distrib]
simp_rw [hfun]
unfold denseTreatedLaw
rw [integral_fintype_prod_eq_prod (fun j : Fin 2 => fun k : ℕ =>
(exp (denseTreatedMean r a 0 j - denseTreatedMean r a t j) *
(denseTreatedMean r a t j / denseTreatedMean r a 0 j) ^ k) *
(exp (denseTreatedMean r a 0 j - denseTreatedMean r a s j) *
(denseTreatedMean r a s j / denseTreatedMean r a 0 j) ^ k))]
simp_rw [poisson_likelihood_gram_scalar _ _ _ (hstar _)]
rw [← Real.exp_sum]
congr 1
simp [Fin.sum_univ_two, denseTreatedMean]
field_simp
ring
theorem measurable_denseTreatedLikelihood
The dense likelihood is jointly measurable in parameter and counts. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
r a :
ℝ
conclusion
Proof (Lean source)
@[fun_prop]
lemma measurable_denseTreatedLikelihood (r a : ℝ) :
Measurable (fun p : ℝ × (Fin 2 → ℕ) => denseTreatedLikelihood r a p.1 p.2) := by
apply measurable_from_prod_countable_left
intro N
unfold denseTreatedLikelihood denseTreatedMean
apply Finset.measurable_prod
intro j _
split_ifs <;> fun_prop
theorem denseTreatedLikelihood_nonneg
A supported parameter yields a nonnegative genuine likelihood. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
0 ≤ denseTreatedLikelihood r a t N
Proof (Lean source)
lemma denseTreatedLikelihood_nonneg {r a t : ℝ} (hr : 0 ≤ r) (ha : 0 ≤ a)
(hahi : a ≤ 1 / 2) (ht : |t| ≤ 1) (N : Fin 2 → ℕ) :
0 ≤ denseTreatedLikelihood r a t N := by
apply Finset.prod_nonneg
intro j _
exact mul_nonneg (Real.exp_nonneg _) (pow_nonneg
(div_nonneg (denseTreatedMean_nonneg hr ha hahi ht j)
(denseTreatedMean_nonneg hr ha hahi (by norm_num) j)) _)
theorem denseTreatedLaw_eq_withDensity
The two-coordinate likelihood recovers the actual Poisson law, even when a supported endpoint makes one alternative intensity zero. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
r a t :
ℝ
hr :
0 < r
ha :
0 ≤ a
hahi :
a ≤ 1 / 2
ht :
|t| ≤ 1
conclusion
denseTreatedLaw r a t
Proof (Lean source)
lemma denseTreatedLaw_eq_withDensity {r a t : ℝ} (hr : 0 < r) (ha : 0 ≤ a)
(hahi : a ≤ 1 / 2) (ht : |t| ≤ 1) :
denseTreatedLaw r a t = (denseTreatedLaw r a 0).withDensity
(fun N => ofReal (denseTreatedLikelihood r a t N)) := by
have hstar (j : Fin 2) : 0 < denseTreatedMean r a 0 j := by
simp [denseTreatedMean]
positivity
have hmean (j : Fin 2) := denseTreatedMean_nonneg hr.le ha hahi ht j
apply Measure.ext_of_singleton
intro N
rw [withDensity_apply _ (measurableSet_singleton N), lintegral_singleton]
simp only [denseTreatedLaw, Measure.pi_singleton]
have hatom (j : Fin 2) := sparse_poisson_likelihood_atom (hmean j) (hstar j) (N j)
simp_rw [hatom]
rw [Finset.prod_mul_distrib]
congr 1
unfold denseTreatedLikelihood
exact (ENNReal.ofReal_prod_of_nonneg (fun j _ =>
mul_nonneg (Real.exp_nonneg _) (pow_nonneg
(div_nonneg (hmean j) (hstar j).le) _))).symm
theorem measurable_denseTreatedLaw
The dense treated counts form a measurable probability experiment. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
Proof (Lean source)
@[fun_prop]
lemma measurable_denseTreatedLaw (r a : ℝ) : Measurable (denseTreatedLaw r a) := by
apply Measure.measurable_of_measurable_coe
intro s hs
have hatom (N : Fin 2 → ℕ) : Measurable (fun t => denseTreatedLaw r a t {N}) := by
simp only [denseTreatedLaw, Measure.pi_singleton, poissonMeasure_singleton]
apply Finset.measurable_prod
intro j _
have hm : Measurable (fun t => denseTreatedMean r a t j) := by
unfold denseTreatedMean
split_ifs <;> fun_prop
fun_prop
simp_rw [← Measure.tsum_indicator_apply_singleton _ s hs]
apply Measurable.tsum
intro N
by_cases hN : N ∈ s
· simpa [Set.indicator_of_mem hN] using hatom N
· simp [Set.indicator_of_notMem hN]
For the displayed parameters, denseTreatedKernel is the object specified by this definition.
Definition (Lean source)
parameters
r a :
ℝ
given by
clause 1
toFun t := denseTreatedLaw r a t
clause 2
measurable' := measurable_denseTreatedLaw r a
The symmetric priors are concentrated on the radius-one support used in the exponential-Gram mixture theorem. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
ν :
hsupp :
conclusion
ν {t | |t| ≤ 1} = 1
Proof (Lean source)
lemma dense_prior_support_radius_one (ν : Measure ℝ) [IsProbabilityMeasure ν]
(hsupp : IsSupportedOnUnitInterval ν) : ν {t | |t| ≤ 1} = 1 := by
apply (mem_ae_iff_prob_eq_one (measurableSet_le (by fun_prop) measurable_const)).1
filter_upwards [(ae_iff).2 hsupp] with t ht
simpa only [symmUnitInterval, Set.mem_Icc, abs_le, neg_le_neg_iff] using ht
theorem dense_product_mixture_tv_le_sqrt_tail
Moment annihilation through K removes all lower terms in (47). The product-mixture testing distance is at most d times the square root of the remaining exponential tail, exactly as in the dense splice. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
tvDist (productPriorPredictive d P.ν₀ (denseTreatedKernel r a)) (productPriorPredictive d P.ν₁ (denseTreatedKernel r a))
≤ d * sqrt (exponentialSeriesTail K (4 * r * a ^ 2))
Proof (Lean source)
lemma dense_product_mixture_tv_le_sqrt_tail {K d : ℕ} (P : AbsMomentMatchedPriors K)
{r a : ℝ} (hr : 0 < r) (ha : 0 ≤ a) (hahi : a ≤ 1 / 2) :
tvDist
(productPriorPredictive d P.ν₀ (denseTreatedKernel r a))
(productPriorPredictive d P.ν₁ (denseTreatedKernel r a)) ≤
d * sqrt (exponentialSeriesTail K (4 * r * a ^ 2)) := by
let : IsProbabilityMeasure P.ν₀ := P.probability₀
let : IsProbabilityMeasure P.ν₁ := P.probability₁
let : IsProbabilityMeasure (denseTreatedLaw r a 0) := by
unfold denseTreatedLaw
infer_instance
have h := momentMatchedProductMixture_tv_le_of_supported
d P.ν₀ P.ν₁ (denseTreatedKernel r a) (denseTreatedLaw r a 0)
(denseTreatedLikelihood r a)
(fun t => by change IsProbabilityMeasure (denseTreatedLaw r a t)
unfold denseTreatedLaw; infer_instance)
(4 * r * a ^ 2) 1 K (by positivity) (by norm_num)
(measurable_denseTreatedLikelihood r a)
(fun t ht N => denseTreatedLikelihood_nonneg hr.le ha hahi ht N)
(fun t ht => denseTreatedLaw_eq_withDensity hr ha hahi ht)
(fun t _ s _ => denseTreatedLikelihood_gram hr a t s)
(dense_prior_support_radius_one P.ν₀ P.supported₀)
(dense_prior_support_radius_one P.ν₁ P.supported₁) P.moments_eq
simpa using h
theorem dense_product_mixture_tv_le_one_eighth
Universal choices of the amplitude coefficient and logarithmic degree make the dense product mixture TV at most 1/8. The calibration condition r a² ≤ η K is provided by the amplitude choice (45). In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ η C : ℝ,
0 < η
0 < C
conclusion 1
K d :
ℕ
P :
r a :
ℝ
2 ≤ d
0 < r
0 ≤ a
a ≤ 1 / 2
C * logAlphabet d ≤ K
r * a ^ 2 ≤ η * K
tvDist (productPriorPredictive d P.ν₀ (denseTreatedKernel r a)) (productPriorPredictive d P.ν₁ (denseTreatedKernel r a))
≤ 1 / 8
Proof (Lean source)
lemma dense_product_mixture_tv_le_one_eighth :
∃ η C : ℝ, 0 < η ∧ 0 < C ∧
∀ (K d : ℕ) (P : AbsMomentMatchedPriors K) (r a : ℝ),
2 ≤ d → 0 < r → 0 ≤ a → a ≤ 1 / 2 →
C * logAlphabet d ≤ K → r * a ^ 2 ≤ η * K →
tvDist
(productPriorPredictive d P.ν₀ (denseTreatedKernel r a))
(productPriorPredictive d P.ν₁ (denseTreatedKernel r a)) ≤ 1 / 8 := by
obtain ⟨b, D, ρ, hb, hD, hρ, htail⟩ :=
FiniteSignedMomentMarkedPoissonMixture.exists_geometric_sqrt_exponentialSeriesTail_bound
1 (by norm_num)
obtain ⟨C, hC, hcal⟩ := sparse_geometric_degree_calibration hD hρ
refine ⟨b / 4, C, by positivity, hC, ?_⟩
intro K d P r a hd hr ha hahi hK hband
have hsmall : 4 * r * a ^ 2 ≤ b * K := by nlinarith
have hdecay : sqrt (exponentialSeriesTail K (4 * r * a ^ 2)) ≤ D * ρ ^ K := by
simpa only [one_mul] using htail K (4 * r * a ^ 2) (by positivity) hsmall
calc
_ ≤ d * sqrt (exponentialSeriesTail K (4 * r * a ^ 2)) :=
dense_product_mixture_tv_le_sqrt_tail P hr ha hahi
_ ≤ d * (D * ρ ^ K) := mul_le_mul_of_nonneg_left hdecay (Nat.cast_nonneg d)
_ ≤ 1 / 8 := hcal d K hd hK
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.LowerDenseFullCounts 15 declarations The untreated counts have a common law.
Dense lower bounds with all observed Poisson counts
The untreated counts have a common law. Mixing the full experiment factors this law out of each predictive mixture, so the treated-count TV bound and the target concentration prove the same lower bound for full-count estimators.
The two untreated outcomes in every cell are independent Poisson counts with the common intensity q = B n (1 - ε)/(2 d).
Full counts, arranged as treated counts followed by untreated counts.
Definition (Lean source)
parameters
d :
ℕ
r a q :
ℝ
def
given by
by letI : IsMarkovKernel (denseProductPoissonKernel d r a) := ⟨denseProductPoissonKernel_probability d r a⟩ exact (denseProductPoissonKernel d r a).prod (Kernel.const _ (denseUntreatedPoissonLaw d q))
theorem denseFullPoissonKernel_apply
The full-count fibre is the product of the treated experiment and its parameter-independent untreated law. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
denseFullPoissonKernel d r a q θ
Proof (Lean source)
-- @node: denseFullPoissonKernel_apply
lemma denseFullPoissonKernel_apply (d : ℕ) (r a q : ℝ) (θ : Fin d → ℝ) :
denseFullPoissonKernel d r a q θ =
(denseProductPoissonKernel d r a θ).prod (denseUntreatedPoissonLaw d q) := by
letI : IsMarkovKernel (denseProductPoissonKernel d r a) :=
⟨denseProductPoissonKernel_probability d r a⟩
letI : IsProbabilityMeasure (denseUntreatedPoissonLaw d q) := by
unfold denseUntreatedPoissonLaw; infer_instance
exact Kernel.prod_apply _ _ _
theorem denseFullPoissonKernel_probability
The full count experiment has probability fibres. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
IsProbabilityMeasure (denseFullPoissonKernel d r a q θ)
Proof (Lean source)
-- @node: denseFullPoissonKernel_probability
lemma denseFullPoissonKernel_probability (d : ℕ) (r a q : ℝ) (θ : Fin d → ℝ) :
IsProbabilityMeasure (denseFullPoissonKernel d r a q θ) := by
rw [denseFullPoissonKernel_apply]
letI := denseProductPoissonKernel_probability d r a θ
unfold denseUntreatedPoissonLaw
infer_instance
theorem dense_full_prior_predictive_eq
The untreated law factors out of the prior-predictive mixture. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
priorPredictive π (denseFullPoissonKernel d r a q)
Proof (Lean source)
-- @node: dense_full_prior_predictive_eq
lemma dense_full_prior_predictive_eq (d : ℕ) (r a q : ℝ)
(π : Measure (Fin d → ℝ)) :
priorPredictive π (denseFullPoissonKernel d r a q) =
(priorPredictive π (denseProductPoissonKernel d r a)).prod
(denseUntreatedPoissonLaw d q) := by
letI : IsProbabilityMeasure (denseUntreatedPoissonLaw d q) := by
unfold denseUntreatedPoissonLaw; infer_instance
apply Measure.ext_of_singleton
rintro ⟨T, U⟩
rw [← Set.singleton_prod_singleton, priorPredictive_apply _ _ ((measurableSet_singleton T).prod (measurableSet_singleton U)),
Measure.prod_prod, priorPredictive_apply _ _ (measurableSet_singleton _)]
simp_rw [denseFullPoissonKernel_apply, Measure.prod_prod]
exact lintegral_mul_const _ ((denseProductPoissonKernel d r a).measurable_coe
(measurableSet_singleton _))
theorem dense_full_prior_predictive_tv_le
Adding all untreated counts cannot increase the predictive TV bound. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
tvDist (priorPredictive π₀ (denseFullPoissonKernel d r a q)) (priorPredictive π₁ (denseFullPoissonKernel d r a q))
≤ tvDist (priorPredictive π₀ (denseProductPoissonKernel d r a)) (priorPredictive π₁ (denseProductPoissonKernel d r a))
Proof (Lean source)
-- @node: dense_full_prior_predictive_tv_le
lemma dense_full_prior_predictive_tv_le (d : ℕ) (r a q : ℝ)
(π₀ π₁ : Measure (Fin d → ℝ)) [IsProbabilityMeasure π₀] [IsProbabilityMeasure π₁] :
tvDist (priorPredictive π₀ (denseFullPoissonKernel d r a q))
(priorPredictive π₁ (denseFullPoissonKernel d r a q)) ≤
tvDist (priorPredictive π₀ (denseProductPoissonKernel d r a))
(priorPredictive π₁ (denseProductPoissonKernel d r a)) := by
letI : IsProbabilityMeasure (denseUntreatedPoissonLaw d q) := by
unfold denseUntreatedPoissonLaw; infer_instance
letI := priorPredictive_isProbability π₀ (denseProductPoissonKernel d r a)
(denseProductPoissonKernel_probability d r a)
letI := priorPredictive_isProbability π₁ (denseProductPoissonKernel d r a)
(denseProductPoissonKernel_probability d r a)
rw [dense_full_prior_predictive_eq, dense_full_prior_predictive_eq]
simpa [tvDist] using tvDist_prod_le_add
(priorPredictive π₀ (denseProductPoissonKernel d r a))
(priorPredictive π₁ (denseProductPoissonKernel d r a))
(denseUntreatedPoissonLaw d q) (denseUntreatedPoissonLaw d q)
def denseCountEquiv
Reorder all four observed counts into treated and untreated blocks, with outcome false at index zero and outcome true at index one.
Definition (Lean source)
parameters
d :
ℕ
given by
clause 2
invFun N := fun o
=> if o.2.1 then N.1 o.1 (if o.2.2 then 1 else 0) else N.2 o.1 (if o.2.2 then 1 else 0)
clause 3
left_inv N := by funext o rcases o with ⟨x, b, y⟩ cases b <;> cases y <;> rfl
clause 5
measurable_toFun := measurable_of_countable _
clause 6
measurable_invFun := measurable_of_countable _
theorem denseFullPoissonKernel_eq_observedCounts
The full dense kernel is exactly the independent Poisson counts of the legal observed law, after a reversible reordering of all four coordinates. No coordinate of the observed experiment is discarded. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
0 < d
ε a B :
ℝ
n :
ℕ
hε :
0 < ε
hεhi :
ε ≤ 1 / 2
ha :
0 ≤ a
hahi :
a ≤ 1 / 2
θ :
Fin d → ℝ
hθ :
∀ x, |θ x| ≤ 1
conclusion
Measure.map (denseCountEquiv d) (Measure.pi fun o : Obs d => poissonMeasure (toNNReal (B * n * jointMass (denseObservedLaw hd ε a hε hεhi ha hahi θ hθ) o.1 o.2.1 o.2.2)))
= denseFullPoissonKernel d (B * n * ε / d) a (B * n * (1 - ε) / (2 * d)) θ
Proof (Lean source)
-- @node: denseFullPoissonKernel_eq_observedCounts
lemma denseFullPoissonKernel_eq_observedCounts {d : ℕ} (hd : 0 < d)
(ε a B : ℝ) (n : ℕ) (hε : 0 < ε) (hεhi : ε ≤ 1 / 2)
(ha : 0 ≤ a) (hahi : a ≤ 1 / 2)
(θ : Fin d → ℝ) (hθ : ∀ x, |θ x| ≤ 1) :
Measure.map (denseCountEquiv d)
(Measure.pi fun o : Obs d => poissonMeasure (toNNReal
(B * n * jointMass (denseObservedLaw hd ε a hε hεhi ha hahi θ hθ)
o.1 o.2.1 o.2.2))) =
denseFullPoissonKernel d (B * n * ε / d) a (B * n * (1 - ε) / (2 * d)) θ := by
apply Measure.ext_of_singleton
rintro ⟨T, U⟩
rw [Measure.map_apply (denseCountEquiv d).measurable (measurableSet_singleton _),
denseFullPoissonKernel_apply]
have hpre : (denseCountEquiv d) ⁻¹' {(T, U)} =
{(denseCountEquiv d).symm (T, U)} := by
ext N
simp only [Set.mem_preimage, Set.mem_singleton_iff]
exact (denseCountEquiv d).apply_eq_iff_eq_symm_apply
rw [hpre, Measure.pi_singleton, ← Set.singleton_prod_singleton, Measure.prod_prod]
change (∏ o : Obs d, _) =
(Measure.pi (fun x => denseTreatedLaw (B * n * ε / d) a (θ x))) {T} *
(denseUntreatedPoissonLaw d (B * n * (1 - ε) / (2 * d))) {U}
simp only [denseUntreatedPoissonLaw, denseTreatedLaw, Measure.pi_singleton]
simp only [Obs, Fintype.prod_prod_type]
have hmean := denseObservedLaw_poissonMeans hd ε a B n hε hεhi ha hahi θ hθ
simp only [Fintype.prod_bool]
dsimp only [denseCountEquiv, symm, MeasurableEquiv.coe_mk, Equiv.coe_fn_mk]
simp_rw [(hmean _ _).1, (hmean _ _).2]
simp [Fin.prod_univ_two, denseTreatedMean]
rw [← Finset.prod_mul_distrib]
apply Finset.prod_congr rfl
intro x _
ring
theorem dense_full_product_prior_bayesRisk_lower
The dense fuzzy-testing step (48): concentration and likelihood TV are both derived; the alphabet condition makes target deviations negligible. In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ η C : ℝ,
0 < η
0 < C
conclusion 1
K d :
ℕ
P :
r a q :
ℝ
2 ≤ d
0 < r
0 < a
a ≤ 1 / 2
C * logAlphabet d ≤ K
r * a ^ 2 ≤ η * K
Delta :
a * ((∫ t, |t| ∂P.ν₁) - (∫ t, |t| ∂P.ν₀)) / 2
128 * a ^ 2 ≤ d * Delta ^ 2
0 < Delta
Measurable est
ofReal (5 * Delta ^ 2 / 256)
≤ max (bayesSquaredRisk (productPrior d P.ν₀) (denseFullPoissonKernel d r a q) (densePriorTarget d a) est) (bayesSquaredRisk (productPrior d P.ν₁) (denseFullPoissonKernel d r a q) (densePriorTarget d a) est)
Proof (Lean source)
-- @node: dense_full_product_prior_bayesRisk_lower
lemma dense_full_product_prior_bayesRisk_lower :
∃ η C : ℝ, 0 < η ∧ 0 < C ∧
∀ (K d : ℕ) (P : AbsMomentMatchedPriors K) (r a q : ℝ),
2 ≤ d → 0 < r → 0 < a → a ≤ 1 / 2 →
C * logAlphabet d ≤ K → r * a ^ 2 ≤ η * K →
let Delta := a * ((∫ t, |t| ∂P.ν₁) - (∫ t, |t| ∂P.ν₀)) / 2
128 * a ^ 2 ≤ d * Delta ^ 2 → 0 < Delta →
∀ est : ((Fin d → (Fin 2 → ℕ)) × (Fin d → (Fin 2 → ℕ))) → ℝ, Measurable est →
ofReal (5 * Delta ^ 2 / 256) ≤
max (bayesSquaredRisk (productPrior d P.ν₀)
(denseFullPoissonKernel d r a q) (densePriorTarget d a) est)
(bayesSquaredRisk (productPrior d P.ν₁)
(denseFullPoissonKernel d r a q) (densePriorTarget d a) est) := by
obtain ⟨η, C, hη, hC, htvbound⟩ := dense_product_mixture_tv_le_one_eighth
refine ⟨η, C, hη, hC, ?_⟩
intro K d P r a q hd hr ha hahi hK hband
dsimp only
intro hsize hDelta est hest
let := P.probability₀
let := P.probability₁
let : IsProbabilityMeasure (productPrior d P.ν₀) := by unfold productPrior; infer_instance
let : IsProbabilityMeasure (productPrior d P.ν₁) := by unfold productPrior; infer_instance
let Delta := a * ((∫ t, |t| ∂P.ν₁) - (∫ t, |t| ∂P.ν₀)) / 2
have ht (ν : Measure ℝ) [IsProbabilityMeasure ν]
(hsupp : IsSupportedOnUnitInterval ν) (hsym : IsSymmetric ν) :
(productPrior d ν).real
{θ | Delta / 4 < |densePriorTarget d a θ - (1 / 2 + a * (∫ t, |t| ∂ν) / 2)|} ≤ 1 / 8 := by
apply (dense_product_prior_target_tail_le (by omega) ν hsupp hsym a
(show 0 < Delta / 4 by positivity)).trans
apply (div_le_iff₀ (by positivity : 0 < (d : ℝ) * (Delta / 4) ^ 2)).2
have hsize' : 128 * a ^ 2 ≤ d * Delta ^ 2 := hsize
nlinarith only [hsize']
have htv : tvDist
(priorPredictive (productPrior d P.ν₀) (denseFullPoissonKernel d r a q))
(priorPredictive (productPrior d P.ν₁) (denseFullPoissonKernel d r a q)) ≤ 1 / 8 := by
apply (dense_full_prior_predictive_tv_le d r a q _ _).trans
rw [dense_product_prior_predictive_eq, dense_product_prior_predictive_eq]
exact htvbound K d P r a hd hr ha.le hahi hK hband
have hb := twoFuzzyHypotheses_bayesRisk_lower (productPrior d P.ν₀) (productPrior d P.ν₁)
(denseFullPoissonKernel d r a q) (densePriorTarget d a)
(denseFullPoissonKernel_probability d r a q) est hest (measurable_densePriorTarget d a)
(1 / 2 + a * (∫ t, |t| ∂P.ν₀) / 2) (1 / 2 + a * (∫ t, |t| ∂P.ν₁) / 2)
Delta (Delta / 4) (1 / 8) (1 / 8) (1 / 8)
(by dsimp [Delta] at *; linarith)
(by dsimp [Delta]; linarith)
(ht P.ν₀ P.supported₀ P.symmetric₀) (ht P.ν₁ P.supported₁ P.symmetric₁) htv
convert hb using 1
congr 1
ring
theorem dense_full_symmetric_bayesRisk_lower
The Fejér symmetric priors attain the dense risk order a²/K². The separation is proved by the approximation certificate, never assumed. In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ η C : ℝ,
0 < η
0 < C
conclusion 1
K d :
ℕ
r a q :
ℝ
0 < K
2 ≤ d
0 < r
0 < a
a ≤ 1 / 2
C * logAlphabet d ≤ K
r * a ^ 2 ≤ η * K
1280000 * (K : ℝ) ^ 2 ≤ d
∃ P : AbsMomentMatchedPriors K,
Measurable est → ofReal ((5 / 2560000 : ℝ) * a ^ 2 / (K : ℝ) ^ 2)
≤ max (bayesSquaredRisk (productPrior d P.ν₀) (denseFullPoissonKernel d r a q) (densePriorTarget d a) est) (bayesSquaredRisk (productPrior d P.ν₁) (denseFullPoissonKernel d r a q) (densePriorTarget d a) est)
Proof (Lean source)
-- @node: dense_full_symmetric_bayesRisk_lower
lemma dense_full_symmetric_bayesRisk_lower :
∃ η C : ℝ, 0 < η ∧ 0 < C ∧
∀ (K d : ℕ) (r a q : ℝ),
0 < K → 2 ≤ d → 0 < r → 0 < a → a ≤ 1 / 2 →
C * logAlphabet d ≤ K → r * a ^ 2 ≤ η * K →
1280000 * (K : ℝ) ^ 2 ≤ d →
∃ P : AbsMomentMatchedPriors K,
∀ est : ((Fin d → (Fin 2 → ℕ)) × (Fin d → (Fin 2 → ℕ))) → ℝ, Measurable est →
ofReal ((5 / 2560000 : ℝ) * a ^ 2 / (K : ℝ) ^ 2) ≤
max (bayesSquaredRisk (productPrior d P.ν₀)
(denseFullPoissonKernel d r a q) (densePriorTarget d a) est)
(bayesSquaredRisk (productPrior d P.ν₁)
(denseFullPoissonKernel d r a q) (densePriorTarget d a) est) := by
obtain ⟨η, C, hη, hC, hrisk⟩ := dense_full_product_prior_bayesRisk_lower
refine ⟨η, C, hη, hC, ?_⟩
intro K d r a q hK hd hr ha hahi hlog hband hsize
obtain ⟨P, hgap⟩ := dense_symmetric_priors_absGap_lower K hK
refine ⟨P, ?_⟩
intro est hest
let Delta := a * ((∫ t, |t| ∂P.ν₁) - (∫ t, |t| ∂P.ν₀)) / 2
have hKR : (0 : ℝ) < K := Nat.cast_pos.mpr hK
have hlo : a / (100 * K) ≤ Delta := by
have hh := mul_le_mul_of_nonneg_left hgap ha.le
dsimp [Delta]
calc
_ = a * ((1 / 50 : ℝ) / K) / 2 := by ring
_ ≤ _ := div_le_div_of_nonneg_right hh (by norm_num)
have hD : 0 < Delta := (by positivity : 0 < a / (100 * K)).trans_le hlo
have hsq : a ^ 2 / (10000 * (K : ℝ) ^ 2) ≤ Delta ^ 2 := by
have h := pow_le_pow_left₀ (by positivity : 0 ≤ a / (100 * K)) hlo 2
convert h using 1 <;> first | ring | rfl
have hsize' : 128 * a ^ 2 ≤ d * Delta ^ 2 := by
have hb : 128 * a ^ 2 ≤ (d : ℝ) * (a ^ 2 / (10000 * (K : ℝ) ^ 2)) := by
rw [← mul_div_assoc]
apply (le_div_iff₀ (by positivity : 0 < 10000 * (K : ℝ) ^ 2)).2
nlinarith only [mul_le_mul_of_nonneg_right hsize (sq_nonneg a)]
exact hb.trans (mul_le_mul_of_nonneg_left hsq (Nat.cast_nonneg d))
have hb := hrisk K d P r a q hd hr ha hahi hlog hband hsize' hD est hest
apply le_trans (ENNReal.ofReal_le_ofReal ?_) hb
calc
_ = (5 / 256 : ℝ) * (a ^ 2 / (10000 * (K : ℝ) ^ 2)) := by ring
_ ≤ (5 / 256 : ℝ) * Delta ^ 2 := mul_le_mul_of_nonneg_left hsq (by norm_num)
_ = _ := by ring
theorem dense_full_logDegree_bayesRisk_lower
Above a universal alphabet cutoff the logarithmic dense experiment has risk at least a universal multiple of a²/K_d², with concentration derived. In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ (η C : ℝ) (D : ℕ),
0 < η
2 ≤ C
2 ≤ D
conclusion 1
d :
ℕ
r a q :
ℝ
D ≤ d
0 < r
0 < a
a ≤ 1 / 2
r * a ^ 2 ≤ η * lowerLogDegree C d
∃ P : AbsMomentMatchedPriors (lowerLogDegree C d),
Measurable est → ofReal ((5 / 2560000 : ℝ) * a ^ 2 / (lowerLogDegree C d : ℝ) ^ 2)
≤ max (bayesSquaredRisk (productPrior d P.ν₀) (denseFullPoissonKernel d r a q) (densePriorTarget d a) est) (bayesSquaredRisk (productPrior d P.ν₁) (denseFullPoissonKernel d r a q) (densePriorTarget d a) est)
Proof (Lean source)
-- @node: dense_full_logDegree_bayesRisk_lower
lemma dense_full_logDegree_bayesRisk_lower :
∃ (η C : ℝ) (D : ℕ), 0 < η ∧ 2 ≤ C ∧ 2 ≤ D ∧
∀ (d : ℕ) (r a q : ℝ), D ≤ d → 0 < r → 0 < a → a ≤ 1 / 2 →
r * a ^ 2 ≤ η * lowerLogDegree C d →
∃ P : AbsMomentMatchedPriors (lowerLogDegree C d),
∀ est : ((Fin d → (Fin 2 → ℕ)) × (Fin d → (Fin 2 → ℕ))) → ℝ, Measurable est →
ofReal ((5 / 2560000 : ℝ) * a ^ 2 /
(lowerLogDegree C d : ℝ) ^ 2) ≤
max (bayesSquaredRisk (productPrior d P.ν₀)
(denseFullPoissonKernel d r a q) (densePriorTarget d a) est)
(bayesSquaredRisk (productPrior d P.ν₁)
(denseFullPoissonKernel d r a q) (densePriorTarget d a) est) := by
obtain ⟨η, C₀, hη, hC₀, hrisk⟩ := dense_full_symmetric_bayesRisk_lower
let C := max C₀ 2
have hC : 2 ≤ C := le_max_right _ _
obtain ⟨D, hD, hcut⟩ := sparse_packet_alphabet_cutoff
(cp := (1 / 100 : ℝ)) (by norm_num) hC
refine ⟨η, C, D, hη, hC, hD, ?_⟩
intro d r a q hd hr ha hahi hband
have hd2 := hD.trans hd
obtain ⟨hK, hlog, _⟩ := lowerLogDegree_bounds hC hd2
have hsmall := hcut d hd 2 0 (by norm_num) (by norm_num) (by norm_num)
norm_num at hsmall
have hsize : 1280000 * (lowerLogDegree C d : ℝ) ^ 2 ≤ d := by linarith
have hL : 0 ≤ logAlphabet d := by
unfold logAlphabet
apply Real.log_nonneg
have hdR : (1 : ℝ) ≤ d := by exact_mod_cast (by omega : 1 ≤ d)
have he : 1 ≤ exp 1 := Real.one_le_exp (by norm_num)
nlinarith
exact hrisk (lowerLogDegree C d) d r a q (by omega) hd2 hr ha hahi
((mul_le_mul_of_nonneg_right (le_max_left C₀ 2) hL).trans hlog) hband hsize
def denseFullObservedPoissonKernel
Return the full Poisson experiment to the usual observed histogram alphabet, using the inverse of the count reordering.
Definition (Lean source)
parameters
d :
ℕ
r a q :
ℝ
given by
theorem denseFullObservedPoissonKernel_eq
On the prior support the full histogram kernel is precisely the independent four-cell observed Poisson experiment. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
0 < d
ε a B :
ℝ
n :
ℕ
hε :
0 < ε
hεhi :
ε ≤ 1 / 2
ha :
0 ≤ a
hahi :
a ≤ 1 / 2
θ :
Fin d → ℝ
hθ :
∀ x, |θ x| ≤ 1
conclusion
denseFullObservedPoissonKernel d (B * n * ε / d) a (B * n * (1 - ε) / (2 * d)) θ
= Measure.pi fun o : Obs d => poissonMeasure (toNNReal (B * n * jointMass (denseObservedLaw hd ε a hε hεhi ha hahi θ hθ) o.1 o.2.1 o.2.2))
Proof (Lean source)
-- @node: denseFullObservedPoissonKernel_eq
lemma denseFullObservedPoissonKernel_eq {d : ℕ} (hd : 0 < d)
(ε a B : ℝ) (n : ℕ) (hε : 0 < ε) (hεhi : ε ≤ 1 / 2)
(ha : 0 ≤ a) (hahi : a ≤ 1 / 2)
(θ : Fin d → ℝ) (hθ : ∀ x, |θ x| ≤ 1) :
denseFullObservedPoissonKernel d (B * n * ε / d) a
(B * n * (1 - ε) / (2 * d)) θ =
Measure.pi fun o : Obs d => poissonMeasure (toNNReal
(B * n * jointMass (denseObservedLaw hd ε a hε hεhi ha hahi θ hθ)
o.1 o.2.1 o.2.2)) := by
rw [denseFullObservedPoissonKernel,
Kernel.map_apply _ (denseCountEquiv d).symm.measurable]
rw [← denseFullPoissonKernel_eq_observedCounts hd ε a B n hε hεhi ha hahi θ hθ]
rw [Measure.map_map (denseCountEquiv d).symm.measurable (denseCountEquiv d).measurable]
simp only [Function.comp_def, MeasurableEquiv.symm_apply_apply, Measure.map_id']
theorem dense_full_observed_bayesRisk_eq
Reordering all observed counts preserves the squared Bayes risk exactly. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
bayesSquaredRisk π (denseFullObservedPoissonKernel d r a q) target est
= bayesSquaredRisk π (denseFullPoissonKernel d r a q) target (fun N => est ((denseCountEquiv d).symm N))
Proof (Lean source)
-- @node: dense_full_observed_bayesRisk_eq
lemma dense_full_observed_bayesRisk_eq (d : ℕ) (r a q : ℝ)
(π : Measure (Fin d → ℝ)) (target : (Fin d → ℝ) → ℝ)
(est : (Obs d → ℕ) → ℝ) (hest : Measurable est) :
bayesSquaredRisk π (denseFullObservedPoissonKernel d r a q) target est =
bayesSquaredRisk π (denseFullPoissonKernel d r a q) target
(fun N => est ((denseCountEquiv d).symm N)) := by
unfold bayesSquaredRisk squaredRisk
apply lintegral_congr
intro θ
rw [denseFullObservedPoissonKernel,
Kernel.map_apply _ (denseCountEquiv d).symm.measurable]
exact lintegral_map (by fun_prop) (denseCountEquiv d).symm.measurable
theorem dense_full_observed_logDegree_bayesRisk_lower
The dense lower bound holds for arbitrary estimators using every observed histogram count, with the legal observed value as its target. The priors, gap, concentration and full-likelihood TV bound are proved. In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ (η C : ℝ) (D : ℕ),
0 < η
2 ≤ C
2 ≤ D
conclusion 1
d :
ℕ
r a q ε :
ℝ
hd :
0 < d
hε :
0 < ε
hεhi :
ε ≤ 1 / 2
ha :
0 ≤ a
hahi :
a ≤ 1 / 2
D ≤ d
0 < r
0 < a
r * a ^ 2 ≤ η * lowerLogDegree C d
∃ P : AbsMomentMatchedPriors (lowerLogDegree C d),
∀ est : (Obs d → ℕ) → ℝ,
Measurable est → ofReal ((5 / 2560000 : ℝ) * a ^ 2 / (lowerLogDegree C d : ℝ) ^ 2)
≤ max (bayesSquaredRisk (productPrior d P.ν₀) (denseFullObservedPoissonKernel d r a q) (fun θ => observedValue (denseObservedModel hd ε a hε hεhi ha hahi θ).1) est) (bayesSquaredRisk (productPrior d P.ν₁) (denseFullObservedPoissonKernel d r a q) (fun θ => observedValue (denseObservedModel hd ε a hε hεhi ha hahi θ).1) est)
Proof (Lean source)
-- @node: dense_full_observed_logDegree_bayesRisk_lower
lemma dense_full_observed_logDegree_bayesRisk_lower :
∃ (η C : ℝ) (D : ℕ), 0 < η ∧ 2 ≤ C ∧ 2 ≤ D ∧
∀ (d : ℕ) (r a q ε : ℝ) (hd : 0 < d)
(hε : 0 < ε) (hεhi : ε ≤ 1 / 2) (ha : 0 ≤ a) (hahi : a ≤ 1 / 2),
D ≤ d → 0 < r → 0 < a →
r * a ^ 2 ≤ η * lowerLogDegree C d →
∃ P : AbsMomentMatchedPriors (lowerLogDegree C d),
∀ est : (Obs d → ℕ) → ℝ, Measurable est →
ofReal ((5 / 2560000 : ℝ) * a ^ 2 /
(lowerLogDegree C d : ℝ) ^ 2) ≤
max (bayesSquaredRisk (productPrior d P.ν₀)
(denseFullObservedPoissonKernel d r a q)
(fun θ => observedValue (denseObservedModel hd ε a hε hεhi ha hahi θ).1) est)
(bayesSquaredRisk (productPrior d P.ν₁)
(denseFullObservedPoissonKernel d r a q)
(fun θ => observedValue (denseObservedModel hd ε a hε hεhi ha hahi θ).1) est) := by
obtain ⟨η, C, D, hη, hC, hD, hbound⟩ := dense_full_logDegree_bayesRisk_lower
refine ⟨η, C, D, hη, hC, hD, ?_⟩
intro d r a q ε hd hε hεhi ha hahi hcut hr hapos hband
obtain ⟨P, hP⟩ := hbound d r a q hcut hr hapos hahi hband
refine ⟨P, ?_⟩
intro est hest
simp_rw [denseObservedModel_value]
rw [dense_full_observed_bayesRisk_eq d r a q _ _ est hest,
dense_full_observed_bayesRisk_eq d r a q _ _ est hest]
exact hP _ (hest.comp (denseCountEquiv d).symm.measurable)
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.LowerDenseObserved 16 declarations Construct the boundary-propensity law in (45) and identify its target with (46), the target used by the proved dense concentration and testing bounds.
Legal observed laws for the dense lower experiment
Construct the boundary-propensity law in (45) and identify its target with (46), the target used by the proved dense concentration and testing bounds.
def denseObservedMass
Four observed atom masses for uniform cells, boundary propensity ε, untreated mean one half and treated mean one half plus a θ.
Definition (Lean source)
parameters
def
denseObservedMass ε a θ o :
ℝ
given by
if o.2.1 then ε * (1 / 2 + if o.2.2 then a * θ o.1 else -(a * θ o.1)) / d else (1 - ε) / (2 * d)
theorem denseObservedMass_nonneg
Legal amplitudes make every observed atom nonnegative. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
ε a :
ℝ
hε :
0 ≤ ε
hεhi :
ε ≤ 1 / 2
ha :
0 ≤ a
hahi :
a ≤ 1 / 2
θ :
Fin d → ℝ
hθ :
∀ x, |θ x| ≤ 1
o :
Obs d
conclusion
0 ≤ denseObservedMass ε a θ o
Proof (Lean source)
-- @node: denseObservedMass_nonneg
lemma denseObservedMass_nonneg {d : ℕ} {ε a : ℝ} (hε : 0 ≤ ε)
(hεhi : ε ≤ 1 / 2) (ha : 0 ≤ a) (hahi : a ≤ 1 / 2)
(θ : Fin d → ℝ) (hθ : ∀ x, |θ x| ≤ 1) (o : Obs d) :
0 ≤ denseObservedMass ε a θ o := by
have hat : |a * θ o.1| ≤ 1 / 2 := by
rw [abs_mul, abs_of_nonneg ha]
exact (mul_le_mul_of_nonneg_left (hθ o.1) ha).trans (by simpa using hahi)
have hb := abs_le.mp hat
unfold denseObservedMass
split_ifs <;> apply div_nonneg <;>
first | positivity | apply mul_nonneg hε <;> linarith | linarith
theorem denseObservedMass_sum
The four atom masses sum to one over the uniform finite alphabet. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
(∑ o : Obs d, denseObservedMass ε a θ o) = 1
Proof (Lean source)
-- @node: denseObservedMass_sum
lemma denseObservedMass_sum {d : ℕ} (hd : 0 < d) (ε a : ℝ)
(θ : Fin d → ℝ) : (∑ o : Obs d, denseObservedMass ε a θ o) = 1 := by
have hdR : (d : ℝ) ≠ 0 := by exact_mod_cast (Nat.ne_of_gt hd)
calc
_ = ∑ x : Fin d, (1 / (d : ℝ)) := by
simp only [Fintype.sum_prod_type]
apply Finset.sum_congr rfl
intro x _
simp [denseObservedMass]
ring
_ = 1 := by simp [hdR]
def denseObservedLaw
The dense construction is an actual probability law, including the endpoint amplitudes where a Bernoulli atom can vanish.
Definition (Lean source)
parameters
d :
ℕ
hd :
0 < d
ε a :
ℝ
hε :
0 < ε
hεhi :
ε ≤ 1 / 2
ha :
0 ≤ a
hahi :
a ≤ 1 / 2
θ :
Fin d → ℝ
hθ :
∀ x, |θ x| ≤ 1
def
given by
⟨PMF.ofFintype (fun o => ofReal (denseObservedMass ε a θ o)) (by rw [← ENNReal.ofReal_sum_of_nonneg (fun o _ => denseObservedMass_nonneg hε.le hεhi ha hahi θ hθ o), denseObservedMass_sum hd] simp)⟩
theorem denseObservedLaw_jointMass
The probability-law construction preserves the specified real masses. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
0 < d
ε a :
ℝ
hε :
0 < ε
hεhi :
ε ≤ 1 / 2
ha :
0 ≤ a
hahi :
a ≤ 1 / 2
θ :
Fin d → ℝ
hθ :
∀ x, |θ x| ≤ 1
x :
Fin d
b y :
conclusion
jointMass (denseObservedLaw hd ε a hε hεhi ha hahi θ hθ) x b y
= denseObservedMass ε a θ (x, b, y)
Proof (Lean source)
-- @node: denseObservedLaw_jointMass
lemma denseObservedLaw_jointMass {d : ℕ} (hd : 0 < d) (ε a : ℝ)
(hε : 0 < ε) (hεhi : ε ≤ 1 / 2) (ha : 0 ≤ a) (hahi : a ≤ 1 / 2)
(θ : Fin d → ℝ) (hθ : ∀ x, |θ x| ≤ 1) (x : Fin d) (b y : Bool) :
jointMass (denseObservedLaw hd ε a hε hεhi ha hahi θ hθ) x b y =
denseObservedMass ε a θ (x, b, y) := by
simp [jointMass, denseObservedLaw, ENNReal.toReal_ofReal
(denseObservedMass_nonneg hε.le hεhi ha hahi θ hθ (x, b, y))]
theorem denseObservedLaw_parameters
Uniform cells, boundary propensity and the two specified outcome means hold exactly; thus the dense construction belongs to the observed class. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
0 < d
ε a :
ℝ
hε :
0 < ε
hεhi :
ε ≤ 1 / 2
ha :
0 ≤ a
hahi :
a ≤ 1 / 2
θ :
Fin d → ℝ
hθ :
∀ x, |θ x| ≤ 1
conclusion
P :
denseObservedLaw hd ε a hε hεhi ha hahi θ hθ
ObservedClass ε P
x :
cellMass P x = 1 / d
propensity P x = ε
outcomeMean P false x = 1 / 2
outcomeMean P true x = 1 / 2 + a * θ x
Proof (Lean source)
-- @node: denseObservedLaw_parameters
lemma denseObservedLaw_parameters {d : ℕ} (hd : 0 < d) (ε a : ℝ)
(hε : 0 < ε) (hεhi : ε ≤ 1 / 2) (ha : 0 ≤ a) (hahi : a ≤ 1 / 2)
(θ : Fin d → ℝ) (hθ : ∀ x, |θ x| ≤ 1) :
let P := denseObservedLaw hd ε a hε hεhi ha hahi θ hθ
ObservedClass ε P ∧ ∀ x,
cellMass P x = 1 / d ∧ propensity P x = ε ∧
outcomeMean P false x = 1 / 2 ∧ outcomeMean P true x = 1 / 2 + a * θ x := by
dsimp only
let P := denseObservedLaw hd ε a hε hεhi ha hahi θ hθ
have hdR : (d : ℝ) ≠ 0 := by exact_mod_cast (Nat.ne_of_gt hd)
have hεne : ε ≠ 0 := hε.ne'
have hone : 1 - ε ≠ 0 := by linarith
have hcell (x : Fin d) : cellMass P x = 1 / d := by
simp [P, cellMass, denseObservedLaw_jointMass, denseObservedMass]
ring
have harm (x : Fin d) : armMass P true x = ε / d := by
simp [P, armMass, denseObservedLaw_jointMass, denseObservedMass]
ring
have hprop (x : Fin d) : propensity P x = ε := by
rw [propensity, harm, hcell]
field_simp
refine ⟨⟨fun x _ => ⟨?_, ?_⟩⟩, fun x => ⟨hcell x, hprop x, ?_, ?_⟩⟩
· rw [hprop]
· rw [hprop]; linarith
· simp [outcomeMean, armMass, denseObservedLaw_jointMass, denseObservedMass]
field_simp
· rw [outcomeMean, harm]
simp [denseObservedLaw_jointMass, denseObservedMass]
field_simp
theorem denseObservedLaw_value
The legal dense observed target is precisely the target already used in the dense Bayes-risk bound, rather than an auxiliary surrogate. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
0 < d
ε a :
ℝ
hε :
0 < ε
hεhi :
ε ≤ 1 / 2
ha :
0 ≤ a
hahi :
a ≤ 1 / 2
θ :
Fin d → ℝ
hθ :
∀ x, |θ x| ≤ 1
conclusion
observedValue (denseObservedLaw hd ε a hε hεhi ha hahi θ hθ) = densePriorTarget d a θ
Proof (Lean source)
-- @node: denseObservedLaw_value
lemma denseObservedLaw_value {d : ℕ} (hd : 0 < d) (ε a : ℝ)
(hε : 0 < ε) (hεhi : ε ≤ 1 / 2) (ha : 0 ≤ a) (hahi : a ≤ 1 / 2)
(θ : Fin d → ℝ) (hθ : ∀ x, |θ x| ≤ 1) :
observedValue (denseObservedLaw hd ε a hε hεhi ha hahi θ hθ) =
densePriorTarget d a θ := by
have hp := (denseObservedLaw_parameters hd ε a hε hεhi ha hahi θ hθ).2
have hdR : (d : ℝ) ≠ 0 := by exact_mod_cast (Nat.ne_of_gt hd)
have hmax (x : Fin d) : max (1 / 2 : ℝ) (1 / 2 + a * θ x) =
1 / 2 + a * densePositivePart (θ x) := by
rw [densePositivePart, min_eq_right
(max_le ((le_abs_self _).trans (hθ x)) (by norm_num))]
by_cases ht : 0 ≤ θ x
· rw [max_eq_left ht, max_eq_right (by nlinarith)]
· rw [max_eq_right (le_of_not_ge ht), max_eq_left (by nlinarith)]
ring
unfold observedValue
simp_rw [(hp _).1, (hp _).2.2.1, (hp _).2.2.2, hmax]
simp only [mul_add, Finset.sum_add_distrib, ← Finset.mul_sum]
simp [densePriorTarget, hdR]
ring
theorem denseObservedLaw_poissonMeans
In the full observed Poisson experiment the treated intensities are exactly those in the proved dense likelihood calculation (47), and the untreated intensities are common to every parameter vector. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
0 < d
ε a B :
ℝ
n :
ℕ
hε :
0 < ε
hεhi :
ε ≤ 1 / 2
ha :
0 ≤ a
hahi :
a ≤ 1 / 2
θ :
Fin d → ℝ
hθ :
∀ x, |θ x| ≤ 1
x :
Fin d
y :
conclusion
conclusion 1
B * n * jointMass (denseObservedLaw hd ε a hε hεhi ha hahi θ hθ) x true y
= denseTreatedMean (B * n * ε / d) a (θ x) (if y then 1 else 0)
conclusion 2
B * n * jointMass (denseObservedLaw hd ε a hε hεhi ha hahi θ hθ) x false y
= B * n * (1 - ε) / (2 * d)
Proof (Lean source)
-- @node: denseObservedLaw_poissonMeans
lemma denseObservedLaw_poissonMeans {d : ℕ} (hd : 0 < d) (ε a B : ℝ) (n : ℕ)
(hε : 0 < ε) (hεhi : ε ≤ 1 / 2) (ha : 0 ≤ a) (hahi : a ≤ 1 / 2)
(θ : Fin d → ℝ) (hθ : ∀ x, |θ x| ≤ 1) (x : Fin d) (y : Bool) :
B * n * jointMass (denseObservedLaw hd ε a hε hεhi ha hahi θ hθ) x true y =
denseTreatedMean (B * n * ε / d) a (θ x) (if y then 1 else 0) ∧
B * n * jointMass (denseObservedLaw hd ε a hε hεhi ha hahi θ hθ) x false y =
B * n * (1 - ε) / (2 * d) := by
rw [denseObservedLaw_jointMass, denseObservedLaw_jointMass]
cases y <;> constructor <;> simp [denseObservedMass, denseTreatedMean] <;> ring
theorem denseObservedLaw_poissonTotal
The full dense Poisson experiment has total mean Bn, with no random normalizer penalty, as required in the transfer following (49). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
0 < d
ε a B :
ℝ
n :
ℕ
hε :
0 < ε
hεhi :
ε ≤ 1 / 2
ha :
0 ≤ a
hahi :
a ≤ 1 / 2
θ :
Fin d → ℝ
hθ :
∀ x, |θ x| ≤ 1
conclusion
(∑ o : Obs d, B * n * jointMass (denseObservedLaw hd ε a hε hεhi ha hahi θ hθ) o.1 o.2.1 o.2.2)
= B * n
Proof (Lean source)
-- @node: denseObservedLaw_poissonTotal
lemma denseObservedLaw_poissonTotal {d : ℕ} (hd : 0 < d) (ε a B : ℝ) (n : ℕ)
(hε : 0 < ε) (hεhi : ε ≤ 1 / 2) (ha : 0 ≤ a) (hahi : a ≤ 1 / 2)
(θ : Fin d → ℝ) (hθ : ∀ x, |θ x| ≤ 1) :
(∑ o : Obs d, B * n *
jointMass (denseObservedLaw hd ε a hε hεhi ha hahi θ hθ) o.1 o.2.1 o.2.2) =
B * n := by
simp_rw [denseObservedLaw_jointMass]
rw [← Finset.mul_sum]
have hmass : (∑ o : Obs d, denseObservedMass ε a θ (o.1, o.2.1, o.2.2)) = 1 :=
denseObservedMass_sum hd ε a θ
rw [hmass, mul_one]
Clip only outside the supported dense prior interval.
theorem denseClippedParameter_abs_le
Every extended parameter gives a legal Bernoulli mean. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
Proof (Lean source)
-- @node: denseClippedParameter_abs_le
lemma denseClippedParameter_abs_le (t : ℝ) : |denseClippedParameter t| ≤ 1 := by
apply abs_le.mpr
exact ⟨le_min (by norm_num) (le_max_left _ _), min_le_left _ _⟩
theorem denseClippedParameter_eq
Clipping leaves all prior-supported parameters unchanged. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
-- @node: denseClippedParameter_eq
lemma denseClippedParameter_eq {t : ℝ} (ht : |t| ≤ 1) : denseClippedParameter t = t := by
rw [denseClippedParameter, max_eq_right (abs_le.mp ht).1,
min_eq_right (abs_le.mp ht).2]
theorem densePositivePart_clipped
Clipping the negative tail does not change the bounded positive part. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
Proof (Lean source)
-- @node: densePositivePart_clipped
lemma densePositivePart_clipped (t : ℝ) :
densePositivePart (denseClippedParameter t) = densePositivePart t := by
unfold densePositivePart denseClippedParameter
by_cases hlo : t ≤ -1
· rw [max_eq_left hlo]
norm_num
rw [max_eq_right (by linarith : t ≤ 0)]
norm_num
· rw [max_eq_right (le_of_lt (lt_of_not_ge hlo))]
by_cases hhi : 1 ≤ t
· rw [min_eq_left hhi, max_eq_left (by linarith : 0 ≤ t)]
norm_num
exact hhi
· rw [min_eq_right (le_of_lt (lt_of_not_ge hhi))]
def denseObservedModel
A legal dense observed law for every parameter vector; on the prior support this is exactly (45).
Definition (Lean source)
parameters
def
given by
⟨denseObservedLaw hd ε a hε hεhi ha hahi (fun x => denseClippedParameter (θ x)) (fun _ => denseClippedParameter_abs_le _), (denseObservedLaw_parameters hd ε a hε hεhi ha hahi (fun x => denseClippedParameter (θ x)) (fun _ => denseClippedParameter_abs_le _)).1⟩
theorem denseObservedModel_value
The all-parameter legal model has exactly the measurable target used in the testing theorem. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
observedValue (denseObservedModel hd ε a hε hεhi ha hahi θ).1 = densePriorTarget d a θ
Proof (Lean source)
-- @node: denseObservedModel_value
lemma denseObservedModel_value {d : ℕ} (hd : 0 < d) (ε a : ℝ)
(hε : 0 < ε) (hεhi : ε ≤ 1 / 2) (ha : 0 ≤ a) (hahi : a ≤ 1 / 2)
(θ : Fin d → ℝ) :
observedValue (denseObservedModel hd ε a hε hεhi ha hahi θ).1 =
densePriorTarget d a θ := by
rw [denseObservedModel, denseObservedLaw_value]
simp only [densePriorTarget, densePositivePart_clipped]
theorem dense_observed_logDegree_bayesRisk_lower
The dense testing bound applies to the actual legal observed value. The Fejér priors and their separation, target concentration, and Poisson mixture TV bound are all supplied by the proved dense lower theorem. In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ (η C : ℝ) (D : ℕ),
0 < η
2 ≤ C
2 ≤ D
conclusion 1
d :
ℕ
r a ε :
ℝ
hd :
0 < d
hε :
0 < ε
hεhi :
ε ≤ 1 / 2
ha :
0 ≤ a
hahi :
a ≤ 1 / 2
D ≤ d
0 < r
0 < a
r * a ^ 2 ≤ η * lowerLogDegree C d
∃ P : AbsMomentMatchedPriors (lowerLogDegree C d),
Measurable est → ofReal ((5 / 2560000 : ℝ) * a ^ 2 / (lowerLogDegree C d : ℝ) ^ 2)
≤ max (bayesSquaredRisk (productPrior d P.ν₀) (denseProductPoissonKernel d r a) (fun θ => observedValue (denseObservedModel hd ε a hε hεhi ha hahi θ).1) est) (bayesSquaredRisk (productPrior d P.ν₁) (denseProductPoissonKernel d r a) (fun θ => observedValue (denseObservedModel hd ε a hε hεhi ha hahi θ).1) est)
Proof (Lean source)
-- @node: dense_observed_logDegree_bayesRisk_lower
lemma dense_observed_logDegree_bayesRisk_lower :
∃ (η C : ℝ) (D : ℕ), 0 < η ∧ 2 ≤ C ∧ 2 ≤ D ∧
∀ (d : ℕ) (r a ε : ℝ) (hd : 0 < d)
(hε : 0 < ε) (hεhi : ε ≤ 1 / 2) (ha : 0 ≤ a) (hahi : a ≤ 1 / 2),
D ≤ d → 0 < r → 0 < a →
r * a ^ 2 ≤ η * lowerLogDegree C d →
∃ P : AbsMomentMatchedPriors (lowerLogDegree C d),
∀ est : (Fin d → (Fin 2 → ℕ)) → ℝ, Measurable est →
ofReal ((5 / 2560000 : ℝ) * a ^ 2 /
(lowerLogDegree C d : ℝ) ^ 2) ≤
max (bayesSquaredRisk (productPrior d P.ν₀)
(denseProductPoissonKernel d r a)
(fun θ => observedValue (denseObservedModel hd ε a hε hεhi ha hahi θ).1) est)
(bayesSquaredRisk (productPrior d P.ν₁)
(denseProductPoissonKernel d r a)
(fun θ => observedValue (denseObservedModel hd ε a hε hεhi ha hahi θ).1) est) := by
obtain ⟨η, C, D, hη, hC, hD, hbound⟩ := dense_logDegree_bayesRisk_lower
refine ⟨η, C, D, hη, hC, hD, ?_⟩
intro d r a ε hd hε hεhi ha hahi hcut hr hapos hband
obtain ⟨P, hP⟩ := hbound d r a hcut hr hapos hahi hband
refine ⟨P, ?_⟩
intro est hest
have htarget :
(fun θ => observedValue (denseObservedModel hd ε a hε hεhi ha hahi θ).1) =
densePriorTarget d a := by
funext θ
exact denseObservedModel_value hd ε a hε hεhi ha hahi θ
rw [htarget]
exact hP est hest
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.LowerDenseRisk 14 declarations The target in (46) is the average positive part of the symmetric prior parameters.
Dense target concentration and fuzzy testing
The target in (46) is the average positive part of the symmetric prior parameters. Its variance is at most a²/d, giving the dense testing step (48). The extension below clips only outside the legal prior support.
For the displayed parameters, densePositivePart is the object specified by this definition.
For the displayed parameters, densePriorTarget is the object specified by this definition.
Definition (Lean source)
theorem measurable_densePositivePart
The positive-part extension is measurable. In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
Proof (Lean source)
@[fun_prop]
lemma measurable_densePositivePart : Measurable densePositivePart := by
unfold densePositivePart
fun_prop
theorem measurable_densePriorTarget
The extended dense target is measurable. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
Proof (Lean source)
@[fun_prop]
lemma measurable_densePriorTarget (d : ℕ) (a : ℝ) : Measurable (densePriorTarget d a) := by
unfold densePriorTarget
fun_prop
theorem densePositivePart_eq
On the legal support the extension gives exactly the summand in (46). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma densePositivePart_eq {t : ℝ} (ht : |t| ≤ 1) :
densePositivePart t = (|t| + t) / 2 := by
rw [densePositivePart, min_eq_right (max_le (le_trans (le_abs_self t) ht) (by norm_num))]
by_cases h : 0 ≤ t
· rw [max_eq_left h, abs_of_nonneg h]; ring
· rw [max_eq_right (le_of_not_ge h), abs_of_nonpos (le_of_not_ge h)]; ring
theorem densePositivePart_integral
Symmetry identifies the positive-part expectation with half the absolute moment. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
(∫ t, densePositivePart t ∂ν) = (∫ t, |t| ∂ν) / 2
Proof (Lean source)
lemma densePositivePart_integral (ν : Measure ℝ) [IsProbabilityMeasure ν]
(hsupp : IsSupportedOnUnitInterval ν) (hsym : IsSymmetric ν) :
(∫ t, densePositivePart t ∂ν) = (∫ t, |t| ∂ν) / 2 := by
have hs : ∀ᵐ t ∂ν, |t| ≤ 1 :=
(mem_ae_iff_prob_eq_one (measurableSet_le (by fun_prop) measurable_const)).2
(dense_prior_support_radius_one ν hsupp)
have hi : MemLp (fun t : ℝ => t) 2 ν :=
MemLp.of_bound (by fun_prop) 1 (hs.mono fun t ht => by simpa using ht)
have ha : Integrable (fun t : ℝ => |t|) ν := (hi.integrable (by norm_num)).abs
calc
_ = ∫ t, (|t| + t) / 2 ∂ν := integral_congr_ae (hs.mono fun t ht => densePositivePart_eq ht)
_ = (∫ t, |t| ∂ν) / 2 := by
rw [integral_div, integral_add ha (hi.integrable (by norm_num)),
dense_symmetric_integral_id_zero ν hsym, add_zero]
theorem dense_product_prior_target_integral_sq_le
Independent dense prior coordinates concentrate the target around its symmetric scalar center, with no assumed concentration inequality. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
0 < d
ν :
hsupp :
hsym :
a :
ℝ
conclusion
(∫ θ, (densePriorTarget d a θ - (1 / 2 + a * (∫ t, |t| ∂ν) / 2)) ^ 2 ∂productPrior d ν)
≤ a ^ 2 / d
Proof (Lean source)
lemma dense_product_prior_target_integral_sq_le {d : ℕ} (hd : 0 < d)
(ν : Measure ℝ) [IsProbabilityMeasure ν] (hsupp : IsSupportedOnUnitInterval ν)
(hsym : IsSymmetric ν) (a : ℝ) :
(∫ θ, (densePriorTarget d a θ - (1 / 2 + a * (∫ t, |t| ∂ν) / 2)) ^ 2
∂productPrior d ν) ≤ a ^ 2 / d := by
let μ := productPrior d ν
let : IsProbabilityMeasure μ := by dsimp [μ, productPrior]; infer_instance
let Z : Fin d → (Fin d → ℝ) → ℝ := fun x θ => densePositivePart (θ x)
have hm (x : Fin d) : Measurable (Z x) := by dsimp [Z]; fun_prop
have hb (x : Fin d) (θ : Fin d → ℝ) : Z x θ ∈ Icc 0 1 :=
⟨le_min (by norm_num) (le_max_right _ _), min_le_left _ _⟩
have hLp (x : Fin d) : MemLp (Z x) 2 μ :=
MemLp.of_bound (hm x).aestronglyMeasurable 1 (ae_of_all _ fun θ => by
rw [Real.norm_eq_abs, abs_of_nonneg (hb x θ).1]; exact (hb x θ).2)
have hv (x : Fin d) : variance (Z x) μ ≤ 1 := by
apply (variance_le_sq_of_bounded (ae_of_all _ (hb x)) (hm x).aemeasurable).trans
norm_num
have hind : iIndepFun Z μ := iIndepFun_pi (fun _ => measurable_densePositivePart.aemeasurable)
let H := fun θ : Fin d → ℝ => (∑ x, Z x θ) / d
have hHi : Integrable H μ :=
(integrable_finsetSum _ (fun x _ => (hLp x).integrable (by norm_num))).div_const _
have hHe : (∫ θ, H θ ∂μ) = (∫ t, |t| ∂ν) / 2 := by
dsimp [H]
rw [integral_div, integral_finsetSum _ (fun x _ => (hLp x).integrable (by norm_num))]
have he (x : Fin d) : (∫ θ, Z x θ ∂μ) = (∫ t, |t| ∂ν) / 2 := by
exact ((measurePreserving_eval (fun _ : Fin d => ν) x).hasLaw.integral_comp
measurable_densePositivePart.aestronglyMeasurable).trans
(densePositivePart_integral ν hsupp hsym)
simp_rw [he]
simp [ne_of_gt (Nat.cast_pos.mpr hd : (0 : ℝ) < d)]
have hTm : Measurable (fun θ => 1 / 2 + a * H θ) := by dsimp [H, Z]; fun_prop
have hTe : (∫ θ, (1 / 2 + a * H θ) ∂μ) = 1 / 2 + a * (∫ t, |t| ∂ν) / 2 := by
rw [integral_add (integrable_const _) (hHi.const_mul _), integral_const_mul, hHe]
simp; ring
have hvar := sparse_independent_average_variance_le μ hd Z 1 hLp hind hv
have heq : variance (fun θ => 1 / 2 + a * H θ) μ =
∫ θ, (densePriorTarget d a θ - (1 / 2 + a * (∫ t, |t| ∂ν) / 2)) ^ 2 ∂μ := by
rw [variance_eq_integral hTm.aemeasurable, hTe]
rfl
change _ ≤ a ^ 2 / d
rw [← heq, variance_const_add (hHi.aestronglyMeasurable.const_mul a), variance_const_mul]
exact (mul_le_mul_of_nonneg_left hvar (sq_nonneg a)).trans_eq (by ring)
theorem dense_product_prior_target_tail_le
Chebyshev supplies the dense prior target-deviation probabilities. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
0 < d
ν :
hsupp :
hsym :
a :
ℝ
r :
ℝ
hr :
0 < r
conclusion
≤ a ^ 2 / (d * r ^ 2)
Proof (Lean source)
lemma dense_product_prior_target_tail_le {d : ℕ} (hd : 0 < d)
(ν : Measure ℝ) [IsProbabilityMeasure ν] (hsupp : IsSupportedOnUnitInterval ν)
(hsym : IsSymmetric ν) (a : ℝ) {r : ℝ} (hr : 0 < r) :
(productPrior d ν).real
{θ | r < |densePriorTarget d a θ - (1 / 2 + a * (∫ t, |t| ∂ν) / 2)|} ≤
a ^ 2 / (d * r ^ 2) := by
let μ := productPrior d ν
let : IsProbabilityMeasure μ := by dsimp [μ, productPrior]; infer_instance
let center := 1 / 2 + a * (∫ t, |t| ∂ν) / 2
have hcoord (x : Fin d) : MemLp (fun θ : Fin d → ℝ => densePositivePart (θ x)) 2 μ :=
MemLp.of_bound (measurable_densePositivePart.comp (measurable_pi_apply x)).aestronglyMeasurable 1 (ae_of_all μ fun θ => by
rw [Real.norm_eq_abs, abs_of_nonneg (le_min (by norm_num) (le_max_right _ _))]
exact min_le_left _ _)
have hsum : MemLp (fun θ : Fin d → ℝ => (∑ x, densePositivePart (θ x)) / d) 2 μ := by
simpa only [div_eq_mul_inv] using
(memLp_finsetSum univ (fun x _ => hcoord x)).mul_const (d : ℝ)⁻¹
have hc : MemLp (fun _ : Fin d → ℝ => (1 / 2 : ℝ)) 2 μ := memLp_const _
have hLp : MemLp (densePriorTarget d a) 2 μ := hc.add (hsum.const_mul a)
have hint := (hLp.sub (memLp_const center)).integrable_sq
have hsub : {θ | r < |densePriorTarget d a θ - center|} ⊆
{θ | r ^ 2 ≤ (densePriorTarget d a θ - center) ^ 2} := by
intro θ hθ
change r < |densePriorTarget d a θ - center| at hθ
change r ^ 2 ≤ (densePriorTarget d a θ - center) ^ 2
nlinarith [sq_abs (densePriorTarget d a θ - center), abs_nonneg (densePriorTarget d a θ - center)]
have hmark := mul_meas_ge_le_integral_of_nonneg
(ae_of_all μ fun θ => sq_nonneg (densePriorTarget d a θ - center)) hint (r ^ 2)
have hb : r ^ 2 * μ.real {θ | r < |densePriorTarget d a θ - center|} ≤ a ^ 2 / d :=
((mul_le_mul_of_nonneg_left (measureReal_mono hsub) (sq_nonneg r)).trans hmark).trans
(dense_product_prior_target_integral_sq_le hd ν hsupp hsym a)
change μ.real {θ | r < |densePriorTarget d a θ - center|} ≤ _
calc
_ ≤ (a ^ 2 / d) / r ^ 2 :=
(le_div_iff₀ (sq_pos_of_pos hr)).2 (by simpa only [mul_comm] using hb)
_ = _ := by ring
For the displayed parameters, denseProductPoissonKernel is the object specified by this definition.
Definition (Lean source)
parameters
d :
ℕ
r a :
ℝ
given by
clause 1
toFun θ := Measure.pi fun x => denseTreatedKernel r a (θ x)
clause 2
measurable' := by let (t : ℝ) : IsProbabilityMeasure (denseTreatedKernel r a t) := by change IsProbabilityMeasure (denseTreatedLaw r a t) unfold denseTreatedLaw; infer_instance apply Measure.measurable_of_measurable_coe intro s hs have hatom (N : Fin d → (Fin 2 → ℕ)) : Measurable (fun θ : Fin d → ℝ => (Measure.pi fun x => denseTreatedKernel r a (θ x)) {N}) := by simp only [Measure.pi_singleton] apply Finset.measurable_prod intro x _ exact ((denseTreatedKernel r a).measurable_coe (measurableSet_singleton _)).comp (measurable_pi_apply x) simp_rw [← Measure.tsum_indicator_apply_singleton _ s hs] apply Measurable.tsum intro N by_cases hN : N ∈ s · simpa [Set.indicator_of_mem hN] using hatom N · simp [Set.indicator_of_notMem hN]
theorem denseProductPoissonKernel_probability
The product kernel has probability fibres. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma denseProductPoissonKernel_probability (d : ℕ) (r a : ℝ) (θ : Fin d → ℝ) :
IsProbabilityMeasure (denseProductPoissonKernel d r a θ) := by
change IsProbabilityMeasure (Measure.pi fun x => denseTreatedLaw r a (θ x))
unfold denseTreatedLaw; infer_instance
theorem dense_product_prior_predictive_eq
Mixing the product kernel gives the predictive law of the proved TV bound. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
= productPriorPredictive d ν (denseTreatedKernel r a)
Proof (Lean source)
lemma dense_product_prior_predictive_eq (d : ℕ) (r a : ℝ)
(ν : Measure ℝ) [IsProbabilityMeasure ν] :
priorPredictive (productPrior d ν) (denseProductPoissonKernel d r a) =
productPriorPredictive d ν (denseTreatedKernel r a) := by
apply priorPredictive_productPrior_eq_productPriorPredictive
· intro t
change IsProbabilityMeasure (denseTreatedLaw r a t)
unfold denseTreatedLaw; infer_instance
· intro θ; rfl
theorem dense_product_prior_bayesRisk_lower
The dense fuzzy-testing step (48): concentration and likelihood TV are both derived; the alphabet condition makes target deviations negligible. In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ η C : ℝ,
0 < η
0 < C
conclusion 1
K d :
ℕ
P :
r a :
ℝ
2 ≤ d
0 < r
0 < a
a ≤ 1 / 2
C * logAlphabet d ≤ K
r * a ^ 2 ≤ η * K
Delta :
a * ((∫ t, |t| ∂P.ν₁) - (∫ t, |t| ∂P.ν₀)) / 2
128 * a ^ 2 ≤ d * Delta ^ 2
0 < Delta
Measurable est
ofReal (5 * Delta ^ 2 / 256)
≤ max (bayesSquaredRisk (productPrior d P.ν₀) (denseProductPoissonKernel d r a) (densePriorTarget d a) est) (bayesSquaredRisk (productPrior d P.ν₁) (denseProductPoissonKernel d r a) (densePriorTarget d a) est)
Proof (Lean source)
lemma dense_product_prior_bayesRisk_lower :
∃ η C : ℝ, 0 < η ∧ 0 < C ∧
∀ (K d : ℕ) (P : AbsMomentMatchedPriors K) (r a : ℝ),
2 ≤ d → 0 < r → 0 < a → a ≤ 1 / 2 →
C * logAlphabet d ≤ K → r * a ^ 2 ≤ η * K →
let Delta := a * ((∫ t, |t| ∂P.ν₁) - (∫ t, |t| ∂P.ν₀)) / 2
128 * a ^ 2 ≤ d * Delta ^ 2 → 0 < Delta →
∀ est : (Fin d → (Fin 2 → ℕ)) → ℝ, Measurable est →
ofReal (5 * Delta ^ 2 / 256) ≤
max (bayesSquaredRisk (productPrior d P.ν₀)
(denseProductPoissonKernel d r a) (densePriorTarget d a) est)
(bayesSquaredRisk (productPrior d P.ν₁)
(denseProductPoissonKernel d r a) (densePriorTarget d a) est) := by
obtain ⟨η, C, hη, hC, htvbound⟩ := dense_product_mixture_tv_le_one_eighth
refine ⟨η, C, hη, hC, ?_⟩
intro K d P r a hd hr ha hahi hK hband
dsimp only
intro hsize hDelta est hest
let := P.probability₀
let := P.probability₁
let : IsProbabilityMeasure (productPrior d P.ν₀) := by unfold productPrior; infer_instance
let : IsProbabilityMeasure (productPrior d P.ν₁) := by unfold productPrior; infer_instance
let Delta := a * ((∫ t, |t| ∂P.ν₁) - (∫ t, |t| ∂P.ν₀)) / 2
have ht (ν : Measure ℝ) [IsProbabilityMeasure ν]
(hsupp : IsSupportedOnUnitInterval ν) (hsym : IsSymmetric ν) :
(productPrior d ν).real
{θ | Delta / 4 < |densePriorTarget d a θ - (1 / 2 + a * (∫ t, |t| ∂ν) / 2)|} ≤ 1 / 8 := by
apply (dense_product_prior_target_tail_le (by omega) ν hsupp hsym a
(show 0 < Delta / 4 by positivity)).trans
apply (div_le_iff₀ (by positivity : 0 < (d : ℝ) * (Delta / 4) ^ 2)).2
have hsize' : 128 * a ^ 2 ≤ d * Delta ^ 2 := hsize
nlinarith only [hsize']
have htv : tvDist
(priorPredictive (productPrior d P.ν₀) (denseProductPoissonKernel d r a))
(priorPredictive (productPrior d P.ν₁) (denseProductPoissonKernel d r a)) ≤ 1 / 8 := by
rw [dense_product_prior_predictive_eq, dense_product_prior_predictive_eq]
exact htvbound K d P r a hd hr ha.le hahi hK hband
have hb := twoFuzzyHypotheses_bayesRisk_lower (productPrior d P.ν₀) (productPrior d P.ν₁)
(denseProductPoissonKernel d r a) (densePriorTarget d a)
(denseProductPoissonKernel_probability d r a) est hest (measurable_densePriorTarget d a)
(1 / 2 + a * (∫ t, |t| ∂P.ν₀) / 2) (1 / 2 + a * (∫ t, |t| ∂P.ν₁) / 2)
Delta (Delta / 4) (1 / 8) (1 / 8) (1 / 8)
(by dsimp [Delta] at *; linarith)
(by dsimp [Delta]; linarith)
(ht P.ν₀ P.supported₀ P.symmetric₀) (ht P.ν₁ P.supported₁ P.symmetric₁) htv
convert hb using 1
congr 1
ring
theorem dense_symmetric_bayesRisk_lower
The Fejér symmetric priors attain the dense risk order a²/K². The separation is proved by the approximation certificate, never assumed. In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ η C : ℝ,
0 < η
0 < C
conclusion 1
K d :
ℕ
r a :
ℝ
0 < K
2 ≤ d
0 < r
0 < a
a ≤ 1 / 2
C * logAlphabet d ≤ K
r * a ^ 2 ≤ η * K
1280000 * (K : ℝ) ^ 2 ≤ d
∃ P : AbsMomentMatchedPriors K,
Measurable est → ofReal ((5 / 2560000 : ℝ) * a ^ 2 / (K : ℝ) ^ 2)
≤ max (bayesSquaredRisk (productPrior d P.ν₀) (denseProductPoissonKernel d r a) (densePriorTarget d a) est) (bayesSquaredRisk (productPrior d P.ν₁) (denseProductPoissonKernel d r a) (densePriorTarget d a) est)
Proof (Lean source)
lemma dense_symmetric_bayesRisk_lower :
∃ η C : ℝ, 0 < η ∧ 0 < C ∧
∀ (K d : ℕ) (r a : ℝ),
0 < K → 2 ≤ d → 0 < r → 0 < a → a ≤ 1 / 2 →
C * logAlphabet d ≤ K → r * a ^ 2 ≤ η * K →
1280000 * (K : ℝ) ^ 2 ≤ d →
∃ P : AbsMomentMatchedPriors K,
∀ est : (Fin d → (Fin 2 → ℕ)) → ℝ, Measurable est →
ofReal ((5 / 2560000 : ℝ) * a ^ 2 / (K : ℝ) ^ 2) ≤
max (bayesSquaredRisk (productPrior d P.ν₀)
(denseProductPoissonKernel d r a) (densePriorTarget d a) est)
(bayesSquaredRisk (productPrior d P.ν₁)
(denseProductPoissonKernel d r a) (densePriorTarget d a) est) := by
obtain ⟨η, C, hη, hC, hrisk⟩ := dense_product_prior_bayesRisk_lower
refine ⟨η, C, hη, hC, ?_⟩
intro K d r a hK hd hr ha hahi hlog hband hsize
obtain ⟨P, hgap⟩ := dense_symmetric_priors_absGap_lower K hK
refine ⟨P, ?_⟩
intro est hest
let Delta := a * ((∫ t, |t| ∂P.ν₁) - (∫ t, |t| ∂P.ν₀)) / 2
have hKR : (0 : ℝ) < K := Nat.cast_pos.mpr hK
have hlo : a / (100 * K) ≤ Delta := by
have hh := mul_le_mul_of_nonneg_left hgap ha.le
dsimp [Delta]
calc
_ = a * ((1 / 50 : ℝ) / K) / 2 := by ring
_ ≤ _ := div_le_div_of_nonneg_right hh (by norm_num)
have hD : 0 < Delta := (by positivity : 0 < a / (100 * K)).trans_le hlo
have hsq : a ^ 2 / (10000 * (K : ℝ) ^ 2) ≤ Delta ^ 2 := by
have h := pow_le_pow_left₀ (by positivity : 0 ≤ a / (100 * K)) hlo 2
convert h using 1 <;> first | ring | rfl
have hsize' : 128 * a ^ 2 ≤ d * Delta ^ 2 := by
have hb : 128 * a ^ 2 ≤ (d : ℝ) * (a ^ 2 / (10000 * (K : ℝ) ^ 2)) := by
rw [← mul_div_assoc]
apply (le_div_iff₀ (by positivity : 0 < 10000 * (K : ℝ) ^ 2)).2
nlinarith only [mul_le_mul_of_nonneg_right hsize (sq_nonneg a)]
exact hb.trans (mul_le_mul_of_nonneg_left hsq (Nat.cast_nonneg d))
have hb := hrisk K d P r a hd hr ha hahi hlog hband hsize' hD est hest
apply le_trans (ENNReal.ofReal_le_ofReal ?_) hb
calc
_ = (5 / 256 : ℝ) * (a ^ 2 / (10000 * (K : ℝ) ^ 2)) := by ring
_ ≤ (5 / 256 : ℝ) * Delta ^ 2 := mul_le_mul_of_nonneg_left hsq (by norm_num)
_ = _ := by ring
theorem dense_logDegree_bayesRisk_lower
Above a universal alphabet cutoff the logarithmic dense experiment has risk at least a universal multiple of a²/K_d², with concentration derived. In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ (η C : ℝ) (D : ℕ),
0 < η
2 ≤ C
2 ≤ D
conclusion 1
d :
ℕ
r a :
ℝ
D ≤ d
0 < r
0 < a
a ≤ 1 / 2
r * a ^ 2 ≤ η * lowerLogDegree C d
∃ P : AbsMomentMatchedPriors (lowerLogDegree C d),
Measurable est → ofReal ((5 / 2560000 : ℝ) * a ^ 2 / (lowerLogDegree C d : ℝ) ^ 2)
≤ max (bayesSquaredRisk (productPrior d P.ν₀) (denseProductPoissonKernel d r a) (densePriorTarget d a) est) (bayesSquaredRisk (productPrior d P.ν₁) (denseProductPoissonKernel d r a) (densePriorTarget d a) est)
Proof (Lean source)
lemma dense_logDegree_bayesRisk_lower :
∃ (η C : ℝ) (D : ℕ), 0 < η ∧ 2 ≤ C ∧ 2 ≤ D ∧
∀ (d : ℕ) (r a : ℝ), D ≤ d → 0 < r → 0 < a → a ≤ 1 / 2 →
r * a ^ 2 ≤ η * lowerLogDegree C d →
∃ P : AbsMomentMatchedPriors (lowerLogDegree C d),
∀ est : (Fin d → (Fin 2 → ℕ)) → ℝ, Measurable est →
ofReal ((5 / 2560000 : ℝ) * a ^ 2 /
(lowerLogDegree C d : ℝ) ^ 2) ≤
max (bayesSquaredRisk (productPrior d P.ν₀)
(denseProductPoissonKernel d r a) (densePriorTarget d a) est)
(bayesSquaredRisk (productPrior d P.ν₁)
(denseProductPoissonKernel d r a) (densePriorTarget d a) est) := by
obtain ⟨η, C₀, hη, hC₀, hrisk⟩ := dense_symmetric_bayesRisk_lower
let C := max C₀ 2
have hC : 2 ≤ C := le_max_right _ _
obtain ⟨D, hD, hcut⟩ := sparse_packet_alphabet_cutoff
(cp := (1 / 100 : ℝ)) (by norm_num) hC
refine ⟨η, C, D, hη, hC, hD, ?_⟩
intro d r a hd hr ha hahi hband
have hd2 := hD.trans hd
obtain ⟨hK, hlog, _⟩ := lowerLogDegree_bounds hC hd2
have hsmall := hcut d hd 2 0 (by norm_num) (by norm_num) (by norm_num)
norm_num at hsmall
have hsize : 1280000 * (lowerLogDegree C d : ℝ) ^ 2 ≤ d := by linarith
have hL : 0 ≤ logAlphabet d := by
unfold logAlphabet
apply Real.log_nonneg
have hdR : (1 : ℝ) ≤ d := by exact_mod_cast (by omega : 1 ≤ d)
have he : 1 ≤ exp 1 := Real.one_le_exp (by norm_num)
nlinarith
exact hrisk (lowerLogDegree C d) d r a (by omega) hd2 hr ha hahi
((mul_le_mul_of_nonneg_right (le_max_left C₀ 2) hL).trans hlog) hband hsize
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.LowerFiniteAlphabet 7 declarations The last step of the lower-bound roadmap uses a common Bernoulli mean shift.
The finite-alphabet boundary-propensity lower experiment
The last step of the lower-bound roadmap uses a common Bernoulli mean shift. Uniform covariate labels carry no information about that shift, so this submodel has the same information as the single occupied cell experiment.
theorem constantDenseObservedLaw_value
The constant dense parameter has exactly the scalar shifted target. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
0 < d
ε a :
ℝ
hε :
0 < ε
hεhi :
ε ≤ 1 / 2
ha :
0 ≤ a
hahi :
a ≤ 1 / 2
conclusion
observedValue (denseObservedLaw hd ε a hε hεhi ha hahi (fun _ => 1) (by intro _; norm_num))
= 1 / 2 + a
Proof (Lean source)
lemma constantDenseObservedLaw_value {d : ℕ} (hd : 0 < d) (ε a : ℝ)
(hε : 0 < ε) (hεhi : ε ≤ 1 / 2) (ha : 0 ≤ a) (hahi : a ≤ 1 / 2) :
observedValue (denseObservedLaw hd ε a hε hεhi ha hahi
(fun _ => 1) (by intro _; norm_num)) = 1 / 2 + a := by
rw [denseObservedLaw_value]
have hdR : (d : ℝ) ≠ 0 := by exact_mod_cast hd.ne'
simp [densePriorTarget, densePositivePart, hdR]
theorem constantDenseObservedLaw_absolutelyContinuous
The central law has positive mass at every atom, so every shifted law is absolutely continuous with respect to it. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
0 < d
ε a :
ℝ
hε :
0 < ε
hεhi :
ε ≤ 1 / 2
ha :
0 ≤ a
hahi :
a ≤ 1 / 2
conclusion
(denseObservedLaw hd ε a hε hεhi ha hahi (fun _ => 1) (by intro _; norm_num)).pmf.toMeasure ≪ (denseObservedLaw hd ε 0 hε hεhi (by norm_num) (by norm_num) (fun _ => 1) (by intro _; norm_num)).pmf.toMeasure
Proof (Lean source)
lemma constantDenseObservedLaw_absolutelyContinuous {d : ℕ} (hd : 0 < d)
(ε a : ℝ) (hε : 0 < ε) (hεhi : ε ≤ 1 / 2) (ha : 0 ≤ a) (hahi : a ≤ 1 / 2) :
(denseObservedLaw hd ε a hε hεhi ha hahi (fun _ => 1)
(by intro _; norm_num)).pmf.toMeasure ≪
(denseObservedLaw hd ε 0 hε hεhi (by norm_num) (by norm_num)
(fun _ => 1) (by intro _; norm_num)).pmf.toMeasure := by
classical
let Q := denseObservedLaw hd ε 0 hε hεhi (by norm_num) (by norm_num)
(fun _ => 1) (by intro _; norm_num)
have hpos (o : Obs d) : 0 < Q.pmf.toMeasure.real {o} := by
have hdR : (0 : ℝ) < d := by exact_mod_cast hd
change 0 < (Q.pmf.toMeasure {o}).toReal
rw [PMF.toMeasure_apply_singleton _ _ (measurableSet_singleton _)]
change 0 < jointMass Q o.1 o.2.1 o.2.2
rw [denseObservedLaw_jointMass]
have he : 0 < 1 - ε := by linarith
cases o.2.1 <;> simp [denseObservedMass] <;> positivity
intro s hs
have hempty : s = ∅ := by
apply Set.eq_empty_iff_forall_notMem.mpr
intro o ho
have hz : Q.pmf.toMeasure {o} = 0 :=
measure_mono_null (Set.singleton_subset_iff.mpr ho) hs
have hp := hpos o
simp [Measure.real_def, hz] at hp
rw [hempty, measure_empty]
theorem discreteLaw_real_singleton
Real singleton masses agree with the observed atom coordinates. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
P.pmf.toMeasure.real {o} = jointMass P o.1 o.2.1 o.2.2
Proof (Lean source)
lemma discreteLaw_real_singleton {d : ℕ} (P : DiscreteLaw d) (o : Obs d) :
P.pmf.toMeasure.real {o} = jointMass P o.1 o.2.1 o.2.2 := by
rw [Measure.real_def, PMF.toMeasure_apply_singleton _ _ (measurableSet_singleton _)]
rfl
theorem constantDenseObservedLaw_chiSqDiv
The exact one-observation information is four times the overlap floor and the squared mean shift, independently of the covariate alphabet. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
0 < d
ε a :
ℝ
hε :
0 < ε
hεhi :
ε ≤ 1 / 2
ha :
0 ≤ a
hahi :
a ≤ 1 / 2
conclusion
chiSqDiv (denseObservedLaw hd ε a hε hεhi ha hahi (fun _ => 1) (by intro _; norm_num)).pmf.toMeasure (denseObservedLaw hd ε 0 hε hεhi (by norm_num) (by norm_num) (fun _ => 1) (by intro _; norm_num)).pmf.toMeasure
= 4 * ε * a ^ 2
Proof (Lean source)
lemma constantDenseObservedLaw_chiSqDiv {d : ℕ} (hd : 0 < d)
(ε a : ℝ) (hε : 0 < ε) (hεhi : ε ≤ 1 / 2) (ha : 0 ≤ a) (hahi : a ≤ 1 / 2) :
chiSqDiv
(denseObservedLaw hd ε a hε hεhi ha hahi (fun _ => 1)
(by intro _; norm_num)).pmf.toMeasure
(denseObservedLaw hd ε 0 hε hεhi (by norm_num) (by norm_num)
(fun _ => 1) (by intro _; norm_num)).pmf.toMeasure = 4 * ε * a ^ 2 := by
classical
have hdR : (d : ℝ) ≠ 0 := by exact_mod_cast hd.ne'
have hεne : ε ≠ 0 := hε.ne'
have hone : 1 - ε ≠ 0 := by linarith
have h := finite_one_add_chiSqDiv _ _
(constantDenseObservedLaw_absolutelyContinuous hd ε a hε hεhi ha hahi)
have hsum :
(∑ o : Obs d,
((denseObservedLaw hd ε a hε hεhi ha hahi (fun _ => 1)
(by intro _; norm_num)).pmf.toMeasure.real {o}) ^ 2 /
(denseObservedLaw hd ε 0 hε hεhi (by norm_num) (by norm_num)
(fun _ => 1) (by intro _; norm_num)).pmf.toMeasure.real {o}) =
1 + 4 * ε * a ^ 2 := by
simp_rw [discreteLaw_real_singleton]
simp_rw [denseObservedLaw_jointMass]
rw [Fintype.sum_prod_type]
have hcell (x : Fin d) :
(∑ b : Bool × Bool,
(denseObservedMass ε a (fun _ : Fin d => 1) (x, b)) ^ 2 /
denseObservedMass ε 0 (fun _ : Fin d => 1) (x, b)) =
(1 + 4 * ε * a ^ 2) / d := by
simp [Fintype.sum_prod_type, denseObservedMass]
field_simp
<;> ring
simp_rw [hcell]
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul]
field_simp
linarith
theorem constantDense_twoPoint_sqRisk_lower
Tensorization and a small information budget give a fixed-sample squared-risk testing floor for the legal constant boundary submodel. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
n d :
ℕ
hd :
0 < d
ε a :
ℝ
hε :
0 < ε
hεhi :
ε ≤ 1 / 2
ha :
0 ≤ a
hahi :
a ≤ 1 / 2
hinfo :
(n : ℝ) * ε * a ^ 2 ≤ 1 / 16
est :
Estimator n d
conclusion
a ^ 2 / 16
≤ max (sqRisk (productLaw (denseObservedLaw hd ε a hε hεhi ha hahi (fun _ => 1) (by intro _; norm_num)) n) est.1 (1 / 2 + a)) (sqRisk (productLaw (denseObservedLaw hd ε 0 hε hεhi (by norm_num) (by norm_num) (fun _ => 1) (by intro _; norm_num)) n) est.1 (1 / 2))
Proof (Lean source)
lemma constantDense_twoPoint_sqRisk_lower {n d : ℕ} (hd : 0 < d)
(ε a : ℝ) (hε : 0 < ε) (hεhi : ε ≤ 1 / 2) (ha : 0 ≤ a) (hahi : a ≤ 1 / 2)
(hinfo : (n : ℝ) * ε * a ^ 2 ≤ 1 / 16) (est : Estimator n d) :
a ^ 2 / 16 ≤ max
(sqRisk
(productLaw (denseObservedLaw hd ε a hε hεhi ha hahi
(fun _ => 1) (by intro _; norm_num)) n) est.1 (1 / 2 + a))
(sqRisk
(productLaw (denseObservedLaw hd ε 0 hε hεhi (by norm_num) (by norm_num)
(fun _ => 1) (by intro _; norm_num)) n) est.1 (1 / 2)) := by
let P := denseObservedLaw hd ε a hε hεhi ha hahi
(fun _ => 1) (by intro _; norm_num)
let Q := denseObservedLaw hd ε 0 hε hεhi (by norm_num) (by norm_num)
(fun _ => 1) (by intro _; norm_num)
letI : IsProbabilityMeasure (productLaw P n) := by unfold productLaw; infer_instance
letI : IsProbabilityMeasure (productLaw Q n) := by unfold productLaw; infer_instance
have hac : P.pmf.toMeasure ≪ Q.pmf.toMeasure :=
constantDenseObservedLaw_absolutelyContinuous hd ε a hε hεhi ha hahi
have hacn : productLaw P n ≪ productLaw Q n :=
Causalean.Mathlib.Probability.ProductAbsolutelyContinuous.pi_iid_absolutelyContinuous _ _ hac n
have hchi : chiSqDiv (productLaw P n) (productLaw Q n) ≤ 1 := by
have ht := Causalean.Stat.one_add_chiSqDiv_pi_iid _ _ hac n
have hsingle : chiSqDiv P.pmf.toMeasure Q.pmf.toMeasure =
4 * ε * a ^ 2 := constantDenseObservedLaw_chiSqDiv hd ε a hε hεhi ha hahi
rw [hsingle] at ht
have hp : (1 + 4 * ε * a ^ 2) ^ n ≤ exp ((n : ℝ) * (4 * ε * a ^ 2)) := by
rw [Real.exp_nat_mul]
exact pow_le_pow_left₀ (by positivity)
(by simpa only [add_comm] using Real.add_one_le_exp (4 * ε * a ^ 2)) n
have he : exp ((n : ℝ) * (4 * ε * a ^ 2)) ≤ 4 / 3 := by
calc
_ ≤ exp (1 / 4) := Real.exp_le_exp.mpr (by nlinarith)
_ ≤ 4 / 3 := by
have h := Real.exp_bound_div_one_sub_of_interval
(by norm_num : (0 : ℝ) ≤ 1 / 4) (by norm_num : (1 / 4 : ℝ) < 1)
norm_num at h ⊢
exact h
change 1 + chiSqDiv (productLaw P n) (productLaw Q n) = _ at ht
linarith
have htest := two_point_lower_bound_of_chiSqDiv_le
(P₀ := productLaw P n) (P₁ := productLaw Q n)
(θ₀ := 1 / 2 + a) (θ₁ := 1 / 2) (s := a / 2) est.2
(by rw [show (1 / 2 + a) - (1 / 2 : ℝ) = a by ring, abs_of_nonneg ha]; linarith)
hacn Integrable.of_finite hchi
norm_num only [Real.sqrt_one] at htest
have htest' : (1 / 4 : ℝ) ≤ max
((productLaw P n).real {o | a / 2 ≤ |est.1 o - (1 / 2 + a)|})
((productLaw Q n).real {o | a / 2 ≤ |est.1 o - 1 / 2|}) := by
convert htest using 1 <;> norm_num
have hmark (μ : Measure (Fin n → Obs d)) [IsProbabilityMeasure μ] (θ : ℝ) :
(a / 2) ^ 2 * μ.real {o | a / 2 ≤ |est.1 o - θ|} ≤
sqRisk μ est.1 θ := by
have hset : {o | a / 2 ≤ |est.1 o - θ|} =
{o | (a / 2) ^ 2 ≤ (est.1 o - θ) ^ 2} := by
ext o
simp only [Set.mem_setOf_eq]
constructor <;> intro h <;>
nlinarith [sq_abs (est.1 o - θ), abs_nonneg (est.1 o - θ)]
rw [hset]
exact mul_meas_ge_le_integral_of_nonneg
(Filter.Eventually.of_forall fun o => sq_nonneg (est.1 o - θ))
Integrable.of_finite _
have hm := max_le_max (hmark (productLaw P n) (1 / 2 + a))
(hmark (productLaw Q n) (1 / 2))
rw [← mul_max_of_nonneg _ _ (sq_nonneg (a / 2))] at hm
have h := mul_le_mul_of_nonneg_left htest' (sq_nonneg (a / 2))
have heq : (a / 2) ^ 2 * (1 / 4) = a ^ 2 / 16 := by ring
rw [heq] at h
exact h.trans hm
theorem finiteAlphabet_parametric_minimaxRisk_lower
The boundary-propensity two-point experiment yields the parametric fallback uniformly over every finite alphabet, with no alphabet cutoff. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
n d :
ℕ
hn :
1 ≤ n
hd :
2 ≤ d
ε :
ℝ
hε :
0 < ε
hεhi :
ε ≤ 1 / 2
conclusion
(1 / 1024 : ℝ) * min 1 (1 / ((n : ℝ) * ε)) ≤ minimaxRisk n d ε
Proof (Lean source)
lemma finiteAlphabet_parametric_minimaxRisk_lower {n d : ℕ} (hn : 1 ≤ n)
(hd : 2 ≤ d) {ε : ℝ} (hε : 0 < ε) (hεhi : ε ≤ 1 / 2) :
(1 / 1024 : ℝ) * min 1 (1 / ((n : ℝ) * ε)) ≤ minimaxRisk n d ε := by
have hdpos : 0 < d := by omega
have hnR : (0 : ℝ) < n := by exact_mod_cast (by omega : 0 < n)
let r := min 1 (1 / ((n : ℝ) * ε))
have hr : 0 < r := lt_min (by norm_num) (by positivity)
let a := sqrt (r / 64)
have ha : 0 ≤ a := Real.sqrt_nonneg _
have hasq : a ^ 2 = r / 64 := Real.sq_sqrt (by positivity)
have hahi : a ≤ 1 / 2 := by
have hcap : r ≤ 1 := min_le_left _ _
nlinarith
have hinfo : (n : ℝ) * ε * a ^ 2 ≤ 1 / 16 := by
rw [hasq]
have h := mul_le_mul_of_nonneg_left (min_le_right 1 (1 / ((n : ℝ) * ε)))
(show 0 ≤ (n : ℝ) * ε by positivity)
have heq : (n : ℝ) * ε * (1 / ((n : ℝ) * ε)) = 1 := by field_simp
rw [heq] at h
dsimp [r]
nlinarith
let P : ModelLaw d ε :=
⟨denseObservedLaw hdpos ε a hε hεhi ha hahi (fun _ => 1) (by intro _; norm_num),
(denseObservedLaw_parameters hdpos ε a hε hεhi ha hahi
(fun _ => 1) (by intro _; norm_num)).1⟩
let Q : ModelLaw d ε :=
⟨denseObservedLaw hdpos ε 0 hε hεhi (by norm_num) (by norm_num)
(fun _ => 1) (by intro _; norm_num),
(denseObservedLaw_parameters hdpos ε 0 hε hεhi (by norm_num) (by norm_num)
(fun _ => 1) (by intro _; norm_num)).1⟩
letI : Nonempty (Estimator n d) := ⟨⟨fun _ => 0, measurable_const⟩⟩
apply le_minimaxValue_of_two_point P Q observedRisk_bddAbove
intro est
have h := constantDense_twoPoint_sqRisk_lower hdpos ε a hε hεhi ha hahi hinfo est
have hP : observedValue P.1 = 1 / 2 + a :=
constantDenseObservedLaw_value hdpos ε a hε hεhi ha hahi
have hQ : observedValue Q.1 = 1 / 2 := by
simpa using constantDenseObservedLaw_value hdpos ε 0 hε hεhi
(by norm_num) (by norm_num)
change (1 / 1024 : ℝ) * r ≤ max
(sqRisk (productLaw P.1 n) est.1 (observedValue P.1))
(sqRisk (productLaw Q.1 n) est.1 (observedValue Q.1))
rw [hP, hQ]
have heq : a ^ 2 / 16 = (1 / 1024 : ℝ) * r := by rw [hasq]; ring
rwa [heq] at h
theorem allAlphabet_minimaxRisk_lower
The parametric fallback covers the finite exceptional set below the large-alphabet cutoff. Together the two experiments give the paper rate for every legal sample size, alphabet and overlap floor. In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ c : ℝ,
0 < c
Proof (Lean source)
lemma allAlphabet_minimaxRisk_lower :
∃ c : ℝ, 0 < c ∧ ∀ (n d : ℕ) (ε : ℝ),
1 ≤ n → 2 ≤ d → 0 < ε → ε ≤ 1 / 2 →
c * rateScale n d ε ≤ minimaxRisk n d ε := by
obtain ⟨c, D, hc, hD, hlarge⟩ := largeAlphabet_minimaxRisk_lower
have hDR : (0 : ℝ) < D := by exact_mod_cast (by omega : 0 < D)
let cstar := min c (1 / (1024 * (D : ℝ)))
refine ⟨cstar, lt_min hc (by positivity), ?_⟩
intro n d ε hn hd hε hεhi
have hr : 0 ≤ rateScale n d ε := rateScale_nonneg (by omega) hε.le
by_cases hcut : D ≤ d
· exact (mul_le_mul_of_nonneg_right (min_le_left _ _) hr).trans
(hlarge n d ε hn hcut hε hεhi)
· have hnR : (0 : ℝ) < n := by exact_mod_cast (by omega : 0 < n)
have hdR : (1 : ℝ) ≤ d := by exact_mod_cast (by omega : 1 ≤ d)
have hdpos : (0 : ℝ) < d := by linarith
have hdD : (d : ℝ) ≤ D := by exact_mod_cast (by omega : d ≤ D)
have hL : 1 ≤ logAlphabet d := by
rw [logAlphabet, Real.log_mul (Real.exp_pos _).ne' hdpos.ne', Real.log_exp]
have hlog : 0 ≤ log (d : ℝ) := Real.log_nonneg hdR
linarith
have hscale : rateScale n d ε ≤
(D : ℝ) * min 1 (1 / ((n : ℝ) * ε)) := by
rw [mul_min_of_nonneg _ _ hDR.le]
apply le_min
· have hD1 : (1 : ℝ) ≤ D := by exact_mod_cast (by omega : 1 ≤ D)
exact (min_le_left _ _).trans (by simpa using hD1)
· calc
_ ≤ (d : ℝ) / ((n : ℝ) * ε * logAlphabet d) := min_le_right _ _
_ ≤ (D : ℝ) / ((n : ℝ) * ε) := by
apply (div_le_div_iff₀ (by positivity) (by positivity)).2
have h := mul_le_mul_of_nonneg_left hL hDR.le
have hbound : (d : ℝ) ≤ D * logAlphabet d := hdD.trans (by simpa using h)
have hm := mul_le_mul_of_nonneg_right hbound
(show 0 ≤ (n : ℝ) * ε by positivity)
nlinarith only [hm]
_ = (D : ℝ) * (1 / ((n : ℝ) * ε)) := by ring
calc
cstar * rateScale n d ε ≤ (1 / (1024 * (D : ℝ))) * rateScale n d ε :=
mul_le_mul_of_nonneg_right (min_le_right _ _) hr
_ ≤ (1 / (1024 * (D : ℝ))) *
((D : ℝ) * min 1 (1 / ((n : ℝ) * ε))) :=
mul_le_mul_of_nonneg_left hscale (by positivity)
_ = (1 / 1024 : ℝ) * min 1 (1 / ((n : ℝ) * ε)) := by field_simp
_ ≤ minimaxRisk n d ε := finiteAlphabet_parametric_minimaxRisk_lower hn hd hε hεhi
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.LowerHistogramTransfer 9 declarations The ordered-prefix law pays only the short-count probability.
Fixed-sample estimators in the lower Poisson count experiment
The ordered-prefix law pays only the short-count probability. Uniform histogram reconstruction and kernel averaging then give a deterministic, parameter-free count estimator with the same risk bound, including parameter-dependent rates.
def lowerPrefixEstimator
Read a fixed prefix and project the fixed-sample estimator to the unit interval. For the displayed parameters, lowerPrefixEstimator is the object specified by this definition.
Definition (Lean source)
parameters
def
lowerPrefixEstimator est fallback s :
ℝ
given by
projectUnit (est.1 (orderedPrefix fallback s))
The exact prefix pushforward bounds its risk by fixed risk plus shortage. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
sqRisk (finitePoissonSampleLaw P.1.pmf.toMeasure lam) (lowerPrefixEstimator est fallback) (observedValue P.1)
Proof (Lean source)
lemma lowerPrefixEstimator_risk_le {n d : ℕ} {ε : ℝ}
(P : ModelLaw d ε) (est : Estimator n d) (fallback : Fin n → Obs d)
(lam : ℝ≥0) :
sqRisk (finitePoissonSampleLaw P.1.pmf.toMeasure lam)
(lowerPrefixEstimator est fallback) (observedValue P.1) ≤
observedRisk n est P + (poissonMeasure lam).real {k : ℕ | k < n} := by
letI : IsProbabilityMeasure (productLaw P.1 n) := by
unfold productLaw
infer_instance
let T : (Fin n → Obs d) → ℝ := fun o => projectUnit (est.1 o)
let loss := fun o => (T o - observedValue P.1) ^ 2
have hm : Measurable T := by dsimp [T]; unfold projectUnit; fun_prop
have hl : Integrable loss (productLaw P.1 n) := Integrable.of_finite
have hdir : Integrable loss (Measure.dirac fallback) := Integrable.of_finite
have hgood : (poissonMeasure lam (Ici n)).toReal ≤ 1 :=
measureReal_le_one
have hfb : loss fallback ≤ 1 := by
have hp := projectUnit_mem_unitInterval (est.1 fallback)
have hv := observedValue_mem_unitInterval P.1
dsimp [loss, T]
nlinarith [hp.1, hp.2, hv.1, hv.2]
have hmap := map_orderedPrefix_finitePoissonSampleLaw P.1.pmf.toMeasure lam fallback
unfold sqRisk lowerPrefixEstimator
change (∫ s, loss (orderedPrefix fallback s) ∂finitePoissonSampleLaw P.1.pmf.toMeasure lam) ≤ _
rw [← integral_map (measurable_orderedPrefix fallback).aemeasurable
(show AEStronglyMeasurable loss
(Measure.map (orderedPrefix fallback) (finitePoissonSampleLaw P.1.pmf.toMeasure lam)) from
(by dsimp [loss]; fun_prop)), hmap]
change (∫ o, loss o ∂((poissonMeasure lam (Ici n)) • productLaw P.1 n +
(poissonMeasure lam (Iio n)) • Measure.dirac fallback)) ≤ _
rw [integral_add_measure (hl.smul_measure (measure_ne_top _ _))
(hdir.smul_measure (measure_ne_top _ _)),
integral_smul_measure, integral_smul_measure, integral_dirac]
have hnonneg : 0 ≤ ∫ o, loss o ∂productLaw P.1 n :=
integral_nonneg (fun _ => sq_nonneg _)
calc
_ ≤ (∫ o, loss o ∂productLaw P.1 n) +
(poissonMeasure lam (Iio n)).toReal := by
apply add_le_add
· exact mul_le_of_le_one_left hnonneg hgood
· exact (mul_le_mul_of_nonneg_left hfb toReal_nonneg).trans_eq (mul_one _)
_ ≤ _ := add_le_add (observedRisk_projectUnit_le est P) le_rfl
def lowerHistogramEstimator
Average the reconstructed prefix over all orderings of a histogram. This estimator depends on the fixed estimator and fallback only.
Definition (Lean source)
parameters
def
lowerHistogramEstimator est fallback :
(Obs d → ℕ) → ℝ
given by
theorem measurable_lowerHistogramEstimator
The deterministic count estimator is measurable on the observed histogram space. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
Measurable (lowerHistogramEstimator est fallback)
Proof (Lean source)
@[fun_prop]
lemma measurable_lowerHistogramEstimator {n d : ℕ} (est : Estimator n d)
(fallback : Fin n → Obs d) :
Measurable (lowerHistogramEstimator est fallback) := by
have hprefix := measurable_orderedPrefix fallback
unfold lowerHistogramEstimator
apply measurable_kernelMean
unfold lowerPrefixEstimator projectUnit
fun_prop
theorem lowerHistogramEstimator_risk_le
Reconstruction followed by Jensen transfers the risk to all observed counts. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
sqRisk (independentPoissonCountLaw P.1.pmf.toMeasure lam) (lowerHistogramEstimator est fallback) (observedValue P.1)
Proof (Lean source)
lemma lowerHistogramEstimator_risk_le {n d : ℕ} {ε : ℝ}
(P : ModelLaw d ε) (est : Estimator n d) (fallback : Fin n → Obs d)
(lam : ℝ≥0) :
sqRisk (independentPoissonCountLaw P.1.pmf.toMeasure lam)
(lowerHistogramEstimator est fallback) (observedValue P.1) ≤
observedRisk n est P + (poissonMeasure lam).real {k : ℕ | k < n} := by
have hm : Measurable (lowerPrefixEstimator est fallback) := by
have hprefix := measurable_orderedPrefix fallback
unfold lowerPrefixEstimator projectUnit
fun_prop
have hb : UniformlyBounded (lowerPrefixEstimator est fallback) := by
refine ⟨1, by norm_num, fun s => ?_⟩
have hp := projectUnit_mem_unitInterval (est.1 (orderedPrefix fallback s))
unfold lowerPrefixEstimator
exact (abs_le.mpr ⟨by linarith [hp.1], hp.2⟩)
have h := sqRisk_kernelMean_le_comp
(independentPoissonCountLaw P.1.pmf.toMeasure lam) (histogramReconstructionKernel (Obs d))
hm hb (observedValue P.1)
rw [independentPoissonCountLaw_comp_reconstruction] at h
exact h.trans (lowerPrefixEstimator_risk_le P est fallback lam)
theorem lowerHistogramEstimator_poisson_risk_le
A mean at least Bn/2 gives the uniform shortage penalty from roadmap (49). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
n d :
ℕ
ε B :
ℝ
P :
ModelLaw d ε
est :
Estimator n d
fallback :
Fin n → Obs d
lam :
ℝ≥0
hn :
1 ≤ n
hB :
2 < B
hmean :
B * (n : ℝ) / 2 ≤ lam
conclusion
sqRisk (independentPoissonCountLaw P.1.pmf.toMeasure lam) (lowerHistogramEstimator est fallback) (observedValue P.1)
≤ observedRisk n est P + exp (-(n : ℝ) * (B - 2) ^ 2 / (4 * B))
Proof (Lean source)
lemma lowerHistogramEstimator_poisson_risk_le {n d : ℕ} {ε B : ℝ}
(P : ModelLaw d ε) (est : Estimator n d) (fallback : Fin n → Obs d)
(lam : ℝ≥0) (hn : 1 ≤ n) (hB : 2 < B)
(hmean : B * (n : ℝ) / 2 ≤ lam) :
sqRisk (independentPoissonCountLaw P.1.pmf.toMeasure lam)
(lowerHistogramEstimator est fallback) (observedValue P.1) ≤
observedRisk n est P + exp (-(n : ℝ) * (B - 2) ^ 2 / (4 * B)) := by
exact (lowerHistogramEstimator_risk_le P est fallback lam).trans
(add_le_add_right (poisson_shortage_uniform_le lam hn hB hmean) _)
theorem observed_independentPoissonCountLaw_eq
Independent count rates agree with the four observed atom masses. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
independentPoissonCountLaw P.pmf.toMeasure lam
= Measure.pi fun o : Obs d => poissonMeasure (toNNReal ((lam : ℝ) * jointMass P o.1 o.2.1 o.2.2))
Proof (Lean source)
lemma observed_independentPoissonCountLaw_eq {d : ℕ} (P : DiscreteLaw d)
(lam : ℝ≥0) :
independentPoissonCountLaw P.pmf.toMeasure lam =
Measure.pi fun o : Obs d => poissonMeasure (toNNReal
((lam : ℝ) * jointMass P o.1 o.2.1 o.2.2)) := by
unfold independentPoissonCountLaw
congr 1
funext o
congr 1
apply NNReal.eq
simp [jointMass, PMF.toMeasure_apply_singleton, ENNReal.coe_toNNReal,
Real.toNNReal_of_nonneg (mul_nonneg lam.coe_nonneg toReal_nonneg),
NNReal.coe_mul]
exact inl rfl
theorem denseHistogram_fixedSample_risk_le
The same parameter-free count estimator transfers every dense fibre. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
n d :
ℕ
hd :
0 < d
ε a B :
ℝ
hε :
0 < ε
hεhi :
ε ≤ 1 / 2
ha :
0 ≤ a
hahi :
a ≤ 1 / 2
hn :
1 ≤ n
hB :
2 < B
θ :
Fin d → ℝ
hθ :
∀ x, |θ x| ≤ 1
est :
Estimator n d
fallback :
Fin n → Obs d
conclusion
sqRisk (denseFullObservedPoissonKernel d (B * n * ε / d) a (B * n * (1 - ε) / (2 * d)) θ) (lowerHistogramEstimator est fallback) (observedValue (denseObservedLaw hd ε a hε hεhi ha hahi θ hθ))
≤ observedRisk n est ⟨denseObservedLaw hd ε a hε hεhi ha hahi θ hθ, (denseObservedLaw_parameters hd ε a hε hεhi ha hahi θ hθ).1⟩
+ exp (-(n : ℝ) * (B - 2) ^ 2 / (4 * B))
Proof (Lean source)
lemma denseHistogram_fixedSample_risk_le {n d : ℕ} (hd : 0 < d)
{ε a B : ℝ} (hε : 0 < ε) (hεhi : ε ≤ 1 / 2)
(ha : 0 ≤ a) (hahi : a ≤ 1 / 2) (hn : 1 ≤ n) (hB : 2 < B)
(θ : Fin d → ℝ) (hθ : ∀ x, |θ x| ≤ 1)
(est : Estimator n d) (fallback : Fin n → Obs d) :
sqRisk
(denseFullObservedPoissonKernel d (B * n * ε / d) a
(B * n * (1 - ε) / (2 * d)) θ)
(lowerHistogramEstimator est fallback)
(observedValue (denseObservedLaw hd ε a hε hεhi ha hahi θ hθ)) ≤
observedRisk n est
⟨denseObservedLaw hd ε a hε hεhi ha hahi θ hθ,
(denseObservedLaw_parameters hd ε a hε hεhi ha hahi θ hθ).1⟩ +
exp (-(n : ℝ) * (B - 2) ^ 2 / (4 * B)) := by
let P : ModelLaw d ε :=
⟨denseObservedLaw hd ε a hε hεhi ha hahi θ hθ,
(denseObservedLaw_parameters hd ε a hε hεhi ha hahi θ hθ).1⟩
have hmean : 0 ≤ B * (n : ℝ) := by positivity
have h := lowerHistogramEstimator_poisson_risk_le P est fallback
(toNNReal (B * n)) hn hB (by rw [Real.coe_toNNReal _ hmean]; nlinarith)
rw [observed_independentPoissonCountLaw_eq,
Real.coe_toNNReal _ hmean] at h
rw [denseFullObservedPoissonKernel_eq hd ε a B n hε hεhi ha hahi θ hθ]
exact h
theorem sparseHistogram_fixedSample_risk_le Lemma sparseHistogram_fixedSample_risk_le in the paper ↗
The transfer also allows the sparse total rate to depend on its normalizer. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
n d :
ℕ
hd :
2 ≤ d
ε M B :
ℝ
hε :
0 < ε
hεhi :
ε ≤ 1 / 2
hM :
2 ≤ M
hn :
1 ≤ n
hB :
2 < B
θ :
Fin d → ℝ
hz :
∀ x, θ x + sparseReference (_hMpaper := hM) M ∈ Icc 0 M
est :
Estimator n d
fallback :
Fin n → Obs d
conclusion
sqRisk (sparseFullObservedPoissonKernel (hMpaper := hM) B n d ε M θ) (lowerHistogramEstimator est fallback) (observedValue (sparseLaw (by omega) ε M ⟨hε, hεhi⟩ (fun x => θ x + sparseReference (_hMpaper := hM) M) hz))
≤ observedRisk n est ⟨sparseLaw (by omega) ε M ⟨hε, hεhi⟩ (fun x => θ x + sparseReference (_hMpaper := hM) M) hz, (sparse_likelihood n d ε M B (fun x => θ x + sparseReference (_hMpaper := hM) M) hn hd hε hεhi hM hB hz).1⟩
+ exp (-(n : ℝ) * (B - 2) ^ 2 / (4 * B))
Proof (Lean source)
lemma sparseHistogram_fixedSample_risk_le {n d : ℕ} (hd : 2 ≤ d)
{ε M B : ℝ} (hε : 0 < ε) (hεhi : ε ≤ 1 / 2)
(hM : 2 ≤ M) (hn : 1 ≤ n) (hB : 2 < B)
(θ : Fin d → ℝ) (hz : ∀ x, θ x + sparseReference (_hMpaper := hM) M ∈ Icc 0 M)
(est : Estimator n d) (fallback : Fin n → Obs d) :
sqRisk (sparseFullObservedPoissonKernel (hMpaper := hM) B n d ε M θ)
(lowerHistogramEstimator est fallback)
(observedValue (sparseLaw (by omega) ε M ⟨hε, hεhi⟩
(fun x => θ x + sparseReference (_hMpaper := hM) M) hz)) ≤
observedRisk n est
⟨sparseLaw (by omega) ε M ⟨hε, hεhi⟩
(fun x => θ x + sparseReference (_hMpaper := hM) M) hz,
(sparse_likelihood n d ε M B (fun x => θ x + sparseReference (_hMpaper := hM) M)
hn hd hε hεhi hM hB hz).1⟩ +
exp (-(n : ℝ) * (B - 2) ^ 2 / (4 * B)) := by
let P : ModelLaw d ε :=
⟨sparseLaw (by omega) ε M ⟨hε, hεhi⟩
(fun x => θ x + sparseReference (_hMpaper := hM) M) hz,
(sparse_likelihood n d ε M B (fun x => θ x + sparseReference (_hMpaper := hM) M)
hn hd hε hεhi hM hB hz).1⟩
let S := sparseNormalizerFormula d ε M (fun x => θ x + sparseReference (_hMpaper := hM) M) hz
have hS : 1 / 2 ≤ S := sparseNormalizer_ge_half hε.le hεhi _ hz
have hmean : 0 ≤ B * (n : ℝ) * S := by positivity
have h := lowerHistogramEstimator_poisson_risk_le P est fallback
(toNNReal (B * n * S)) hn hB (by
rw [Real.coe_toNNReal _ hmean]
nlinarith [mul_nonneg (show 0 ≤ B by linarith) (Nat.cast_nonneg n)])
rw [observed_independentPoissonCountLaw_eq,
Real.coe_toNNReal _ hmean] at h
rw [sparseFullObservedPoissonKernel_eq (hMpaper := hM) (by omega) ε M B n hε hεhi θ hz]
exact h
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.LowerPriorTransfer 8 declarations Integrate the parameter-free histogram comparison over the legal packet and Fejér priors.
Integrating the fixed-sample lower comparison
Integrate the parameter-free histogram comparison over the legal packet and Fejér priors. Both full-count Bayes bounds give fixed-sample minimax lower bounds, with only the explicit Poisson shortage penalty from roadmap (49).
theorem observedRisk_bddAbove
Finite sample estimators have uniformly bounded observed risk, because the sample alphabet is finite and the observed value lies in the unit interval. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma observedRisk_bddAbove {n d : ℕ} {ε : ℝ} (est : Estimator n d) :
BddAbove (range (observedRisk n (ε := ε) est)) := by
obtain ⟨R, hR⟩ := (Set.finite_range (fun o => |est.1 o|)).bddAbove
refine ⟨(max R 0 + 1) ^ 2, ?_⟩
rintro _ ⟨P, rfl⟩
letI : IsProbabilityMeasure (productLaw P.1 n) := by unfold productLaw; infer_instance
have hpoint (o : Fin n → Obs d) :
(est.1 o - observedValue P.1) ^ 2 ≤ (max R 0 + 1) ^ 2 := by
have he : |est.1 o| ≤ max R 0 := (hR (Set.mem_range_self o)).trans (le_max_left _ _)
have hv := observedValue_mem_unitInterval P.1
have hb : |est.1 o - observedValue P.1| ≤ max R 0 + 1 := by
calc
_ ≤ |est.1 o| + |observedValue P.1| := abs_sub _ _
_ ≤ _ := add_le_add he (by rw [abs_of_nonneg hv.1]; exact hv.2)
simpa only [sq_abs] using pow_le_pow_left₀ (abs_nonneg _) hb 2
unfold observedRisk sqRisk
exact (integral_mono Integrable.of_finite (integrable_const _) hpoint).trans_eq (by simp)
theorem lowerHistogramEstimator_abs_le_one
Averaging a projected prefix leaves the histogram statistic bounded by one. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
|lowerHistogramEstimator est fallback N| ≤ 1
Proof (Lean source)
lemma lowerHistogramEstimator_abs_le_one {n d : ℕ} (est : Estimator n d)
(fallback : Fin n → Obs d) (N : Obs d → ℕ) :
|lowerHistogramEstimator est fallback N| ≤ 1 := by
apply abs_kernelMean_le _
intro s
have hp := projectUnit_mem_unitInterval (est.1 (orderedPrefix fallback s))
exact abs_le.mpr ⟨by dsimp [lowerPrefixEstimator]; linarith [hp.1], hp.2⟩
theorem lowerHistogram_squaredRisk_eq
A bounded histogram estimator's nonnegative integral loss agrees exactly with the ENNReal squared risk used by fuzzy testing. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
Θ :
Type*
n d :
ℕ
K :
Kernel Θ (Obs d → ℕ)
θ :
Θ
IsProbabilityMeasure (K θ)
target :
Θ → ℝ
est :
Estimator n d
fallback :
Fin n → Obs d
conclusion
squaredRisk K target (lowerHistogramEstimator est fallback) θ
Proof (Lean source)
lemma lowerHistogram_squaredRisk_eq {Θ : Type*} [MeasurableSpace Θ]
{n d : ℕ} (K : Kernel Θ (Obs d → ℕ)) (θ : Θ)
[IsProbabilityMeasure (K θ)] (target : Θ → ℝ)
(est : Estimator n d) (fallback : Fin n → Obs d) :
squaredRisk K target (lowerHistogramEstimator est fallback) θ =
ofReal (sqRisk (K θ)
(lowerHistogramEstimator est fallback) (target θ)) := by
have hi : Integrable (fun N =>
(lowerHistogramEstimator est fallback N - target θ) ^ 2) (K θ) := by
refine (integrable_const ((1 + |target θ|) ^ 2)).mono' (by fun_prop) ?_
filter_upwards [] with N
rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)]
have hb : |lowerHistogramEstimator est fallback N - target θ| ≤ 1 + |target θ| :=
(abs_sub _ _).trans (add_le_add (lowerHistogramEstimator_abs_le_one est fallback N) le_rfl)
simpa only [sq_abs] using pow_le_pow_left₀ (abs_nonneg _) hb 2
exact (ofReal_integral_eq_lintegral_ofReal hi
(Filter.Eventually.of_forall fun _ => sq_nonneg _)).symm
theorem sparseHistogram_bayesRisk_le_fixedWorstRisk Lemma sparseHistogram_bayesRisk_le_fixedWorstRisk in the paper ↗
On legal sparse vectors, integrating the fibre transfer costs at most the fixed estimator's worst-case risk plus the uniform shortage penalty. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
n d :
ℕ
hd :
2 ≤ d
ε M B :
ℝ
hε :
0 < ε
hεhi :
ε ≤ 1 / 2
hM :
2 ≤ M
hn :
1 ≤ n
hB :
2 < B
ν :
hsupp :
ν (Icc 0 M)ᶜ = 0
est :
Estimator n d
fallback :
Fin n → Obs d
conclusion
bayesSquaredRisk (productPrior d (Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) ν)) (sparseFullObservedPoissonKernel (hMpaper := hM) B n d ε M) (fun θ => observedValue (sparseObservedModel hd ε M hε hεhi hM θ).1) (lowerHistogramEstimator est fallback)
Proof (Lean source)
lemma sparseHistogram_bayesRisk_le_fixedWorstRisk {n d : ℕ} (hd : 2 ≤ d)
{ε M B : ℝ} (hε : 0 < ε) (hεhi : ε ≤ 1 / 2)
(hM : 2 ≤ M) (hn : 1 ≤ n) (hB : 2 < B)
(ν : Measure ℝ) [IsProbabilityMeasure ν] (hsupp : ν (Icc 0 M)ᶜ = 0)
(est : Estimator n d) (fallback : Fin n → Obs d) :
bayesSquaredRisk
(productPrior d (Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) ν))
(sparseFullObservedPoissonKernel (hMpaper := hM) B n d ε M)
(fun θ => observedValue (sparseObservedModel hd ε M hε hεhi hM θ).1)
(lowerHistogramEstimator est fallback) ≤
ofReal (worstCaseRiskReal (observedRisk n (ε := ε)) est +
exp (-(n : ℝ) * (B - 2) ^ 2 / (4 * B))) := by
letI : IsProbabilityMeasure (Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) ν) :=
Measure.isProbabilityMeasure_map (by fun_prop)
letI : IsProbabilityMeasure (productPrior d (Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) ν)) := by
unfold productPrior
infer_instance
unfold bayesSquaredRisk
calc
_ ≤ ∫⁻ _θ,
ofReal (worstCaseRiskReal (observedRisk n (ε := ε)) est +
exp (-(n : ℝ) * (B - 2) ^ 2 / (4 * B)))
∂productPrior d (Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) ν) := by
apply lintegral_mono_ae
filter_upwards [sparse_product_prior_ae_legal (hMpaper := hM) ν hsupp] with θ hθ
letI := sparseFullObservedPoissonKernel_probability (hMpaper := hM) B n d ε M θ
rw [lowerHistogram_squaredRisk_eq]
apply ENNReal.ofReal_le_ofReal
have hclip (x : Fin d) : sparseClippedIntensity (hMpaper := hM) M (θ x) = θ x + sparseReference (_hMpaper := hM) M := by
simpa only [add_sub_cancel_right] using sparseClippedIntensity_sub_reference (hMpaper := hM) (hθ x)
have hmodel : (sparseObservedModel hd ε M hε hεhi hM θ).1 =
sparseLaw (by omega) ε M ⟨hε, hεhi⟩ (fun x => θ x + sparseReference (_hMpaper := hM) M) hθ := by
unfold sparseObservedModel
simp_rw [hclip]
rw [hmodel]
exact (sparseHistogram_fixedSample_risk_le hd hε hεhi hM hn hB θ hθ est fallback).trans
(add_le_add (le_worstCaseRisk (observedRisk_bddAbove est) _) le_rfl)
_ = _ := by simp
theorem sparse_fixedSample_minimaxRisk_lower
The packet lower experiment yields a fixed-sample sparse minimax bound. Both prior risks are bounded by the same fixed estimator's worst-case risk; thus only the uniform Poisson shortage term is lost in the comparison. In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ (c η C : ℝ) (D : ℕ),
0 < c
0 < η
2 ≤ C
2 ≤ D
conclusion 1
n d :
ℕ
ε M B :
ℝ
1 ≤ n
D ≤ d
0 < ε
ε ≤ 1 / 2
2 ≤ M
M ≤ (lowerLogDegree C d : ℝ) ^ 2
2 < B
M ≤ η * d * lowerLogDegree C d / (B * n * ε)
c * M / (lowerLogDegree C d : ℝ) ^ 2 - exp (-(n : ℝ) * (B - 2) ^ 2 / (4 * B))
≤ minimaxRisk n d ε
Proof (Lean source)
lemma sparse_fixedSample_minimaxRisk_lower :
∃ (c η C : ℝ) (D : ℕ), 0 < c ∧ 0 < η ∧ 2 ≤ C ∧ 2 ≤ D ∧
∀ (n d : ℕ) (ε M B : ℝ),
1 ≤ n → D ≤ d → 0 < ε → ε ≤ 1 / 2 → 2 ≤ M →
M ≤ (lowerLogDegree C d : ℝ) ^ 2 → 2 < B →
M ≤ η * d * lowerLogDegree C d / (B * n * ε) →
c * M / (lowerLogDegree C d : ℝ) ^ 2 -
exp (-(n : ℝ) * (B - 2) ^ 2 / (4 * B)) ≤ minimaxRisk n d ε := by
obtain ⟨c, η, C, A, D, hc, hη, hC, hA, hD, hbound⟩ :=
sparse_full_observed_logDegree_bayesRisk_lower
refine ⟨c, η, C, D, hc, hη, hC, hD, ?_⟩
intro n d ε M B hn hd hε hεhi hM hMhi hB hband
have hd2 : 2 ≤ d := hD.trans hd
obtain ⟨P, hdom, hcompletion, hP⟩ :=
hbound n d ε M B hd2 hε hεhi hM hn hd hMhi hB hband
letI := P.prob₀
letI := P.prob₁
letI : Nonempty (Estimator n d) := ⟨⟨fun _ => 0, measurable_const⟩⟩
apply Causalean.Stat.le_minimaxValue
intro est
let fallback : Fin n → Obs d := fun _ => (⟨0, by omega⟩, false, false)
have hl := hP (lowerHistogramEstimator est fallback) (by fun_prop)
have hu := max_le
(sparseHistogram_bayesRisk_le_fixedWorstRisk hd2 hε hεhi hM hn hB P.ν₀ P.supp₀ est fallback)
(sparseHistogram_bayesRisk_le_fixedWorstRisk hd2 hε hεhi hM hn hB P.ν₁ P.supp₁ est fallback)
have hr := ENNReal.ofReal_le_ofReal_iff (p := c * M / (lowerLogDegree C d : ℝ) ^ 2)
(show 0 ≤ worstCaseRiskReal (observedRisk n (ε := ε)) est +
exp (-(n : ℝ) * (B - 2) ^ 2 / (4 * B)) from
add_nonneg (worstCaseRisk_nonneg (observedRisk_nonneg est)) (Real.exp_nonneg _))
have hreal := hr.mp (hl.trans hu)
linarith
Supported scalar dense priors put every product coordinate in the legal parameter interval, so clipping has no effect on the comparison experiment. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
∀ᵐ θ ∂productPrior d ν, ∀ x, |θ x| ≤ 1
Proof (Lean source)
lemma dense_product_prior_ae_legal {d : ℕ} (ν : Measure ℝ) [IsProbabilityMeasure ν]
(hsupp : IsSupportedOnUnitInterval ν) :
∀ᵐ θ ∂productPrior d ν, ∀ x, |θ x| ≤ 1 := by
have hs : ∀ᵐ t ∂ν, |t| ≤ 1 :=
(mem_ae_iff_prob_eq_one (measurableSet_le (by fun_prop) measurable_const)).2
(dense_prior_support_radius_one ν hsupp)
rw [Filter.eventually_all]
intro x
exact ((measurePreserving_eval (fun _ : Fin d => ν) x).hasLaw.ae_iff
(by fun_prop)).2 hs
theorem denseHistogram_bayesRisk_le_fixedWorstRisk
The dense full-count prior comparison integrates to the same fixed-sample worst risk and shortage penalty as the sparse comparison. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
n d :
ℕ
hd :
0 < d
ε a B :
ℝ
hε :
0 < ε
hεhi :
ε ≤ 1 / 2
ha :
0 ≤ a
hahi :
a ≤ 1 / 2
hn :
1 ≤ n
hB :
2 < B
ν :
hsupp :
est :
Estimator n d
fallback :
Fin n → Obs d
conclusion
bayesSquaredRisk (productPrior d ν) (denseFullObservedPoissonKernel d (B * n * ε / d) a (B * n * (1 - ε) / (2 * d))) (fun θ => observedValue (denseObservedModel hd ε a hε hεhi ha hahi θ).1) (lowerHistogramEstimator est fallback)
Proof (Lean source)
lemma denseHistogram_bayesRisk_le_fixedWorstRisk {n d : ℕ} (hd : 0 < d)
{ε a B : ℝ} (hε : 0 < ε) (hεhi : ε ≤ 1 / 2)
(ha : 0 ≤ a) (hahi : a ≤ 1 / 2) (hn : 1 ≤ n) (hB : 2 < B)
(ν : Measure ℝ) [IsProbabilityMeasure ν]
(hsupp : IsSupportedOnUnitInterval ν)
(est : Estimator n d) (fallback : Fin n → Obs d) :
bayesSquaredRisk (productPrior d ν)
(denseFullObservedPoissonKernel d (B * n * ε / d) a (B * n * (1 - ε) / (2 * d)))
(fun θ => observedValue (denseObservedModel hd ε a hε hεhi ha hahi θ).1)
(lowerHistogramEstimator est fallback) ≤
ofReal (worstCaseRiskReal (observedRisk n (ε := ε)) est +
exp (-(n : ℝ) * (B - 2) ^ 2 / (4 * B))) := by
letI : IsProbabilityMeasure (productPrior d ν) := by unfold productPrior; infer_instance
unfold bayesSquaredRisk
calc
_ ≤ ∫⁻ _θ, ofReal
(worstCaseRiskReal (observedRisk n (ε := ε)) est +
exp (-(n : ℝ) * (B - 2) ^ 2 / (4 * B))) ∂productPrior d ν := by
apply lintegral_mono_ae
filter_upwards [dense_product_prior_ae_legal ν hsupp] with θ hθ
letI : IsProbabilityMeasure
(denseFullObservedPoissonKernel d (B * n * ε / d) a
(B * n * (1 - ε) / (2 * d)) θ) := by
rw [denseFullObservedPoissonKernel_eq hd ε a B n hε hεhi ha hahi θ hθ]
infer_instance
rw [lowerHistogram_squaredRisk_eq]
apply ENNReal.ofReal_le_ofReal
have hmodel : (denseObservedModel hd ε a hε hεhi ha hahi θ).1 =
denseObservedLaw hd ε a hε hεhi ha hahi θ hθ := by
unfold denseObservedModel
simp_rw [denseClippedParameter_eq (hθ _)]
rw [hmodel]
exact (denseHistogram_fixedSample_risk_le hd hε hεhi ha hahi hn hB θ hθ est fallback).trans
(add_le_add (le_worstCaseRisk (observedRisk_bddAbove est) _) le_rfl)
_ = _ := by simp
theorem dense_fixedSample_minimaxRisk_lower
The Fejér dense lower experiment transfers to fixed-sample minimax risk, retaining its a²/K² order and subtracting only the explicit shortage penalty. In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ (η C : ℝ) (D : ℕ),
0 < η
2 ≤ C
2 ≤ D
conclusion 1
n d :
ℕ
ε a B :
ℝ
1 ≤ n
D ≤ d
0 < ε
ε ≤ 1 / 2
0 < a
a ≤ 1 / 2
2 < B
(B * n * ε / d) * a ^ 2 ≤ η * lowerLogDegree C d
(5 / 2560000 : ℝ) * a ^ 2 / (lowerLogDegree C d : ℝ) ^ 2
- exp (-(n : ℝ) * (B - 2) ^ 2 / (4 * B))
≤ minimaxRisk n d ε
Proof (Lean source)
lemma dense_fixedSample_minimaxRisk_lower :
∃ (η C : ℝ) (D : ℕ), 0 < η ∧ 2 ≤ C ∧ 2 ≤ D ∧
∀ (n d : ℕ) (ε a B : ℝ),
1 ≤ n → D ≤ d → 0 < ε → ε ≤ 1 / 2 → 0 < a → a ≤ 1 / 2 → 2 < B →
(B * n * ε / d) * a ^ 2 ≤ η * lowerLogDegree C d →
(5 / 2560000 : ℝ) * a ^ 2 / (lowerLogDegree C d : ℝ) ^ 2 -
exp (-(n : ℝ) * (B - 2) ^ 2 / (4 * B)) ≤ minimaxRisk n d ε := by
obtain ⟨η, C, D, hη, hC, hD, hbound⟩ := dense_full_observed_logDegree_bayesRisk_lower
refine ⟨η, C, D, hη, hC, hD, ?_⟩
intro n d ε a B hn hd hε hεhi ha hahi hB hband
have hdpos : 0 < d := by omega
have hr : 0 < B * (n : ℝ) * ε / d := by positivity
obtain ⟨P, hP⟩ := hbound d (B * n * ε / d) a
(B * n * (1 - ε) / (2 * d)) ε hdpos hε hεhi ha.le hahi hd hr ha hband
letI := P.probability₀
letI := P.probability₁
letI : Nonempty (Estimator n d) := ⟨⟨fun _ => 0, measurable_const⟩⟩
apply Causalean.Stat.le_minimaxValue
intro est
let fallback : Fin n → Obs d := fun _ => (⟨0, hdpos⟩, false, false)
have hl := hP (lowerHistogramEstimator est fallback) (by fun_prop)
have hu := max_le
(denseHistogram_bayesRisk_le_fixedWorstRisk hdpos hε hεhi ha.le hahi hn hB P.ν₀ P.supported₀ est fallback)
(denseHistogram_bayesRisk_le_fixedWorstRisk hdpos hε hεhi ha.le hahi hn hB P.ν₁ P.supported₁ est fallback)
have hreal := (ENNReal.ofReal_le_ofReal_iff
(show 0 ≤ worstCaseRiskReal (observedRisk n (ε := ε)) est +
exp (-(n : ℝ) * (B - 2) ^ 2 / (4 * B)) from
add_nonneg (worstCaseRisk_nonneg (observedRisk_nonneg est))
(Real.exp_nonneg _))).mp (hl.trans hu)
linarith
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.LowerRateCalibration 2 declarations Roadmap (33), (40), and (45)--(48) optimize the supported sparse endpoint and the legal dense amplitude.
Choosing the lower-experiment endpoint and amplitude
Roadmap (33), (40), and (45)--(48) optimize the supported sparse endpoint and the legal dense amplitude. These bounds retain the explicit fixed-sample shortage penalty; no testing or approximation conclusion is assumed.
theorem sparse_fixedSample_calibrated_lower
Choosing the endpoint in (33) gives the capped sparse rate in (40), whenever the intensity-dependent endpoint is at least two. In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ (c η C : ℝ) (D : ℕ),
0 < c
0 < η
2 ≤ C
2 ≤ D
conclusion 1
c * min 1 (η * d / (B * n * ε * lowerLogDegree C d))
- exp (-(n : ℝ) * (B - 2) ^ 2 / (4 * B))
≤ minimaxRisk n d ε
Proof (Lean source)
lemma sparse_fixedSample_calibrated_lower :
∃ (c η C : ℝ) (D : ℕ), 0 < c ∧ 0 < η ∧ 2 ≤ C ∧ 2 ≤ D ∧
∀ (n d : ℕ) (ε B : ℝ),
1 ≤ n → D ≤ d → 0 < ε → ε ≤ 1 / 2 → 2 < B →
2 ≤ η * d * lowerLogDegree C d / (B * n * ε) →
c * min 1 (η * d / (B * n * ε * lowerLogDegree C d)) -
exp (-(n : ℝ) * (B - 2) ^ 2 / (4 * B)) ≤ minimaxRisk n d ε := by
obtain ⟨c, η, C, D, hc, hη, hC, hD, hbound⟩ := sparse_fixedSample_minimaxRisk_lower
refine ⟨c, η, C, D, hc, hη, hC, hD, ?_⟩
intro n d ε B hn hd hε hεhi hB hband
have hK := (lowerLogDegree_bounds hC (hD.trans hd)).1
have hKpos : (0 : ℝ) < lowerLogDegree C d := by exact_mod_cast (by omega : 0 < lowerLogDegree C d)
have hnpos : (0 : ℝ) < n := by exact_mod_cast (by omega : 0 < n)
have hden : 0 < B * (n : ℝ) * ε := by positivity
let M := min ((lowerLogDegree C d : ℝ) ^ 2)
(η * d * lowerLogDegree C d / (B * n * ε))
have hKtwo : (2 : ℝ) ≤ lowerLogDegree C d := by exact_mod_cast hK
have hM : 2 ≤ M := le_min (by nlinarith) hband
have hl := hbound n d ε M B hn hd hε hεhi hM
(min_le_left _ _) hB (min_le_right _ _)
have heq : M / (lowerLogDegree C d : ℝ) ^ 2 =
min 1 (η * d / (B * n * ε * lowerLogDegree C d)) := by
dsimp [M]
rw [← min_div_div_right (sq_nonneg _)]
congr 1
· exact div_self (ne_of_gt (sq_pos_of_pos hKpos))
· field_simp
simpa only [mul_div_assoc, heq] using hl
theorem dense_fixedSample_calibrated_lower
Taking the square root of the largest legal dense squared amplitude optimizes (48), including the amplitude cap needed for Bernoulli legality. In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ (η C : ℝ) (D : ℕ),
0 < η
2 ≤ C
2 ≤ D
conclusion 1
n d :
ℕ
ε B :
ℝ
1 ≤ n
D ≤ d
0 < ε
ε ≤ 1 / 2
2 < B
(5 / 2560000 : ℝ) * min (1 / (4 * (lowerLogDegree C d : ℝ) ^ 2)) (η * d / (B * n * ε * lowerLogDegree C d))
- exp (-(n : ℝ) * (B - 2) ^ 2 / (4 * B))
≤ minimaxRisk n d ε
Proof (Lean source)
lemma dense_fixedSample_calibrated_lower :
∃ (η C : ℝ) (D : ℕ), 0 < η ∧ 2 ≤ C ∧ 2 ≤ D ∧
∀ (n d : ℕ) (ε B : ℝ),
1 ≤ n → D ≤ d → 0 < ε → ε ≤ 1 / 2 → 2 < B →
(5 / 2560000 : ℝ) *
min (1 / (4 * (lowerLogDegree C d : ℝ) ^ 2))
(η * d / (B * n * ε * lowerLogDegree C d)) -
exp (-(n : ℝ) * (B - 2) ^ 2 / (4 * B)) ≤ minimaxRisk n d ε := by
obtain ⟨η, C, D, hη, hC, hD, hbound⟩ := dense_fixedSample_minimaxRisk_lower
refine ⟨η, C, D, hη, hC, hD, ?_⟩
intro n d ε B hn hd hε hεhi hB
have hK := (lowerLogDegree_bounds hC (hD.trans hd)).1
have hKpos : (0 : ℝ) < lowerLogDegree C d := by exact_mod_cast (by omega : 0 < lowerLogDegree C d)
have hnpos : (0 : ℝ) < n := by exact_mod_cast (by omega : 0 < n)
have hdpos : (0 : ℝ) < d := by exact_mod_cast (by omega : 0 < d)
have hden : 0 < B * (n : ℝ) * ε := by positivity
let q := min (1 / 4 : ℝ) (η * d * lowerLogDegree C d / (B * n * ε))
have hq : 0 < q := lt_min (by norm_num) (by positivity)
have hsquare : (sqrt q) ^ 2 = q := Real.sq_sqrt hq.le
have hcap : sqrt q ≤ 1 / 2 := by
have hqcap : q ≤ 1 / 4 := min_le_left _ _
nlinarith [Real.sqrt_nonneg q]
have hband : (B * n * ε / d) * (sqrt q) ^ 2 ≤ η * lowerLogDegree C d := by
rw [hsquare]
calc
_ ≤ (B * n * ε / d) * (η * d * lowerLogDegree C d / (B * n * ε)) :=
mul_le_mul_of_nonneg_left (min_le_right _ _) (by positivity)
_ = _ := by field_simp
have hl := hbound n d ε (sqrt q) B hn hd hε hεhi
(Real.sqrt_pos.2 hq) hcap hB hband
have heq : q / (lowerLogDegree C d : ℝ) ^ 2 =
min (1 / (4 * (lowerLogDegree C d : ℝ) ^ 2))
(η * d / (B * n * ε * lowerLogDegree C d)) := by
dsimp [q]
rw [← min_div_div_right (sq_nonneg _)]
congr 1 <;> field_simp
simpa only [hsquare, mul_div_assoc, heq] using hl
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.LowerRateSplice 4 declarations The complementary endpoint ranges in roadmap (40) and (48) give one large-alphabet lower bound.
Splicing the sparse and dense lower experiments
The complementary endpoint ranges in roadmap (40) and (48) give one large-alphabet lower bound. Different universal logarithmic degrees in the two experiments are compared explicitly. The Poisson shortage penalty is absorbed by a universal inflation, yielding the exact paper rate.
theorem sparse_dense_fixedSample_spliced_lower
The sparse packet experiment and the dense boundary experiment together cover every intensity, with a single positive lower constant and alphabet cutoff. This is the range splice in (40)--(48), before absorbing (49). In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ (c C : ℝ) (D : ℕ),
0 < c
2 ≤ C
2 ≤ D
conclusion 1
n d :
ℕ
ε B :
ℝ
1 ≤ n
D ≤ d
0 < ε
ε ≤ 1 / 2
2 < B
c * min 1 ((d : ℝ) / (B * n * ε * lowerLogDegree C d))
- exp (-(n : ℝ) * (B - 2) ^ 2 / (4 * B))
≤ minimaxRisk n d ε
Proof (Lean source)
lemma sparse_dense_fixedSample_spliced_lower :
∃ (c C : ℝ) (D : ℕ), 0 < c ∧ 2 ≤ C ∧ 2 ≤ D ∧
∀ (n d : ℕ) (ε B : ℝ),
1 ≤ n → D ≤ d → 0 < ε → ε ≤ 1 / 2 → 2 < B →
c * min 1 ((d : ℝ) / (B * n * ε * lowerLogDegree C d)) -
exp (-(n : ℝ) * (B - 2) ^ 2 / (4 * B)) ≤ minimaxRisk n d ε := by
obtain ⟨cs, ηs, Cs, Ds, hcs, hηs, hCs, hDs, hs⟩ :=
sparse_fixedSample_calibrated_lower
obtain ⟨ηd, Cd, Dd, hηd, hCd, hDd, hdense⟩ :=
dense_fixedSample_calibrated_lower
let q := (Cd + 2) / Cs
have hCspos : 0 < Cs := by linarith
have hq : 0 < q := by dsimp [q]; positivity
let b := min (ηs / (8 * q ^ 2)) (ηd / q)
have hb : 0 < b := lt_min (by positivity) (by positivity)
let c := min (cs * min 1 ηs) ((5 / 2560000 : ℝ) * b)
have hc : 0 < c := lt_min (mul_pos hcs (lt_min (by norm_num) hηs)) (by positivity)
refine ⟨c, Cs, max Ds Dd, hc, hCs, hDs.trans (le_max_left _ _), ?_⟩
intro n d ε B hn hd hε hεhi hB
have hds : Ds ≤ d := (le_max_left _ _).trans hd
have hdd : Dd ≤ d := (le_max_right _ _).trans hd
have hd2 := hDs.trans hds
have hnpos : (0 : ℝ) < n := by exact_mod_cast (by omega : 0 < n)
have hdpos : (0 : ℝ) < d := by exact_mod_cast (by omega : 0 < d)
have hBpos : 0 < B := by linarith
let Ks : ℝ := lowerLogDegree Cs d
let Kd : ℝ := lowerLogDegree Cd d
have hKs : 0 < Ks := by
have h := (lowerLogDegree_bounds hCs hd2).1
dsimp [Ks]; exact_mod_cast (by omega : 0 < lowerLogDegree Cs d)
have hKd : 0 < Kd := by
have h := (lowerLogDegree_bounds hCd hd2).1
dsimp [Kd]; exact_mod_cast (by omega : 0 < lowerLogDegree Cd d)
have hdegrees : Kd ≤ q * Ks := by
have hlo := (lowerLogDegree_bounds hCs hd2).2.1
have hhi := (lowerLogDegree_bounds hCd hd2).2.2
have hm := mul_le_mul_of_nonneg_left hlo hq.le
have heq : q * (Cs * logAlphabet d) = (Cd + 2) * logAlphabet d := by
dsimp [q]; field_simp
rw [heq] at hm
exact hhi.trans hm
let x : ℝ := (d : ℝ) / (B * n * ε)
have hx : 0 < x := by dsimp [x]; positivity
have hrate : (d : ℝ) / (B * n * ε * lowerLogDegree Cs d) = x / Ks := by
dsimp [x, Ks]; rw [div_div]
rw [hrate]
by_cases hband : 2 ≤ ηs * d * lowerLogDegree Cs d / (B * n * ε)
· have hl := hs n d ε B hn hds hε hεhi hB hband
have heq : ηs * d / (B * n * ε * lowerLogDegree Cs d) = ηs * (x / Ks) := by
dsimp [x, Ks]; field_simp
rw [heq] at hl
have hmin : min 1 ηs * min 1 (x / Ks) ≤ min 1 (ηs * (x / Ks)) := by
apply le_min
· calc
_ ≤ (1 : ℝ) * 1 := mul_le_mul (min_le_left 1 ηs) (min_le_left 1 (x / Ks))
(le_min (by norm_num) (by positivity)) (by norm_num)
_ = 1 := by ring
· exact mul_le_mul (min_le_right _ _) (min_le_right _ _)
(le_min (by norm_num) (by positivity)) hηs.le
have hbound : c * min 1 (x / Ks) ≤ cs * min 1 (ηs * (x / Ks)) := by
calc
_ ≤ (cs * min 1 ηs) * min 1 (x / Ks) :=
mul_le_mul_of_nonneg_right (min_le_left _ _) (by positivity)
_ = cs * (min 1 ηs * min 1 (x / Ks)) := by ring
_ ≤ _ := mul_le_mul_of_nonneg_left hmin hcs.le
exact (sub_le_sub_right hbound _).trans hl
· have hsmall : ηs * x * Ks ≤ 2 := by
have hlt := le_of_lt (lt_of_not_ge hband)
have heq : ηs * (d : ℝ) * lowerLogDegree Cs d / (B * n * ε) = ηs * x * Ks := by
dsimp [x, Ks]; field_simp
rwa [heq] at hlt
have hl := hdense n d ε B hn hdd hε hεhi hB
have heq : ηd * d / (B * n * ε * lowerLogDegree Cd d) = ηd * x / Kd := by
dsimp [x, Kd]; field_simp
rw [heq] at hl
have hcap : b * (x / Ks) ≤ 1 / (4 * Kd ^ 2) := by
have hsq : Kd ^ 2 ≤ q ^ 2 * Ks ^ 2 := by
have h := pow_le_pow_left₀ hKd.le hdegrees 2
simpa only [mul_pow] using h
have hbcap : b ≤ ηs / (8 * q ^ 2) := min_le_left _ _
have hprod : (ηs / (8 * q ^ 2)) * (x / Ks) * (4 * Kd ^ 2) ≤ 1 := by
calc
_ ≤ (ηs / (8 * q ^ 2)) * (x / Ks) * (4 * (q ^ 2 * Ks ^ 2)) :=
mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_left hsq (by norm_num))
(by positivity)
_ = ηs * x * Ks / 2 := by field_simp; ring
_ ≤ 1 := by linarith
apply (le_div_iff₀ (by positivity : 0 < 4 * Kd ^ 2)).2
exact (mul_le_mul_of_nonneg_right
(mul_le_mul_of_nonneg_right hbcap (by positivity)) (by positivity)).trans hprod
have hlinear : b * (x / Ks) ≤ ηd * x / Kd := by
have hblinear : b ≤ ηd / q := min_le_right _ _
calc
_ ≤ (ηd / q) * (x / Ks) := mul_le_mul_of_nonneg_right hblinear (by positivity)
_ = ηd * x / (q * Ks) := by field_simp
_ ≤ ηd * x / Kd := div_le_div_of_nonneg_left (by positivity) hKd hdegrees
have hbound : c * min 1 (x / Ks) ≤
(5 / 2560000 : ℝ) * min (1 / (4 * Kd ^ 2)) (ηd * x / Kd) := by
calc
_ ≤ ((5 / 2560000 : ℝ) * b) * min 1 (x / Ks) :=
mul_le_mul_of_nonneg_right (min_le_right _ _) (by positivity)
_ ≤ ((5 / 2560000 : ℝ) * b) * (x / Ks) :=
mul_le_mul_of_nonneg_left (min_le_right _ _) (by positivity)
_ = (5 / 2560000 : ℝ) * (b * (x / Ks)) := by ring
_ ≤ _ := mul_le_mul_of_nonneg_left (le_min hcap hlinear) (by norm_num)
exact (sub_le_sub_right hbound _).trans hl
theorem largeAlphabet_lower_with_shortage
Comparing the rounded degree with the paper logarithm converts the spliced bound to the exact capped rate, uniformly in the inflation B. In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ (c : ℝ) (D : ℕ),
0 < c
2 ≤ D
Proof (Lean source)
lemma largeAlphabet_lower_with_shortage :
∃ (c : ℝ) (D : ℕ), 0 < c ∧ 2 ≤ D ∧
∀ (n d : ℕ) (ε B : ℝ),
1 ≤ n → D ≤ d → 0 < ε → ε ≤ 1 / 2 → 2 < B →
(c / B) * rateScale n d ε -
exp (-(n : ℝ) * (B - 2) ^ 2 / (4 * B)) ≤ minimaxRisk n d ε := by
obtain ⟨c, C, D, hc, hC, hD, hl⟩ := sparse_dense_fixedSample_spliced_lower
refine ⟨c / (C + 2), D, by positivity, hD, ?_⟩
intro n d ε B hn hd hε hεhi hB
have hd2 := hD.trans hd
have hL := logAlphabet_pos (by omega : 1 ≤ d)
have hnpos : (0 : ℝ) < n := by exact_mod_cast (by omega : 0 < n)
have hBpos : 0 < B := by linarith
have hKpos : (0 : ℝ) < lowerLogDegree C d := by
have h := (lowerLogDegree_bounds hC hd2).1
exact_mod_cast (by omega : 0 < lowerLogDegree C d)
let r := (d : ℝ) / (n * ε * logAlphabet d)
let a := 1 / (B * (C + 2))
have ha : 0 < a := by dsimp [a]; positivity
have ha1 : a ≤ 1 := by
dsimp [a]
apply (div_le_iff₀ (by positivity : 0 < B * (C + 2))).2
nlinarith
have hr : 0 ≤ r := by dsimp [r]; positivity
have hratio : a * r ≤ (d : ℝ) / (B * n * ε * lowerLogDegree C d) := by
have hdegree := (lowerLogDegree_bounds hC hd2).2.2
calc
_ = (d : ℝ) / (B * n * ε * ((C + 2) * logAlphabet d)) := by
dsimp [a, r]; field_simp
_ ≤ _ := div_le_div_of_nonneg_left (Nat.cast_nonneg d) (by positivity)
(mul_le_mul_of_nonneg_left hdegree (by positivity))
have hmin : a * min 1 r ≤ min 1 ((d : ℝ) / (B * n * ε * lowerLogDegree C d)) := by
apply le_min
· exact (mul_le_mul_of_nonneg_left (min_le_left _ _) ha.le).trans
(by simpa using ha1)
· exact (mul_le_mul_of_nonneg_left (min_le_right _ _) ha.le).trans hratio
have hbound : (c / (C + 2) / B) * rateScale n d ε ≤
c * min 1 ((d : ℝ) / (B * n * ε * lowerLogDegree C d)) := by
calc
_ = c * (a * min 1 r) := by dsimp [a, r, rateScale]; field_simp
_ ≤ _ := mul_le_mul_of_nonneg_left hmin hc.le
exact (sub_le_sub_right hbound _).trans (hl n d ε B hn hd hε hεhi hB)
theorem shortageExponential_le_inverse_inflation_sq Lemma shortageExponential_le_inverse_inflation_sq in the paper ↗
Squaring the inverse-inflation exponential bound makes the shortage penalty negligible compared with the lower constant's 1/B scaling. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
B :
ℝ
hB :
4 ≤ B
n :
ℕ
hn :
1 ≤ n
conclusion
exp (-(n : ℝ) * (B - 2) ^ 2 / (4 * B)) ≤ 1024 / (B ^ 2 * n)
Proof (Lean source)
lemma shortageExponential_le_inverse_inflation_sq {B : ℝ} (hB : 4 ≤ B)
{n : ℕ} (hn : 1 ≤ n) :
exp (-(n : ℝ) * (B - 2) ^ 2 / (4 * B)) ≤ 1024 / (B ^ 2 * n) := by
have hBpos : 0 < B := by linarith
let a := (B - 2) ^ 2 / (8 * B)
have ha : B / 32 ≤ a := by
dsimp [a]
apply (le_div_iff₀ (by positivity : 0 < 8 * B)).2
nlinarith
have hδ : 0 < 32 / B := by positivity
have hcoef : 1 / (32 / B) ≤ a := by simpa using ha
have hfirst := shortageExponential_le_inverse_sample hn hδ hcoef
have hsecond := shortageExponential_le_inverse_sample (n := 1) (by omega) hδ hcoef
simp only [Nat.cast_one, mul_one, div_one] at hsecond
have hmono : exp (-a * (n : ℝ)) ≤ exp (-a) := by
apply Real.exp_le_exp.mpr
have hnR : (1 : ℝ) ≤ n := by exact_mod_cast hn
have ha0 : 0 ≤ a := (by positivity : 0 ≤ B / 32).trans ha
nlinarith
calc
_ = exp (-a * (n : ℝ)) * exp (-a * (n : ℝ)) := by
rw [← Real.exp_add]
congr 1
dsimp [a]; ring
_ ≤ (32 / B / n) * (32 / B) :=
mul_le_mul hfirst (hmono.trans hsecond) (Real.exp_nonneg _) (by positivity)
_ = _ := by ring
theorem largeAlphabet_minimaxRisk_lower
One universal inflation absorbs the Poisson shortage penalty, completing the large-alphabet fixed-sample lower bound in the exact paper rate. In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ (c : ℝ) (D : ℕ),
0 < c
2 ≤ D
Proof (Lean source)
lemma largeAlphabet_minimaxRisk_lower :
∃ (c : ℝ) (D : ℕ), 0 < c ∧ 2 ≤ D ∧
∀ (n d : ℕ) (ε : ℝ),
1 ≤ n → D ≤ d → 0 < ε → ε ≤ 1 / 2 →
c * rateScale n d ε ≤ minimaxRisk n d ε := by
obtain ⟨c, D, hc, hD, hl⟩ := largeAlphabet_lower_with_shortage
let B := max 4 (2048 / c)
have hB4 : 4 ≤ B := le_max_left _ _
have hB : 2 < B := by linarith
have hBpos : 0 < B := by linarith
have hBc : 2048 ≤ B * c := (div_le_iff₀ hc).mp (le_max_right _ _)
refine ⟨c / (2 * B), D, by positivity, hD, ?_⟩
intro n d ε hn hd hε hεhi
have hd1 : 1 ≤ d := by omega
have hdpos : (0 : ℝ) < d := by exact_mod_cast (by omega : 0 < d)
have hnpos : (0 : ℝ) < n := by exact_mod_cast (by omega : 0 < n)
have hL := logAlphabet_pos hd1
have hLhi : logAlphabet d ≤ d := by
have hlog := Real.log_le_sub_one_of_pos hdpos
rw [logAlphabet, Real.log_mul (Real.exp_pos _).ne' hdpos.ne', Real.log_exp]
linarith
have hrate : 1 / (n : ℝ) ≤ rateScale n d ε := by
apply le_min
· exact (div_le_one hnpos).2 (by exact_mod_cast hn)
· apply (div_le_div_iff₀ hnpos (by positivity : 0 < (n : ℝ) * ε * logAlphabet d)).2
have he : ε ≤ 1 := by linarith
have hprod : ε * logAlphabet d ≤ d :=
(mul_le_mul_of_nonneg_right he hL.le).trans (by simpa using hLhi)
nlinarith
have hpenalty : exp (-(n : ℝ) * (B - 2) ^ 2 / (4 * B)) ≤
(c / (2 * B)) * rateScale n d ε := by
calc
_ ≤ 1024 / (B ^ 2 * n) := shortageExponential_le_inverse_inflation_sq hB4 hn
_ ≤ (c / (2 * B)) * (1 / n) := by
apply (div_le_iff₀ (by positivity : 0 < B ^ 2 * (n : ℝ))).2
have heq : (c / (2 * B) * (1 / (n : ℝ))) * (B ^ 2 * n) = B * c / 2 := by
field_simp
rw [heq]
linarith
_ ≤ _ := mul_le_mul_of_nonneg_left hrate (by positivity)
have hbound := hl n d ε B hn hd hε hεhi hB
have heq : c / B = 2 * (c / (2 * B)) := by ring
rw [heq] at hbound
linarith
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.LowerSparse 9 declarations Pointwise envelope and normalization bounds used in equations (37)--(39) of the weighted-separation proof roadmap.
Sparse target normalization
Pointwise envelope and normalization bounds used in equations (37)--(39) of the weighted-separation proof roadmap.
The positive-part functional is nonnegative and bounded by its cell normalizer. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
ε z :
ℝ
hε :
0 ≤ ε
hεhalf :
ε ≤ 1 / 2
hz :
0 ≤ z
Proof (Lean source)
lemma phiEps_nonneg_le_weight {ε z : ℝ} (hε : 0 ≤ ε) (hεhalf : ε ≤ 1 / 2)
(hz : 0 ≤ z) : 0 ≤ phiEpsFormula ε z ∧ phiEpsFormula ε z ≤ 1 - ε + ε * z := by
have hbase : 0 ≤ 1 - ε := by linarith
have hw : 0 ≤ 1 - ε + ε * z := add_nonneg hbase (mul_nonneg hε hz)
have hd : 0 < 1 + z := by linarith
have hp : max (z - 1) 0 ≤ 1 + z := max_le (by linarith) (by linarith)
constructor
· unfold phiEpsFormula
exact div_nonneg (mul_nonneg hw (le_max_right _ _)) hd.le
· unfold phiEpsFormula
apply (div_le_iff₀ hd).2
exact mul_le_mul_of_nonneg_left hp hw
A bounded nonnegative intensity has second moment scale at most M times its mean. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma sparse_intensity_sq_le {M z : ℝ} (hz : z ∈ Icc 0 M) :
z ^ 2 ≤ M * z := by
have h := mul_le_mul_of_nonneg_right hz.2 hz.1
nlinarith
The scalar envelope gives the second-moment scale in roadmap equation (37). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
(phiEpsFormula ε z) ^ 2 ≤ 2 * (1 + ε ^ 2 * M * z)
Proof (Lean source)
lemma phiEps_sq_le {ε M z : ℝ} (hε : 0 ≤ ε) (hεhalf : ε ≤ 1 / 2)
(hz : z ∈ Icc 0 M) :
(phiEpsFormula ε z) ^ 2 ≤ 2 * (1 + ε ^ 2 * M * z) := by
obtain ⟨hlo, hhi⟩ := phiEps_nonneg_le_weight hε hεhalf hz.1
have hsq : (phiEpsFormula ε z) ^ 2 ≤ (1 - ε + ε * z) ^ 2 := by
nlinarith
have hi := sparse_intensity_sq_le hz
have hscaled := mul_le_mul_of_nonneg_left hi (sq_nonneg ε)
have hbase : (1 - ε) ^ 2 ≤ 1 := by nlinarith
nlinarith [sq_nonneg (1 - ε - ε * z)]
The sparse denominator stays above one half throughout the closed overlap range. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
1 / 2 ≤ sparseNormalizerFormula d ε M z hz
Proof (Lean source)
lemma sparseNormalizer_ge_half {d : ℕ} {ε M : ℝ} (hε : 0 ≤ ε)
(hεhalf : ε ≤ 1 / 2) (z : Fin d → ℝ) (hz : ∀ x, z x ∈ Icc 0 M) :
1 / 2 ≤ sparseNormalizerFormula d ε M z hz := by
have hs : 0 ≤ ∑ x : Fin d, z x := Finset.sum_nonneg (fun x _ => (hz x).1)
have ht : 0 ≤ ε / (d : ℝ) * ∑ x : Fin d, z x := by positivity
unfold sparseNormalizerFormula
linarith
The empirical average of the scalar functional is between zero and S_d. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Proof (Lean source)
lemma sparse_average_phi_bounds {d : ℕ} (hd : 0 < d) {ε M : ℝ}
(hε : 0 ≤ ε) (hεhalf : ε ≤ 1 / 2)
(z : Fin d → ℝ) (hz : ∀ x, z x ∈ Icc 0 M) :
0 ≤ (∑ x : Fin d, phiEpsFormula ε (z x)) / d ∧
(∑ x : Fin d, phiEpsFormula ε (z x)) / d ≤ sparseNormalizerFormula d ε M z hz := by
have hdR : (0 : ℝ) < d := Nat.cast_pos.mpr hd
have hb (x : Fin d) := phiEps_nonneg_le_weight hε hεhalf (hz x).1
constructor
· exact div_nonneg (Finset.sum_nonneg (fun x _ => (hb x).1)) hdR.le
· apply (div_le_iff₀ hdR).2
calc
_ ≤ ∑ x : Fin d, (1 - ε + ε * z x) :=
Finset.sum_le_sum (fun x _ => (hb x).2)
_ = sparseNormalizerFormula d ε M z hz * d := by
simp [sparseNormalizerFormula, Finset.sum_add_distrib, ← Finset.mul_sum]
field_simp
Normalization perturbs H_d by at most the denominator error, as in equation (38). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
H S :
ℝ
hS :
0 < S
hH :
0 ≤ H
hHS :
H ≤ S
conclusion
|H / S - H| ≤ |1 - S|
Proof (Lean source)
lemma sparse_ratio_normalization_le {H S : ℝ} (hS : 0 < S)
(hH : 0 ≤ H) (hHS : H ≤ S) : |H / S - H| ≤ |1 - S| := by
have hq : 0 ≤ H / S := div_nonneg hH hS.le
have hq1 : H / S ≤ 1 := (div_le_one hS).2 hHS
have hid : H / S - H = (H / S) * (1 - S) := by field_simp
rw [hid, abs_mul, abs_of_nonneg hq]
simpa using mul_le_mul_of_nonneg_right hq1 (abs_nonneg (1 - S))
The normalized target is close to a deterministic prior center by two errors. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
H S h :
ℝ
hS :
0 < S
hH :
0 ≤ H
hHS :
H ≤ S
conclusion
(1 / 2 + H / (2 * S) - (1 / 2 + h / 2)) ^ 2 ≤ ((S - 1) ^ 2 + (H - h) ^ 2) / 2
Proof (Lean source)
lemma sparse_target_center_sq_le {H S h : ℝ} (hS : 0 < S)
(hH : 0 ≤ H) (hHS : H ≤ S) :
(1 / 2 + H / (2 * S) - (1 / 2 + h / 2)) ^ 2 ≤
((S - 1) ^ 2 + (H - h) ^ 2) / 2 := by
have hb := sparse_ratio_normalization_le hS hH hHS
have hsq : (H / S - H) ^ 2 ≤ (S - 1) ^ 2 := by
have hp := abs_nonneg (H / S - H)
nlinarith [sq_abs (H / S - H), sq_abs (1 - S)]
have hid : 1 / 2 + H / (2 * S) - (1 / 2 + h / 2) =
((H / S - H) + (H - h)) / 2 := by ring
rw [hid]
nlinarith [sq_nonneg ((H / S - H) - (H - h))]
Applying the exact sparse likelihood identity gives the target-center estimate for the actual observed law, including arbitrary boundary intensities. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
n d :
ℕ
ε M B :
ℝ
hn :
1 ≤ n
hd :
2 ≤ d
hε :
0 < ε
hεhalf :
ε ≤ 1 / 2
hM :
2 ≤ M
hB :
B > 2
z :
Fin d → ℝ
hz :
∀ x, z x ∈ Icc 0 M
h :
ℝ
conclusion
(observedValue (sparseLaw (by omega : 0 < d) ε M ⟨hε, hεhalf⟩ z hz) - (1 / 2 + h / 2)) ^ 2
≤ ((sparseNormalizerFormula d ε M z hz - 1) ^ 2 + ((∑ x : Fin d, phiEpsFormula ε (z x)) / d - h) ^ 2) / 2
Proof (Lean source)
lemma sparseLaw_target_center_sq_le (n d : ℕ) (ε M B : ℝ)
(hn : 1 ≤ n) (hd : 2 ≤ d) (hε : 0 < ε) (hεhalf : ε ≤ 1 / 2)
(hM : 2 ≤ M) (hB : B > 2) (z : Fin d → ℝ)
(hz : ∀ x, z x ∈ Icc 0 M) (h : ℝ) :
(observedValue (sparseLaw (by omega : 0 < d) ε M ⟨hε, hεhalf⟩ z hz) -
(1 / 2 + h / 2)) ^ 2 ≤
((sparseNormalizerFormula d ε M z hz - 1) ^ 2 +
((∑ x : Fin d, phiEpsFormula ε (z x)) / d - h) ^ 2) / 2 := by
have hval := (sparse_likelihood n d ε M B z hn hd hε hεhalf hM hB hz).2.2.1
rw [hval]
have hs : 0 < sparseNormalizerFormula d ε M z hz :=
lt_of_lt_of_le (by norm_num) (sparseNormalizer_ge_half hε.le hεhalf z hz)
obtain ⟨hlo, hhi⟩ := sparse_average_phi_bounds (by omega : 0 < d)
hε.le hεhalf z hz
convert sparse_target_center_sq_le (h := h) hs hlo hhi using 1
ring
theorem sparse_target_center_integral_sq_le Lemma sparse_target_center_integral_sq_le in the paper ↗
Integrating the pointwise estimate reduces target concentration to the two second moments appearing in roadmap equations (37)--(39). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
H S :
Ω → ℝ
h :
ℝ
hS :
∀ ω, 0 < S ω
hH :
∀ ω, 0 ≤ H ω
hHS :
∀ ω, H ω ≤ S ω
hSsecond :
Integrable (fun ω => (S ω - 1) ^ 2) μ
hHsecond :
Integrable (fun ω => (H ω - h) ^ 2) μ
conclusion
(∫ ω, (1 / 2 + H ω / (2 * S ω) - (1 / 2 + h / 2)) ^ 2 ∂μ)
≤ ((∫ ω, (S ω - 1) ^ 2 ∂μ) + (∫ ω, (H ω - h) ^ 2 ∂μ)) / 2
Proof (Lean source)
lemma sparse_target_center_integral_sq_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (H S : Ω → ℝ) (h : ℝ)
(hS : ∀ ω, 0 < S ω) (hH : ∀ ω, 0 ≤ H ω) (hHS : ∀ ω, H ω ≤ S ω)
(hSsecond : Integrable (fun ω => (S ω - 1) ^ 2) μ)
(hHsecond : Integrable (fun ω => (H ω - h) ^ 2) μ) :
(∫ ω, (1 / 2 + H ω / (2 * S ω) - (1 / 2 + h / 2)) ^ 2 ∂μ) ≤
((∫ ω, (S ω - 1) ^ 2 ∂μ) + (∫ ω, (H ω - h) ^ 2 ∂μ)) / 2 := by
calc
_ ≤ ∫ ω, ((S ω - 1) ^ 2 + (H ω - h) ^ 2) / 2 ∂μ := by
apply integral_mono_of_nonneg
(Filter.Eventually.of_forall (fun _ => sq_nonneg _))
((hSsecond.add hHsecond).div_const 2)
exact Filter.Eventually.of_forall fun ω =>
sparse_target_center_sq_le (hS ω) (hH ω) (hHS ω)
_ = _ := by rw [integral_div, integral_add hSsecond hHsecond]
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.LowerSparseCalibration 5 declarations The even logarithmic degree from (32) makes the target concentration condition (39) hold above a universal alphabet cutoff.
Universal sparse alphabet calibration
The even logarithmic degree from (32) makes the target concentration condition (39) hold above a universal alphabet cutoff. This removes the numerical concentration premise from the explicit packet Bayes lower bound.
For the displayed parameters, lowerLogDegree is the object specified by this definition.
Definition (Lean source)
parameters
C :
ℝ
d :
ℕ
def
lowerLogDegree C d :
ℕ
given by
2 * ⌈C * logAlphabet d / 2⌉₊
theorem lowerLogDegree_bounds
Rounding the degree preserves its logarithmic lower bound and costs at most two logarithmic units, uniformly for every legal alphabet. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
C :
ℝ
hC :
2 ≤ C
d :
ℕ
hd :
2 ≤ d
conclusion
conclusion 1
2 ≤ lowerLogDegree C d
conclusion 2
C * logAlphabet d ≤ (lowerLogDegree C d : ℝ)
conclusion 3
(lowerLogDegree C d : ℝ) ≤ (C + 2) * logAlphabet d
Proof (Lean source)
lemma lowerLogDegree_bounds {C : ℝ} (hC : 2 ≤ C) {d : ℕ} (hd : 2 ≤ d) :
2 ≤ lowerLogDegree C d ∧
C * logAlphabet d ≤ (lowerLogDegree C d : ℝ) ∧
(lowerLogDegree C d : ℝ) ≤ (C + 2) * logAlphabet d := by
have hdR : (1 : ℝ) ≤ d := by exact_mod_cast (by omega : 1 ≤ d)
have hdpos : (0 : ℝ) < d := by linarith
have hL : 1 ≤ logAlphabet d := by
rw [logAlphabet, Real.log_mul (Real.exp_pos _).ne' hdpos.ne', Real.log_exp]
linarith [Real.log_nonneg hdR]
have hnonneg : 0 ≤ C * logAlphabet d / 2 := by positivity
have hlo := Nat.le_ceil (C * logAlphabet d / 2)
have hhi := Nat.ceil_lt_add_one hnonneg
have hcast : (lowerLogDegree C d : ℝ) = 2 * (⌈C * logAlphabet d / 2⌉₊ : ℝ) := by
simp [lowerLogDegree]
have hdeg : (2 : ℝ) ≤ lowerLogDegree C d := by
rw [hcast]
nlinarith
refine ⟨by exact_mod_cast hdeg, ?_, ?_⟩ <;> rw [hcast] <;> nlinarith
theorem logAlphabet_sq_div_tendsto_zero
Logarithmic degree squared is negligible relative to alphabet size. In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Proof (Lean source)
lemma logAlphabet_sq_div_tendsto_zero :
Tendsto (fun d : ℕ => logAlphabet d ^ 2 / (d : ℝ)) atTop (𝓝 0) := by
have ht := (Real.tendsto_pow_log_div_mul_add_atTop
(exp 1)⁻¹ 0 2 (inv_ne_zero (Real.exp_ne_zero 1))).comp
(tendsto_natCast_atTop_atTop.const_mul_atTop (Real.exp_pos 1))
simpa only [Function.comp_def, logAlphabet, add_zero,
← mul_assoc, inv_mul_cancel₀ (Real.exp_ne_zero 1), one_mul] using ht
theorem sparse_packet_alphabet_cutoff
A single alphabet cutoff enforces the target-deviation bound (39) for all endpoints and overlap floors in the theorem's range. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
cp C :
ℝ
hcp :
0 < cp
hC :
2 ≤ C
conclusion
∃ D : ℕ,
2 ≤ D
conclusion 1
d :
ℕ
D ≤ d
M ε :
ℝ
2 ≤ M
0 ≤ ε
ε ≤ 1 / 2
256 * (2 + 3 * ε ^ 2 * M) * (lowerLogDegree C d : ℝ) ^ 2 ≤ cp ^ 2 * M * d
Proof (Lean source)
lemma sparse_packet_alphabet_cutoff {cp C : ℝ} (hcp : 0 < cp) (hC : 2 ≤ C) :
∃ D : ℕ, 2 ≤ D ∧ ∀ d : ℕ, D ≤ d → ∀ M ε : ℝ,
2 ≤ M → 0 ≤ ε → ε ≤ 1 / 2 →
256 * (2 + 3 * ε ^ 2 * M) * (lowerLogDegree C d : ℝ) ^ 2 ≤
cp ^ 2 * M * d := by
have hδ : 0 < cp ^ 2 / (448 * (C + 2) ^ 2) := by positivity
obtain ⟨D, hD⟩ := eventually_atTop.mp
(logAlphabet_sq_div_tendsto_zero.eventually_lt_const hδ)
refine ⟨max D 2, le_max_right _ _, ?_⟩
intro d hd M ε hM hε hεhalf
have hd2 : 2 ≤ d := (le_max_right _ _).trans hd
have hdpos : (0 : ℝ) < d := by exact_mod_cast (by omega : 0 < d)
have hsmall := hD d ((le_max_left _ _).trans hd)
have hL := (lowerLogDegree_bounds hC hd2).2.2
have hLpos : 0 ≤ (C + 2) * logAlphabet d := by
exact (Nat.cast_nonneg (lowerLogDegree C d)).trans hL
have hsq := pow_le_pow_left₀ (Nat.cast_nonneg (lowerLogDegree C d)) hL 2
have hbound : 448 * (lowerLogDegree C d : ℝ) ^ 2 ≤ cp ^ 2 * d := by
have hscaled := (div_lt_div_iff₀ hdpos
(by positivity : 0 < 448 * (C + 2) ^ 2)).mp hsmall
rw [mul_pow] at hsq
nlinarith
have hεsq : ε ^ 2 ≤ 1 / 4 := by nlinarith
have hfactor : 256 * (2 + 3 * ε ^ 2 * M) ≤ 448 * M := by
have hm := mul_le_mul_of_nonneg_right hεsq (by linarith : 0 ≤ M)
nlinarith
calc
_ ≤ (448 * M) * (lowerLogDegree C d : ℝ) ^ 2 :=
mul_le_mul_of_nonneg_right hfactor (sq_nonneg _)
_ = M * (448 * (lowerLogDegree C d : ℝ) ^ 2) := by ring
_ ≤ M * (cp ^ 2 * d) := mul_le_mul_of_nonneg_left hbound (by linarith)
_ = _ := by ring
theorem sparse_packet_bayesRisk_lower_largeAlphabet Lemma sparse_packet_bayesRisk_lower_largeAlphabet in the paper ↗
Above a universal alphabet cutoff the explicit packet yields the sparse Bayes lower order without any assumed concentration or testing bound. All numerical side conditions are derived from the logarithmic degree. In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ (c η C : ℝ) (A D : ℕ),
0 < c
0 < η
2 ≤ C
1 ≤ A
2 ≤ D
conclusion 1
n d :
ℕ
ε M B :
ℝ
1 ≤ n
D ≤ d
0 < ε
ε ≤ 1 / 2
hM :
2 ≤ M
M ≤ (lowerLogDegree C d : ℝ) ^ 2
2 < B
M ≤ η * d * lowerLogDegree C d / (B * n * ε)
∃ (P : ConstrainedPriorPair (lowerLogDegree C d) M),
∃ hdom : CommonAtomDomain (packetPositive A (lowerLogDegree C d) M) (packetNegative A (lowerLogDegree C d) M),
commonAtomCompletion (packetPositive A (lowerLogDegree C d) M) (packetNegative A (lowerLogDegree C d) M) hdom
= (P.ν₀, P.ν₁)
Measurable est
ofReal (c * M / (lowerLogDegree C d : ℝ) ^ 2)
≤ max (bayesSquaredRisk (productPrior d (Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) P.ν₀)) (sparseProductPoissonKernel (hMpaper := hM) B n d ε M) (sparsePriorTarget (hMpaper := hM) d ε M) est) (bayesSquaredRisk (productPrior d (Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) P.ν₁)) (sparseProductPoissonKernel (hMpaper := hM) B n d ε M) (sparsePriorTarget (hMpaper := hM) d ε M) est)
Proof (Lean source)
lemma sparse_packet_bayesRisk_lower_largeAlphabet :
∃ (c η C : ℝ) (A D : ℕ),
0 < c ∧ 0 < η ∧ 2 ≤ C ∧ 1 ≤ A ∧ 2 ≤ D ∧
∀ (n d : ℕ) (ε M B : ℝ),
1 ≤ n → D ≤ d → 0 < ε → ε ≤ 1 / 2 →
∀ hM : 2 ≤ M, M ≤ (lowerLogDegree C d : ℝ) ^ 2 → 2 < B →
M ≤ η * d * lowerLogDegree C d / (B * n * ε) →
∃ (P : ConstrainedPriorPair (lowerLogDegree C d) M),
∃ hdom : CommonAtomDomain
(packetPositive A (lowerLogDegree C d) M)
(packetNegative A (lowerLogDegree C d) M),
commonAtomCompletion
(packetPositive A (lowerLogDegree C d) M)
(packetNegative A (lowerLogDegree C d) M) hdom = (P.ν₀, P.ν₁) ∧
∀ est : (Fin d → (Fin 4 → ℕ)) → ℝ, Measurable est →
ofReal (c * M / (lowerLogDegree C d : ℝ) ^ 2) ≤
max
(bayesSquaredRisk
(productPrior d (Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) P.ν₀))
(sparseProductPoissonKernel (hMpaper := hM) B n d ε M) (sparsePriorTarget (hMpaper := hM) d ε M) est)
(bayesSquaredRisk
(productPrior d (Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) P.ν₁))
(sparseProductPoissonKernel (hMpaper := hM) B n d ε M) (sparsePriorTarget (hMpaper := hM) d ε M) est) := by
obtain ⟨c, cp, η, C₀, A, hc, hcp, hη, _, hA, hrisk⟩ := sparse_packet_bayesRisk_lower
let C := max C₀ 2
have hC : 2 ≤ C := le_max_right _ _
obtain ⟨D, hD, hcut⟩ := sparse_packet_alphabet_cutoff hcp hC
refine ⟨c, η, C, A, D, hc, hη, hC, hA, hD, ?_⟩
intro n d ε M B hn hd hε hεhalf hM hMhi hB hband
have hd2 : 2 ≤ d := hD.trans hd
obtain ⟨hK, hlog, _⟩ := lowerLogDegree_bounds hC hd2
have hL : 0 ≤ logAlphabet d := by
rw [logAlphabet, Real.log_mul (Real.exp_pos _).ne'
(show (d : ℝ) ≠ 0 by exact_mod_cast (by omega : d ≠ 0)), Real.log_exp]
have hdR : (1 : ℝ) ≤ d := by exact_mod_cast (by omega : 1 ≤ d)
linarith [Real.log_nonneg hdR]
exact hrisk (lowerLogDegree C d) n d ε M B hK hn hd2 hε hεhalf hM hMhi hB
((mul_le_mul_of_nonneg_right (le_max_left C₀ 2) hL).trans hlog)
hband (hcut d hd M ε hM hε.le hεhalf)
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.LowerSparseFullCounts 10 declarations Reorder the four counts in each cell without losing any observations.
Sparse lower bounds for the full observed histogram
Reorder the four counts in each cell without losing any observations. Identify this experiment with the inflated Poisson counts of the normalized sparse law, and transport the packet Bayes lower bound to arbitrary histogram estimators.
def sparseCountEquiv
All observed histogram counts, reversibly arranged into four-count cells.
Definition (Lean source)
parameters
d :
ℕ
given by
clause 1
toFun N := fun x j => N (x, j.val / 2 = 1, j.val % 2 = 1)
clause 2
invFun N := fun o => N o.1 (cellIdx o.2.1 o.2.2)
clause 3
left_inv N := by funext o rcases o with ⟨x, a, y⟩ cases a <;> cases y <;> simp [cellIdx]
clause 4
right_inv N := by funext x j fin_cases j <;> rfl
clause 5
measurable_toFun := measurable_of_countable _
clause 6
measurable_invFun := measurable_of_countable _
The observed histogram version of the full sparse four-count experiment.
Definition (Lean source)
parameters
B :
ℝ
n d :
ℕ
ε M :
ℝ
hMpaper :
2 ≤ M
given by
theorem sparseFullObservedPoissonKernel_probability Lemma sparseFullObservedPoissonKernel_probability in the paper ↗
The full observed experiment has probability fibres. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
IsProbabilityMeasure (sparseFullObservedPoissonKernel (hMpaper := hMpaper) B n d ε M θ)
Proof (Lean source)
lemma sparseFullObservedPoissonKernel_probability (B : ℝ) (n d : ℕ) (ε M : ℝ)
(θ : Fin d → ℝ) (hMpaper : 2 ≤ M) :
IsProbabilityMeasure (sparseFullObservedPoissonKernel (hMpaper := hMpaper) B n d ε M θ) := by
rw [sparseFullObservedPoissonKernel, Kernel.map_apply _ (sparseCountEquiv d).symm.measurable]
letI := sparseProductPoissonKernel_probability (hMpaper := hMpaper) B n d ε M θ
exact Measure.isProbabilityMeasure_map (sparseCountEquiv d).symm.measurable.aemeasurable
Normalization of the observed law cancels the total-mean inflation, yielding exactly the four unnormalized Poisson intensities in (34). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
0 < d
ε M B :
ℝ
n :
ℕ
hε :
0 < ε
hεhi :
ε ≤ 1 / 2
z :
Fin d → ℝ
hz :
∀ x, z x ∈ Icc 0 M
x :
Fin d
a y :
conclusion
(B * n * sparseNormalizerFormula d ε M z hz) * jointMass (sparseLaw hd ε M ⟨hε, hεhi⟩ z hz) x a y
= poissonCellMean B n d ε (z x) (cellIdx a y)
Proof (Lean source)
lemma sparseLaw_poissonMeans {d : ℕ} (hd : 0 < d) (ε M B : ℝ) (n : ℕ)
(hε : 0 < ε) (hεhi : ε ≤ 1 / 2) (z : Fin d → ℝ)
(hz : ∀ x, z x ∈ Icc 0 M) (x : Fin d) (a y : Bool) :
(B * n * sparseNormalizerFormula d ε M z hz) *
jointMass (sparseLaw hd ε M ⟨hε, hεhi⟩ z hz) x a y =
poissonCellMean B n d ε (z x) (cellIdx a y) := by
have hS : 0 < sparseNormalizerFormula d ε M z hz := by
have h := sparseNormalizer_ge_half hε.le hεhi z hz
linarith
rw [sparseLaw_jointMass]
unfold sparseMass
rw [mul_assoc, mul_div_cancel₀ _ hS.ne']
cases a <;> cases y <;> simp [sparseUnnormalizedMass, poissonCellMean, cellIdx]
theorem sparseProductPoissonKernel_eq_observedCounts Lemma sparseProductPoissonKernel_eq_observedCounts in the paper ↗
Reordering the independent observed counts gives precisely the full sparse product experiment. The Poisson total mean is B n S, as in (49). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
0 < d
ε M B :
ℝ
n :
ℕ
hε :
0 < ε
hεhi :
ε ≤ 1 / 2
hMpaper :
2 ≤ M
θ :
Fin d → ℝ
hz :
∀ x, θ x + sparseReference (_hMpaper := hMpaper) M ∈ Icc 0 M
conclusion
Measure.map (sparseCountEquiv d) (Measure.pi fun o : Obs d => poissonMeasure (toNNReal ((B * n * sparseNormalizerFormula d ε M (fun x => θ x + sparseReference (_hMpaper := hMpaper) M) hz) * jointMass (sparseLaw hd ε M ⟨hε, hεhi⟩ (fun x => θ x + sparseReference (_hMpaper := hMpaper) M) hz) o.1 o.2.1 o.2.2)))
= sparseProductPoissonKernel (hMpaper := hMpaper) B n d ε M θ
Proof (Lean source)
lemma sparseProductPoissonKernel_eq_observedCounts {d : ℕ} (hd : 0 < d)
(ε M B : ℝ) (n : ℕ) (hε : 0 < ε) (hεhi : ε ≤ 1 / 2) (hMpaper : 2 ≤ M)
(θ : Fin d → ℝ) (hz : ∀ x, θ x + sparseReference (_hMpaper := hMpaper) M ∈ Icc 0 M) :
Measure.map (sparseCountEquiv d)
(Measure.pi fun o : Obs d => poissonMeasure (toNNReal
((B * n * sparseNormalizerFormula d ε M (fun x => θ x + sparseReference (_hMpaper := hMpaper) M) hz) *
jointMass (sparseLaw hd ε M ⟨hε, hεhi⟩
(fun x => θ x + sparseReference (_hMpaper := hMpaper) M) hz) o.1 o.2.1 o.2.2))) =
sparseProductPoissonKernel (hMpaper := hMpaper) B n d ε M θ := by
apply Measure.ext_of_singleton
intro N
rw [Measure.map_apply (sparseCountEquiv d).measurable (measurableSet_singleton _)]
have hpre : (sparseCountEquiv d) ⁻¹' {N} = {(sparseCountEquiv d).symm N} := by
ext T
simp only [Set.mem_preimage, Set.mem_singleton_iff]
exact (sparseCountEquiv d).apply_eq_iff_eq_symm_apply
rw [hpre, Measure.pi_singleton]
change _ = (Measure.pi fun x => poissonCellLaw B n d ε (θ x + sparseReference (_hMpaper := hMpaper) M)) {N}
simp only [poissonCellLaw, Measure.pi_singleton, Obs, Fintype.prod_prod_type]
apply Finset.prod_congr rfl
intro x _
simp_rw [sparseLaw_poissonMeans hd ε M B n hε hεhi]
simp [Fintype.prod_bool, Fin.prod_univ_four, sparseCountEquiv, cellIdx]
let f (j : Fin 4) := (poissonMeasure (toNNReal
(poissonCellMean B n d ε (θ x + sparseReference (_hMpaper := hMpaper) M) j))) {N x j}
change f 3 * f 2 * (f 1 * f 0) = f 0 * f 1 * f 2 * f 3
ac_rfl
Undoing the count reordering recovers the exact independent Poisson histogram law of the legal sparse observed model on the prior support. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
0 < d
ε M B :
ℝ
n :
ℕ
hε :
0 < ε
hεhi :
ε ≤ 1 / 2
hMpaper :
2 ≤ M
θ :
Fin d → ℝ
hz :
∀ x, θ x + sparseReference (_hMpaper := hMpaper) M ∈ Icc 0 M
conclusion
sparseFullObservedPoissonKernel (hMpaper := hMpaper) B n d ε M θ
= Measure.pi fun o : Obs d => poissonMeasure (toNNReal ((B * n * sparseNormalizerFormula d ε M (fun x => θ x + sparseReference (_hMpaper := hMpaper) M) hz) * jointMass (sparseLaw hd ε M ⟨hε, hεhi⟩ (fun x => θ x + sparseReference (_hMpaper := hMpaper) M) hz) o.1 o.2.1 o.2.2))
Proof (Lean source)
lemma sparseFullObservedPoissonKernel_eq {d : ℕ} (hd : 0 < d)
(ε M B : ℝ) (n : ℕ) (hε : 0 < ε) (hεhi : ε ≤ 1 / 2) (hMpaper : 2 ≤ M)
(θ : Fin d → ℝ) (hz : ∀ x, θ x + sparseReference (_hMpaper := hMpaper) M ∈ Icc 0 M) :
sparseFullObservedPoissonKernel (hMpaper := hMpaper) B n d ε M θ =
Measure.pi fun o : Obs d => poissonMeasure (toNNReal
((B * n * sparseNormalizerFormula d ε M (fun x => θ x + sparseReference (_hMpaper := hMpaper) M) hz) *
jointMass (sparseLaw hd ε M ⟨hε, hεhi⟩
(fun x => θ x + sparseReference (_hMpaper := hMpaper) M) hz) o.1 o.2.1 o.2.2)) := by
rw [sparseFullObservedPoissonKernel,
Kernel.map_apply _ (sparseCountEquiv d).symm.measurable]
rw [← sparseProductPoissonKernel_eq_observedCounts (hMpaper := hMpaper) hd ε M B n hε hεhi θ hz]
rw [Measure.map_map (sparseCountEquiv d).symm.measurable (sparseCountEquiv d).measurable]
simp only [Function.comp_def, MeasurableEquiv.symm_apply_apply, Measure.map_id']
Every histogram estimator has exactly the same Bayes risk after the reversible arrangement into cells; none of the four counts is discarded. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
B :
ℝ
n d :
ℕ
ε M :
ℝ
target :
(Fin d → ℝ) → ℝ
est :
(Obs d → ℕ) → ℝ
hest :
Measurable est
hMpaper :
2 ≤ M
conclusion
bayesSquaredRisk π (sparseFullObservedPoissonKernel (hMpaper := hMpaper) B n d ε M) target est
= bayesSquaredRisk π (sparseProductPoissonKernel (hMpaper := hMpaper) B n d ε M) target (fun N => est ((sparseCountEquiv d).symm N))
Proof (Lean source)
lemma sparse_full_observed_bayesRisk_eq (B : ℝ) (n d : ℕ) (ε M : ℝ)
(π : Measure (Fin d → ℝ)) (target : (Fin d → ℝ) → ℝ)
(est : (Obs d → ℕ) → ℝ) (hest : Measurable est) (hMpaper : 2 ≤ M) :
bayesSquaredRisk π (sparseFullObservedPoissonKernel (hMpaper := hMpaper) B n d ε M) target est =
bayesSquaredRisk π (sparseProductPoissonKernel (hMpaper := hMpaper) B n d ε M) target
(fun N => est ((sparseCountEquiv d).symm N)) := by
unfold bayesSquaredRisk squaredRisk
apply lintegral_congr
intro θ
rw [sparseFullObservedPoissonKernel,
Kernel.map_apply _ (sparseCountEquiv d).symm.measurable]
exact lintegral_map (by fun_prop) (sparseCountEquiv d).symm.measurable
def sparseObservedModel
Clip only outside the prior support to obtain a legal normalized sparse observed model at every centered parameter vector. For the displayed parameters, sparseObservedModel is the object specified by this definition.
Definition (Lean source)
parameters
def
given by
⟨sparseLaw (by omega) ε M ⟨hε, hεhi⟩ (fun x => sparseClippedIntensity (hMpaper := hM) M (θ x)) (fun _ => sparseClippedIntensity_mem (hMpaper := hM) (by linarith) _), (sparse_likelihood 1 d ε M 4 (fun x => sparseClippedIntensity (hMpaper := hM) M (θ x)) (by norm_num) hd hε hεhi hM (by norm_num) (fun _ => sparseClippedIntensity_mem (hMpaper := hM) (by linarith) _)).1⟩
The legal observed model has exactly the normalized target used in sparse concentration and fuzzy testing. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
observedValue (sparseObservedModel hd ε M hε hεhi hM θ).1
= sparsePriorTarget (hMpaper := hM) d ε M θ
Proof (Lean source)
lemma sparseObservedModel_value {d : ℕ} (hd : 2 ≤ d) (ε M : ℝ)
(hε : 0 < ε) (hεhi : ε ≤ 1 / 2) (hM : 2 ≤ M) (θ : Fin d → ℝ) :
observedValue (sparseObservedModel hd ε M hε hεhi hM θ).1 =
sparsePriorTarget (hMpaper := hM) d ε M θ := by
have hval := (sparse_likelihood 1 d ε M 4
(fun x => sparseClippedIntensity (hMpaper := hM) M (θ x)) (by norm_num) hd hε hεhi hM
(by norm_num) (fun _ => sparseClippedIntensity_mem (hMpaper := hM) (by linarith) _)).2.2.1
rw [sparseObservedModel, hval]
unfold sparsePriorTarget sparseNormalizerFormula
rw [div_mul_eq_mul_div]
simp only [div_eq_mul_inv, mul_inv_rev]
ring
theorem sparse_full_observed_logDegree_bayesRisk_lower Lemma sparse_full_observed_logDegree_bayesRisk_lower in the paper ↗
The explicit packet sparse lower order holds for every estimator using all observed histogram counts and the actual legal observed value as target, with concentration and likelihood closeness derived from the packet construction and the logarithmic alphabet cutoff. In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ (c η C : ℝ) (A D : ℕ),
0 < c
0 < η
2 ≤ C
1 ≤ A
2 ≤ D
conclusion 1
n d :
ℕ
ε M B :
ℝ
hd2 :
2 ≤ d
hε :
0 < ε
hεhi :
ε ≤ 1 / 2
hM :
2 ≤ M
1 ≤ n
D ≤ d
M ≤ (lowerLogDegree C d : ℝ) ^ 2
2 < B
M ≤ η * d * lowerLogDegree C d / (B * n * ε)
∃ (P : ConstrainedPriorPair (lowerLogDegree C d) M),
∃ hdom : CommonAtomDomain (packetPositive A (lowerLogDegree C d) M) (packetNegative A (lowerLogDegree C d) M),
commonAtomCompletion (packetPositive A (lowerLogDegree C d) M) (packetNegative A (lowerLogDegree C d) M) hdom
= (P.ν₀, P.ν₁)
est :
(Obs d → ℕ) → ℝ
Measurable est
ofReal (c * M / (lowerLogDegree C d : ℝ) ^ 2)
≤ max (bayesSquaredRisk (productPrior d (Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) P.ν₀)) (sparseFullObservedPoissonKernel (hMpaper := hM) B n d ε M) (fun θ => observedValue (sparseObservedModel hd2 ε M hε hεhi hM θ).1) est) (bayesSquaredRisk (productPrior d (Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) P.ν₁)) (sparseFullObservedPoissonKernel (hMpaper := hM) B n d ε M) (fun θ => observedValue (sparseObservedModel hd2 ε M hε hεhi hM θ).1) est)
Proof (Lean source)
lemma sparse_full_observed_logDegree_bayesRisk_lower :
∃ (c η C : ℝ) (A D : ℕ),
0 < c ∧ 0 < η ∧ 2 ≤ C ∧ 1 ≤ A ∧ 2 ≤ D ∧
∀ (n d : ℕ) (ε M B : ℝ) (hd2 : 2 ≤ d)
(hε : 0 < ε) (hεhi : ε ≤ 1 / 2) (hM : 2 ≤ M),
1 ≤ n → D ≤ d →
M ≤ (lowerLogDegree C d : ℝ) ^ 2 → 2 < B →
M ≤ η * d * lowerLogDegree C d / (B * n * ε) →
∃ (P : ConstrainedPriorPair (lowerLogDegree C d) M),
∃ hdom : CommonAtomDomain
(packetPositive A (lowerLogDegree C d) M)
(packetNegative A (lowerLogDegree C d) M),
commonAtomCompletion
(packetPositive A (lowerLogDegree C d) M)
(packetNegative A (lowerLogDegree C d) M) hdom = (P.ν₀, P.ν₁) ∧
∀ est : (Obs d → ℕ) → ℝ, Measurable est →
ofReal (c * M / (lowerLogDegree C d : ℝ) ^ 2) ≤
max
(bayesSquaredRisk
(productPrior d (Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) P.ν₀))
(sparseFullObservedPoissonKernel (hMpaper := hM) B n d ε M)
(fun θ => observedValue (sparseObservedModel hd2 ε M hε hεhi hM θ).1) est)
(bayesSquaredRisk
(productPrior d (Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) P.ν₁))
(sparseFullObservedPoissonKernel (hMpaper := hM) B n d ε M)
(fun θ => observedValue (sparseObservedModel hd2 ε M hε hεhi hM θ).1) est) := by
obtain ⟨c, η, C, A, D, hc, hη, hC, hA, hD, hbound⟩ :=
sparse_packet_bayesRisk_lower_largeAlphabet
refine ⟨c, η, C, A, D, hc, hη, hC, hA, hD, ?_⟩
intro n d ε M B hd2 hε hεhi hM hn hd hMhi hB hband
obtain ⟨P, hdom, hcompletion, hP⟩ := hbound n d ε M B hn hd hε hεhi hM hMhi hB hband
refine ⟨P, hdom, hcompletion, ?_⟩
intro est hest
simp_rw [sparseObservedModel_value]
rw [sparse_full_observed_bayesRisk_eq (hMpaper := hM) B n d ε M _ _ est hest,
sparse_full_observed_bayesRisk_eq (hMpaper := hM) B n d ε M _ _ est hest]
exact hP _ (hest.comp (sparseCountEquiv d).symm.measurable)
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.LowerSparseMoments 7 declarations Roadmap equation (37) follows from bounded intensities with common first moment one and independence across cells.
Sparse prior concentration
Roadmap equation (37) follows from bounded intensities with common first moment one and independence across cells. Square integrability is derived from the support envelope, rather than imposed as an extra premise.
A supported intensity of mean one has variance at most its support endpoint. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Proof (Lean source)
lemma sparse_intensity_memLp_variance_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (Z : Ω → ℝ) (M : ℝ)
(hZ : Measurable Z) (hsupp : ∀ ω, Z ω ∈ Icc 0 M)
(hmean : (∫ ω, Z ω ∂μ) = 1) :
MemLp Z 2 μ ∧ variance Z μ ≤ M := by
have hLp : MemLp Z 2 μ := MemLp.of_bound hZ.aestronglyMeasurable M
(ae_of_all _ fun ω => by
rw [Real.norm_eq_abs, abs_of_nonneg (hsupp ω).1]
exact (hsupp ω).2)
refine ⟨hLp, (variance_le_expectation_sq hLp.aestronglyMeasurable).trans ?_⟩
calc
(∫ ω, Z ω ^ 2 ∂μ) ≤ ∫ ω, M * Z ω ∂μ :=
integral_mono hLp.integrable_sq
((hLp.integrable (by norm_num)).const_mul M)
(fun ω => sparse_intensity_sq_le (hsupp ω))
_ = M := by rw [integral_const_mul, hmean, mul_one]
The functional envelope yields the scalar variance bound in (37), uniformly including zero overlap and the upper boundary of the intensity support. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
Z :
Ω → ℝ
M ε :
ℝ
hZ :
hsupp :
∀ ω, Z ω ∈ Icc 0 M
hmean :
(∫ ω, Z ω ∂μ) = 1
hε :
0 ≤ ε
hεhalf :
ε ≤ 1 / 2
conclusion
conclusion 1
MemLp (fun ω => phiEpsFormula ε (Z ω)) 2 μ
conclusion 2
variance (fun ω => phiEpsFormula ε (Z ω)) μ ≤ 2 * (1 + ε ^ 2 * M)
Proof (Lean source)
lemma sparse_phi_memLp_variance_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (Z : Ω → ℝ) (M ε : ℝ)
(hZ : Measurable Z) (hsupp : ∀ ω, Z ω ∈ Icc 0 M)
(hmean : (∫ ω, Z ω ∂μ) = 1) (hε : 0 ≤ ε) (hεhalf : ε ≤ 1 / 2) :
MemLp (fun ω => phiEpsFormula ε (Z ω)) 2 μ ∧
variance (fun ω => phiEpsFormula ε (Z ω)) μ ≤ 2 * (1 + ε ^ 2 * M) := by
have hm : Measurable (fun ω => phiEpsFormula ε (Z ω)) := by
unfold phiEpsFormula
fun_prop
have hLp : MemLp (fun ω => phiEpsFormula ε (Z ω)) 2 μ :=
MemLp.of_bound hm.aestronglyMeasurable (1 - ε + ε * M)
(ae_of_all _ fun ω => by
have hb := phiEps_nonneg_le_weight hε hεhalf (hsupp ω).1
rw [Real.norm_eq_abs, abs_of_nonneg hb.1]
exact hb.2.trans (by gcongr; exact (hsupp ω).2))
have hZi := (sparse_intensity_memLp_variance_le μ Z M hZ hsupp hmean).1.integrable
(by norm_num : (1 : ENNReal) ≤ 2)
refine ⟨hLp, (variance_le_expectation_sq hLp.aestronglyMeasurable).trans ?_⟩
calc
(∫ ω, phiEpsFormula ε (Z ω) ^ 2 ∂μ) ≤
∫ ω, 2 * (1 + ε ^ 2 * M * Z ω) ∂μ :=
integral_mono hLp.integrable_sq
(((integrable_const 1).add (hZi.const_mul (ε ^ 2 * M))).const_mul 2)
(fun ω => phiEps_sq_le hε hεhalf (hsupp ω))
_ = 2 * (1 + ε ^ 2 * M) := by
rw [integral_const_mul, integral_add (integrable_const 1)
(hZi.const_mul _), integral_const_mul, hmean]
simp
theorem sparse_independent_average_variance_le Lemma sparse_independent_average_variance_le in the paper ↗
Independent cell variances give the inverse-alphabet concentration factor for their average; the cells need not have identical distributions. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma sparse_independent_average_variance_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] {d : ℕ} (hd : 0 < d)
(Z : Fin d → Ω → ℝ) (v : ℝ) (hZ : ∀ x, MemLp (Z x) 2 μ)
(hind : iIndepFun Z μ) (hv : ∀ x, variance (Z x) μ ≤ v) :
variance (fun ω => (∑ x, Z x ω) / d) μ ≤ v / d := by
have hdR : (0 : ℝ) < d := Nat.cast_pos.mpr hd
have he : (∑ x, Z x) = (fun ω => ∑ x, Z x ω) := by funext ω; simp
have hsum : variance (fun ω => ∑ x, Z x ω) μ = ∑ x, variance (Z x) μ := by
rw [← he]
exact IndepFun.variance_sum (fun x _ => hZ x)
(fun x _ y _ hxy => hind.indepFun hxy)
simp_rw [div_eq_mul_inv]
rw [variance_mul_const, hsum]
calc
_ ≤ (∑ _x : Fin d, v) * (d : ℝ)⁻¹ ^ 2 :=
mul_le_mul_of_nonneg_right (Finset.sum_le_sum fun x _ => hv x) (sq_nonneg _)
_ = _ := by simp; field_simp
The empirical average of the positive-part functional has the H_d variance bound of (37), derived from the intensity support and first moments. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
d :
ℕ
hd :
0 < d
Z :
Fin d → Ω → ℝ
M ε :
ℝ
hZ :
∀ x, Measurable (Z x)
hsupp :
∀ x ω, Z x ω ∈ Icc 0 M
hmean :
∀ x, (∫ ω, Z x ω ∂μ) = 1
hind :
iIndepFun Z μ
hε :
0 ≤ ε
hεhalf :
ε ≤ 1 / 2
conclusion
variance (fun ω => (∑ x, phiEpsFormula ε (Z x ω)) / d) μ ≤ 2 * (1 + ε ^ 2 * M) / d
Proof (Lean source)
lemma sparse_average_phi_variance_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] {d : ℕ} (hd : 0 < d)
(Z : Fin d → Ω → ℝ) (M ε : ℝ) (hZ : ∀ x, Measurable (Z x))
(hsupp : ∀ x ω, Z x ω ∈ Icc 0 M)
(hmean : ∀ x, (∫ ω, Z x ω ∂μ) = 1) (hind : iIndepFun Z μ)
(hε : 0 ≤ ε) (hεhalf : ε ≤ 1 / 2) :
variance (fun ω => (∑ x, phiEpsFormula ε (Z x ω)) / d) μ ≤
2 * (1 + ε ^ 2 * M) / d := by
have hphi : Measurable (phiEpsFormula ε) := by unfold phiEpsFormula; fun_prop
have hb (x : Fin d) := sparse_phi_memLp_variance_le μ (Z x) M ε
(hZ x) (hsupp x) (hmean x) hε hεhalf
exact sparse_independent_average_variance_le μ hd
(fun x ω => phiEpsFormula ε (Z x ω)) _ (fun x => (hb x).1)
(hind.comp (fun _ => phiEpsFormula ε) (fun _ => hphi)) (fun x => (hb x).2)
The sparse denominator has expectation one, so its squared error about one is its variance. This proves the S_d assertion of roadmap (37). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
d :
ℕ
hd :
0 < d
Z :
Fin d → Ω → ℝ
M ε :
ℝ
hZ :
∀ x, Measurable (Z x)
hsupp :
∀ x ω, Z x ω ∈ Icc 0 M
hmean :
∀ x, (∫ ω, Z x ω ∂μ) = 1
hind :
iIndepFun Z μ
conclusion
(∫ ω, (1 - ε + ε * ((∑ x, Z x ω) / d) - 1) ^ 2 ∂μ) ≤ ε ^ 2 * M / d
Proof (Lean source)
lemma sparse_normalizer_integral_sq_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] {d : ℕ} (hd : 0 < d)
(Z : Fin d → Ω → ℝ) (M ε : ℝ) (hZ : ∀ x, Measurable (Z x))
(hsupp : ∀ x ω, Z x ω ∈ Icc 0 M)
(hmean : ∀ x, (∫ ω, Z x ω ∂μ) = 1) (hind : iIndepFun Z μ) :
(∫ ω, (1 - ε + ε * ((∑ x, Z x ω) / d) - 1) ^ 2 ∂μ) ≤ ε ^ 2 * M / d := by
have hdR : (0 : ℝ) < d := Nat.cast_pos.mpr hd
have hb (x : Fin d) := sparse_intensity_memLp_variance_le μ (Z x) M
(hZ x) (hsupp x) (hmean x)
let A : Ω → ℝ := fun ω => (∑ x, Z x ω) / d
have hAm : Measurable A := by dsimp [A]; fun_prop
have hAi : Integrable A μ :=
(integrable_finsetSum _ (fun x _ => (hb x).1.integrable (by norm_num))).div_const _
have hAe : (∫ ω, A ω ∂μ) = 1 := by
dsimp [A]
rw [integral_div, integral_finsetSum _
(fun x _ => (hb x).1.integrable (by norm_num))]
simp [hmean, ne_of_gt hdR]
have hSm : Measurable (fun ω => ε * A ω + (1 - ε)) := by fun_prop
have hSe : (∫ ω, ε * A ω + (1 - ε) ∂μ) = 1 := by
rw [integral_add (hAi.const_mul _) (integrable_const _), integral_const_mul, hAe]
simp
have hv := sparse_independent_average_variance_le μ hd Z M
(fun x => (hb x).1) hind (fun x => (hb x).2)
have hid (ω : Ω) : 1 - ε + ε * A ω = ε * A ω + (1 - ε) := by ring
have hvar : variance (fun ω => ε * A ω + (1 - ε)) μ =
∫ ω, (ε * A ω + (1 - ε) - 1) ^ 2 ∂μ := by
rw [variance_eq_integral hSm.aemeasurable, hSe]
change (∫ ω, (1 - ε + ε * A ω - 1) ^ 2 ∂μ) ≤ _
simp_rw [hid]
rw [← hvar, variance_add_const (hAm.aestronglyMeasurable.const_mul ε),
variance_const_mul]
exact (mul_le_mul_of_nonneg_left hv (sq_nonneg ε)).trans_eq (by ring)
The H_d variance estimate is also its integrated squared deviation from its prior mean, the second assertion of (37) in precisely its roadmap form. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
d :
ℕ
hd :
0 < d
Z :
Fin d → Ω → ℝ
M ε :
ℝ
hZ :
∀ x, Measurable (Z x)
hsupp :
∀ x ω, Z x ω ∈ Icc 0 M
hmean :
∀ x, (∫ ω, Z x ω ∂μ) = 1
hind :
iIndepFun Z μ
hε :
0 ≤ ε
hεhalf :
ε ≤ 1 / 2
conclusion
H :
fun ω => (∑ x, phiEpsFormula ε (Z x ω)) / d
(∫ ω, (H ω - (∫ η, H η ∂μ)) ^ 2 ∂μ) ≤ 2 * (1 + ε ^ 2 * M) / d
Proof (Lean source)
lemma sparse_average_phi_integral_sq_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] {d : ℕ} (hd : 0 < d)
(Z : Fin d → Ω → ℝ) (M ε : ℝ) (hZ : ∀ x, Measurable (Z x))
(hsupp : ∀ x ω, Z x ω ∈ Icc 0 M)
(hmean : ∀ x, (∫ ω, Z x ω ∂μ) = 1) (hind : iIndepFun Z μ)
(hε : 0 ≤ ε) (hεhalf : ε ≤ 1 / 2) :
let H := fun ω => (∑ x, phiEpsFormula ε (Z x ω)) / d
(∫ ω, (H ω - (∫ η, H η ∂μ)) ^ 2 ∂μ) ≤
2 * (1 + ε ^ 2 * M) / d := by
dsimp only
have hm : Measurable (fun ω => (∑ x, phiEpsFormula ε (Z x ω)) / d) := by
unfold phiEpsFormula
fun_prop
rw [← variance_eq_integral hm.aemeasurable]
exact sparse_average_phi_variance_le μ hd Z M ε hZ hsupp hmean hind hε hεhalf
Equations (37)--(39) control the normalized sparse target around its nonrandom prior center. This retains both denominator and numerator errors. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
d :
ℕ
hd :
0 < d
Z :
Fin d → Ω → ℝ
M ε :
ℝ
hZ :
∀ x, Measurable (Z x)
hsupp :
∀ x ω, Z x ω ∈ Icc 0 M
hmean :
∀ x, (∫ ω, Z x ω ∂μ) = 1
hind :
iIndepFun Z μ
hε :
0 ≤ ε
hεhalf :
ε ≤ 1 / 2
conclusion
H :
fun ω => (∑ x, phiEpsFormula ε (Z x ω)) / d
S :
fun ω => 1 - ε + ε * ((∑ x, Z x ω) / d)
(∫ ω, (1 / 2 + H ω / (2 * S ω) - (1 / 2 + (∫ η, H η ∂μ) / 2)) ^ 2 ∂μ)
≤ (2 + 3 * ε ^ 2 * M) / (2 * d)
Proof (Lean source)
lemma sparse_prior_target_integral_sq_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] {d : ℕ} (hd : 0 < d)
(Z : Fin d → Ω → ℝ) (M ε : ℝ) (hZ : ∀ x, Measurable (Z x))
(hsupp : ∀ x ω, Z x ω ∈ Icc 0 M)
(hmean : ∀ x, (∫ ω, Z x ω ∂μ) = 1) (hind : iIndepFun Z μ)
(hε : 0 ≤ ε) (hεhalf : ε ≤ 1 / 2) :
let H := fun ω => (∑ x, phiEpsFormula ε (Z x ω)) / d
let S := fun ω => 1 - ε + ε * ((∑ x, Z x ω) / d)
(∫ ω, (1 / 2 + H ω / (2 * S ω) -
(1 / 2 + (∫ η, H η ∂μ) / 2)) ^ 2 ∂μ) ≤
(2 + 3 * ε ^ 2 * M) / (2 * d) := by
dsimp only
let H := fun ω => (∑ x, phiEpsFormula ε (Z x ω)) / d
let S := fun ω => 1 - ε + ε * ((∑ x, Z x ω) / d)
have hdR : (0 : ℝ) < d := Nat.cast_pos.mpr hd
have hA : MemLp (fun ω => (∑ x, Z x ω) / d) 2 μ := by
simpa only [div_eq_mul_inv] using
(memLp_finsetSum univ (fun x _ =>
(sparse_intensity_memLp_variance_le μ (Z x) M (hZ x)
(hsupp x) (hmean x)).1)).mul_const (d : ℝ)⁻¹
have hS : MemLp S 2 μ := by
exact (memLp_const (1 - ε)).add (hA.const_mul ε)
have hH : MemLp H 2 μ := by
simpa only [H, div_eq_mul_inv] using
(memLp_finsetSum univ (fun x _ =>
(sparse_phi_memLp_variance_le μ (Z x) M ε (hZ x)
(hsupp x) (hmean x) hε hεhalf).1)).mul_const (d : ℝ)⁻¹
have hSpos (ω : Ω) : 0 < S ω := by
have hb := sparseNormalizer_ge_half hε hεhalf (fun x => Z x ω)
(fun x => hsupp x ω)
have he : sparseNormalizerFormula d ε M (fun x => Z x ω) (fun x => hsupp x ω) = S ω := by
dsimp [sparseNormalizerFormula, S]
ring
rw [he] at hb
linarith
have hbounds (ω : Ω) : 0 ≤ H ω ∧ H ω ≤ S ω := by
have hb := sparse_average_phi_bounds hd hε hεhalf (fun x => Z x ω)
(fun x => hsupp x ω)
have he : sparseNormalizerFormula d ε M (fun x => Z x ω) (fun x => hsupp x ω) = S ω := by
dsimp [sparseNormalizerFormula, S]
ring
simpa only [H, he] using hb
have hnorm := sparse_normalizer_integral_sq_le μ hd Z M ε hZ hsupp hmean hind
have hnum := sparse_average_phi_integral_sq_le μ hd Z M ε hZ hsupp hmean hind hε hεhalf
have htarget := sparse_target_center_integral_sq_le μ H S (∫ η, H η ∂μ)
hSpos (fun ω => (hbounds ω).1) (fun ω => (hbounds ω).2)
((hS.sub (memLp_const 1)).integrable_sq)
((hH.sub (memLp_const _)).integrable_sq)
apply htarget.trans
calc
_ ≤ (ε ^ 2 * M / d + 2 * (1 + ε ^ 2 * M) / d) / 2 := by
exact div_le_div_of_nonneg_right (add_le_add hnorm hnum) (by norm_num)
_ = _ := by ring
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.LowerSparseRisk 17 declarations The supported, mean-one product priors concentrate the normalized target around their scalar functional centers.
Sparse prior testing risk
The supported, mean-one product priors concentrate the normalized target around their scalar functional centers. Chebyshev and the full sparse product-mixture TV bound then give the auxiliary squared-risk lower bound in roadmap (40), before the fixed-sample transfer.
Extend a centered sparse intensity to the legal support by clipping. On either supported centered prior this equals the original intensity. For the displayed parameters, sparseClippedIntensity is the object specified by this definition.
Definition (Lean source)
parameters
M θ :
ℝ
hMpaper :
2 ≤ M
def
sparseClippedIntensity M θ hMpaper :
ℝ
given by
The normalized sparse value, extended measurably to all centered vectors. The extension agrees with the actual sparse observed value on legal vectors. For the displayed parameters, sparsePriorTarget is the object specified by this definition.
Definition (Lean source)
parameters
def
sparsePriorTarget d ε M θ hMpaper :
ℝ
given by
1 / 2
+ ((∑ x, phiEpsFormula ε (sparseClippedIntensity (hMpaper := hMpaper) M (θ x))) / d) / (2 * (1 - ε + ε * ((∑ x, sparseClippedIntensity (hMpaper := hMpaper) M (θ x)) / d)))
Clipping puts every extended intensity in the legal interval. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
M :
ℝ
hM :
0 ≤ M
θ :
ℝ
hMpaper :
2 ≤ M
conclusion
sparseClippedIntensity (hMpaper := hMpaper) M θ ∈ Icc 0 M
Proof (Lean source)
lemma sparseClippedIntensity_mem {M : ℝ} (hM : 0 ≤ M) (θ : ℝ) (hMpaper : 2 ≤ M) :
sparseClippedIntensity (hMpaper := hMpaper) M θ ∈ Icc 0 M := by
exact ⟨le_min hM (le_max_left _ _), min_le_left _ _⟩
theorem sparseClippedIntensity_sub_reference Lemma sparseClippedIntensity_sub_reference in the paper ↗
On the supported prior, translation followed by clipping is the identity. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
sparseClippedIntensity (hMpaper := hMpaper) M (z - sparseReference (_hMpaper := hMpaper) M)
= z
Proof (Lean source)
lemma sparseClippedIntensity_sub_reference {M z : ℝ} (hz : z ∈ Icc 0 M) (hMpaper : 2 ≤ M) :
sparseClippedIntensity (hMpaper := hMpaper) M (z - sparseReference (_hMpaper := hMpaper) M) = z := by
simp [sparseClippedIntensity, max_eq_right hz.1, min_eq_right hz.2]
The full independent sparse Poisson experiment as a parameter kernel. For the displayed parameters, sparseProductPoissonKernel is the object specified by this definition.
Definition (Lean source)
parameters
B :
ℝ
n d :
ℕ
ε M :
ℝ
hMpaper :
2 ≤ M
given by
clause 1
toFun θ := Measure.pi fun x
=> sparseCenteredPoissonKernel (hMpaper := hMpaper) B n d ε M (θ x)
clause 2
measurable' := by let (θ : ℝ) : IsProbabilityMeasure (sparseCenteredPoissonKernel (hMpaper := hMpaper) B n d ε M θ) := by change IsProbabilityMeasure (poissonCellLaw B n d ε (θ + sparseReference (_hMpaper := hMpaper) M)) unfold poissonCellLaw infer_instance apply Measure.measurable_of_measurable_coe intro s hs have hatom (N : Fin d → (Fin 4 → ℕ)) : Measurable (fun θ : Fin d → ℝ => (Measure.pi fun x => sparseCenteredPoissonKernel (hMpaper := hMpaper) B n d ε M (θ x)) {N}) := by simp only [Measure.pi_singleton] apply Finset.measurable_prod intro x _ exact ((sparseCenteredPoissonKernel (hMpaper := hMpaper) B n d ε M).measurable_coe (measurableSet_singleton _)).comp (measurable_pi_apply x) simp_rw [← Measure.tsum_indicator_apply_singleton _ s hs] apply Measurable.tsum intro N by_cases hN : N ∈ s · simpa [Set.indicator_of_mem hN] using hatom N · simp [Set.indicator_of_notMem hN]
Clipping is measurable, including the endpoints of the intensity support. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
M :
ℝ
hMpaper :
2 ≤ M
conclusion
Measurable (sparseClippedIntensity (hMpaper := hMpaper) M)
Proof (Lean source)
@[fun_prop]
lemma measurable_sparseClippedIntensity (M : ℝ) (hMpaper : 2 ≤ M) :
Measurable (sparseClippedIntensity (hMpaper := hMpaper) M) := by
unfold sparseClippedIntensity
fun_prop
The extended target is a measurable function of the centered vector. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
ε M :
ℝ
hMpaper :
2 ≤ M
conclusion
Measurable (sparsePriorTarget (hMpaper := hMpaper) d ε M)
Proof (Lean source)
@[fun_prop]
lemma measurable_sparsePriorTarget (d : ℕ) (ε M : ℝ) (hMpaper : 2 ≤ M) :
Measurable (sparsePriorTarget (hMpaper := hMpaper) d ε M) := by
unfold sparsePriorTarget phiEpsFormula
fun_prop
Translation and clipping preserve every measurable prior integral on the support. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
(∫ θ, f (sparseClippedIntensity (hMpaper := hMpaper) M θ) ∂Measure.map (fun z => z - sparseReference (_hMpaper := hMpaper) M) ν)
= ∫ z, f z ∂ν
Proof (Lean source)
lemma sparse_centered_clipped_integral (ν : Measure ℝ) {M : ℝ}
(hsupp : ν (Icc 0 M)ᶜ = 0) (f : ℝ → ℝ) (hf : Measurable f) (hMpaper : 2 ≤ M) :
(∫ θ, f (sparseClippedIntensity (hMpaper := hMpaper) M θ)
∂Measure.map (fun z => z - sparseReference (_hMpaper := hMpaper) M) ν) = ∫ z, f z ∂ν := by
rw [integral_map (by fun_prop)
(show AEStronglyMeasurable (fun θ => f (sparseClippedIntensity (hMpaper := hMpaper) M θ)) _ from
(by fun_prop : Measurable (fun θ => f (sparseClippedIntensity (hMpaper := hMpaper) M θ))).aestronglyMeasurable)]
apply integral_congr_ae
filter_upwards [show ∀ᵐ z ∂ν, z ∈ Icc 0 M from (ae_iff).2 hsupp] with z hz
rw [sparseClippedIntensity_sub_reference (hMpaper := hMpaper) hz]
The extended sparse target stays in the unit interval at every vector. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
sparsePriorTarget (hMpaper := hMpaper) d ε M θ ∈ Icc 0 1
Proof (Lean source)
lemma sparsePriorTarget_mem_unit {d : ℕ} (hd : 0 < d) {ε M : ℝ}
(hε : 0 ≤ ε) (hεhalf : ε ≤ 1 / 2) (hM : 0 ≤ M) (θ : Fin d → ℝ) (hMpaper : 2 ≤ M) :
sparsePriorTarget (hMpaper := hMpaper) d ε M θ ∈ Icc 0 1 := by
let z := fun x => sparseClippedIntensity (hMpaper := hMpaper) M (θ x)
have hz (x : Fin d) := sparseClippedIntensity_mem (hMpaper := hMpaper) hM (θ x)
have hs := sparseNormalizer_ge_half hε hεhalf z hz
obtain ⟨hlo, hhi⟩ := sparse_average_phi_bounds hd hε hεhalf z hz
let H := (∑ x, phiEpsFormula ε (z x)) / d
let S := 1 - ε + ε * ((∑ x, z x) / d)
have he : sparseNormalizerFormula d ε M z hz = S := by
dsimp [sparseNormalizerFormula, S]
ring
rw [he] at hs hhi
have hS : 0 < S := by linarith
have hq : 0 ≤ H / (2 * S) := div_nonneg hlo (by positivity)
have hqhi : H / (2 * S) ≤ 1 / 2 := by
apply (div_le_iff₀ (by positivity : 0 < 2 * S)).2
dsimp [H]
linarith
change 0 ≤ 1 / 2 + H / (2 * S) ∧ 1 / 2 + H / (2 * S) ≤ 1
constructor <;> linarith
theorem sparse_product_prior_target_integral_sq_le Lemma sparse_product_prior_target_integral_sq_le in the paper ↗
Product copies of a supported mean-one prior satisfy the normalized target second-moment estimate (39), with the center expressed as a scalar prior integral. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
0 < d
ν :
ε M :
ℝ
hsupp :
ν (Icc 0 M)ᶜ = 0
hmean :
(∫ z, z ∂ν) = 1
hε :
0 ≤ ε
hεhalf :
ε ≤ 1 / 2
hM :
0 ≤ M
hMpaper :
2 ≤ M
conclusion
(∫ θ, (sparsePriorTarget (hMpaper := hMpaper) d ε M θ - (1 / 2 + (∫ z, phiEpsFormula ε z ∂ν) / 2)) ^ 2 ∂productPrior d (Measure.map (fun z => z - sparseReference (_hMpaper := hMpaper) M) ν))
≤ (2 + 3 * ε ^ 2 * M) / (2 * d)
Proof (Lean source)
lemma sparse_product_prior_target_integral_sq_le {d : ℕ} (hd : 0 < d)
(ν : Measure ℝ) [IsProbabilityMeasure ν] {ε M : ℝ}
(hsupp : ν (Icc 0 M)ᶜ = 0) (hmean : (∫ z, z ∂ν) = 1)
(hε : 0 ≤ ε) (hεhalf : ε ≤ 1 / 2) (hM : 0 ≤ M) (hMpaper : 2 ≤ M) :
(∫ θ, (sparsePriorTarget (hMpaper := hMpaper) d ε M θ -
(1 / 2 + (∫ z, phiEpsFormula ε z ∂ν) / 2)) ^ 2
∂productPrior d (Measure.map (fun z => z - sparseReference (_hMpaper := hMpaper) M) ν)) ≤
(2 + 3 * ε ^ 2 * M) / (2 * d) := by
let π := Measure.map (fun z => z - sparseReference (_hMpaper := hMpaper) M) ν
let : IsProbabilityMeasure π := Measure.isProbabilityMeasure_map (by fun_prop)
let μ := productPrior d π
let : IsProbabilityMeasure μ := by dsimp [μ, productPrior]; infer_instance
let Z : Fin d → (Fin d → ℝ) → ℝ := fun x θ => sparseClippedIntensity (hMpaper := hMpaper) M (θ x)
have hZ (x : Fin d) : Measurable (Z x) := by dsimp [Z]; fun_prop
have hz (x : Fin d) (θ : Fin d → ℝ) := sparseClippedIntensity_mem (hMpaper := hMpaper) hM (θ x)
have hscalar : (∫ θ, sparseClippedIntensity (hMpaper := hMpaper) M θ ∂π) = 1 := by
simpa only [π, id_eq] using
(sparse_centered_clipped_integral (hMpaper := hMpaper) ν hsupp id measurable_id).trans hmean
have hzm (x : Fin d) : (∫ θ, Z x θ ∂μ) = 1 := by
change (∫ θ, sparseClippedIntensity (hMpaper := hMpaper) M (θ x) ∂Measure.pi (fun _ : Fin d => π)) = 1
simpa only [Function.comp_apply, Function.eval] using
((measurePreserving_eval (fun _ : Fin d => π) x).hasLaw.integral_comp
(measurable_sparseClippedIntensity (hMpaper := hMpaper) M).aestronglyMeasurable).trans hscalar
have hind : iIndepFun Z μ := by
exact iIndepFun_pi (fun _ => (measurable_sparseClippedIntensity (hMpaper := hMpaper) M).aemeasurable)
have hphi : Measurable (phiEpsFormula ε) := by unfold phiEpsFormula; fun_prop
have hpi (x : Fin d) : Integrable (fun θ => phiEpsFormula ε (Z x θ)) μ :=
(sparse_phi_memLp_variance_le μ (Z x) M ε (hZ x) (hz x) (hzm x)
hε hεhalf).1.integrable (by norm_num)
have hcenter : (∫ θ, (∑ x, phiEpsFormula ε (Z x θ)) / d ∂μ) =
∫ z, phiEpsFormula ε z ∂ν := by
rw [integral_div, integral_finsetSum _ (fun x _ => hpi x)]
have hi (x : Fin d) : (∫ θ, phiEpsFormula ε (Z x θ) ∂μ) = ∫ z, phiEpsFormula ε z ∂ν := by
change (∫ θ, phiEpsFormula ε (sparseClippedIntensity (hMpaper := hMpaper) M (θ x))
∂Measure.pi (fun _ : Fin d => π)) = _
simpa only [Function.comp_apply, Function.eval] using
((measurePreserving_eval (fun _ : Fin d => π) x).hasLaw.integral_comp
(show AEStronglyMeasurable (fun θ => phiEpsFormula ε (sparseClippedIntensity (hMpaper := hMpaper) M θ)) π from
(by fun_prop : Measurable
(fun θ => phiEpsFormula ε (sparseClippedIntensity (hMpaper := hMpaper) M θ))).aestronglyMeasurable)).trans
(sparse_centered_clipped_integral (hMpaper := hMpaper) ν hsupp _ hphi)
simp_rw [hi]
simp [ne_of_gt (Nat.cast_pos.mpr hd : (0 : ℝ) < d)]
have hb := sparse_prior_target_integral_sq_le μ hd Z M ε hZ hz hzm hind hε hεhalf
dsimp only at hb
rw [hcenter] at hb
exact hb
theorem sparse_product_prior_target_tail_le Lemma sparse_product_prior_target_tail_le in the paper ↗
Chebyshev converts the derived sparse target second moment into the prior-tail bound used by fuzzy testing; no target concentration is assumed. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
0 < d
ν :
ε M r :
ℝ
hsupp :
ν (Icc 0 M)ᶜ = 0
hmean :
(∫ z, z ∂ν) = 1
hε :
0 ≤ ε
hεhalf :
ε ≤ 1 / 2
hM :
0 ≤ M
hr :
0 < r
hMpaper :
2 ≤ M
conclusion
(productPrior d (Measure.map (fun z => z - sparseReference (_hMpaper := hMpaper) M) ν)).real {θ | r < |sparsePriorTarget (hMpaper := hMpaper) d ε M θ - (1 / 2 + (∫ z, phiEpsFormula ε z ∂ν) / 2)|}
≤ (2 + 3 * ε ^ 2 * M) / (2 * d * r ^ 2)
Proof (Lean source)
lemma sparse_product_prior_target_tail_le {d : ℕ} (hd : 0 < d)
(ν : Measure ℝ) [IsProbabilityMeasure ν] {ε M r : ℝ}
(hsupp : ν (Icc 0 M)ᶜ = 0) (hmean : (∫ z, z ∂ν) = 1)
(hε : 0 ≤ ε) (hεhalf : ε ≤ 1 / 2) (hM : 0 ≤ M) (hr : 0 < r) (hMpaper : 2 ≤ M) :
(productPrior d (Measure.map (fun z => z - sparseReference (_hMpaper := hMpaper) M) ν)).real
{θ | r < |sparsePriorTarget (hMpaper := hMpaper) d ε M θ -
(1 / 2 + (∫ z, phiEpsFormula ε z ∂ν) / 2)|} ≤
(2 + 3 * ε ^ 2 * M) / (2 * d * r ^ 2) := by
let π := Measure.map (fun z => z - sparseReference (_hMpaper := hMpaper) M) ν
let : IsProbabilityMeasure π := Measure.isProbabilityMeasure_map (by fun_prop)
let μ := productPrior d π
let : IsProbabilityMeasure μ := by dsimp [μ, productPrior]; infer_instance
let center := 1 / 2 + (∫ z, phiEpsFormula ε z ∂ν) / 2
have hLp : MemLp (sparsePriorTarget (hMpaper := hMpaper) d ε M) 2 μ :=
MemLp.of_bound (measurable_sparsePriorTarget (hMpaper := hMpaper) d ε M).aestronglyMeasurable 1
(ae_of_all _ fun θ => by
have hb := sparsePriorTarget_mem_unit (hMpaper := hMpaper) hd hε hεhalf hM θ
rw [Real.norm_eq_abs, abs_of_nonneg hb.1]
exact hb.2)
have hint : Integrable (fun θ => (sparsePriorTarget (hMpaper := hMpaper) d ε M θ - center) ^ 2) μ :=
(hLp.sub (memLp_const center)).integrable_sq
have hsub : {θ | r < |sparsePriorTarget (hMpaper := hMpaper) d ε M θ - center|} ⊆
{θ | r ^ 2 ≤ (sparsePriorTarget (hMpaper := hMpaper) d ε M θ - center) ^ 2} := by
intro θ hθ
change r < |sparsePriorTarget (hMpaper := hMpaper) d ε M θ - center| at hθ
change r ^ 2 ≤ (sparsePriorTarget (hMpaper := hMpaper) d ε M θ - center) ^ 2
have ha := abs_nonneg (sparsePriorTarget (hMpaper := hMpaper) d ε M θ - center)
nlinarith [sq_abs (sparsePriorTarget (hMpaper := hMpaper) d ε M θ - center)]
have hmark := mul_meas_ge_le_integral_of_nonneg
(ae_of_all μ fun θ => sq_nonneg (sparsePriorTarget (hMpaper := hMpaper) d ε M θ - center)) hint (r ^ 2)
have hmoment := sparse_product_prior_target_integral_sq_le (hMpaper := hMpaper) hd ν hsupp hmean hε hεhalf hM
have hbound : r ^ 2 * μ.real {θ | r < |sparsePriorTarget (hMpaper := hMpaper) d ε M θ - center|} ≤
(2 + 3 * ε ^ 2 * M) / (2 * d) := by
exact ((mul_le_mul_of_nonneg_left (measureReal_mono hsub) (sq_nonneg r)).trans
hmark).trans hmoment
change μ.real {θ | r < |sparsePriorTarget (hMpaper := hMpaper) d ε M θ - center|} ≤ _
calc
_ ≤ ((2 + 3 * ε ^ 2 * M) / (2 * d)) / r ^ 2 :=
(le_div_iff₀ (sq_pos_of_pos hr)).2 (by simpa only [mul_comm] using hbound)
_ = _ := by ring
theorem sparseProductPoissonKernel_probability Lemma sparseProductPoissonKernel_probability in the paper ↗
The product sparse experiment has probability fibres at every centered vector. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
IsProbabilityMeasure (sparseProductPoissonKernel (hMpaper := hMpaper) B n d ε M θ)
Proof (Lean source)
lemma sparseProductPoissonKernel_probability (B : ℝ) (n d : ℕ) (ε M : ℝ)
(θ : Fin d → ℝ) (hMpaper : 2 ≤ M) : IsProbabilityMeasure (sparseProductPoissonKernel (hMpaper := hMpaper) B n d ε M θ) := by
change IsProbabilityMeasure
(Measure.pi fun x => poissonCellLaw B n d ε (θ x + sparseReference (_hMpaper := hMpaper) M))
unfold poissonCellLaw
infer_instance
Mixing the full sparse product kernel against the centered product prior is exactly the product predictive law bounded in (36). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
priorPredictive (productPrior d (Measure.map (fun z => z - sparseReference (_hMpaper := hMpaper) M) ν)) (sparseProductPoissonKernel (hMpaper := hMpaper) B n d ε M)
= productPriorPredictive d (Measure.map (fun z => z - sparseReference (_hMpaper := hMpaper) M) ν) (sparseCenteredPoissonKernel (hMpaper := hMpaper) B n d ε M)
Proof (Lean source)
lemma sparse_product_prior_predictive_eq (B : ℝ) (n d : ℕ) (ε M : ℝ)
(ν : Measure ℝ) [IsProbabilityMeasure ν] (hMpaper : 2 ≤ M) :
priorPredictive
(productPrior d (Measure.map (fun z => z - sparseReference (_hMpaper := hMpaper) M) ν))
(sparseProductPoissonKernel (hMpaper := hMpaper) B n d ε M) =
productPriorPredictive d (Measure.map (fun z => z - sparseReference (_hMpaper := hMpaper) M) ν)
(sparseCenteredPoissonKernel (hMpaper := hMpaper) B n d ε M) := by
let : IsProbabilityMeasure (Measure.map (fun z => z - sparseReference (_hMpaper := hMpaper) M) ν) :=
Measure.isProbabilityMeasure_map (by fun_prop)
apply priorPredictive_productPrior_eq_productPriorPredictive
· intro θ
change IsProbabilityMeasure (poissonCellLaw B n d ε (θ + sparseReference (_hMpaper := hMpaper) M))
unfold poissonCellLaw
infer_instance
· intro θ
rfl
theorem sparse_product_prior_bayesRisk_lower Lemma sparse_product_prior_bayesRisk_lower in the paper ↗
Equations (36)--(40), before fixed-sample transfer: the exact supported priors and their derived concentration give an estimator-wise Bayes-risk lower bound. The numerical alphabet condition is (39); the likelihood closeness is proved by the sparse full-likelihood theorem, not assumed. In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ η C : ℝ,
0 < η
0 < C
conclusion 1
K n d :
ℕ
ε M B :
ℝ
P :
1 ≤ n
2 ≤ d
0 < ε
ε ≤ 1 / 2
hM :
2 ≤ M
2 < B
C * logAlphabet d ≤ K
M ≤ η * d * K / (B * n * ε)
Delta :
|(∫ z, phiEpsFormula ε z ∂P.ν₁) - (∫ z, phiEpsFormula ε z ∂P.ν₀)| / 2
64 * (2 + 3 * ε ^ 2 * M) ≤ d * Delta ^ 2
Measurable est
ofReal (5 * Delta ^ 2 / 256)
≤ max (bayesSquaredRisk (productPrior d (Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) P.ν₀)) (sparseProductPoissonKernel (hMpaper := hM) B n d ε M) (sparsePriorTarget (hMpaper := hM) d ε M) est) (bayesSquaredRisk (productPrior d (Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) P.ν₁)) (sparseProductPoissonKernel (hMpaper := hM) B n d ε M) (sparsePriorTarget (hMpaper := hM) d ε M) est)
Proof (Lean source)
lemma sparse_product_prior_bayesRisk_lower :
∃ η C : ℝ, 0 < η ∧ 0 < C ∧
∀ (K n d : ℕ) (ε M B : ℝ) (P : ConstrainedPriorPair K M),
1 ≤ n → 2 ≤ d → 0 < ε → ε ≤ 1 / 2 → ∀ hM : 2 ≤ M,
2 < B → C * logAlphabet d ≤ K →
M ≤ η * d * K / (B * n * ε) →
let Delta := |(∫ z, phiEpsFormula ε z ∂P.ν₁) - (∫ z, phiEpsFormula ε z ∂P.ν₀)| / 2
64 * (2 + 3 * ε ^ 2 * M) ≤ d * Delta ^ 2 →
∀ est : (Fin d → (Fin 4 → ℕ)) → ℝ, Measurable est →
ofReal (5 * Delta ^ 2 / 256) ≤
max
(bayesSquaredRisk
(productPrior d (Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) P.ν₀))
(sparseProductPoissonKernel (hMpaper := hM) B n d ε M) (sparsePriorTarget (hMpaper := hM) d ε M) est)
(bayesSquaredRisk
(productPrior d (Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) P.ν₁))
(sparseProductPoissonKernel (hMpaper := hM) B n d ε M) (sparsePriorTarget (hMpaper := hM) d ε M) est) := by
obtain ⟨η, C, hη, hC, htvbound⟩ := sparse_product_mixture_tv_le_one_eighth
refine ⟨η, C, hη, hC, ?_⟩
intro K n d ε M B P hn hd hε hεhalf hM hB hK hband
dsimp only
intro hsize est hest
let := P.prob₀
let := P.prob₁
let π₀ := productPrior d (Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) P.ν₀)
let π₁ := productPrior d (Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) P.ν₁)
let : IsProbabilityMeasure (Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) P.ν₀) :=
Measure.isProbabilityMeasure_map (by fun_prop)
let : IsProbabilityMeasure (Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) P.ν₁) :=
Measure.isProbabilityMeasure_map (by fun_prop)
let : IsProbabilityMeasure π₀ := by dsimp [π₀, productPrior]; infer_instance
let : IsProbabilityMeasure π₁ := by dsimp [π₁, productPrior]; infer_instance
let a := ∫ z, phiEpsFormula ε z ∂P.ν₀
let b := ∫ z, phiEpsFormula ε z ∂P.ν₁
let Delta := |b - a| / 2
have hM0 : 0 ≤ M := by linarith
have hdR : (0 : ℝ) < d := by exact_mod_cast (by omega : 0 < d)
have hsize' : 64 * (2 + 3 * ε ^ 2 * M) ≤ d * Delta ^ 2 := hsize
have hDelta : 0 < Delta := by
have hnonneg : 0 ≤ Delta := by dsimp [Delta]; positivity
have hprod : 0 ≤ ε ^ 2 * M := by positivity
by_contra hnpos
have hz : Delta = 0 := le_antisymm (le_of_not_gt hnpos) hnonneg
rw [hz] at hsize'
nlinarith
have htail (ν : Measure ℝ) [IsProbabilityMeasure ν]
(hsupp : ν (Icc 0 M)ᶜ = 0) (hmean : (∫ z, z ∂ν) = 1) :
(productPrior d (Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) ν)).real
{θ | Delta / 4 < |sparsePriorTarget (hMpaper := hM) d ε M θ -
(1 / 2 + (∫ z, phiEpsFormula ε z ∂ν) / 2)|} ≤ 1 / 8 := by
apply (sparse_product_prior_target_tail_le (hMpaper := hM) (by omega) ν hsupp hmean
hε.le hεhalf hM0 (by positivity : 0 < Delta / 4)).trans
apply (div_le_iff₀ (by positivity : 0 < 2 * (d : ℝ) * (Delta / 4) ^ 2)).2
nlinarith [hsize']
have ht0 := htail P.ν₀ P.supp₀ P.mean₀
have ht1 := htail P.ν₁ P.supp₁ P.mean₁
have htv : tvDist
(priorPredictive π₀ (sparseProductPoissonKernel (hMpaper := hM) B n d ε M))
(priorPredictive π₁ (sparseProductPoissonKernel (hMpaper := hM) B n d ε M)) ≤ 1 / 8 := by
rw [sparse_product_prior_predictive_eq, sparse_product_prior_predictive_eq]
exact htvbound K n d ε M B P hn hd hε hεhalf hM hB hK hband
have hnum : ((Delta / 2 - Delta / 4) ^ 2 *
(1 - (1 / 8 : ℝ) - 1 / 8 - 1 / 8)) / 2 = 5 * Delta ^ 2 / 256 := by ring
by_cases hab : a ≤ b
· have hsep : Delta ≤ (1 / 2 + b / 2) - (1 / 2 + a / 2) := by
dsimp [Delta]
rw [abs_of_nonneg (by linarith : 0 ≤ b - a)]
linarith
have hb := twoFuzzyHypotheses_bayesRisk_lower π₀ π₁
(sparseProductPoissonKernel (hMpaper := hM) B n d ε M) (sparsePriorTarget (hMpaper := hM) d ε M)
(sparseProductPoissonKernel_probability (hMpaper := hM) B n d ε M) est hest
(measurable_sparsePriorTarget (hMpaper := hM) d ε M)
(1 / 2 + a / 2) (1 / 2 + b / 2) Delta (Delta / 4) (1 / 8) (1 / 8) (1 / 8)
(by linarith) hsep
ht0 ht1 htv
rw [hnum] at hb
exact hb
· have hsep : Delta ≤ (1 / 2 + a / 2) - (1 / 2 + b / 2) := by
dsimp [Delta]
rw [abs_of_nonpos (by linarith : b - a ≤ 0)]
linarith
have htv' : tvDist
(priorPredictive π₁ (sparseProductPoissonKernel (hMpaper := hM) B n d ε M))
(priorPredictive π₀ (sparseProductPoissonKernel (hMpaper := hM) B n d ε M)) ≤ 1 / 8 := by
rw [tvDist_symm]
exact htv
have hb := twoFuzzyHypotheses_bayesRisk_lower π₁ π₀
(sparseProductPoissonKernel (hMpaper := hM) B n d ε M) (sparsePriorTarget (hMpaper := hM) d ε M)
(sparseProductPoissonKernel_probability (hMpaper := hM) B n d ε M) est hest
(measurable_sparsePriorTarget (hMpaper := hM) d ε M)
(1 / 2 + b / 2) (1 / 2 + a / 2) Delta (Delta / 4) (1 / 8) (1 / 8) (1 / 8)
(by linarith) hsep
ht1 ht0 htv'
rw [hnum, max_comm] at hb
exact hb
The explicit Jackson-packet witness attains the auxiliary sparse testing lower order M/K². Both its expectation gap and its mixture TV bound are proved, and (39) is exposed as a numerical alphabet condition to be calibrated in the final sparse/dense assembly. In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ (c cp η C : ℝ) (A : ℕ),
0 < c
0 < cp
0 < η
0 < C
1 ≤ A
conclusion 1
K n d :
ℕ
ε M B :
ℝ
2 ≤ K
1 ≤ n
2 ≤ d
0 < ε
ε ≤ 1 / 2
hM :
2 ≤ M
M ≤ (K : ℝ) ^ 2
2 < B
C * logAlphabet d ≤ K
M ≤ η * d * K / (B * n * ε)
256 * (2 + 3 * ε ^ 2 * M) * (K : ℝ) ^ 2 ≤ cp ^ 2 * M * d
∃ (P : ConstrainedPriorPair K M),
Measurable est
ofReal (c * M / (K : ℝ) ^ 2)
≤ max (bayesSquaredRisk (productPrior d (Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) P.ν₀)) (sparseProductPoissonKernel (hMpaper := hM) B n d ε M) (sparsePriorTarget (hMpaper := hM) d ε M) est) (bayesSquaredRisk (productPrior d (Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) P.ν₁)) (sparseProductPoissonKernel (hMpaper := hM) B n d ε M) (sparsePriorTarget (hMpaper := hM) d ε M) est)
Proof (Lean source)
lemma sparse_packet_bayesRisk_lower :
∃ (c cp η C : ℝ) (A : ℕ),
0 < c ∧ 0 < cp ∧ 0 < η ∧ 0 < C ∧ 1 ≤ A ∧
∀ (K n d : ℕ) (ε M B : ℝ),
2 ≤ K → 1 ≤ n → 2 ≤ d → 0 < ε → ε ≤ 1 / 2 →
∀ hM : 2 ≤ M, M ≤ (K : ℝ) ^ 2 → 2 < B → C * logAlphabet d ≤ K →
M ≤ η * d * K / (B * n * ε) →
256 * (2 + 3 * ε ^ 2 * M) * (K : ℝ) ^ 2 ≤ cp ^ 2 * M * d →
∃ (P : ConstrainedPriorPair K M),
∃ hdom : CommonAtomDomain (packetPositive A K M) (packetNegative A K M),
commonAtomCompletion (packetPositive A K M) (packetNegative A K M) hdom =
(P.ν₀, P.ν₁) ∧
∀ est : (Fin d → (Fin 4 → ℕ)) → ℝ, Measurable est →
ofReal (c * M / (K : ℝ) ^ 2) ≤
max
(bayesSquaredRisk
(productPrior d (Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) P.ν₀))
(sparseProductPoissonKernel (hMpaper := hM) B n d ε M) (sparsePriorTarget (hMpaper := hM) d ε M) est)
(bayesSquaredRisk
(productPrior d (Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) P.ν₁))
(sparseProductPoissonKernel (hMpaper := hM) B n d ε M) (sparsePriorTarget (hMpaper := hM) d ε M) est) := by
obtain ⟨cp, A, hcp, hA, hpacket⟩ := packetCompletion_separation_lower
obtain ⟨η, C, hη, hC, hrisk⟩ := sparse_product_prior_bayesRisk_lower
refine ⟨5 * cp ^ 2 / 1024, cp, η, C, A, by positivity, hcp, hη, hC, hA, ?_⟩
intro K n d ε M B hK hn hd hε hεhalf hM hMhi hB hlog hband hsize
obtain ⟨P, hdom, _, _, _, hcompletion, hgap⟩ :=
hpacket K M ε hK hM hMhi hε.le hεhalf
refine ⟨P, hdom, hcompletion, ?_⟩
intro est hest
let Delta := |(∫ z, phiEpsFormula ε z ∂P.ν₁) - (∫ z, phiEpsFormula ε z ∂P.ν₀)| / 2
have hKR : (0 : ℝ) < K := by exact_mod_cast (by omega : 0 < K)
have hdR : (0 : ℝ) < d := by exact_mod_cast (by omega : 0 < d)
have hM0 : 0 ≤ M := by linarith
have hlower : cp ^ 2 * M / (4 * (K : ℝ) ^ 2) ≤ Delta ^ 2 := by
have hhalf : cp * sqrt M / (2 * K) ≤ Delta := by
dsimp [Delta]
simpa only [div_div, mul_comm] using
div_le_div_of_nonneg_right hgap (by norm_num : (0 : ℝ) ≤ 2)
have hnonneg : 0 ≤ cp * sqrt M / (2 * K) := by positivity
have hs := pow_le_pow_left₀ hnonneg hhalf 2
have he : (cp * sqrt M / (2 * K)) ^ 2 = cp ^ 2 * M / (4 * (K : ℝ) ^ 2) := by
rw [div_pow, mul_pow, Real.sq_sqrt hM0]
ring
rwa [he] at hs
have hsize' : 64 * (2 + 3 * ε ^ 2 * M) ≤ d * Delta ^ 2 := by
have hscaled : 64 * (2 + 3 * ε ^ 2 * M) ≤
d * (cp ^ 2 * M / (4 * (K : ℝ) ^ 2)) := by
calc
_ ≤ ((d : ℝ) * cp ^ 2 * M) / (4 * (K : ℝ) ^ 2) := by
apply (le_div_iff₀ (by positivity : 0 < 4 * (K : ℝ) ^ 2)).2
nlinarith only [hsize]
_ = _ := by ring
exact hscaled.trans (mul_le_mul_of_nonneg_left hlower hdR.le)
have hb := hrisk K n d ε M B P hn hd hε hεhalf hM hB hlog hband hsize' est hest
apply le_trans (ENNReal.ofReal_le_ofReal ?_) hb
calc
(5 * cp ^ 2 / 1024) * M / (K : ℝ) ^ 2 =
5 / 256 * (cp ^ 2 * M / (4 * (K : ℝ) ^ 2)) := by ring
_ ≤ 5 / 256 * Delta ^ 2 := mul_le_mul_of_nonneg_left hlower (by norm_num)
_ = 5 * Delta ^ 2 / 256 := by ring
The centered product priors put all their mass on legal sparse vectors. Thus the extensions used in testing do not introduce illegal prior parameters. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
∀ᵐ θ ∂productPrior d (Measure.map (fun z => z - sparseReference (_hMpaper := hMpaper) M) ν), ∀ x, θ x
+ sparseReference (_hMpaper := hMpaper) M ∈ Icc 0 M
Proof (Lean source)
lemma sparse_product_prior_ae_legal {d : ℕ} (ν : Measure ℝ) [IsProbabilityMeasure ν]
{M : ℝ} (hsupp : ν (Icc 0 M)ᶜ = 0) (hMpaper : 2 ≤ M) :
∀ᵐ θ ∂productPrior d (Measure.map (fun z => z - sparseReference (_hMpaper := hMpaper) M) ν),
∀ x, θ x + sparseReference (_hMpaper := hMpaper) M ∈ Icc 0 M := by
let π := Measure.map (fun z => z - sparseReference (_hMpaper := hMpaper) M) ν
let : IsProbabilityMeasure π := Measure.isProbabilityMeasure_map (by fun_prop)
have hmeas : MeasurableSet {θ : ℝ | θ + sparseReference (_hMpaper := hMpaper) M ∈ Icc 0 M} :=
measurableSet_Icc.preimage (by fun_prop)
have hscalar : ∀ᵐ θ ∂π, θ + sparseReference (_hMpaper := hMpaper) M ∈ Icc 0 M := by
rw [ae_map_iff (by fun_prop) hmeas]
simpa only [sub_add_cancel] using (show ∀ᵐ z ∂ν, z ∈ Icc 0 M from (ae_iff).2 hsupp)
rw [Filter.eventually_all]
intro x
exact ((measurePreserving_eval (fun _ : Fin d => π) x).hasLaw.ae_iff
(measurableSet_setOfPred.mp hmeas)).2 hscalar
On every legal vector the testing target is exactly the value of the normalized sparse observed law in the paper, by the full likelihood identity. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
ε M :
ℝ
hε :
0 < ε
hεhalf :
ε ≤ 1 / 2
hM :
2 ≤ M
θ :
Fin d → ℝ
hz :
∀ x, θ x + sparseReference (_hMpaper := hM) M ∈ Icc 0 M
conclusion
sparsePriorTarget (hMpaper := hM) d ε M θ
= observedValue (sparseLaw (by omega : 0 < d) ε M ⟨hε, hεhalf⟩ (fun x => θ x + sparseReference (_hMpaper := hM) M) hz)
Proof (Lean source)
lemma sparsePriorTarget_eq_observedValue {d : ℕ} (hd : 2 ≤ d) {ε M : ℝ}
(hε : 0 < ε) (hεhalf : ε ≤ 1 / 2) (hM : 2 ≤ M) (θ : Fin d → ℝ)
(hz : ∀ x, θ x + sparseReference (_hMpaper := hM) M ∈ Icc 0 M) :
sparsePriorTarget (hMpaper := hM) d ε M θ =
observedValue (sparseLaw (by omega : 0 < d) ε M ⟨hε, hεhalf⟩
(fun x => θ x + sparseReference (_hMpaper := hM) M) hz) := by
have hval := (sparse_likelihood 1 d ε M 4 (fun x => θ x + sparseReference (_hMpaper := hM) M)
(by norm_num) hd hε hεhalf hM (by norm_num) hz).2.2.1
rw [hval]
have hclip (x : Fin d) : sparseClippedIntensity (hMpaper := hM) M (θ x) = θ x + sparseReference (_hMpaper := hM) M := by
simpa only [add_sub_cancel_right] using sparseClippedIntensity_sub_reference (hMpaper := hM) (hz x)
unfold sparsePriorTarget
simp_rw [hclip]
have he : 1 - ε + ε * ((∑ x, (θ x + sparseReference (_hMpaper := hM) M)) / d) =
sparseNormalizerFormula d ε M (fun x => θ x + sparseReference (_hMpaper := hM) M) hz := by
unfold sparseNormalizerFormula
ring
rw [he]
ring
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.LowerSparseTV 5 declarations The four-coordinate Poisson laws form a measurable probability kernel.
Total variation for the sparse product-Poisson mixtures
The four-coordinate Poisson laws form a measurable probability kernel. The supported moment-matching API converts their exact Gram formula into the product total-variation estimate in roadmap equation (36).
The sparse Poisson experiment is measurable as a function of intensity. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
Proof (Lean source)
@[fun_prop]
lemma measurable_poissonCellLaw (B : ℝ) (n d : ℕ) (ε : ℝ) :
Measurable (poissonCellLaw B n d ε) := by
apply Measure.measurable_of_measurable_coe
intro s hs
have hatom (N : Fin 4 → ℕ) :
Measurable (fun z => poissonCellLaw B n d ε z {N}) := by
simp only [poissonCellLaw, Measure.pi_singleton, poissonMeasure_singleton]
apply Finset.measurable_prod
intro j _
have hm : Measurable (fun z => poissonCellMean B n d ε z j) := by
unfold poissonCellMean
split_ifs <;> fun_prop
fun_prop
simp_rw [← Measure.tsum_indicator_apply_singleton _ s hs]
apply Measurable.tsum
intro N
by_cases hN : N ∈ s
· simpa [Set.indicator_of_mem hN] using hatom N
· simp [Set.indicator_of_notMem hN]
The centered sparse experiment, with probability fibres at every real parameter. For the displayed parameters, sparseCenteredPoissonKernel is the object specified by this definition.
Definition (Lean source)
parameters
B :
ℝ
n d :
ℕ
ε M :
ℝ
hMpaper :
2 ≤ M
given by
clause 1
toFun θ := poissonCellLaw B n d ε (θ + sparseReference (_hMpaper := hMpaper) M)
clause 2
measurable' := (measurable_poissonCellLaw B n d ε).comp (by fun_prop)
theorem sparse_product_mixture_tv_le_sqrt_tail Lemma sparse_product_mixture_tv_le_sqrt_tail in the paper ↗
Moment-matched packet priors have product-mixture TV bounded by the alphabet size times the square root of the tail from equation (34). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
K n d :
ℕ
ε M B :
ℝ
P :
hn :
1 ≤ n
hd :
2 ≤ d
hε :
0 < ε
hεhalf :
ε ≤ 1 / 2
hM :
2 ≤ M
hB :
2 < B
conclusion
tvDist (productPriorPredictive d (Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) P.ν₀) (sparseCenteredPoissonKernel (hMpaper := hM) B n d ε M)) (productPriorPredictive d (Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) P.ν₁) (sparseCenteredPoissonKernel (hMpaper := hM) B n d ε M))
≤ d * sqrt (exponentialSeriesTail K (sparseGramCoefficient B hB n d ε M ⟨hε, hεhalf⟩ hM * M ^ 2 / 4))
Proof (Lean source)
lemma sparse_product_mixture_tv_le_sqrt_tail {K n d : ℕ} {ε M B : ℝ}
(P : ConstrainedPriorPair K M) (hn : 1 ≤ n) (hd : 2 ≤ d)
(hε : 0 < ε) (hεhalf : ε ≤ 1 / 2) (hM : 2 ≤ M) (hB : 2 < B) :
tvDist
(productPriorPredictive d (Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) P.ν₀)
(sparseCenteredPoissonKernel (hMpaper := hM) B n d ε M))
(productPriorPredictive d (Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) P.ν₁)
(sparseCenteredPoissonKernel (hMpaper := hM) B n d ε M)) ≤
d * sqrt (exponentialSeriesTail K
(sparseGramCoefficient B hB n d ε M ⟨hε, hεhalf⟩ hM * M ^ 2 / 4)) := by
let := P.prob₀
let := P.prob₁
let π₀ := Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) P.ν₀
let π₁ := Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) P.ν₁
let Q := poissonCellLaw B n d ε (sparseReference (_hMpaper := hM) M)
let L := fun θ N => oneCellLikelihood (hMpaper := hM) B n d ε M (θ + sparseReference (_hMpaper := hM) M) N
have hcenter : Measurable (fun z => z - sparseReference (_hMpaper := hM) M) := by fun_prop
let : IsProbabilityMeasure π₀ := Measure.isProbabilityMeasure_map hcenter.aemeasurable
let : IsProbabilityMeasure π₁ := Measure.isProbabilityMeasure_map hcenter.aemeasurable
let : IsProbabilityMeasure Q := by dsimp [Q, poissonCellLaw]; infer_instance
have hα : 0 ≤ sparseGramCoefficient B hB n d ε M ⟨hε, hεhalf⟩ hM := by
have hB0 : 0 ≤ B := by linarith
have hbase : 0 ≤ 1 - ε := by linarith
have hM0 : 0 ≤ M := by linarith
unfold sparseGramCoefficient sparseReference
positivity
have hlegal (θ : ℝ) (hθ : |θ| ≤ M / 2) :
θ + sparseReference (_hMpaper := hM) M ∈ Icc 0 M := by
rw [abs_le] at hθ
dsimp [sparseReference]
constructor <;> linarith [hθ.1, hθ.2]
have hmeas : Measurable (fun p : ℝ × (Fin 4 → ℕ) => L p.1 p.2) := by
change Measurable (fun p : ℝ × (Fin 4 → ℕ) =>
oneCellLikelihood (hMpaper := hM) B n d ε M (p.1 + sparseReference (_hMpaper := hM) M) p.2)
have ht : Measurable (fun p : ℝ × (Fin 4 → ℕ) =>
(p.1 + sparseReference (_hMpaper := hM) M, p.2)) := by fun_prop
exact comp
(g := fun p : ℝ × (Fin 4 → ℕ) => oneCellLikelihood (hMpaper := hM) B n d ε M p.1 p.2)
(f := fun p : ℝ × (Fin 4 → ℕ) => (p.1 + sparseReference (_hMpaper := hM) M, p.2))
(measurable_oneCellLikelihood (hMpaper := hM) B n d ε M) ht
have hgram := (sparse_likelihood n d ε M B (fun _ => 0) hn hd hε hεhalf hM hB
(fun _ => ⟨le_rfl, by linarith⟩)).2.2.2.1
have hinner : ∀ θ, |θ| ≤ M / 2 → ∀ θ', |θ'| ≤ M / 2 →
(∫ N, L θ N * L θ' N ∂Q) =
exp (sparseGramCoefficient B hB n d ε M ⟨hε, hεhalf⟩ hM * θ * θ') := by
intro θ hθ θ' hθ'
simpa [L, Q] using hgram _ _ (hlegal θ hθ) (hlegal θ' hθ')
have hdensity : ∀ θ, |θ| ≤ M / 2 →
sparseCenteredPoissonKernel (hMpaper := hM) B n d ε M θ =
Q.withDensity (fun N => ofReal (L θ N)) := by
intro θ hθ
exact poissonCellLaw_eq_withDensity_oneCellLikelihood
hB hn hd hε hεhalf hM (hlegal θ hθ)
have hbound := momentMatchedProductMixture_tv_le_of_supported
d π₀ π₁ (sparseCenteredPoissonKernel (hMpaper := hM) B n d ε M) Q L
(fun θ => by
change IsProbabilityMeasure (poissonCellLaw B n d ε (θ + sparseReference (_hMpaper := hM) M))
unfold poissonCellLaw
infer_instance)
(sparseGramCoefficient B hB n d ε M ⟨hε, hεhalf⟩ hM) (M / 2) K hα (by linarith)
hmeas (fun θ hθ N => oneCellLikelihood_nonneg hB hε hεhalf hM (hlegal θ hθ) N)
hdensity hinner (sparse_centered_prior_support (hMpaper := hM) P.ν₀ P.supp₀)
(sparse_centered_prior_support (hMpaper := hM) P.ν₁ P.supp₁)
(constrainedPriorPair_centered_moments (hMpaper := hM) P)
simpa only [π₀, π₁, show (M / 2) ^ 2 = M ^ 2 / 4 by ring,
mul_div_assoc] using hbound
theorem sparse_geometric_degree_calibration
A geometric one-cell error becomes at most 1/8 after tensorization when the degree exceeds a universal multiple of log(ed). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
∃ C : ℝ,
0 < C
conclusion 1
d K :
ℕ
2 ≤ d
C * logAlphabet d ≤ K
(d : ℝ) * (D * ρ ^ K) ≤ 1 / 8
Proof (Lean source)
lemma sparse_geometric_degree_calibration {D ρ : ℝ}
(hD : 0 < D) (hρ : ρ ∈ Ioo (0 : ℝ) 1) :
∃ C : ℝ, 0 < C ∧ ∀ (d K : ℕ), 2 ≤ d →
C * logAlphabet d ≤ K → (d : ℝ) * (D * ρ ^ K) ≤ 1 / 8 := by
let a := -Real.log ρ
have ha : 0 < a := neg_pos.mpr (Real.log_neg hρ.1 hρ.2)
let C := (|Real.log (8 * D)| + 2) / a
refine ⟨C, by dsimp [C]; positivity, ?_⟩
intro d K hd hK
have hd1 : (1 : ℝ) ≤ d := by exact_mod_cast (show 1 ≤ d by omega)
have hd0 : (0 : ℝ) < d := lt_of_lt_of_le (by norm_num) hd1
have hld : 0 ≤ Real.log d := Real.log_nonneg hd1
have hL : logAlphabet d = 1 + Real.log d := by
rw [logAlphabet, Real.log_mul (Real.exp_pos _).ne' hd0.ne', Real.log_exp]
have hKa : (|Real.log (8 * D)| + 2) * (1 + Real.log d) ≤ a * K := by
rw [hL] at hK
have := (mul_le_mul_of_nonneg_left hK ha.le)
dsimp [C] at this
field_simp at this
nlinarith
have hlog : Real.log (8 * D * d) ≤ a * K := by
rw [Real.log_mul (by positivity) hd0.ne']
nlinarith [le_abs_self (Real.log (8 * D)), abs_nonneg (Real.log (8 * D)),
mul_nonneg (abs_nonneg (Real.log (8 * D))) hld]
have hp : ρ ^ K = exp (-a * K) := by
rw [← Real.exp_log (pow_pos hρ.1 K), Real.log_pow]
congr 1
dsimp [a]
ring
calc
(d : ℝ) * (D * ρ ^ K) = (d * D) * exp (-a * K) := by rw [hp]; ring
_ ≤ (d * D) * exp (-Real.log (8 * D * d)) := by
gcongr
linarith
_ = 1 / 8 := by
rw [Real.exp_neg, Real.exp_log (by positivity)]
field_simp
<;> ring
theorem sparse_product_mixture_tv_le_one_eighth Lemma sparse_product_mixture_tv_le_one_eighth in the paper ↗
Roadmap (34)--(36): one universal small support scale and one universal logarithmic degree scale make the full sparse product experiments indistinguishable. The support constraint is precisely the second branch of (33). In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ η C : ℝ,
0 < η
0 < C
conclusion 1
K n d :
ℕ
ε M B :
ℝ
P :
1 ≤ n
2 ≤ d
0 < ε
ε ≤ 1 / 2
hM :
2 ≤ M
hB :
2 < B
C * logAlphabet d ≤ K
M ≤ η * d * K / (B * n * ε)
tvDist (productPriorPredictive d (Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) P.ν₀) (sparseCenteredPoissonKernel (hMpaper := hM) B n d ε M)) (productPriorPredictive d (Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) P.ν₁) (sparseCenteredPoissonKernel (hMpaper := hM) B n d ε M))
≤ 1 / 8
Proof (Lean source)
lemma sparse_product_mixture_tv_le_one_eighth :
∃ η C : ℝ, 0 < η ∧ 0 < C ∧
∀ (K n d : ℕ) (ε M B : ℝ) (P : ConstrainedPriorPair K M),
1 ≤ n → 2 ≤ d → 0 < ε → ε ≤ 1 / 2 → ∀ hM : 2 ≤ M,
∀ hB : 2 < B, C * logAlphabet d ≤ K →
M ≤ η * d * K / (B * n * ε) →
tvDist
(productPriorPredictive d (Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) P.ν₀)
(sparseCenteredPoissonKernel (hMpaper := hM) B n d ε M))
(productPriorPredictive d (Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) P.ν₁)
(sparseCenteredPoissonKernel (hMpaper := hM) B n d ε M)) ≤ 1 / 8 := by
obtain ⟨b, D, ρ, hb, hD, hρ, htail⟩ :=
FiniteSignedMomentMarkedPoissonMixture.exists_geometric_sqrt_exponentialSeriesTail_bound
1 (by norm_num)
obtain ⟨C, hC, hcal⟩ := sparse_geometric_degree_calibration hD hρ
refine ⟨2 * b, C, by positivity, hC, ?_⟩
intro K n d ε M B P hn hd hε hεhalf hM hB hK hband
have hn0 : (0 : ℝ) < n := by exact_mod_cast (show 0 < n by omega)
have hd0 : (0 : ℝ) < d := by exact_mod_cast (show 0 < d by omega)
have hB0 : 0 < B := by linarith
have hden : 0 < B * n * ε := by positivity
have hsmall : B * n * ε * M / (2 * d) ≤ b * K := by
apply (div_le_iff₀ (by positivity : (0 : ℝ) < 2 * d)).2
have h := (le_div_iff₀ hden).1 hband
nlinarith
let t := sparseGramCoefficient B hB n d ε M ⟨hε, hεhalf⟩ hM * M ^ 2 / 4
have ht0 : 0 ≤ t := by
have hbase : 0 ≤ 1 - ε := by linarith
have hM0 : 0 ≤ M := by linarith
dsimp [t, sparseGramCoefficient, sparseReference]
positivity
have ht : t ≤ b * K :=
(sparse_mixture_series_argument_le hn hd hε hεhalf hM hB).trans hsmall
have hdecay : sqrt (exponentialSeriesTail K t) ≤ D * ρ ^ K := by
simpa only [one_mul] using htail K t ht0 ht
calc
_ ≤ d * sqrt (exponentialSeriesTail K t) :=
sparse_product_mixture_tv_le_sqrt_tail P hn hd hε hεhalf hM hB
_ ≤ d * (D * ρ ^ K) := mul_le_mul_of_nonneg_left hdecay (Nat.cast_nonneg d)
_ ≤ 1 / 8 := hcal d K hd hK
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.LowerSparseTesting 8 declarations Centering the explicit constrained priors preserves their matched moments and places their support in the radius-M/2 interval.
Sparse moment-matched likelihood separation
Centering the explicit constrained priors preserves their matched moments and places their support in the radius-M/2 interval. The full four-coordinate Poisson Gram identity then gives the unmatched exponential-series bound (34).
Translating a supported prior by its reference midpoint gives support radius M/2. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
(Measure.map (fun z => z - sparseReference (_hMpaper := hMpaper) M) ν) {z | |z| ≤ M / 2}
= 1
Proof (Lean source)
lemma sparse_centered_prior_support (ν : Measure ℝ) [IsProbabilityMeasure ν]
{M : ℝ} (hsupp : ν (Icc 0 M)ᶜ = 0) (hMpaper : 2 ≤ M) :
(Measure.map (fun z => z - sparseReference (_hMpaper := hMpaper) M) ν) {z | |z| ≤ M / 2} = 1 := by
have hm : Measurable (fun z => z - sparseReference (_hMpaper := hMpaper) M) := by fun_prop
rw [Measure.map_apply hm (measurableSet_le (by fun_prop) measurable_const)]
apply (mem_ae_iff_prob_eq_one
((measurableSet_le (by fun_prop) measurable_const).preimage hm)).1
filter_upwards [show ∀ᵐ z ∂ν, z ∈ Icc 0 M from
(ae_iff).2 hsupp] with z hz
change |z - sparseReference (_hMpaper := hMpaper) M| ≤ M / 2
rw [abs_le]
dsimp [sparseReference]
constructor <;> linarith [hz.1, hz.2]
theorem constrainedPriorPair_centered_moments Lemma constrainedPriorPair_centered_moments in the paper ↗
Every centered moment through K agrees for the completed packet priors. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
(∫ z, z ^ j ∂Measure.map (fun z => z - sparseReference (_hMpaper := hMpaper) M) P.ν₀)
= (∫ z, z ^ j ∂Measure.map (fun z => z - sparseReference (_hMpaper := hMpaper) M) P.ν₁)
Proof (Lean source)
lemma constrainedPriorPair_centered_moments {K : ℕ} {M : ℝ}
(P : ConstrainedPriorPair K M) (j : ℕ) (hj : j ≤ K) (hMpaper : 2 ≤ M) :
(∫ z, z ^ j ∂Measure.map (fun z => z - sparseReference (_hMpaper := hMpaper) M) P.ν₀) =
(∫ z, z ^ j ∂Measure.map (fun z => z - sparseReference (_hMpaper := hMpaper) M) P.ν₁) := by
rw [integral_map (by fun_prop) (by fun_prop),
integral_map (by fun_prop) (by fun_prop)]
let p : Polynomial ℝ := (X - C (sparseReference (_hMpaper := hMpaper) M)) ^ j
have hp : p.natDegree ≤ K := by
calc
p.natDegree ≤ (X - C (sparseReference (_hMpaper := hMpaper) M)).natDegree * j := by
simpa only [Nat.mul_comm] using
(Polynomial.natDegree_pow_le
(p := X - C (sparseReference (_hMpaper := hMpaper) M)) (n := j))
_ ≤ 1 * j := by
gcongr
exact (Polynomial.natDegree_sub_le _ _).trans (by simp)
_ ≤ K := by simpa using hj
simpa [p] using constrainedPriorPair_polynomial_integrals P p hp
The full sparse likelihood is jointly measurable in its intensity and counts. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
B :
ℝ
n d :
ℕ
ε M :
ℝ
hMpaper :
2 ≤ M
conclusion
Proof (Lean source)
@[fun_prop]
lemma measurable_oneCellLikelihood (B : ℝ) (n d : ℕ) (ε M : ℝ) (hMpaper : 2 ≤ M) :
Measurable (fun p : ℝ × (Fin 4 → ℕ) => oneCellLikelihood (hMpaper := hMpaper) B n d ε M p.1 p.2) := by
apply measurable_from_prod_countable_left
intro N
unfold oneCellLikelihood poissonCellMean
apply Finset.measurable_prod
intro j _
split_ifs <;> fun_prop
Legal sparse intensities give nonnegative likelihoods on every count vector. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
B ε M :
ℝ
n d :
ℕ
hB :
2 < B
hε :
0 < ε
hεhalf :
ε ≤ 1 / 2
hM :
2 ≤ M
z :
ℝ
hz :
z ∈ Icc 0 M
N :
Fin 4 → ℕ
conclusion
0 ≤ oneCellLikelihood (hMpaper := hM) B n d ε M z N
Proof (Lean source)
lemma oneCellLikelihood_nonneg {B ε M : ℝ} {n d : ℕ}
(hB : 2 < B) (hε : 0 < ε) (hεhalf : ε ≤ 1 / 2) (hM : 2 ≤ M)
{z : ℝ} (hz : z ∈ Icc 0 M) (N : Fin 4 → ℕ) :
0 ≤ oneCellLikelihood (hMpaper := hM) B n d ε M z N := by
have hbase : 0 ≤ 1 - ε := by linarith
have hB0 : 0 ≤ B := by linarith
have hM0 : 0 ≤ sparseReference (_hMpaper := hM) M := by dsimp [sparseReference]; linarith
have hmean (w : ℝ) (hw : 0 ≤ w) (j : Fin 4) :
0 ≤ poissonCellMean B n d ε w j := by
unfold poissonCellMean
split_ifs <;> positivity
apply Finset.prod_nonneg
intro j _
exact mul_nonneg (Real.exp_nonneg _) (pow_nonneg
(div_nonneg (hmean z hz.1 j) (hmean _ hM0 j)) _)
The scalar Poisson likelihood recovers each atom, including zero intensity. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
μ μstar :
ℝ
hμ :
0 ≤ μ
hstar :
0 < μstar
k :
ℕ
conclusion
poissonMeasure (toNNReal μ) {k}
Proof (Lean source)
lemma sparse_poisson_likelihood_atom {μ μstar : ℝ} (hμ : 0 ≤ μ)
(hstar : 0 < μstar) (k : ℕ) :
poissonMeasure (toNNReal μ) {k} =
ofReal (exp (μstar - μ) * (μ / μstar) ^ k) *
poissonMeasure (toNNReal μstar) {k} := by
rw [poissonMeasure_singleton, poissonMeasure_singleton,
Real.coe_toNNReal _ hμ, Real.coe_toNNReal _ hstar.le,
← ENNReal.ofReal_mul (by positivity)]
congr 1
rw [div_pow]
have he : exp (μstar - μ) * exp (-μstar) = exp (-μ) := by
rw [← Real.exp_add]
congr 1
ring
symm
calc
_ = (exp (μstar - μ) * exp (-μstar)) * μ ^ k / (k.factorial : ℝ) := by
field_simp [ne_of_gt hstar]
<;> ring
_ = _ := by rw [he]
theorem poissonCellLaw_eq_withDensity_oneCellLikelihood Lemma poissonCellLaw_eq_withDensity_oneCellLikelihood in the paper ↗
The full four-coordinate sparse likelihood is the actual Poisson density, so the Gram calculation applies to probability experiments rather than merely formal functions of the counts. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
B ε M :
ℝ
n d :
ℕ
hB :
2 < B
hn :
1 ≤ n
hd :
2 ≤ d
hε :
0 < ε
hεhalf :
ε ≤ 1 / 2
hM :
2 ≤ M
z :
ℝ
hz :
z ∈ Icc 0 M
conclusion
poissonCellLaw B n d ε z
= (poissonCellLaw B n d ε (sparseReference (_hMpaper := hM) M)).withDensity (fun N => ofReal (oneCellLikelihood (hMpaper := hM) B n d ε M z N))
Proof (Lean source)
lemma poissonCellLaw_eq_withDensity_oneCellLikelihood {B ε M : ℝ} {n d : ℕ}
(hB : 2 < B) (hn : 1 ≤ n) (hd : 2 ≤ d) (hε : 0 < ε)
(hεhalf : ε ≤ 1 / 2) (hM : 2 ≤ M) {z : ℝ} (hz : z ∈ Icc 0 M) :
poissonCellLaw B n d ε z =
(poissonCellLaw B n d ε (sparseReference (_hMpaper := hM) M)).withDensity
(fun N => ofReal (oneCellLikelihood (hMpaper := hM) B n d ε M z N)) := by
have hdR : (0 : ℝ) < d := by exact_mod_cast (show 0 < d by omega)
have hnR : (0 : ℝ) < n := by exact_mod_cast (show 0 < n by omega)
have hε1 : 0 < 1 - ε := by linarith
have hMstar : 0 < sparseReference (_hMpaper := hM) M := by unfold sparseReference; linarith
have hstar (j : Fin 4) : 0 < poissonCellMean B n d ε (sparseReference (_hMpaper := hM) M) j := by
fin_cases j <;> simp [poissonCellMean, cellIdx] <;> positivity
have hz0 : 0 ≤ z := hz.1
have hmean (j : Fin 4) : 0 ≤ poissonCellMean B n d ε z j := by
unfold poissonCellMean
split_ifs <;> positivity
apply Measure.ext_of_singleton
intro N
rw [withDensity_apply _ (measurableSet_singleton N), lintegral_singleton]
simp only [poissonCellLaw, Measure.pi_singleton]
have hatom (j : Fin 4) := sparse_poisson_likelihood_atom (hmean j) (hstar j) (N j)
simp_rw [hatom]
rw [Finset.prod_mul_distrib]
congr 1
unfold oneCellLikelihood
exact (ENNReal.ofReal_prod_of_nonneg (fun j _ => by
exact mul_nonneg (Real.exp_nonneg _) (pow_nonneg
(div_nonneg (hmean j) (hstar j).le) _))).symm
theorem sparse_mixture_likelihood_sq_le_tail Lemma sparse_mixture_likelihood_sq_le_tail in the paper ↗
Equation (34): moment matching removes every low-degree term in the full four-coordinate Poisson mixture, with the exact midpoint radius. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
K n d :
ℕ
ε M B :
ℝ
P :
hn :
1 ≤ n
hd :
2 ≤ d
hε :
0 < ε
hεhalf :
ε ≤ 1 / 2
hM :
2 ≤ M
hB :
2 < B
conclusion
(∫ N : Fin 4 → ℕ, ((∫ z, oneCellLikelihood (hMpaper := hM) B n d ε M z N ∂P.ν₀) - (∫ z, oneCellLikelihood (hMpaper := hM) B n d ε M z N ∂P.ν₁)) ^ 2 ∂poissonCellLaw B n d ε (sparseReference (_hMpaper := hM) M))
≤ 4 * exponentialSeriesTail K (sparseGramCoefficient B hB n d ε M ⟨hε, hεhalf⟩ hM * M ^ 2 / 4)
Proof (Lean source)
lemma sparse_mixture_likelihood_sq_le_tail {K n d : ℕ} {ε M B : ℝ}
(P : ConstrainedPriorPair K M) (hn : 1 ≤ n) (hd : 2 ≤ d)
(hε : 0 < ε) (hεhalf : ε ≤ 1 / 2) (hM : 2 ≤ M) (hB : 2 < B) :
(∫ N : Fin 4 → ℕ,
((∫ z, oneCellLikelihood (hMpaper := hM) B n d ε M z N ∂P.ν₀) -
(∫ z, oneCellLikelihood (hMpaper := hM) B n d ε M z N ∂P.ν₁)) ^ 2
∂poissonCellLaw B n d ε (sparseReference (_hMpaper := hM) M)) ≤
4 * exponentialSeriesTail K (sparseGramCoefficient B hB n d ε M ⟨hε, hεhalf⟩ hM * M ^ 2 / 4) := by
let := P.prob₀
let := P.prob₁
let π₀ := Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) P.ν₀
let π₁ := Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) P.ν₁
let Q := poissonCellLaw B n d ε (sparseReference (_hMpaper := hM) M)
let L := fun θ N => oneCellLikelihood (hMpaper := hM) B n d ε M (θ + sparseReference (_hMpaper := hM) M) N
have hcenter : Measurable (fun z => z - sparseReference (_hMpaper := hM) M) := by fun_prop
let : IsProbabilityMeasure π₀ := Measure.isProbabilityMeasure_map hcenter.aemeasurable
let : IsProbabilityMeasure π₁ := Measure.isProbabilityMeasure_map hcenter.aemeasurable
let : IsProbabilityMeasure Q := by dsimp [Q, poissonCellLaw]; infer_instance
have hα : 0 ≤ sparseGramCoefficient B hB n d ε M ⟨hε, hεhalf⟩ hM := by
have hB0 : 0 ≤ B := by linarith
have hbase : 0 ≤ 1 - ε := by linarith
have hM0 : 0 ≤ M := by linarith
unfold sparseGramCoefficient sparseReference
positivity
have hlegal (θ : ℝ) (hθ : |θ| ≤ M / 2) :
θ + sparseReference (_hMpaper := hM) M ∈ Icc 0 M := by
rw [abs_le] at hθ
dsimp [sparseReference]
constructor <;> linarith [hθ.1, hθ.2]
have hmeas : Measurable (fun p : ℝ × (Fin 4 → ℕ) => L p.1 p.2) := by
change Measurable (fun p : ℝ × (Fin 4 → ℕ) =>
oneCellLikelihood (hMpaper := hM) B n d ε M (p.1 + sparseReference (_hMpaper := hM) M) p.2)
have ht : Measurable (fun p : ℝ × (Fin 4 → ℕ) =>
(p.1 + sparseReference (_hMpaper := hM) M, p.2)) := by fun_prop
exact comp
(g := fun p : ℝ × (Fin 4 → ℕ) => oneCellLikelihood (hMpaper := hM) B n d ε M p.1 p.2)
(f := fun p : ℝ × (Fin 4 → ℕ) => (p.1 + sparseReference (_hMpaper := hM) M, p.2))
(measurable_oneCellLikelihood (hMpaper := hM) B n d ε M) ht
have hgram := (sparse_likelihood n d ε M B (fun _ => 0) hn hd hε hεhalf hM hB
(fun _ => ⟨le_rfl, by linarith⟩)).2.2.2.1
have hinner : ∀ θ, |θ| ≤ M / 2 → ∀ θ', |θ'| ≤ M / 2 →
(∫ N, L θ N * L θ' N ∂Q) =
exp (sparseGramCoefficient B hB n d ε M ⟨hε, hεhalf⟩ hM * θ * θ') := by
intro θ hθ θ' hθ'
simpa [L, Q] using hgram _ _ (hlegal θ hθ) (hlegal θ' hθ')
have hbound := integral_sq_mixtureLikelihood_sub_le_tail_of_supported
π₀ π₁ Q L (sparseGramCoefficient B hB n d ε M ⟨hε, hεhalf⟩ hM) (M / 2) K hα (by linarith)
hmeas (fun θ hθ N => oneCellLikelihood_nonneg hB hε hεhalf hM (hlegal θ hθ) N)
hinner (sparse_centered_prior_support (hMpaper := hM) P.ν₀ P.supp₀)
(sparse_centered_prior_support (hMpaper := hM) P.ν₁ P.supp₁)
(constrainedPriorPair_centered_moments (hMpaper := hM) P)
have hmix (ν : Measure ℝ) (N : Fin 4 → ℕ) :
mixtureLikelihood (Measure.map (fun z => z - sparseReference (_hMpaper := hM) M) ν) L N =
∫ z, oneCellLikelihood (hMpaper := hM) B n d ε M z N ∂ν := by
unfold mixtureLikelihood
have hf : Measurable (fun θ => L θ N) :=
comp (g := fun p : ℝ × (Fin 4 → ℕ) => L p.1 p.2)
(f := fun θ : ℝ => (θ, N)) hmeas measurable_prodMk_right
rw [integral_map hcenter.aemeasurable hf.aestronglyMeasurable]
simp [L]
simpa only [π₀, π₁, Q, hmix, show (M / 2) ^ 2 = M ^ 2 / 4 by ring,
mul_div_assoc] using hbound
The exact full-likelihood series argument obeys the scalar upper bound in (34), including the two untreated-outcome coordinates. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
n d :
ℕ
ε M B :
ℝ
hn :
1 ≤ n
hd :
2 ≤ d
hε :
0 < ε
hεhalf :
ε ≤ 1 / 2
hM :
2 ≤ M
hB :
2 < B
conclusion
sparseGramCoefficient B hB n d ε M ⟨hε, hεhalf⟩ hM * M ^ 2 / 4 ≤ B * n * ε * M / (2 * d)
Proof (Lean source)
lemma sparse_mixture_series_argument_le {n d : ℕ} {ε M B : ℝ}
(hn : 1 ≤ n) (hd : 2 ≤ d) (hε : 0 < ε) (hεhalf : ε ≤ 1 / 2)
(hM : 2 ≤ M) (hB : 2 < B) :
sparseGramCoefficient B hB n d ε M ⟨hε, hεhalf⟩ hM * M ^ 2 / 4 ≤ B * n * ε * M / (2 * d) := by
have hd0 : (0 : ℝ) < d := by exact_mod_cast (by omega : 0 < d)
have hM0 : 0 < M := by linarith
calc
_ ≤ (2 * B * n * ε / (d * M)) * M ^ 2 / 4 := by
gcongr
exact sparseGramCoefficient_bound B hB n d hn hd ε M hε hεhalf hM
_ = _ := by field_simp; ring
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.LowerTransfer 18 declarations Identification and causal completion preserve the attained risks, their worst-case supremum, and the all-estimator minimax value.
Lower Transfer scaffold
Identification and causal completion preserve the attained risks, their worst-case supremum, and the all-estimator minimax value. The remaining Poisson shortage estimate in roadmap (49) is proved uniformly over the sparse normalizer. Projection and the first-label risk transfer below implement the loss comparison after (49); constructing the label map from the count experiment remains open.
theorem causalRisk_eq_observedRisk
Identification makes the causal risk equal the risk at the observed margin. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
causalRisk n est Q = observedRisk n est ⟨observedMarginal Q.1, ⟨Q.2.overlap⟩⟩
Proof (Lean source)
lemma causalRisk_eq_observedRisk {n d : ℕ} {ε : ℝ}
(hd : 2 ≤ d) (hε : 0 < ε) (hεhalf : ε ≤ 1 / 2)
(est : Estimator n d) (Q : CausalModelLaw d ε) :
causalRisk n est Q =
observedRisk n est ⟨observedMarginal Q.1, ⟨Q.2.overlap⟩⟩ := by
unfold causalRisk observedRisk
rw [identification.2.1 d ε Q.1 hd hε hεhalf Q.2]
theorem causalRisk_range_eq_observedRisk_range
Every risk attained in either model class is attained in the other: observed margins give one inclusion and causal completion gives the reverse. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma causalRisk_range_eq_observedRisk_range {n d : ℕ} {ε : ℝ}
(hd : 2 ≤ d) (hε : 0 < ε) (hεhalf : ε ≤ 1 / 2)
(est : Estimator n d) :
range (causalRisk n (ε := ε) est) =
range (observedRisk n (ε := ε) est) := by
ext r
constructor
· rintro ⟨Q, rfl⟩
exact ⟨⟨observedMarginal Q.1, ⟨Q.2.overlap⟩⟩,
(causalRisk_eq_observedRisk hd hε hεhalf est Q).symm⟩
· rintro ⟨P, rfl⟩
obtain ⟨Q, hQ, hmargin⟩ := identification.1 d ε P.1 hd hε hεhalf P.2
refine ⟨⟨Q, hQ⟩, ?_⟩
rw [causalRisk_eq_observedRisk hd hε hεhalf]
change sqRisk (productLaw (observedMarginal Q) n) est.1
(observedValue (observedMarginal Q)) =
sqRisk (productLaw P.1 n) est.1 (observedValue P.1)
rw [hmargin]
theorem causalWorstRisk_eq_observedWorstRisk
Each estimator has the same worst-case risk for the observed and causal targets. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
worstCaseRiskReal (causalRisk n (ε := ε)) est
= worstCaseRiskReal (observedRisk n (ε := ε)) est
Proof (Lean source)
lemma causalWorstRisk_eq_observedWorstRisk {n d : ℕ} {ε : ℝ}
(hd : 2 ≤ d) (hε : 0 < ε) (hεhalf : ε ≤ 1 / 2)
(est : Estimator n d) :
worstCaseRiskReal (causalRisk n (ε := ε)) est =
worstCaseRiskReal (observedRisk n (ε := ε)) est := by
rw [worstCaseRisk_eq_sSup_range,
worstCaseRisk_eq_sSup_range,
causalRisk_range_eq_observedRisk_range hd hε hεhalf est]
theorem causalArmwiseWorstRisk_eq_armwiseWorstRisk
Identification and completion preserve the armwise estimator's worst-case risk. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
H₀ κ :
ℝ
D₀ :
ℕ
hH₀ :
0 < H₀
hκ :
0 < κ
hD₀ :
2 ≤ D₀
n d :
ℕ
ε :
ℝ
hd :
2 ≤ d
hε :
0 < ε
hεhalf :
ε ≤ 1 / 2
conclusion
causalArmwiseWorstRisk H₀ κ D₀ hH₀ hκ hD₀ n d ε
= armwiseWorstRisk H₀ κ D₀ hH₀ hκ hD₀ n d ε
Proof (Lean source)
lemma causalArmwiseWorstRisk_eq_armwiseWorstRisk (H₀ κ : ℝ) (D₀ : ℕ)
(hH₀ : 0 < H₀) (hκ : 0 < κ) (hD₀ : 2 ≤ D₀)
(n d : ℕ) (ε : ℝ) (hd : 2 ≤ d) (hε : 0 < ε) (hεhalf : ε ≤ 1 / 2) :
causalArmwiseWorstRisk H₀ κ D₀ hH₀ hκ hD₀ n d ε =
armwiseWorstRisk H₀ κ D₀ hH₀ hκ hD₀ n d ε := by
exact causalWorstRisk_eq_observedWorstRisk hd hε hεhalf
(armwiseEstimatorWitness H₀ κ D₀ hH₀ hκ hD₀ n d ε)
theorem causalMinimaxRisk_eq_minimaxRisk
Taking the infimum over the shared estimator class preserves the exact risk equality. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
n d :
ℕ
ε :
ℝ
hd :
2 ≤ d
hε :
0 < ε
hεhalf :
ε ≤ 1 / 2
conclusion
causalMinimaxRisk n d ε = minimaxRisk n d ε
Proof (Lean source)
lemma causalMinimaxRisk_eq_minimaxRisk {n d : ℕ} {ε : ℝ}
(hd : 2 ≤ d) (hε : 0 < ε) (hεhalf : ε ≤ 1 / 2) :
causalMinimaxRisk n d ε = minimaxRisk n d ε := by
unfold causalMinimaxRisk minimaxRisk minimaxValueReal
congr 1
funext est
exact causalWorstRisk_eq_observedWorstRisk hd hε hεhalf est
theorem causalMinimaxRisk_bounds_of_observed
A two-sided observed minimax bound transfers unchanged to the causal target. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
n d :
ℕ
ε c C :
ℝ
hd :
2 ≤ d
hε :
0 < ε
hεhalf :
ε ≤ 1 / 2
hlower :
c * rateScale n d ε ≤ minimaxRisk n d ε
hupper :
minimaxRisk n d ε ≤ C * rateScale n d ε
conclusion
conclusion 1
c * rateScale n d ε ≤ causalMinimaxRisk n d ε
conclusion 2
causalMinimaxRisk n d ε ≤ C * rateScale n d ε
Proof (Lean source)
lemma causalMinimaxRisk_bounds_of_observed {n d : ℕ} {ε c C : ℝ}
(hd : 2 ≤ d) (hε : 0 < ε) (hεhalf : ε ≤ 1 / 2)
(hlower : c * rateScale n d ε ≤ minimaxRisk n d ε)
(hupper : minimaxRisk n d ε ≤ C * rateScale n d ε) :
c * rateScale n d ε ≤ causalMinimaxRisk n d ε ∧
causalMinimaxRisk n d ε ≤ C * rateScale n d ε := by
rw [causalMinimaxRisk_eq_minimaxRisk hd hε hεhalf]
exact ⟨hlower, hupper⟩
theorem observedRisk_nonneg
Squared observed risk is nonnegative for every estimator and observed law. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
Proof (Lean source)
lemma observedRisk_nonneg {n d : ℕ} {ε : ℝ}
(est : Estimator n d) (P : ModelLaw d ε) :
0 ≤ observedRisk n est P := by
exact MeasureTheory.integral_nonneg fun _ => sq_nonneg _
theorem minimaxRisk_le_armwiseWorstRisk
The admissible deterministic armwise estimator upper-bounds the minimax infimum. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
H₀ κ :
ℝ
D₀ :
ℕ
hH₀ :
0 < H₀
hκ :
0 < κ
hD₀ :
2 ≤ D₀
n d :
ℕ
ε :
ℝ
conclusion
minimaxRisk n d ε ≤ armwiseWorstRisk H₀ κ D₀ hH₀ hκ hD₀ n d ε
Proof (Lean source)
lemma minimaxRisk_le_armwiseWorstRisk (H₀ κ : ℝ) (D₀ : ℕ)
(hH₀ : 0 < H₀) (hκ : 0 < κ) (hD₀ : 2 ≤ D₀)
(n d : ℕ) (ε : ℝ) :
minimaxRisk n d ε ≤ armwiseWorstRisk H₀ κ D₀ hH₀ hκ hD₀ n d ε := by
exact minimaxValue_le_worstCaseRisk_of_nonneg
(fun est P => observedRisk_nonneg est P)
(armwiseEstimatorWitness H₀ κ D₀ hH₀ hκ hD₀ n d ε)
The lower Gaussian Poisson tail gives the exact shortage exponent before uniformizing over the normalizer in roadmap (49). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
lam :
ℝ≥0
n :
ℕ
hlam :
(n : ℝ) < lam
conclusion
Proof (Lean source)
lemma poisson_shortage_prob_le (lam : ℝ≥0) (n : ℕ)
(hlam : (n : ℝ) < lam) :
(poissonMeasure lam).real {k : ℕ | k < n} ≤
exp (-((lam : ℝ) - n) ^ 2 / (2 * lam)) := by
have hlampos : 0 < (lam : ℝ) := lt_of_le_of_lt (Nat.cast_nonneg n) hlam
have hs : 0 ≤ (lam : ℝ) - n := by linarith
have hsqrt : sqrt (2 * (lam : ℝ) * (((lam : ℝ) - n) ^ 2 / (2 * lam))) =
(lam : ℝ) - n := by
rw [mul_div_cancel₀ _ (ne_of_gt (mul_pos (by norm_num) hlampos)),
Real.sqrt_sq_eq_abs, abs_of_nonneg hs]
have hsub : {k : ℕ | k < n} ⊆
{k : ℕ | (lam : ℝ) - k >
sqrt (2 * (lam : ℝ) * (((lam : ℝ) - n) ^ 2 / (2 * lam)))} := by
intro k hk
change (lam : ℝ) - (k : ℝ) > _
rw [hsqrt]
have hkR : (k : ℝ) < n := by exact_mod_cast hk
linarith
have ht := poisson_lower_bernstein lam
(z := ((lam : ℝ) - n) ^ 2 / (2 * lam)) (by positivity)
have h := ENNReal.toReal_mono ENNReal.ofReal_ne_top ((measure_mono hsub).trans ht)
simpa only [Measure.real_def, ENNReal.toReal_ofReal (Real.exp_nonneg _), neg_div] using h
The shortage exponent increases above n. Evaluating it at Bn/2 gives exactly the exponent in roadmap (49). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
lam B r :
ℝ
hr :
0 < r
hB :
2 < B
hlam :
B * r / 2 ≤ lam
conclusion
r * (B - 2) ^ 2 / (4 * B) ≤ (lam - r) ^ 2 / (2 * lam)
Proof (Lean source)
lemma shortage_exponent_ge {lam B r : ℝ} (hr : 0 < r) (hB : 2 < B)
(hlam : B * r / 2 ≤ lam) :
r * (B - 2) ^ 2 / (4 * B) ≤ (lam - r) ^ 2 / (2 * lam) := by
have hBpos : 0 < B := by linarith
have ha : 0 < B * r / 2 := by positivity
have hlampos : 0 < lam := ha.trans_le hlam
have hmono : (B * r / 2 - r) ^ 2 / (2 * (B * r / 2)) ≤
(lam - r) ^ 2 / (2 * lam) := by
apply (div_le_div_iff₀ (by positivity) (by positivity)).2
have hprod : 0 ≤ (lam - B * r / 2) * (lam * (B * r / 2) - r ^ 2) := by
apply mul_nonneg (by linarith)
have hfirst := mul_le_mul_of_nonneg_right hlam ha.le
have hbase : r ≤ B * r / 2 := by nlinarith
have hsq := pow_le_pow_left₀ hr.le hbase 2
nlinarith
nlinarith
calc
_ = (B * r / 2 - r) ^ 2 / (2 * (B * r / 2)) := by field_simp; ring
_ ≤ _ := hmono
Any Poisson experiment with total mean at least Bn/2 satisfies the uniform shortage probability in roadmap (49). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
lam :
ℝ≥0
n :
ℕ
B :
ℝ
hn :
1 ≤ n
hB :
2 < B
hlam :
B * (n : ℝ) / 2 ≤ lam
conclusion
Proof (Lean source)
lemma poisson_shortage_uniform_le (lam : ℝ≥0) {n : ℕ} {B : ℝ}
(hn : 1 ≤ n) (hB : 2 < B) (hlam : B * (n : ℝ) / 2 ≤ lam) :
(poissonMeasure lam).real {k : ℕ | k < n} ≤
exp (-(n : ℝ) * (B - 2) ^ 2 / (4 * B)) := by
have hnpos : (0 : ℝ) < n := by exact_mod_cast (by omega : 0 < n)
have hlamn : (n : ℝ) < lam := by nlinarith
apply (poisson_shortage_prob_le lam n hlamn).trans
apply Real.exp_le_exp.mpr
have he := neg_le_neg (shortage_exponent_ge hnpos hB hlam)
simpa only [neg_div, neg_mul] using he
The sparse normalizer is at least one half, so the conditional total count of every legal sparse experiment obeys the same shortage bound. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
n d :
ℕ
ε M B :
ℝ
hn :
1 ≤ n
hε :
0 ≤ ε
hεhalf :
ε ≤ 1 / 2
hB :
2 < B
z :
Fin d → ℝ
hz :
∀ x, z x ∈ Icc 0 M
lam :
ℝ≥0
hmean :
(lam : ℝ) = B * n * sparseNormalizerFormula d ε M z hz
conclusion
Proof (Lean source)
lemma sparse_poisson_shortage_le {n d : ℕ} {ε M B : ℝ}
(hn : 1 ≤ n) (hε : 0 ≤ ε) (hεhalf : ε ≤ 1 / 2) (hB : 2 < B)
(z : Fin d → ℝ) (hz : ∀ x, z x ∈ Icc 0 M)
(lam : ℝ≥0) (hmean : (lam : ℝ) = B * n * sparseNormalizerFormula d ε M z hz) :
(poissonMeasure lam).real {k : ℕ | k < n} ≤
exp (-(n : ℝ) * (B - 2) ^ 2 / (4 * B)) := by
apply poisson_shortage_uniform_le lam hn hB
rw [hmean]
have hs := mul_le_mul_of_nonneg_left (sparseNormalizer_ge_half hε hεhalf z hz)
(show 0 ≤ B * (n : ℝ) by positivity)
nlinarith
The uniform shortage estimate applies to a count on any probability space once its conditional Poisson law has been established. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
N :
Ω → ℕ
lam :
ℝ≥0
hN :
HasLaw N (poissonMeasure lam) μ
n :
ℕ
B :
ℝ
hn :
1 ≤ n
hB :
2 < B
hmean :
B * (n : ℝ) / 2 ≤ lam
conclusion
μ.real {ω | N ω < n} ≤ exp (-(n : ℝ) * (B - 2) ^ 2 / (4 * B))
Proof (Lean source)
lemma poissonCount_shortage_uniform_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (N : Ω → ℕ) (lam : ℝ≥0)
(hN : HasLaw N (poissonMeasure lam) μ) {n : ℕ} {B : ℝ}
(hn : 1 ≤ n) (hB : 2 < B) (hmean : B * (n : ℝ) / 2 ≤ lam) :
μ.real {ω | N ω < n} ≤
exp (-(n : ℝ) * (B - 2) ^ 2 / (4 * B)) := by
rw [hN.measureReal_eq (show MeasurableSet {k : ℕ | k < n} from measurableSet_Iio)]
exact poisson_shortage_uniform_le lam hn hB hmean
theorem shortageExponential_le_inverse_sample Lemma shortageExponential_le_inverse_sample in the paper ↗
Exponential shortage can be made an arbitrarily small inverse-sample penalty by increasing its positive exponent coefficient. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
a δ :
ℝ
n :
ℕ
hn :
1 ≤ n
hδ :
0 < δ
hcoef :
1 / δ ≤ a
conclusion
exp (-a * (n : ℝ)) ≤ δ / n
Proof (Lean source)
lemma shortageExponential_le_inverse_sample {a δ : ℝ} {n : ℕ}
(hn : 1 ≤ n) (hδ : 0 < δ) (hcoef : 1 / δ ≤ a) :
exp (-a * (n : ℝ)) ≤ δ / n := by
have hnpos : (0 : ℝ) < n := by exact_mod_cast (by omega : 0 < n)
have hexp : a * (n : ℝ) * exp (-(a * n)) ≤ 1 :=
(Real.mul_exp_neg_le_exp_neg_one (a * n)).trans
(Real.exp_le_one_iff.mpr (by norm_num))
have hδa : 1 ≤ δ * a := by
simpa only [mul_comm] using (div_le_iff₀ hδ).mp hcoef
have hb := mul_le_mul_of_nonneg_left hexp hδ.le
have hsmall : (n : ℝ) * exp (-(a * n)) ≤ δ := by
have hm := mul_le_mul_of_nonneg_right hδa
(show 0 ≤ (n : ℝ) * exp (-(a * n)) by positivity)
nlinarith
apply (le_div_iff₀ hnpos).2
rw [neg_mul, mul_comm]
exact hsmall
A single universal inflation B makes the shortage penalty smaller than any specified positive multiple of 1/n, uniformly over all sample sizes. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
δ :
ℝ
hδ :
0 < δ
Proof (Lean source)
lemma shortage_inflation_exists {δ : ℝ} (hδ : 0 < δ) :
∃ B : ℝ, 2 < B ∧ ∀ n : ℕ, 1 ≤ n →
exp (-(n : ℝ) * (B - 2) ^ 2 / (4 * B)) ≤ δ / n := by
let B : ℝ := 4 * (1 / δ + 2)
have hq : 0 < 1 / δ := by positivity
have hB : 2 < B := by dsimp [B]; linarith
have hBpos : 0 < B := by linarith
have hcoef : 1 / δ ≤ (B - 2) ^ 2 / (4 * B) := by
apply (le_div_iff₀ (by positivity : 0 < 4 * B)).2
have he : B = 4 * (1 / δ + 2) := rfl
nlinarith
refine ⟨B, hB, fun n hn => ?_⟩
have ht := shortageExponential_le_inverse_sample hn hδ hcoef
convert ht using 1
congr 1
ring
theorem observedRisk_projectUnit_le
Every finite observed estimator can be projected without increasing its risk. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
sqRisk (productLaw P.1 n) (fun o => projectUnit (est.1 o)) (observedValue P.1)
≤ observedRisk n est P
Proof (Lean source)
lemma observedRisk_projectUnit_le {n d : ℕ} {ε : ℝ}
(est : Estimator n d) (P : ModelLaw d ε) :
sqRisk (productLaw P.1 n) (fun o => projectUnit (est.1 o))
(observedValue P.1) ≤ observedRisk n est P := by
let : IsProbabilityMeasure (productLaw P.1 n) := by
unfold productLaw
infer_instance
apply integral_mono Integrable.of_finite Integrable.of_finite
intro o
exact projectUnit_sq_error_le (observedValue_mem_unitInterval P.1)
theorem firstLabels_shortage_sqRisk_le
Use the first n labels on a sufficiently large count and zero on shortage. If the label block has the fixed-sample law, projection and an indicator bound transfer its risk with at most the shortage probability as an additive penalty. The label block may include unobserved continuation labels on shortage; the transferred statistic uses none of them there. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
n d :
ℕ
ε :
ℝ
P :
ModelLaw d ε
est :
Estimator n d
N :
Ω → ℕ
hN :
X :
Ω → (Fin n → Obs d)
hX :
HasLaw X (productLaw P.1 n) μ
conclusion
sqRisk μ (fun ω => if N ω < n then 0 else projectUnit (est.1 (X ω))) (observedValue P.1)
≤ observedRisk n est P + μ.real {ω | N ω < n}
Proof (Lean source)
lemma firstLabels_shortage_sqRisk_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] {n d : ℕ} {ε : ℝ}
(P : ModelLaw d ε) (est : Estimator n d) (N : Ω → ℕ) (hN : Measurable N)
(X : Ω → (Fin n → Obs d)) (hX : HasLaw X (productLaw P.1 n) μ) :
sqRisk μ
(fun ω => if N ω < n then 0 else projectUnit (est.1 (X ω)))
(observedValue P.1) ≤ observedRisk n est P + μ.real {ω | N ω < n} := by
classical
let loss : (Fin n → Obs d) → ℝ := fun o => (est.1 o - observedValue P.1) ^ 2
have hloss : AEStronglyMeasurable loss (productLaw P.1 n) := by fun_prop
have hsource : Integrable loss (Measure.map X μ) := by
rw [hX.map_eq]
let : IsProbabilityMeasure (productLaw P.1 n) := by
unfold productLaw
infer_instance
exact Integrable.of_finite
have hcomp : Integrable (fun ω => loss (X ω)) μ :=
hsource.comp_aemeasurable hX.aemeasurable
have hs : MeasurableSet {ω | N ω < n} := hN measurableSet_Iio
have hi : Integrable ({ω | N ω < n}.indicator (fun _ => (1 : ℝ))) μ :=
(integrable_const 1).indicator hs
have ht := observedValue_mem_unitInterval P.1
have hpoint (ω : Ω) :
((if N ω < n then 0 else projectUnit (est.1 (X ω))) - observedValue P.1) ^ 2 ≤
loss (X ω) + {ω | N ω < n}.indicator (fun _ => (1 : ℝ)) ω := by
by_cases hshort : N ω < n
· simp only [indicator, Set.mem_ofPred_eq, hshort, ↓reduceIte, zero_sub, neg_sq]
have hsq : (observedValue P.1) ^ 2 ≤ 1 := by nlinarith [ht.1, ht.2]
exact hsq.trans (le_add_of_nonneg_left (sq_nonneg _))
· simp only [indicator, Set.mem_ofPred_eq, hshort, ↓reduceIte, add_zero]
exact projectUnit_sq_error_le ht
have hmeas : AEStronglyMeasurable
(fun ω => ((if N ω < n then 0 else projectUnit (est.1 (X ω))) -
observedValue P.1) ^ 2) μ := by
have hv : AEStronglyMeasurable (fun ω => projectUnit (est.1 (X ω))) μ := by
have hm : Measurable (fun o : Fin n → Obs d => projectUnit (est.1 o)) := by
unfold projectUnit
fun_prop
exact (hm.comp_aemeasurable hX.aemeasurable).aestronglyMeasurable
have he : AEStronglyMeasurable
(fun ω => if N ω < n then 0 else projectUnit (est.1 (X ω))) μ := by
apply (AEStronglyMeasurable.piecewise hs
(aestronglyMeasurable_const (b := (0 : ℝ))) hv.restrict).congr
exact Filter.Eventually.of_forall fun ω => by
by_cases h : N ω < n <;> simp [Set.piecewise, h]
fun_prop
have hint : Integrable
(fun ω => ((if N ω < n then 0 else projectUnit (est.1 (X ω))) -
observedValue P.1) ^ 2) μ := by
apply (hcomp.add hi).mono' hmeas
exact Filter.Eventually.of_forall fun ω => by
rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)]
exact hpoint ω
unfold sqRisk
calc
_ ≤ ∫ ω, (loss (X ω) + {ω | N ω < n}.indicator (fun _ => (1 : ℝ)) ω) ∂μ :=
integral_mono hint (hcomp.add hi) hpoint
_ = observedRisk n est P + μ.real {ω | N ω < n} := by
rw [integral_add hcomp hi, integral_indicator_const 1 hs]
rw [show (∫ ω, loss (X ω) ∂μ) = ∫ o, loss o ∂productLaw P.1 n from
hX.integral_comp hloss]
simp only [smul_eq_mul, mul_one]
rfl
theorem firstLabels_poisson_sqRisk_le
Combining the first-label transfer with (49) gives the uniform Poisson penalty used in both the sparse and dense lower experiments. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
n d :
ℕ
ε B :
ℝ
P :
ModelLaw d ε
est :
Estimator n d
N :
Ω → ℕ
hN :
X :
Ω → (Fin n → Obs d)
hX :
HasLaw X (productLaw P.1 n) μ
lam :
ℝ≥0
hlaw :
HasLaw N (poissonMeasure lam) μ
hn :
1 ≤ n
hB :
2 < B
hmean :
B * (n : ℝ) / 2 ≤ lam
conclusion
sqRisk μ (fun ω => if N ω < n then 0 else projectUnit (est.1 (X ω))) (observedValue P.1)
≤ observedRisk n est P + exp (-(n : ℝ) * (B - 2) ^ 2 / (4 * B))
Proof (Lean source)
lemma firstLabels_poisson_sqRisk_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] {n d : ℕ} {ε B : ℝ}
(P : ModelLaw d ε) (est : Estimator n d) (N : Ω → ℕ) (hN : Measurable N)
(X : Ω → (Fin n → Obs d)) (hX : HasLaw X (productLaw P.1 n) μ)
(lam : ℝ≥0) (hlaw : HasLaw N (poissonMeasure lam) μ)
(hn : 1 ≤ n) (hB : 2 < B) (hmean : B * (n : ℝ) / 2 ≤ lam) :
sqRisk μ
(fun ω => if N ω < n then 0 else projectUnit (est.1 (X ω)))
(observedValue P.1) ≤ observedRisk n est P +
exp (-(n : ℝ) * (B - 2) ^ 2 / (4 * B)) := by
exact (firstLabels_shortage_sqRisk_le μ P est N hN X hX).trans
(add_le_add_right (poissonCount_shortage_uniform_le μ N lam hlaw hn hB hmean) _)
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.PacketCompletion 4 declarations Moment annihilation and weighted normalization give the exact identity t + u = 1/2.
Common-atom completion of the explicit Jackson packet
Moment annihilation and weighted normalization give the exact identity
t + u = 1/2. Consequently the packet parts admit the common-atom
completion from roadmap (6)--(7), with no loss in their kink response.
Strict positivity of the weight is proved for the paper's range, so the
explicit prior witness has no normalization premise. The uniform kink-response
bound remains open.
theorem packetParts_mass_match
Degree-zero annihilation equates the actual masses of the two packet parts. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
A₀ K :
ℕ
hA :
1 ≤ A₀
hK :
2 ≤ K
M :
ℝ
conclusion
Proof (Lean source)
lemma packetParts_mass_match {A₀ K : ℕ} (hA : 1 ≤ A₀) (hK : 2 ≤ K)
(M : ℝ) :
packetPositive A₀ K M univ = packetNegative A₀ K M univ := by
let := packetPositive_finite A₀ K M
let := packetNegative_finite A₀ K M
have hm := packetParts_moments_match hA hK M (j := 0) (by omega)
simp only [pow_zero, integral_const, smul_eq_mul, mul_one, Measure.real] at hm
exact (ENNReal.toReal_eq_toReal_iff' (measure_ne_top _ _) (measure_ne_top _ _)).mp hm
theorem packetParts_half_normalization
Equal degree-zero and degree-one moments split the combined weighted normalization equally between the two Jordan parts, as in roadmap (6). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma packetParts_half_normalization {A₀ K : ℕ} (hA : 1 ≤ A₀) (hK : 2 ≤ K)
{M : ℝ} (hM : 0 ≤ M) (hW : 0 < packetWeight A₀ K M) :
(packetPositive A₀ K M univ).toReal +
(∫ z, z ∂packetPositive A₀ K M) = 1 / 2 := by
have hn := packetParts_mass_add_moment A₀ K hM hW
have hm := packetParts_mass_match hA hK M
have hu := packetParts_moments_match hA hK M (j := 1) (by omega)
simp only [pow_one] at hu
rw [← hm, ← hu] at hn
linarith
theorem packetParts_commonAtomDomain
Positive weight makes the concrete packet admissible for the common-atom map: its positive mass is at most one half, hence strictly less than one. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma packetParts_commonAtomDomain {A₀ K : ℕ} (hA : 1 ≤ A₀) (hK : 2 ≤ K)
{M : ℝ} (hM : 0 ≤ M) (hW : 0 < packetWeight A₀ K M) :
CommonAtomDomain (packetPositive A₀ K M) (packetNegative A₀ K M) := by
let := packetPositive_finite A₀ K M
have hn := packetParts_half_normalization hA hK hM hW
have hu : 0 ≤ ∫ z, z ∂packetPositive A₀ K M := by
apply integral_nonneg_of_ae
filter_upwards [show ∀ᵐ z ∂packetPositive A₀ K M, z ∈ Icc 0 M from
mem_ae_iff.mpr (packetPositive_support A₀ K hM)] with z hz
exact hz.1
apply (packetParts_commonAtomDomain_iff A₀ K hM).mpr
apply (ENNReal.toReal_lt_toReal (measure_ne_top _ _) ENNReal.one_ne_top).mp
simp only [ENNReal.toReal_one]
linarith
The explicit packet completes to probability priors supported on [0,M], with mean one and matching moments through K. The common atom lies in [1/2,2], and the objective gap is exactly the normalized circle response. This assembles the construction portion of roadmap (30), using proved positivity of the normalization weight. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
A₀ K :
ℕ
hA :
1 ≤ A₀
hK :
2 ≤ K
M :
ℝ
hM :
2 ≤ M
ε :
ℝ
conclusion
∃ (p : ConstrainedPriorPair K M),
conclusion 1
conclusion 2
(1 / 2 : ℝ)
conclusion 3
conclusion 4
conclusion 5
(∫ z, phiEpsFormula ε z ∂p.ν₁) - (∫ z, phiEpsFormula ε z ∂p.ν₀)
= ∫ t in Icc (-pi) pi, (packetWave A₀ K M t / packetWeight A₀ K M / (2 * pi)) * phiEpsFormula ε (packetIntensity M t)
Proof (Lean source)
lemma packetCompletion_witness {A₀ K : ℕ} (hA : 1 ≤ A₀) (hK : 2 ≤ K)
{M : ℝ} (hM : 2 ≤ M) (ε : ℝ) :
∃ (p : ConstrainedPriorPair K M),
∃ hdom : CommonAtomDomain (packetPositive A₀ K M) (packetNegative A₀ K M),
(packetPositive A₀ K M univ).toReal ≤ 1 / 2 ∧
(1 / 2 : ℝ) ≤ (1 - ∫ z, z ∂packetPositive A₀ K M) /
(1 - (packetPositive A₀ K M univ).toReal) ∧
(1 - ∫ z, z ∂packetPositive A₀ K M) /
(1 - (packetPositive A₀ K M univ).toReal) ≤ 2 ∧
commonAtomCompletion (packetPositive A₀ K M) (packetNegative A₀ K M) hdom =
(p.ν₀, p.ν₁) ∧
(∫ z, phiEpsFormula ε z ∂p.ν₁) - (∫ z, phiEpsFormula ε z ∂p.ν₀) =
∫ t in Icc (-pi) pi,
(packetWave A₀ K M t / packetWeight A₀ K M / (2 * pi)) *
phiEpsFormula ε (packetIntensity M t) := by
have hW := packetWeight_pos hA hK hM
have hM0 : 0 ≤ M := by linarith
have hdom := packetParts_commonAtomDomain hA hK hM0 hW
have hn := packetParts_half_normalization hA hK hM0 hW
have hu : 0 ≤ ∫ z, z ∂packetPositive A₀ K M := by
apply integral_nonneg_of_ae
filter_upwards [show ∀ᵐ z ∂packetPositive A₀ K M, z ∈ Icc 0 M from
mem_ae_iff.mpr (packetPositive_support A₀ K hM0)] with z hz
exact hz.1
have hloc := commonAtom_location _ _ toReal_nonneg hu hn
obtain ⟨p, hp⟩ := commonAtomCompletion_constrained K M (by omega) hM
_ _ hdom (packetPositive_support A₀ K hM0) (packetNegative_support A₀ K hM0)
(packetParts_mass_match hA hK M) (fun j hj => packetParts_moments_match hA hK M hj) hn
refine ⟨p, hdom, hloc.1, hloc.2.1, hloc.2.2, hp, ?_⟩
have hg := commonAtomCompletion_phiEps_gap _ _ hdom M ε
(packetPositive_support A₀ K hM0) (packetNegative_support A₀ K hM0)
(by linarith [hloc.1])
rw [hp] at hg
exact hg.trans (packetParts_phiEps_sub A₀ K hM0 ε)
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.PacketIntegrationByParts 8 declarations The two integrations retain the endpoint slopes on each smooth arc.
Integration by parts for the localized packet
The two integrations retain the endpoint slopes on each smooth arc. These boundary terms supply the kink atoms in roadmap (25).
Finite Fourier construction supplies all differentiability needed for integration by parts, without extra packet regularity hypotheses. The stated conclusion holds.
Formal statement
Proof (Lean source)
-- @node: contDiff_packetAntideriv
@[fun_prop] lemma contDiff_packetAntideriv (N : ℕ) :
ContDiff ℝ 2 (packetAntideriv N) := by
unfold packetAntideriv
fun_prop
Twice integration by parts on one smooth arc, with the actual translated Jackson packet. The endpoint slopes are retained rather than discarded. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
N :
ℕ
hN :
2 ≤ N
θ a b :
ℝ
f f' f'' :
ℝ → ℝ
hf :
∀ t, HasDerivAt f (f' t) t
hf' :
∀ t, HasDerivAt f' (f'' t) t
hc :
Continuous f''
Proof (Lean source)
lemma packet_arc_integration_by_parts {N : ℕ} (hN : 2 ≤ N)
(θ a b : ℝ) (f f' f'' : ℝ → ℝ)
(hf : ∀ t, HasDerivAt f (f' t) t)
(hf' : ∀ t, HasDerivAt f' (f'' t) t)
(hc : Continuous f'') :
(∫ t in a..b, f t * packetOscillation N (t - θ)) =
f b * deriv (packetAntideriv N) (b - θ) -
f a * deriv (packetAntideriv N) (a - θ) -
f' b * packetAntideriv N (b - θ) +
f' a * packetAntideriv N (a - θ) +
∫ t in a..b, f'' t * packetAntideriv N (t - θ) := by
have hG := contDiff_packetAntideriv N
have hdG := hG.differentiable (by norm_num)
have hdG' := hG.differentiable_deriv_two
have hcG' := hG.continuous_deriv (by norm_num)
have hfirst (t : ℝ) : HasDerivAt (fun s => packetAntideriv N (s - θ))
(deriv (packetAntideriv N) (t - θ)) t := by
simpa only [Function.comp_def, mul_one, id_eq] using
(hdG (t - θ)).hasDerivAt.comp t ((hasDerivAt_id t).sub_const θ)
have hsecond (t : ℝ) : HasDerivAt (fun s => deriv (packetAntideriv N) (s - θ))
(packetOscillation N (t - θ)) t := by
obtain ⟨c, C, hc, hcC, hp⟩ := jackson_packet_localization
have he := (hp N hN).2.2.2.2.2.2.1 (t - θ)
simpa only [he, mul_one, Function.comp_def, id_eq] using
(hdG' (t - θ)).hasDerivAt.comp t ((hasDerivAt_id t).sub_const θ)
have hcF' : Continuous f' := continuous_iff_continuousAt.mpr
(fun t => (hf' t).continuousAt)
have hi1 := intervalIntegral.integral_mul_deriv_eq_deriv_mul
(fun t ht => hf t) (fun t ht => hsecond t)
(hcF'.intervalIntegrable a b)
((show Continuous (fun t => packetOscillation N (t - θ)) from
by unfold packetOscillation packetKernel; fun_prop (disch := exact
continuous_jackson _)
).intervalIntegrable a b)
have hi2 := intervalIntegral.integral_mul_deriv_eq_deriv_mul
(fun t ht => hf' t) (fun t ht => hfirst t)
(hc.intervalIntegrable a b)
((hcG'.comp (by fun_prop)).intervalIntegrable a b)
rw [hi1, hi2]
ring
The exact intensity and speed at the two kinks. The speed determines the atomic mass, uniformly in the overlap parameter. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
M :
ℝ
hM :
2 ≤ M
Proof (Lean source)
lemma packetKink_intensity_speed {M : ℝ} (hM : 2 ≤ M) :
let θ := arccos (2 / M - 1)
packetIntensity M θ = 1 ∧
(M / 2) * sin θ = sqrt (M - 1) := by
dsimp only
let θ := arccos (2 / M - 1)
have hm : 0 < M := by linarith
have hr : 2 / M ≤ 1 := (div_le_one hm).2 hM
have hc : cos θ = 2 / M - 1 :=
Real.cos_arccos (by linarith [div_pos (by norm_num : (0 : ℝ) < 2) hm])
(by linarith)
have hz : packetIntensity M θ = 1 := by
unfold packetIntensity
rw [hc]
field_simp
ring
refine ⟨hz, ?_⟩
have hv := packetIntensity_speed_sq M θ
rw [hz] at hv
have hs := Real.sin_nonneg_of_nonneg_of_le_pi
(Real.arccos_nonneg (2 / M - 1)) (Real.arccos_le_pi _)
have hp : 0 ≤ (M / 2) * sin θ := mul_nonneg (by positivity) hs
have hroot := Real.sq_sqrt (show 0 ≤ M - 1 by linarith)
have hrootpos := Real.sqrt_nonneg (M - 1)
change (M / 2) * sin θ = sqrt (M - 1)
nlinarith
On the central arc the intensity is strictly above one; on the outer arcs it is strictly below one. Endpoints are excluded to retain kink atoms. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
M :
ℝ
hM :
2 ≤ M
conclusion
θ :
arccos (2 / M - 1)
t ∈ Set.Ioo (-θ) θ :
1 < packetIntensity M t
t ∈ Set.Ioo (-Real.pi) (-θ) :
packetIntensity M t < 1
t ∈ Set.Ioo θ Real.pi :
packetIntensity M t < 1
Proof (Lean source)
lemma packetKink_smooth_arcs {M : ℝ} (hM : 2 ≤ M) :
let θ := arccos (2 / M - 1)
(∀ t ∈ Ioo (-θ) θ, 1 < packetIntensity M t) ∧
(∀ t ∈ Ioo (-pi) (-θ), packetIntensity M t < 1) ∧
(∀ t ∈ Ioo θ pi, packetIntensity M t < 1) := by
dsimp only
let θ := arccos (2 / M - 1)
have hθ0 : 0 ≤ θ := Real.arccos_nonneg _
have hθpi : θ ≤ pi := Real.arccos_le_pi _
have hz := (packetKink_intensity_speed hM).1
change packetIntensity M θ = 1 at hz
have hpos : 0 < M / 2 := by linarith
have heven (t : ℝ) : packetIntensity M (-t) = packetIntensity M t := by
simp only [packetIntensity, Real.cos_neg]
have hright (t : ℝ) (ht : t ∈ Ioo θ pi) : packetIntensity M t < 1 := by
have hc := Real.cos_lt_cos_of_nonneg_of_le_pi hθ0 ht.2.le ht.1
rw [← hz]
exact mul_lt_mul_of_pos_left (by linarith) hpos
refine ⟨?_, ?_, fun t ht => hright t ht⟩
· intro t ht
have habs : |t| < θ := abs_lt.mpr ⟨ht.1, ht.2⟩
have hc := Real.cos_lt_cos_of_nonneg_of_le_pi (abs_nonneg t) hθpi habs
rw [Real.cos_abs] at hc
rw [← hz]
exact mul_lt_mul_of_pos_left (by linarith) hpos
· intro t ht
rw [← heven t]
exact hright (-t) ⟨by linarith [ht.2], by linarith [ht.1]⟩
Differentiating the periodic Fourier series retains the circle period, so the outer boundary terms cancel in integration by parts. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Proof (Lean source)
lemma packetAntideriv_deriv_periodic (N : ℕ) :
Periodic (deriv (packetAntideriv N)) (2 * pi) := by
intro t
have he : (fun s => packetAntideriv N (s + 2 * pi)) = packetAntideriv N :=
funext (packetAntideriv_periodic N)
have hd := congrArg (fun f : ℝ → ℝ => deriv f t) he
rw [deriv_comp_add_const] at hd
exact hd
theorem packetRemainder_integration_by_parts
Twice integration by parts on the three smooth arcs, with both derivative jumps retained. This is roadmap (25), for any translated packet center. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
N :
ℕ
hN :
2 ≤ N
M ε :
ℝ
hM :
2 ≤ M
hε :
0 ≤ ε
hεhi :
ε ≤ 1 / 2
center :
ℝ
conclusion
θ :
arccos (2 / M - 1)
(∫ t in Icc (-pi) pi, packetRemainder ε (packetIntensity M t) * packetOscillation N (t - center) / (2 * pi))
Proof (Lean source)
lemma packetRemainder_integration_by_parts {N : ℕ} (hN : 2 ≤ N)
{M ε : ℝ} (hM : 2 ≤ M) (hε : 0 ≤ ε) (hεhi : ε ≤ 1 / 2) (center : ℝ) :
let θ := arccos (2 / M - 1)
(∫ t in Icc (-pi) pi,
packetRemainder ε (packetIntensity M t) * packetOscillation N (t - center) /
(2 * pi)) =
(∫ t in Icc (-pi) pi,
packetSmoothCurvature M ε t * packetAntideriv N (t - center) /
(2 * pi)) +
(sqrt (M - 1) / 2) / (2 * pi) *
(packetAntideriv N (θ - center) + packetAntideriv N (-θ - center)) := by
dsimp only
let θ := arccos (2 / M - 1)
let f₀ := fun t => ε * (1 - packetIntensity M t)
let v₀ := fun t => ε * (M / 2) * sin t
let a₀ := fun t => ε * (M / 2) * cos t
let f₁ := fun t => (1 - 2 * ε) * (packetIntensity M t - 1) /
(1 + packetIntensity M t)
let v₁ := fun t => 2 * (1 - 2 * ε) * (-(M / 2) * sin t) /
(1 + packetIntensity M t) ^ 2
let F := fun t => packetRemainder ε (packetIntensity M t) *
packetOscillation N (t - center)
let R := fun t => packetSmoothCurvature M ε t * packetAntideriv N (t - center)
have hm : 0 ≤ M := by linarith
have hd (t : ℝ) : 1 + packetIntensity M t ≠ 0 := by
have hz := (packetIntensity_mem_Icc hm t).1
linarith
have hf₀ (t : ℝ) : HasDerivAt f₀ (v₀ t) t := by
dsimp [f₀, v₀]
convert ((packetIntensity_hasDerivAt M t).const_sub 1).const_mul ε using 1 <;>
first | rfl | ring
have hv₀ (t : ℝ) : HasDerivAt v₀ (a₀ t) t :=
(Real.hasDerivAt_sin t).const_mul (ε * (M / 2))
have hf₁ (t : ℝ) : HasDerivAt f₁ (v₁ t) t := packetUpperBranch_hasDerivAt hm ε t
have hv₁ (t : ℝ) : HasDerivAt v₁ (packetUpperCurvature M ε t) t :=
packetUpperSlope_hasDerivAt hm ε t
have ha₁ : Continuous (packetUpperCurvature M ε) := by
unfold packetUpperCurvature
fun_prop (disch := intro t; exact pow_ne_zero _ (hd t))
have hleft := packet_arc_integration_by_parts hN center (-pi) (-θ)
f₀ v₀ a₀ hf₀ hv₀ (by fun_prop)
have hmiddle := packet_arc_integration_by_parts hN center (-θ) θ
f₁ v₁ (packetUpperCurvature M ε) hf₁ hv₁ ha₁
have hright := packet_arc_integration_by_parts hN center θ pi
f₀ v₀ a₀ hf₀ hv₀ (by fun_prop)
have hθ0 : 0 ≤ θ := Real.arccos_nonneg _
have hθpi : θ ≤ pi := Real.arccos_le_pi _
obtain ⟨hcentral, houterleft, houterright⟩ := packetKink_smooth_arcs hM
have hlo : -pi ≤ -θ := by linarith
have hmid : -θ ≤ θ := by linarith
have hhi : θ ≤ pi := hθpi
have hF : Continuous F := by
dsimp [F, packetRemainder]
unfold packetOscillation packetKernel
fun_prop (disch := first | exact hd _ | exact
continuous_jackson _)
have hRi : IntervalIntegrable R volume (-pi) pi := by
have hi : IntervalIntegrable (packetSmoothCurvature M ε) volume (-pi) pi :=
(show IntegrableOn (packetSmoothCurvature M ε) (uIcc (-pi) pi) volume
by
rw [Set.uIcc_of_le (by linarith [Real.pi_pos] : -pi ≤ pi)]
exact packetSmoothCurvature_integrable hm hε hεhi).intervalIntegrable
exact hi.mul_continuousOn
(show ContinuousOn (fun t => packetAntideriv N (t - center))
(uIcc (-pi) pi) by fun_prop)
have he₀ (t : ℝ) (ht : packetIntensity M t < 1) :
packetRemainder ε (packetIntensity M t) = f₀ t ∧
packetSmoothCurvature M ε t = a₀ t := by
simp [packetRemainder, packetSmoothCurvature, f₀, a₀, ht,
max_eq_left (by linarith : 0 ≤ 1 - packetIntensity M t),
max_eq_right (by linarith : packetIntensity M t - 1 ≤ 0)]
have he₁ (t : ℝ) (ht : 1 < packetIntensity M t) :
packetRemainder ε (packetIntensity M t) = f₁ t ∧
packetSmoothCurvature M ε t = packetUpperCurvature M ε t := by
simp [packetRemainder, packetSmoothCurvature, f₁, ht, not_lt.mpr ht.le,
max_eq_right (by linarith : 1 - packetIntensity M t ≤ 0),
max_eq_left (by linarith : 0 ≤ packetIntensity M t - 1)]
have hFl : (∫ t in -Real.pi..-θ, F t) =
∫ t in -Real.pi..-θ, f₀ t * packetOscillation N (t - center) :=
intervalIntegral.integral_congr_Ioo_of_le hlo
(fun t ht => congrArg (fun x => x * packetOscillation N (t - center))
(he₀ t (houterleft t ht)).1)
have hFm : (∫ t in -θ..θ, F t) =
∫ t in -θ..θ, f₁ t * packetOscillation N (t - center) :=
intervalIntegral.integral_congr_Ioo_of_le hmid
(fun t ht => congrArg (fun x => x * packetOscillation N (t - center))
(he₁ t (hcentral t ht)).1)
have hFr : (∫ t in θ..Real.pi, F t) =
∫ t in θ..Real.pi, f₀ t * packetOscillation N (t - center) :=
intervalIntegral.integral_congr_Ioo_of_le hhi
(fun t ht => congrArg (fun x => x * packetOscillation N (t - center))
(he₀ t (houterright t ht)).1)
have hRl : (∫ t in -Real.pi..-θ, R t) =
∫ t in -Real.pi..-θ, a₀ t * packetAntideriv N (t - center) :=
intervalIntegral.integral_congr_Ioo_of_le hlo
(fun t ht => congrArg (fun x => x * packetAntideriv N (t - center))
(he₀ t (houterleft t ht)).2)
have hRm : (∫ t in -θ..θ, R t) =
∫ t in -θ..θ, packetUpperCurvature M ε t * packetAntideriv N (t - center) :=
intervalIntegral.integral_congr_Ioo_of_le hmid
(fun t ht => congrArg (fun x => x * packetAntideriv N (t - center))
(he₁ t (hcentral t ht)).2)
have hRr : (∫ t in θ..Real.pi, R t) =
∫ t in θ..Real.pi, a₀ t * packetAntideriv N (t - center) :=
intervalIntegral.integral_congr_Ioo_of_le hhi
(fun t ht => congrArg (fun x => x * packetAntideriv N (t - center))
(he₀ t (houterright t ht)).2)
have hFsplit : (∫ t in -Real.pi..Real.pi, F t) =
(∫ t in -Real.pi..-θ, F t) + (∫ t in -θ..θ, F t) +
(∫ t in θ..Real.pi, F t) := by
rw [intervalIntegral.integral_add_adjacent_intervals
(hF.intervalIntegrable (-pi) (-θ)) (hF.intervalIntegrable (-θ) θ),
intervalIntegral.integral_add_adjacent_intervals
(hF.intervalIntegrable (-pi) θ) (hF.intervalIntegrable θ pi)]
have hRsplit : (∫ t in -Real.pi..Real.pi, R t) =
(∫ t in -Real.pi..-θ, R t) + (∫ t in -θ..θ, R t) +
(∫ t in θ..Real.pi, R t) := by
have hil := hRi.mono_set (show uIcc (-pi) (-θ) ⊆
uIcc (-pi) pi by
rw [Set.uIcc_of_le hlo, Set.uIcc_of_le (by linarith [Real.pi_pos])]
exact Set.Icc_subset_Icc le_rfl (by linarith))
have him := hRi.mono_set (show uIcc (-θ) θ ⊆ uIcc (-pi) pi by
rw [Set.uIcc_of_le hmid, Set.uIcc_of_le (by linarith [Real.pi_pos])]
exact Set.Icc_subset_Icc hlo hhi)
have hir := hRi.mono_set (show uIcc θ pi ⊆ uIcc (-pi) pi by
rw [Set.uIcc_of_le hhi, Set.uIcc_of_le (by linarith [Real.pi_pos])]
exact Set.Icc_subset_Icc (by linarith) le_rfl)
rw [intervalIntegral.integral_add_adjacent_intervals hil him,
intervalIntegral.integral_add_adjacent_intervals (hil.trans him) hir]
obtain ⟨hz, hs⟩ := packetKink_intensity_speed hM
change packetIntensity M θ = 1 at hz
change (M / 2) * sin θ = sqrt (M - 1) at hs
have hzneg : packetIntensity M (-θ) = 1 := by
simpa only [packetIntensity, Real.cos_neg] using hz
have hperiod := packetAntideriv_deriv_periodic N (-pi - center)
have harg : -pi - center + 2 * pi = pi - center := by ring
rw [harg] at hperiod
have hboundary : f₀ θ = 0 ∧ f₀ (-θ) = 0 ∧ f₁ θ = 0 ∧ f₁ (-θ) = 0 := by
simp [f₀, f₁, hz, hzneg]
have hvpos : v₀ θ - v₁ θ = sqrt (M - 1) / 2 := by
dsimp [v₀, v₁]
rw [hz]
norm_num
nlinarith only [hs]
have hvneg : v₁ (-θ) - v₀ (-θ) = sqrt (M - 1) / 2 := by
dsimp [v₀, v₁]
rw [hzneg, Real.sin_neg]
norm_num
nlinarith only [hs]
have hmain : (∫ t in -Real.pi..Real.pi, F t) =
(∫ t in -Real.pi..Real.pi, R t) +
(sqrt (M - 1) / 2) *
(packetAntideriv N (θ - center) + packetAntideriv N (-θ - center)) := by
rw [hFsplit, hFl, hFm, hFr, hleft, hmiddle, hright,
hRsplit, hRl, hRm, hRr]
rcases hboundary with ⟨hb₀, hb₀n, hb₁, hb₁n⟩
rw [hb₀, hb₀n, hb₁, hb₁n]
have hep : f₀ pi = ε := by simp [f₀, packetIntensity]
have hen : f₀ (-pi) = ε := by simp [f₀, packetIntensity]
have hsp : v₀ pi = 0 := by simp [v₀]
have hsn : v₀ (-pi) = 0 := by simp [v₀]
rw [hep, hen, hsp, hsn, hperiod]
linear_combination packetAntideriv N (θ - center) * hvpos +
packetAntideriv N (-θ - center) * hvneg
rw [integral_div, integral_div, integral_Icc_eq_integral_Ioc,
integral_Icc_eq_integral_Ioc,
← intervalIntegral.integral_of_le (by linarith [Real.pi_pos] : -pi ≤ pi),
← intervalIntegral.integral_of_le (by linarith [Real.pi_pos] : -pi ≤ pi)]
change (∫ t in -Real.pi..Real.pi, F t) / (2 * pi) =
(∫ t in -Real.pi..Real.pi, R t) / (2 * pi) + _
rw [hmain]
ring
Evenness of the finite cosine antiderivative identifies the opposite kink in the exact response formula. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
N :
ℕ
u :
ℝ
conclusion
packetAntideriv N (-u) = packetAntideriv N u
Proof (Lean source)
lemma packetAntideriv_even (N : ℕ) (u : ℝ) :
packetAntideriv N (-u) = packetAntideriv N u := by
unfold packetAntideriv
simp only [mul_neg, Real.cos_neg]
theorem packetRemainder_response_lower
The uniform lower response of one localized packet, roadmap (28). The universal scale suppresses both the opposite kink and the smooth term; no response bound or integration identity is assumed. In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ (c : ℝ) (A : ℕ),
0 < c
1 ≤ A
conclusion 1
K :
ℕ
M ε :
ℝ
2 ≤ K
2 ≤ M
M ≤ (K : ℝ) ^ 2
0 ≤ ε
ε ≤ 1 / 2
c * sqrt M / K
≤ |∫ t in Icc (-pi) pi, packetRemainder ε (packetIntensity M t) * packetOscillation (A * K) (t - arccos (2 / M - 1)) / (2 * pi)|
Proof (Lean source)
lemma packetRemainder_response_lower :
∃ (c : ℝ) (A : ℕ), 0 < c ∧ 1 ≤ A ∧
∀ (K : ℕ) (M ε : ℝ), 2 ≤ K → 2 ≤ M → M ≤ (K : ℝ) ^ 2 →
0 ≤ ε → ε ≤ 1 / 2 →
c * sqrt M / K ≤
|∫ t in Icc (-pi) pi,
packetRemainder ε (packetIntensity M t) *
packetOscillation (A * K) (t - arccos (2 / M - 1)) /
(2 * pi)| := by
obtain ⟨c, Ccenter, hc, _, hpacket⟩ := jackson_packet_localization
obtain ⟨Co, hCo, hopposite⟩ := packetKink_antideriv_bound
obtain ⟨Cs, hCs, hsmooth⟩ := packetSmoothRemainder_bound
obtain ⟨A, hA⟩ := exists_nat_gt
(max 1 (max (2 * Co / c) (32 * pi * Cs / c)))
have hA1R : (1 : ℝ) ≤ A := (le_max_left _ _).trans hA.le
have hA1 : 1 ≤ A := by exact_mod_cast hA1R
have hAR : (0 : ℝ) < A := by linarith only [hA1R]
have hAo : 2 * Co ≤ c * (A : ℝ) ^ 2 := by
have ha : 2 * Co / c ≤ A :=
(le_max_left _ _).trans ((le_max_right _ _).trans hA.le)
have hb := (div_le_iff₀ hc).mp ha
have hsq : (A : ℝ) ≤ (A : ℝ) ^ 2 := by nlinarith only [hA1R]
nlinarith only [hb, mul_le_mul_of_nonneg_left hsq hc.le]
have hAs : 32 * pi * Cs ≤ c * (A : ℝ) := by
have ha : 32 * pi * Cs / c ≤ A :=
(le_max_right _ _).trans ((le_max_right _ _).trans hA.le)
simpa only [mul_comm] using (div_le_iff₀ hc).mp ha
refine ⟨c / (32 * pi * A), A, by positivity, hA1, ?_⟩
intro K M ε hK hM hMhi hε hεhi
let N := A * K
let θ := arccos (2 / M - 1)
have hN : 2 ≤ N := by dsimp [N]; nlinarith only [hA1, hK]
have hn : (0 : ℝ) < N := by exact_mod_cast (show 0 < N by omega)
have hk : (0 : ℝ) < K := by exact_mod_cast (show 0 < K by omega)
have hNcast : (N : ℝ) = (A : ℝ) * K := by simp only [N, Nat.cast_mul]
have hm : 0 ≤ M := by linarith only [hM]
have hroot : 0 < sqrt M := Real.sqrt_pos.2 (by linarith only [hM])
have hrootK : sqrt M ≤ K := by
nlinarith only [Real.sq_sqrt hm, hMhi, Real.sqrt_nonneg M, hk]
have hcenter : c / (N : ℝ) ≤ |packetAntideriv N 0| :=
(hpacket N hN).2.2.2.2.2.2.2.2.1
have hsmall : |packetAntideriv N (2 * θ)| ≤ c / (2 * N) := by
apply (hopposite N M hN hM).trans
apply (div_le_div_iff₀ (by positivity : (0 : ℝ) < (N : ℝ) ^ 3)
(by positivity : (0 : ℝ) < 2 * N)).mpr
have hh : 2 * Co * M ≤ c * (N : ℝ) ^ 2 := by
calc
_ ≤ 2 * Co * (K : ℝ) ^ 2 :=
mul_le_mul_of_nonneg_left hMhi (by positivity)
_ ≤ (c * (A : ℝ) ^ 2) * (K : ℝ) ^ 2 :=
mul_le_mul_of_nonneg_right hAo (sq_nonneg _)
_ = _ := by rw [hNcast]; ring
nlinarith only [mul_le_mul_of_nonneg_right hh hn.le]
have hsum : c / (2 * N) ≤ |packetAntideriv N 0 + packetAntideriv N (2 * θ)| := by
have hh := abs_add_le (packetAntideriv N 0 + packetAntideriv N (2 * θ))
(-packetAntideriv N (2 * θ))
simp only [add_neg_cancel_right, abs_neg] at hh
have he : c / (N : ℝ) = 2 * (c / (2 * N)) := by ring
rw [he] at hcenter
linarith only [hh, hcenter, hsmall]
have hrootlo : sqrt M / 2 ≤ sqrt (M - 1) := by
nlinarith only [Real.sq_sqrt hm, Real.sq_sqrt (show 0 ≤ M - 1 by linarith),
Real.sqrt_nonneg M, Real.sqrt_nonneg (M - 1), hM]
have hatom : c * sqrt M / (16 * pi * N) ≤
|(sqrt (M - 1) / 2) / (2 * pi) *
(packetAntideriv N 0 + packetAntideriv N (2 * θ))| := by
rw [abs_mul, abs_of_nonneg (by positivity : 0 ≤ (sqrt (M - 1) / 2) /
(2 * pi))]
have hcoeff : sqrt M / (8 * pi) ≤
(sqrt (M - 1) / 2) / (2 * pi) := by
calc
_ = (sqrt M / 4) / (2 * pi) := by ring
_ ≤ _ := div_le_div_of_nonneg_right (by linarith only [hrootlo]) (by positivity)
calc
_ = (sqrt M / (8 * pi)) * (c / (2 * N)) := by ring
_ ≤ _ := mul_le_mul hcoeff hsum (by positivity) (by positivity)
have hrem := hsmooth N M ε θ hN hm hε hεhi
have hremSmall : Cs * M / (N : ℝ) ^ 2 ≤
c * sqrt M / (32 * pi * N) := by
apply (div_le_div_iff₀ (by positivity : (0 : ℝ) < (N : ℝ) ^ 2)
(by positivity : (0 : ℝ) < 32 * pi * N)).mpr
have hh : 32 * pi * Cs * sqrt M ≤ c * (N : ℝ) := by
calc
_ ≤ 32 * pi * Cs * (K : ℝ) :=
mul_le_mul_of_nonneg_left hrootK (by positivity)
_ ≤ (c * A) * (K : ℝ) := mul_le_mul_of_nonneg_right hAs hk.le
_ = _ := by rw [hNcast]; ring
have hh' := mul_le_mul_of_nonneg_right
(mul_le_mul_of_nonneg_right hh (Real.sqrt_nonneg M)) hn.le
rw [mul_assoc (32 * pi * Cs), Real.mul_self_sqrt hm] at hh'
nlinarith only [hh']
have he := packetRemainder_integration_by_parts hN hM hε hεhi θ
change (∫ t in Icc (-pi) pi,
packetRemainder ε (packetIntensity M t) * packetOscillation N (t - θ) /
(2 * pi)) =
(∫ t in Icc (-pi) pi,
packetSmoothCurvature M ε t * packetAntideriv N (t - θ) / (2 * pi)) +
(sqrt (M - 1) / 2) / (2 * pi) *
(packetAntideriv N (θ - θ) + packetAntideriv N (-θ - θ)) at he
have harg : -θ - θ = -(2 * θ) := by ring
simp only [sub_self, harg, packetAntideriv_even] at he
rw [he]
have hh := abs_add_le
((∫ t in Icc (-pi) pi,
packetSmoothCurvature M ε t * packetAntideriv N (t - θ) / (2 * pi)) +
(sqrt (M - 1) / 2) / (2 * pi) *
(packetAntideriv N 0 + packetAntideriv N (2 * θ)))
(-(∫ t in Icc (-pi) pi,
packetSmoothCurvature M ε t * packetAntideriv N (t - θ) / (2 * pi)))
simp only [add_neg_cancel_comm, abs_neg] at hh
have hscale : (c / (32 * pi * A)) * sqrt M / K =
c * sqrt M / (32 * pi * N) := by rw [hNcast]; ring
have htwice : c * sqrt M / (16 * pi * N) =
2 * (c * sqrt M / (32 * pi * N)) := by ring
rw [hscale]
rw [htwice] at hatom
linarith only [hh, hatom, hrem, hremSmall]
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.PacketKinkSeparation 5 declarations Roadmap (26)--(27): the circular distance between the kinks stays at least four times the inverse square root of M.
Separation of the two packet kinks
Roadmap (26)--(27): the circular distance between the kinks stays at least four times the inverse square root of M. Jackson antiderivative localization therefore suppresses the opposite atomic contribution even when M = K².
The shortest arc between the two kinks passes through the endpoint π. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma packetKink_distance_eq {M : ℝ} (hM : 2 ≤ M) :
torusDistance (2 * arccos (2 / M - 1)) =
2 * (pi - arccos (2 / M - 1)) := by
let θ := arccos (2 / M - 1)
have hM0 : 0 < M := by linarith
have hr : 2 / M ≤ 1 := (div_le_one hM0).2 hM
have hlo : pi / 2 ≤ θ := by
have h := Real.arccos_le_arccos (show 2 / M - 1 ≤ 0 by linarith)
simpa only [Real.arccos_zero] using h
have hhi : θ ≤ pi := Real.arccos_le_pi _
have hθ0 : 0 ≤ θ := Real.arccos_nonneg _
change sInf {r : ℝ | ∃ j : ℤ, r = |2 * θ - 2 * pi * j|} = _
apply le_antisymm
· apply csInf_le
· exact ⟨0, by rintro r ⟨j, rfl⟩; exact abs_nonneg _⟩
· refine ⟨1, ?_⟩
simp only [Int.cast_one, mul_one, abs_of_nonpos (show 2 * θ - 2 * pi ≤ 0 by linarith)]
dsimp [θ]
ring
· have hne : Nonempty {r : ℝ | ∃ j : ℤ, r = |2 * θ - 2 * pi * j|} :=
⟨|2 * θ|, 0, by simp⟩
apply le_csInf hne
rintro r ⟨j, rfl⟩
by_cases hj : j ≤ 0
· have hjR : (j : ℝ) ≤ 0 := by exact_mod_cast hj
have hshift : 0 ≤ 2 * θ - 2 * pi * j := by
nlinarith [Real.pi_pos]
rw [abs_of_nonneg hshift]
change 2 * (pi - θ) ≤ _
nlinarith [Real.pi_pos]
· have hjR : (1 : ℝ) ≤ j := by exact_mod_cast (show 1 ≤ j by omega)
have hshift : 2 * θ - 2 * pi * j ≤ 0 := by
nlinarith [Real.pi_pos]
rw [abs_of_nonpos hshift]
change 2 * (pi - θ) ≤ _
nlinarith [Real.pi_pos]
The opposite kink remains at circular distance at least 4/√M, including at the dense endpoint. This is the lower bound in roadmap (26). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma packetKink_distance_lower {M : ℝ} (hM : 2 ≤ M) :
4 / sqrt M ≤ torusDistance (2 * arccos (2 / M - 1)) := by
rw [packetKink_distance_eq hM]
let α := pi - arccos (2 / M - 1)
have hM0 : 0 < M := by linarith
have hα : 0 ≤ α := sub_nonneg.mpr (Real.arccos_le_pi _)
have hr : 2 / M ≤ 1 := (div_le_one hM0).2 hM
have hc : cos α = 1 - 2 / M := by
dsimp [α]
rw [Real.cos_pi_sub, Real.cos_arccos
(by linarith [div_pos (by norm_num : (0 : ℝ) < 2) hM0]) (by linarith)]
ring
have ht := Real.one_sub_sq_div_two_le_cos (x := α)
rw [hc] at ht
have hs : 4 / M ≤ α ^ 2 := by
rw [show 4 / M = 2 * (2 / M) by ring]
linarith
have hroot : 0 < sqrt M := Real.sqrt_pos.2 hM0
have hsq : (2 / sqrt M) ^ 2 = 4 / M := by
rw [div_pow, Real.sq_sqrt hM0.le]
norm_num
have hq : 0 ≤ 2 / sqrt M := by positivity
have ha : 2 / sqrt M ≤ α := by
nlinarith [hsq]
change 4 / sqrt M ≤ 2 * α
convert mul_le_mul_of_nonneg_left ha (by norm_num : (0 : ℝ) ≤ 2) using 1 <;> first | rfl | ring
Localization and kink separation give the opposite-center response bound C M/N³. This is the antiderivative estimate used in roadmap (27). In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ C : ℝ,
0 < C
conclusion 1
N :
ℕ
M :
ℝ
2 ≤ N
2 ≤ M
|packetAntideriv N (2 * arccos (2 / M - 1))| ≤ C * M / (N : ℝ) ^ 3
Proof (Lean source)
lemma packetKink_antideriv_bound :
∃ C : ℝ, 0 < C ∧ ∀ (N : ℕ) (M : ℝ), 2 ≤ N → 2 ≤ M →
|packetAntideriv N (2 * arccos (2 / M - 1))| ≤
C * M / (N : ℝ) ^ 3 := by
obtain ⟨c, C, hc, hcC, hpacket⟩ := jackson_packet_localization
refine ⟨C / 16, by linarith, ?_⟩
intro N M hN hM
have hC : 0 < C := hc.trans_le hcC
have hn : (0 : ℝ) < N := by exact_mod_cast (show 0 < N by omega)
have hm : 0 < M := by linarith
have hs : 0 < sqrt M := Real.sqrt_pos.2 hm
have hd := packetKink_distance_lower hM
have hloc := (hpacket N hN).2.2.2.2.2.2.2.2.2.2.1
let d := torusDistance (2 * arccos (2 / M - 1))
have hd0 : 0 ≤ d := le_trans (by positivity) hd
have hbase : 4 * (N : ℝ) / sqrt M ≤ 1 + (N : ℝ) * d := by
have hh := mul_le_mul_of_nonneg_left hd hn.le
dsimp [d]
calc
_ = (N : ℝ) * (4 / sqrt M) := by ring
_ ≤ (N : ℝ) * torusDistance (2 * arccos (2 / M - 1)) := hh
_ ≤ _ := by linarith
calc
_ ≤ C / ((N : ℝ) * (1 + (N : ℝ) * d) ^ 2) := hloc _
_ ≤ C / ((N : ℝ) * (4 * (N : ℝ) / sqrt M) ^ 2) := by
apply div_le_div_of_nonneg_left hC.le (by positivity)
exact mul_le_mul_of_nonneg_left
(pow_le_pow_left₀ (by positivity) hbase 2) hn.le
_ = (C / 16) * M / (N : ℝ) ^ 3 := by
rw [div_pow, Real.sq_sqrt hm.le]
field_simp
ring
The opposite atomic contribution has order M^(3/2)/N³, uniformly in M. Its coefficient is the actual derivative jump √(M-1)/2 of roadmap (24). In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
Proof (Lean source)
lemma packetKink_opposite_atom_bound :
∃ C : ℝ, 0 < C ∧ ∀ (N : ℕ) (M : ℝ), 2 ≤ N → 2 ≤ M →
(sqrt (M - 1) / 2) *
|packetAntideriv N (2 * arccos (2 / M - 1))| ≤
C * M * sqrt M / (N : ℝ) ^ 3 := by
obtain ⟨C, hC, hbound⟩ := packetKink_antideriv_bound
refine ⟨C / 2, by positivity, ?_⟩
intro N M hN hM
calc
_ ≤ (sqrt M / 2) * (C * M / (N : ℝ) ^ 3) := by
exact mul_le_mul
(div_le_div_of_nonneg_right (Real.sqrt_le_sqrt (by linarith)) (by norm_num))
(hbound N M hN hM) (abs_nonneg _) (by positivity)
_ = _ := by ring
A universal packet scale prevents the opposite kink from canceling even half of the center response, uniformly over the full range M ≤ K². In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
Proof (Lean source)
lemma packetKink_atomic_response_lower :
∃ (c : ℝ) (A₀ : ℕ), 0 < c ∧ 1 ≤ A₀ ∧
∀ (K : ℕ) (M : ℝ), 2 ≤ K → 2 ≤ M → M ≤ (K : ℝ) ^ 2 →
c * sqrt M / K ≤
(sqrt (M - 1) / 2) *
|packetAntideriv (A₀ * K) 0 +
packetAntideriv (A₀ * K) (2 * arccos (2 / M - 1))| := by
obtain ⟨c, Ccenter, hc, _, hpacket⟩ := jackson_packet_localization
obtain ⟨C, hC, hopposite⟩ := packetKink_antideriv_bound
obtain ⟨A, hA⟩ := exists_nat_gt (max 1 (2 * C / c))
have hA1R : (1 : ℝ) ≤ A := (le_max_left _ _).trans hA.le
have hA1 : 1 ≤ A := by exact_mod_cast hA1R
have hAR : (0 : ℝ) < A := by linarith
have hdom : 2 * C ≤ c * (A : ℝ) ^ 2 := by
have hratio : 2 * C / c ≤ (A : ℝ) := (le_max_right _ _).trans hA.le
have hh := (div_le_iff₀ hc).mp hratio
have hsq : (A : ℝ) ≤ (A : ℝ) ^ 2 := by nlinarith
have hh' := mul_le_mul_of_nonneg_left hsq hc.le
nlinarith
refine ⟨c / (8 * A), A, by positivity, hA1, ?_⟩
intro K M hK hM hMhi
let N := A * K
have hN : 2 ≤ N := by dsimp [N]; nlinarith
have hn : (0 : ℝ) < N := by exact_mod_cast (show 0 < N by omega)
have hk : (0 : ℝ) < K := by exact_mod_cast (show 0 < K by omega)
have hNcast : (N : ℝ) = (A : ℝ) * K := by simp [N]
have hcenter : c / (N : ℝ) ≤ |packetAntideriv N 0| :=
(hpacket N hN).2.2.2.2.2.2.2.2.1
have hsmall : |packetAntideriv N (2 * arccos (2 / M - 1))| ≤
c / (2 * N) := by
apply (hopposite N M hN hM).trans
apply (div_le_div_iff₀ (by positivity : (0 : ℝ) < (N : ℝ) ^ 3)
(by positivity : (0 : ℝ) < 2 * N)).mpr
have hh : 2 * C * M ≤ c * (N : ℝ) ^ 2 := by
calc
_ ≤ 2 * C * (K : ℝ) ^ 2 :=
mul_le_mul_of_nonneg_left hMhi (by positivity)
_ ≤ (c * (A : ℝ) ^ 2) * (K : ℝ) ^ 2 :=
mul_le_mul_of_nonneg_right hdom (sq_nonneg _)
_ = _ := by rw [hNcast]; ring
have hh' := mul_le_mul_of_nonneg_right hh hn.le
nlinarith only [hh']
have hsum : c / (2 * N) ≤
|packetAntideriv N 0 + packetAntideriv N (2 * arccos (2 / M - 1))| := by
have hh := abs_add_le
(packetAntideriv N 0 + packetAntideriv N (2 * arccos (2 / M - 1)))
(-packetAntideriv N (2 * arccos (2 / M - 1)))
simp only [add_neg_cancel_right, abs_neg] at hh
have he : c / (N : ℝ) = 2 * (c / (2 * N)) := by ring
rw [he] at hcenter
linarith
have hroot : sqrt M / 2 ≤ sqrt (M - 1) := by
have hm : 0 ≤ M := by linarith
have hm1 : 0 ≤ M - 1 := by linarith
have hs := Real.sq_sqrt hm
have hs1 := Real.sq_sqrt hm1
have hp := Real.sqrt_nonneg M
have hp1 := Real.sqrt_nonneg (M - 1)
nlinarith
calc
(c / (8 * A)) * sqrt M / K =
(sqrt M / 4) * (c / (2 * N)) := by
rw [hNcast]
field_simp
ring
_ ≤ _ := mul_le_mul
(by linarith : sqrt M / 4 ≤ sqrt (M - 1) / 2)
hsum (by positivity) (by positivity)
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.PacketLowerOrder 29 declarations The positive and negative packet parts are finite measures supported on [0,M], with every polynomial moment integrable.
Concrete Jackson packet measures
The positive and negative packet parts are finite measures supported on [0,M],
with every polynomial moment integrable. These construction properties implement
roadmap (30). Exact pushforward, signed-pairing, and absolute-density formulas
recover the circle packet and its weighted normalization when the weight is
positive. The Fourier support argument proves matching of every moment through
degree K. Strict positivity and the kink-response lower bound remain open.
def packetIntensity
The cosine coordinate in the Jackson-packet lower construction.
Definition (Lean source)
def packetWave
The two translated, high-frequency Jackson waves centered at the kink.
Definition (Lean source)
parameters
A₀ K :
ℕ
M t :
ℝ
def
packetWave A₀ K M t :
ℝ
given by
N :
A₀ * K
θ :
arccos (2 / M - 1)
packetOscillation N (t - θ) + packetOscillation N (t + θ)
def packetWeight
The exact weighted variation used to normalize the signed packet.
Definition (Lean source)
parameters
A₀ K :
ℕ
M :
ℝ
def
packetWeight A₀ K M :
ℝ
given by
Positive Jordan part of the normalized Jackson packet, pushed to [0,M].
Definition (Lean source)
parameters
A₀ K :
ℕ
M :
ℝ
def
given by
Measure.map (packetIntensity M) ((volume.restrict (Icc (-pi) pi)).withDensity (fun t => ofReal (max (packetWave A₀ K M t / packetWeight A₀ K M / (2 * pi)) 0)))
Negative Jordan part of the same signed packet, pushed to [0,M].
Definition (Lean source)
parameters
A₀ K :
ℕ
M :
ℝ
def
given by
Measure.map (packetIntensity M) ((volume.restrict (Icc (-pi) pi)).withDensity (fun t => ofReal (max (-(packetWave A₀ K M t / packetWeight A₀ K M / (2 * pi))) 0)))
The cosine coordinate is continuous on the entire circle parameter line. The stated conclusion holds.
Formal statement
Proof (Lean source)
-- @node: continuous_packetIntensity
@[fun_prop] lemma continuous_packetIntensity (M : ℝ) :
Continuous (packetIntensity M) := by
unfold packetIntensity
fun_prop
Nonnegative endpoints put every cosine intensity in the desired support. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
-- @node: packetIntensity_mem_Icc
lemma packetIntensity_mem_Icc {M : ℝ} (hM : 0 ≤ M) (t : ℝ) :
packetIntensity M t ∈ Icc 0 M := by
have hlo := Real.neg_one_le_cos t
have hhi := Real.cos_le_one t
unfold packetIntensity
constructor
· exact mul_nonneg (by positivity) (by linarith)
· nlinarith
Pushing any measure through the cosine coordinate concentrates it on [0,M]; in particular no Jordan-part mass escapes the paper's interval. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
(Measure.map (packetIntensity M) μ) (Icc 0 M)ᶜ = 0
Proof (Lean source)
-- @node: packetIntensity_map_support
lemma packetIntensity_map_support {M : ℝ} (hM : 0 ≤ M) (μ : Measure ℝ) :
(Measure.map (packetIntensity M) μ) (Icc 0 M)ᶜ = 0 := by
rw [Measure.map_apply (by fun_prop) measurableSet_Icc.compl]
have he : packetIntensity M ⁻¹' (Icc 0 M)ᶜ = ∅ := by
ext t
simp only [Set.mem_preimage, Set.mem_compl_iff, Set.mem_empty_iff_false,
iff_false, not_not]
exact packetIntensity_mem_Icc hM t
rw [he, measure_empty]
theorem packetPositive_support
The concrete positive packet is supported on the constrained-prior interval. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
-- @node: packetPositive_support
lemma packetPositive_support (A₀ K : ℕ) {M : ℝ} (hM : 0 ≤ M) :
packetPositive A₀ K M (Icc 0 M)ᶜ = 0 := by
unfold packetPositive
exact packetIntensity_map_support hM _
theorem packetNegative_support
The concrete negative packet has the same constrained-prior support. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
-- @node: packetNegative_support
lemma packetNegative_support (A₀ K : ℕ) {M : ℝ} (hM : 0 ≤ M) :
packetNegative A₀ K M (Icc 0 M)ᶜ = 0 := by
unfold packetNegative
exact packetIntensity_map_support hM _
The translated packet wave is continuous, including zero degree. The stated conclusion holds.
Formal statement
Proof (Lean source)
-- @node: continuous_packetWave
@[fun_prop] lemma continuous_packetWave (A₀ K : ℕ) (M : ℝ) :
Continuous (packetWave A₀ K M) := by
unfold packetWave packetOscillation packetKernel
fun_prop (disch := exact
continuous_jackson _)
Either Jordan density is integrable on the compact circle interval, even before positivity of the normalization weight is proved. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
A₀ K :
ℕ
M sign :
ℝ
conclusion
Integrable (fun t => max (sign * (packetWave A₀ K M t / packetWeight A₀ K M / (2 * pi))) 0) (volume.restrict (Icc (-pi) pi))
Proof (Lean source)
-- @node: packetDensity_integrable
lemma packetDensity_integrable (A₀ K : ℕ) (M sign : ℝ) :
Integrable (fun t => max
(sign * (packetWave A₀ K M t / packetWeight A₀ K M / (2 * pi))) 0)
(volume.restrict (Icc (-pi) pi)) := by
apply ContinuousOn.integrableOn_compact (K := Icc (-pi) pi)
(μ := volume) isCompact_Icc
fun_prop
theorem packetPositive_finite
The compactly supported positive density defines a finite measure. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
Proof (Lean source)
-- @node: packetPositive_finite
lemma packetPositive_finite (A₀ K : ℕ) (M : ℝ) :
IsFiniteMeasure (packetPositive A₀ K M) := by
have hi := packetDensity_integrable A₀ K M 1
simp only [one_mul] at hi
let hm := isFiniteMeasure_withDensity
((lintegral_ofReal_ne_top_iff_integrable hi.aestronglyMeasurable
(Filter.Eventually.of_forall fun t => le_max_right _ _)).mpr hi)
unfold packetPositive
infer_instance
theorem packetNegative_finite
The compactly supported negative density also defines a finite measure. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
Proof (Lean source)
-- @node: packetNegative_finite
lemma packetNegative_finite (A₀ K : ℕ) (M : ℝ) :
IsFiniteMeasure (packetNegative A₀ K M) := by
have hi := packetDensity_integrable A₀ K M (-1)
simp only [neg_one_mul] at hi
let hm := isFiniteMeasure_withDensity
((lintegral_ofReal_ne_top_iff_integrable hi.aestronglyMeasurable
(Filter.Eventually.of_forall fun t => le_max_right _ _)).mpr hi)
unfold packetNegative
infer_instance
theorem packetParts_integrable_moments
Compact support makes every polynomial moment of both concrete packet parts integrable; no moment regularity premise is necessary. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
A₀ K :
ℕ
M :
ℝ
hM :
0 ≤ M
j :
ℕ
conclusion
conclusion 1
Integrable (fun z => z ^ j) (packetPositive A₀ K M)
conclusion 2
Integrable (fun z => z ^ j) (packetNegative A₀ K M)
Proof (Lean source)
-- @node: packetParts_integrable_moments
lemma packetParts_integrable_moments (A₀ K : ℕ) {M : ℝ} (hM : 0 ≤ M)
(j : ℕ) :
Integrable (fun z => z ^ j) (packetPositive A₀ K M) ∧
Integrable (fun z => z ^ j) (packetNegative A₀ K M) := by
let := packetPositive_finite A₀ K M
let := packetNegative_finite A₀ K M
exact ⟨integrable_of_supported_Icc _ M (packetPositive_support A₀ K hM)
_ (by fun_prop),
integrable_of_supported_Icc _ M (packetNegative_support A₀ K hM)
_ (by fun_prop)⟩
theorem packetParts_commonAtomDomain_iff
The only remaining common-atom domain obligation for these concrete packet parts is the positive part's strict mass bound. Finiteness and first-moment integrability follow from the construction. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
-- @node: packetParts_commonAtomDomain_iff
lemma packetParts_commonAtomDomain_iff (A₀ K : ℕ) {M : ℝ} (hM : 0 ≤ M) :
CommonAtomDomain (packetPositive A₀ K M) (packetNegative A₀ K M) ↔
packetPositive A₀ K M univ < 1 := by
constructor
· intro h
exact h.2.2.2.2
· intro hmass
have hi := packetParts_integrable_moments A₀ K hM 1
simp only [pow_one] at hi
exact ⟨packetPositive_finite A₀ K M, packetNegative_finite A₀ K M,
hi.1, hi.2, hmass⟩
Change of variables and the real density formula recover either packet part exactly, without a positivity assumption on the normalization weight. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
(∫ z, f z ∂Measure.map (packetIntensity M) ((volume.restrict (Icc (-pi) pi)).withDensity (fun t => ofReal (max (sign * (packetWave A₀ K M t / packetWeight A₀ K M / (2 * pi))) 0))))
= ∫ t in Icc (-pi) pi, max (sign * (packetWave A₀ K M t / packetWeight A₀ K M / (2 * pi))) 0 * f (packetIntensity M t)
Proof (Lean source)
-- @node: packetPart_integral
lemma packetPart_integral (A₀ K : ℕ) (M sign : ℝ) (f : ℝ → ℝ)
(hf : Measurable f) :
(∫ z, f z ∂Measure.map (packetIntensity M)
((volume.restrict (Icc (-pi) pi)).withDensity
(fun t => ofReal (max
(sign * (packetWave A₀ K M t / packetWeight A₀ K M / (2 * pi))) 0)))) =
∫ t in Icc (-pi) pi,
max (sign * (packetWave A₀ K M t / packetWeight A₀ K M / (2 * pi))) 0 *
f (packetIntensity M t) := by
rw [integral_map (by fun_prop) hf.aestronglyMeasurable]
rw [integral_withDensity_eq_integral_toReal_smul (by fun_prop)
(Filter.Eventually.of_forall fun t => ENNReal.ofReal_lt_top)]
simp only [ENNReal.toReal_ofReal (le_max_right _ _), smul_eq_mul]
theorem packetParts_integral_sub
The signed difference of the pushed-forward packet parts is exactly the normalized circle pairing, as used in roadmap (30). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
A₀ K :
ℕ
M :
ℝ
f :
ℝ → ℝ
hf :
hfc :
conclusion
(∫ z, f z ∂packetPositive A₀ K M) - (∫ z, f z ∂packetNegative A₀ K M)
= ∫ t in Icc (-pi) pi, (packetWave A₀ K M t / packetWeight A₀ K M / (2 * pi)) * f (packetIntensity M t)
Proof (Lean source)
-- @node: packetParts_integral_sub
lemma packetParts_integral_sub (A₀ K : ℕ) (M : ℝ) (f : ℝ → ℝ)
(hf : Measurable f)
(hfc : ContinuousOn (fun t => f (packetIntensity M t))
(Icc (-pi) pi)) :
(∫ z, f z ∂packetPositive A₀ K M) -
(∫ z, f z ∂packetNegative A₀ K M) =
∫ t in Icc (-pi) pi,
(packetWave A₀ K M t / packetWeight A₀ K M / (2 * pi)) *
f (packetIntensity M t) := by
have hpformula := packetPart_integral A₀ K M 1 f hf
have hmformula := packetPart_integral A₀ K M (-1) f hf
simp only [one_mul, neg_one_mul] at hpformula hmformula
rw [packetPositive, packetNegative, hpformula, hmformula]
have hp : Integrable (fun t => max
(packetWave A₀ K M t / packetWeight A₀ K M / (2 * pi)) 0 *
f (packetIntensity M t)) (volume.restrict (Icc (-pi) pi)) := by
apply ContinuousOn.integrableOn_compact isCompact_Icc
exact (by fun_prop : ContinuousOn _ (Icc (-pi) pi)).mul hfc
have hm : Integrable (fun t => max
(-(packetWave A₀ K M t / packetWeight A₀ K M / (2 * pi))) 0 *
f (packetIntensity M t)) (volume.restrict (Icc (-pi) pi)) := by
apply ContinuousOn.integrableOn_compact isCompact_Icc
exact (by fun_prop : ContinuousOn _ (Icc (-pi) pi)).mul hfc
rw [← integral_sub hp hm]
apply integral_congr_ae
filter_upwards with t
by_cases h : 0 ≤ packetWave A₀ K M t / packetWeight A₀ K M / (2 * pi)
· rw [max_eq_left h, max_eq_right (neg_nonpos.mpr h)]
ring
· rw [max_eq_right (le_of_not_ge h), max_eq_left (by linarith)]
ring
theorem packetParts_integral_add
The sum of the two packet-part integrals is the absolute-density pairing; there is no loss of variation in the explicit positive/negative construction. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
A₀ K :
ℕ
M :
ℝ
f :
ℝ → ℝ
hf :
hfc :
conclusion
(∫ z, f z ∂packetPositive A₀ K M) + (∫ z, f z ∂packetNegative A₀ K M)
= ∫ t in Icc (-pi) pi, |packetWave A₀ K M t / packetWeight A₀ K M / (2 * pi)| * f (packetIntensity M t)
Proof (Lean source)
-- @node: packetParts_integral_add
lemma packetParts_integral_add (A₀ K : ℕ) (M : ℝ) (f : ℝ → ℝ)
(hf : Measurable f)
(hfc : ContinuousOn (fun t => f (packetIntensity M t))
(Icc (-pi) pi)) :
(∫ z, f z ∂packetPositive A₀ K M) +
(∫ z, f z ∂packetNegative A₀ K M) =
∫ t in Icc (-pi) pi,
|packetWave A₀ K M t / packetWeight A₀ K M / (2 * pi)| *
f (packetIntensity M t) := by
have hpformula := packetPart_integral A₀ K M 1 f hf
have hmformula := packetPart_integral A₀ K M (-1) f hf
simp only [one_mul, neg_one_mul] at hpformula hmformula
rw [packetPositive, packetNegative, hpformula, hmformula]
have hp : Integrable (fun t => max
(packetWave A₀ K M t / packetWeight A₀ K M / (2 * pi)) 0 *
f (packetIntensity M t)) (volume.restrict (Icc (-pi) pi)) := by
apply ContinuousOn.integrableOn_compact isCompact_Icc
exact (by fun_prop : ContinuousOn _ (Icc (-pi) pi)).mul hfc
have hm : Integrable (fun t => max
(-(packetWave A₀ K M t / packetWeight A₀ K M / (2 * pi))) 0 *
f (packetIntensity M t)) (volume.restrict (Icc (-pi) pi)) := by
apply ContinuousOn.integrableOn_compact isCompact_Icc
exact (by fun_prop : ContinuousOn _ (Icc (-pi) pi)).mul hfc
rw [← integral_add hp hm]
apply integral_congr_ae
filter_upwards with t
by_cases h : 0 ≤ packetWave A₀ K M t / packetWeight A₀ K M / (2 * pi)
· rw [max_eq_left h, max_eq_right (neg_nonpos.mpr h), abs_of_nonneg h]
ring
· rw [max_eq_right (le_of_not_ge h), max_eq_left (by linarith),
abs_of_neg (lt_of_not_ge h)]
ring
The circle normalization weight is nonnegative for the paper's support range. Its strict positivity is a separate Fourier/kink-response obligation. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
-- @node: packetWeight_nonneg
lemma packetWeight_nonneg (A₀ K : ℕ) {M : ℝ} (hM : 0 ≤ M) :
0 ≤ packetWeight A₀ K M := by
apply integral_nonneg
intro t
have hz := (packetIntensity_mem_Icc hM t).1
exact div_nonneg (mul_nonneg (by linarith) (abs_nonneg _)) (by positivity)
theorem packetParts_weighted_normalization
With positive normalization weight, the two explicit packet parts have combined weighted mass exactly one, the normalization in roadmap (30). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
(∫ z, (1 + z) ∂packetPositive A₀ K M) + (∫ z, (1 + z) ∂packetNegative A₀ K M) = 1
Proof (Lean source)
-- @node: packetParts_weighted_normalization
lemma packetParts_weighted_normalization (A₀ K : ℕ) (M : ℝ)
(hW : 0 < packetWeight A₀ K M) :
(∫ z, (1 + z) ∂packetPositive A₀ K M) +
(∫ z, (1 + z) ∂packetNegative A₀ K M) = 1 := by
rw [packetParts_integral_add A₀ K M (fun z => 1 + z) (by fun_prop) (by fun_prop)]
have hpi : 0 < (2 * Real.pi : ℝ) := by positivity
calc
_ = ∫ t in Icc (-Real.pi) Real.pi,
((1 + packetIntensity M t) * |packetWave A₀ K M t| /
(2 * Real.pi)) / packetWeight A₀ K M := by
apply integral_congr_ae
filter_upwards with t
rw [abs_div, abs_div, abs_of_pos hW, abs_of_pos hpi]
ring
_ = packetWeight A₀ K M / packetWeight A₀ K M := by
rw [integral_div]
rfl
_ = 1 := div_self (ne_of_gt hW)
theorem packetParts_mass_add_moment
The normalization can be read directly as the sum of the two masses and first moments, ready for the degree-zero/one annihilation identities. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
+ (∫ z, z ∂packetPositive A₀ K M)
= 1
Proof (Lean source)
-- @node: packetParts_mass_add_moment
lemma packetParts_mass_add_moment (A₀ K : ℕ) {M : ℝ} (hM : 0 ≤ M)
(hW : 0 < packetWeight A₀ K M) :
((packetPositive A₀ K M) univ).toReal +
(∫ z, z ∂packetPositive A₀ K M) +
(((packetNegative A₀ K M) univ).toReal +
(∫ z, z ∂packetNegative A₀ K M)) = 1 := by
let := packetPositive_finite A₀ K M
let := packetNegative_finite A₀ K M
have hi := packetParts_integrable_moments A₀ K hM 1
simp only [pow_one] at hi
have hnorm := packetParts_weighted_normalization A₀ K M hW
rw [integral_add (integrable_const 1) hi.1,
integral_add (integrable_const 1) hi.2] at hnorm
simpa only [integral_const, smul_eq_mul, mul_one, Measure.real] using hnorm
The actual kink objective has the exact signed-packet response, despite its irrelevant singular encoding outside the nonnegative supporting interval. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
A₀ K :
ℕ
M :
ℝ
hM :
0 ≤ M
ε :
ℝ
conclusion
(∫ z, phiEpsFormula ε z ∂packetPositive A₀ K M)
- (∫ z, phiEpsFormula ε z ∂packetNegative A₀ K M)
= ∫ t in Icc (-pi) pi, (packetWave A₀ K M t / packetWeight A₀ K M / (2 * pi)) * phiEpsFormula ε (packetIntensity M t)
Proof (Lean source)
lemma packetParts_phiEps_sub (A₀ K : ℕ) {M : ℝ} (hM : 0 ≤ M) (ε : ℝ) :
(∫ z, phiEpsFormula ε z ∂packetPositive A₀ K M) -
(∫ z, phiEpsFormula ε z ∂packetNegative A₀ K M) =
∫ t in Icc (-pi) pi,
(packetWave A₀ K M t / packetWeight A₀ K M / (2 * pi)) *
phiEpsFormula ε (packetIntensity M t) := by
apply packetParts_integral_sub
· unfold phiEpsFormula
fun_prop
· unfold phiEpsFormula
apply ContinuousOn.div
· fun_prop
· fun_prop
· intro t ht
have hz := (packetIntensity_mem_Icc hM t).1
linarith
Integer cosine modes are unchanged after a full circle turn. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Proof (Lean source)
-- @node: packet_cos_periodic
lemma packet_cos_periodic (r : ℤ) :
Periodic (fun t : ℝ => cos ((r : ℝ) * t)) (2 * pi) := by
intro t
change cos ((r : ℝ) * (t + 2 * pi)) = cos ((r : ℝ) * t)
rw [mul_add]
exact Real.cos_add_int_mul_two_pi _ r
Fourier reconstruction makes the packet periodic before translation. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Proof (Lean source)
-- @node: packetOscillation_periodic
lemma packetOscillation_periodic {N : ℕ} (hN : 0 < N) :
Periodic (packetOscillation N) (2 * pi) := by
intro t
change kernel N
(t + 2 * pi) * cos (4 * (N : ℝ) * (t + 2 * pi)) =
kernel N t *
cos (4 * (N : ℝ) * t)
rw [kernel_fourier
N hN (t + 2 * pi),
kernel_fourier
N hN t]
have hsum : (∑ j ∈ Icc (-((2 * N - 2 : ℕ) : ℤ)) ((2 * N - 2 : ℕ) : ℤ),
normalizedCoeff N j *
cos ((j : ℝ) * (t + 2 * pi))) =
∑ j ∈ Icc (-((2 * N - 2 : ℕ) : ℤ)) ((2 * N - 2 : ℕ) : ℤ),
normalizedCoeff N j *
cos ((j : ℝ) * t) := by
apply Finset.sum_congr rfl
intro j hj
exact congrArg (fun x =>
normalizedCoeff N j * x)
(packet_cos_periodic j t)
rw [hsum]
congr 1
convert packet_cos_periodic (4 * (N : ℤ)) t using 1 <;> push_cast <;> rfl
Translation preserves the integral of a periodic function over one turn. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
-- @node: packet_periodic_integral_translate
lemma packet_periodic_integral_translate (f : ℝ → ℝ)
(hf : Periodic f (2 * pi)) (θ : ℝ) :
(∫ t in Icc (-pi) pi, f (t - θ)) =
∫ t in Icc (-pi) pi, f t := by
rw [integral_Icc_eq_integral_Ioc, integral_Icc_eq_integral_Ioc,
← intervalIntegral.integral_of_le (by linarith [Real.pi_pos] : -pi ≤ pi),
← intervalIntegral.integral_of_le (by linarith [Real.pi_pos] : -pi ≤ pi),
intervalIntegral.integral_comp_sub_right]
convert hf.intervalIntegral_add_eq (-pi - θ) (-pi) using 1 <;>
congr 1 <;> ring
theorem packetOscillation_trig_annihilation Lemma packetOscillation_trig_annihilation in the paper ↗
A translated high-frequency packet annihilates every low-frequency real trigonometric polynomial. This is the Fourier-support argument in (16). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Proof (Lean source)
-- @node: packetOscillation_trig_annihilation
lemma packetOscillation_trig_annihilation {N K : ℕ} (hN : 2 ≤ N)
(hKN : K < 2 * N + 2) (f : ℝ → ℝ)
(hf : IsTrigPolyLE K f)
(θ : ℝ) :
(∫ t in Icc (-pi) pi, packetOscillation N (t - θ) * f t) = 0 := by
obtain ⟨a, b, hab⟩ := hf
have hshift (t : ℝ) : f (t + θ) =
∑ r ∈ range (K + 1),
((a r * cos ((r : ℝ) * θ) + b r * sin ((r : ℝ) * θ)) *
cos ((r : ℝ) * t) +
(b r * cos ((r : ℝ) * θ) - a r * sin ((r : ℝ) * θ)) *
sin ((r : ℝ) * t)) := by
rw [hab]
apply Finset.sum_congr rfl
intro r hr
rw [mul_add, Real.cos_add, Real.sin_add]
ring
have hperiod : Periodic (fun t => packetOscillation N t * f (t + θ))
(2 * pi) := by
intro t
change packetOscillation N (t + 2 * pi) * f (t + 2 * pi + θ) =
packetOscillation N t * f (t + θ)
rw [packetOscillation_periodic (N := N) (by omega) t]
congr 1
rw [hab, hab]
apply Finset.sum_congr rfl
intro r hr
have hc := packet_cos_periodic (r : ℤ) (t + θ)
have hs : sin ((r : ℝ) * ((t + θ) + 2 * pi)) =
sin ((r : ℝ) * (t + θ)) := by
rw [mul_add]
exact Real.sin_add_int_mul_two_pi _ (r : ℤ)
convert congrArg₂ (fun x y => a r * x + b r * y) hc hs using 1 <;>
congr 2 <;> push_cast <;> ring
calc
_ = ∫ t in Icc (-pi) pi, packetOscillation N t * f (t + θ) := by
convert packet_periodic_integral_translate _ hperiod θ using 1
simp only [sub_add_cancel]
_ = 0 := by
simp_rw [hshift, Finset.mul_sum]
rw [integral_finsetSum _ (fun r hr =>
(show Continuous (fun t => packetOscillation N t *
((a r * cos ((r : ℝ) * θ) + b r * sin ((r : ℝ) * θ)) *
cos ((r : ℝ) * t) +
(b r * cos ((r : ℝ) * θ) - a r * sin ((r : ℝ) * θ)) *
sin ((r : ℝ) * t))) by
unfold packetOscillation packetKernel
fun_prop (disch := exact
continuous_jackson _)
).continuousOn.integrableOn_compact isCompact_Icc)]
apply Finset.sum_eq_zero
intro r hr
have hrK : r ≤ K := le_of_lt_succ (Finset.mem_range.mp hr)
have hz := packet_shifted_support
N hN (r : ℤ) (inl (by
rw [abs_of_nonneg (by positivity : (0 : ℤ) ≤ r)]
exact_mod_cast (lt_of_le_of_lt hrK hKN)))
change (∫ t in Icc (-pi) pi,
packetOscillation N t * cos ((r : ℝ) * t)) = 0 ∧
(∫ t in Icc (-pi) pi,
packetOscillation N t * sin ((r : ℝ) * t)) = 0 at hz
simp_rw [mul_add, mul_left_comm (packetOscillation N _) _]
rw [integral_add, integral_const_mul, integral_const_mul, hz.1, hz.2]
· ring
all_goals
apply ContinuousOn.integrableOn_compact isCompact_Icc
unfold packetOscillation packetKernel
fun_prop (disch := exact
continuous_jackson _)
Every degree-at-most-K polynomial in the intensity has zero circle pairing with the explicit two-center packet, exactly as in roadmap (16). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
= 0
Proof (Lean source)
-- @node: packetWave_polynomial_annihilation
lemma packetWave_polynomial_annihilation {A₀ K : ℕ} (hA : 1 ≤ A₀)
(hK : 2 ≤ K) (M : ℝ) (p : Polynomial ℝ) (hp : p.natDegree ≤ K) :
(∫ t in Icc (-pi) pi,
packetWave A₀ K M t * p.eval (packetIntensity M t)) = 0 := by
let q : Polynomial ℝ := p.comp (C (M / 2) * (1 + X))
have hlin : (C (M / 2) * (1 + X)).natDegree ≤ 1 := by
calc
_ ≤ (C (M / 2)).natDegree + (1 + X).natDegree :=
Polynomial.natDegree_mul_le
_ ≤ 1 := by
simp only [Polynomial.natDegree_C, zero_add]
exact (Polynomial.natDegree_add_le _ _).trans (by simp)
have hq : q.natDegree ≤ K := by
exact Polynomial.natDegree_comp_le.trans
((Nat.mul_le_mul_right _ hp).trans (by simpa using Nat.mul_le_mul_left K hlin))
have heval (t : ℝ) : q.eval (cos t) = p.eval (packetIntensity M t) := by
simp [q, Polynomial.eval_comp, packetIntensity]
have ht := (polynomial_cos_is_even_trigPoly
q hq).1
have hN : 2 ≤ A₀ * K := by nlinarith
have hKN : K < 2 * (A₀ * K) + 2 := by nlinarith
have hplus := packetOscillation_trig_annihilation hN hKN _ ht
(arccos (2 / M - 1))
have hminus := packetOscillation_trig_annihilation hN hKN _ ht
(-arccos (2 / M - 1))
simp_rw [heval] at hplus hminus
simp only [sub_neg_eq_add] at hminus
unfold packetWave
simp only [add_mul]
rw [integral_add, hplus, hminus, add_zero]
all_goals
apply ContinuousOn.integrableOn_compact isCompact_Icc
unfold packetOscillation packetKernel
fun_prop (disch := exact
continuous_jackson _)
theorem packetParts_moments_match
The actual pushed-forward positive and negative packets match every moment through degree K; no positivity or moment-matching premise is assumed. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
A₀ K :
ℕ
hA :
1 ≤ A₀
hK :
2 ≤ K
M :
ℝ
j :
ℕ
hj :
j ≤ K
conclusion
(∫ z, z ^ j ∂packetPositive A₀ K M) = ∫ z, z ^ j ∂packetNegative A₀ K M
Proof (Lean source)
-- @node: packetParts_moments_match
lemma packetParts_moments_match {A₀ K : ℕ} (hA : 1 ≤ A₀) (hK : 2 ≤ K)
(M : ℝ) {j : ℕ} (hj : j ≤ K) :
(∫ z, z ^ j ∂packetPositive A₀ K M) =
∫ z, z ^ j ∂packetNegative A₀ K M := by
apply sub_eq_zero.mp
rw [packetParts_integral_sub A₀ K M (fun z => z ^ j) (by fun_prop) (by fun_prop)]
have hz := packetWave_polynomial_annihilation hA hK M
(X ^ j) (by simpa using hj)
simp only [Polynomial.eval_pow, Polynomial.eval_X] at hz
calc
_ = (∫ t in Icc (-pi) pi,
packetWave A₀ K M t * packetIntensity M t ^ j) /
packetWeight A₀ K M / (2 * pi) := by
rw [← integral_div, ← integral_div]
apply integral_congr_ae
filter_upwards with t
ring
_ = 0 := by rw [hz]; simp
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.PacketPositivity 5 declarations Nonnegative Fourier coefficients, including a strictly positive coefficient at frequency one, make the Jackson kernel strictly maximal at its center.
Strict positivity of the explicit packet normalization
Nonnegative Fourier coefficients, including a strictly positive coefficient at frequency one, make the Jackson kernel strictly maximal at its center. The other translated wave therefore cannot cancel the centered wave at the kink. Continuity then gives strictly positive weighted variation.
Every normalized Jackson coefficient is nonnegative. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
Proof (Lean source)
lemma packetNormalizedCoeff_nonneg (N : ℕ) (j : ℤ) :
0 ≤ normalizedCoeff N j := by
unfold normalizedCoeff
apply div_nonneg <;>
unfold convolution <;>
apply Finset.sum_nonneg <;> intros <;>
unfold triangle <;>
positivity
At order at least two, the frequency-one coefficient is strictly positive. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma packetNormalizedCoeff_one_pos {N : ℕ} (hN : 2 ≤ N) :
0 < normalizedCoeff N 1 := by
open Causalean.Mathlib.Analysis.Approximation.Chebyshev.Jackson in
apply div_pos ?_ (convolution_zero_pos N (by omega))
unfold convolution
apply Finset.sum_pos'
· intro j hj
unfold triangle
positivity
· refine ⟨0, by simp, ?_⟩
simp only [triangle, Int.natAbs_zero, Nat.sub_zero, sub_zero, Int.natAbs_one]
have h0 : 0 < N := by omega
have h1 : 0 < N - 1 := by omega
exact mul_pos (by exact_mod_cast h0) (by exact_mod_cast h1)
A cosine strictly below one forces a strict drop from the kernel's center. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma packetKernel_lt_center {N : ℕ} (hN : 2 ≤ N) {u : ℝ}
(hu : cos u < 1) : packetKernel N u < packetKernel N 0 := by
open Causalean.Mathlib.Analysis.Approximation.Chebyshev.Jackson in
have hN0 : 0 < N := by omega
change kernel N u < kernel N 0
rw [kernel_fourier N hN0 u, kernel_fourier N hN0 0]
simp only [mul_zero, Real.cos_zero, mul_one]
apply Finset.sum_lt_sum
· intro j hj
exact mul_le_of_le_one_right (packetNormalizedCoeff_nonneg N j) (Real.cos_le_one _)
· refine ⟨1, ?_, ?_⟩
· simp only [Finset.mem_Icc]
constructor <;> omega
· simpa using mul_lt_of_lt_one_right (packetNormalizedCoeff_one_pos hN) hu
At the kink, the centered wave dominates the other translated wave. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
A₀ K :
ℕ
hA :
1 ≤ A₀
hK :
2 ≤ K
M :
ℝ
hM :
2 ≤ M
conclusion
0 < packetWave A₀ K M (arccos (2 / M - 1))
Proof (Lean source)
lemma packetWave_at_kink_pos {A₀ K : ℕ} (hA : 1 ≤ A₀) (hK : 2 ≤ K)
{M : ℝ} (hM : 2 ≤ M) :
0 < packetWave A₀ K M (arccos (2 / M - 1)) := by
let θ := arccos (2 / M - 1)
have hM0 : 0 < M := by linarith
have hratio : 0 < 2 / M := div_pos (by norm_num) hM0
have hratio_le : 2 / M ≤ 1 := (div_le_one hM0).2 hM
have hcos : cos θ = 2 / M - 1 :=
Real.cos_arccos (by linarith) (by linarith)
have hcos2 : cos (2 * θ) < 1 := by
rw [Real.cos_two_mul, hcos]
nlinarith
have hN : 2 ≤ A₀ * K := by nlinarith
have hdrop := packetKernel_lt_center hN hcos2
have hker : 0 ≤ packetKernel (A₀ * K) (2 * θ) := by
unfold packetKernel
exact mul_nonneg (by positivity)
(jackson_nonneg _ (by omega) _)
have hw : -packetKernel (A₀ * K) (2 * θ) ≤ packetOscillation (A₀ * K) (2 * θ) := by
unfold packetOscillation
simpa using mul_le_mul_of_nonneg_left
(Real.neg_one_le_cos (4 * (↑(A₀ * K) : ℝ) * (2 * θ))) hker
change 0 < packetOscillation (A₀ * K) (θ - θ) + packetOscillation (A₀ * K) (θ + θ)
rw [sub_self, show θ + θ = 2 * θ by ring]
simp only [packetOscillation, mul_zero, Real.cos_zero, mul_one] at *
linarith
The explicit packet weight is positive throughout the paper's support range. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma packetWeight_pos {A₀ K : ℕ} (hA : 1 ≤ A₀) (hK : 2 ≤ K)
{M : ℝ} (hM : 2 ≤ M) : 0 < packetWeight A₀ K M := by
let f : ℝ → ℝ := fun t =>
(1 + packetIntensity M t) * |packetWave A₀ K M t| / (2 * pi)
have hf : Continuous f := by
dsimp [f]
fun_prop
have hnonneg (t : ℝ) : 0 ≤ f t := by
have hz := (packetIntensity_mem_Icc (by linarith : 0 ≤ M) t).1
dsimp [f]
exact div_nonneg (mul_nonneg (by linarith) (abs_nonneg _)) (by positivity)
have hpos : 0 < f (arccos (2 / M - 1)) := by
have hz := (packetIntensity_mem_Icc (by linarith : 0 ≤ M)
(arccos (2 / M - 1))).1
have hw := packetWave_at_kink_pos hA hK hM
dsimp [f]
exact div_pos (mul_pos (by linarith) (abs_pos.mpr (ne_of_gt hw))) (by positivity)
have hi := intervalIntegral.integral_pos (by linarith [Real.pi_pos] : -pi < pi)
hf.continuousOn (fun t _ => hnonneg t)
⟨Real.arccos (2 / M - 1),
⟨by linarith [Real.arccos_nonneg (2 / M - 1), Real.pi_pos],
Real.arccos_le_pi _⟩, hpos⟩
rw [intervalIntegral.integral_of_le (by linarith [Real.pi_pos]),
← integral_Icc_eq_integral_Ioc] at hi
exact hi
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.PacketResponseLower 8 declarations Evenness adds the two localized responses in roadmap (29).
Separation attained by the completed Jackson packet
Evenness adds the two localized responses in roadmap (29). Fourier annihilation removes the affine term, and normalization followed by common-atom completion gives the explicit lower-prior witness in (30).
The modulated Jackson kernel is even, so reflection preserves its response. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
N :
ℕ
u :
ℝ
conclusion
packetOscillation N (-u) = packetOscillation N u
Proof (Lean source)
lemma packetOscillation_even (N : ℕ) (u : ℝ) :
packetOscillation N (-u) = packetOscillation N u := by
unfold packetOscillation packetKernel
rw [jackson_even N u]
simp only [mul_neg, Real.cos_neg]
theorem packetRemainder_reflected_response
The two kink centers give exactly equal responses, by reflection on the circle. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
N :
ℕ
M ε θ :
ℝ
conclusion
(∫ t in Icc (-pi) pi, packetRemainder ε (packetIntensity M t) * packetOscillation N (t + θ) / (2 * pi))
= ∫ t in Icc (-pi) pi, packetRemainder ε (packetIntensity M t) * packetOscillation N (t - θ) / (2 * pi)
Proof (Lean source)
lemma packetRemainder_reflected_response (N : ℕ) (M ε θ : ℝ) :
(∫ t in Icc (-pi) pi,
packetRemainder ε (packetIntensity M t) * packetOscillation N (t + θ) /
(2 * pi)) =
∫ t in Icc (-pi) pi,
packetRemainder ε (packetIntensity M t) * packetOscillation N (t - θ) /
(2 * pi) := by
let f := fun t => packetRemainder ε (packetIntensity M t) *
packetOscillation N (t - θ) / (2 * pi)
have he (t : ℝ) : f (-t) = packetRemainder ε (packetIntensity M t) *
packetOscillation N (t + θ) / (2 * pi) := by
dsimp [f]
have ht : -t - θ = -(t + θ) := by ring
rw [ht, packetOscillation_even]
simp only [packetIntensity, Real.cos_neg]
simp_rw [← he]
rw [integral_Icc_eq_integral_Ioc, integral_Icc_eq_integral_Ioc,
← intervalIntegral.integral_of_le (by linarith [Real.pi_pos] : -pi ≤ pi),
← intervalIntegral.integral_of_le (by linarith [Real.pi_pos] : -pi ≤ pi),
intervalIntegral.integral_comp_neg]
simp only [neg_neg]
rfl
theorem packetWave_remainder_response
The whole packet pairs with the nonlinear remainder as twice one response. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
A K :
ℕ
M :
ℝ
hM :
0 ≤ M
ε :
ℝ
conclusion
= 2 * ∫ t in Icc (-pi) pi, packetRemainder ε (packetIntensity M t) * packetOscillation (A * K) (t - arccos (2 / M - 1)) / (2 * pi)
Proof (Lean source)
lemma packetWave_remainder_response (A K : ℕ) {M : ℝ} (hM : 0 ≤ M) (ε : ℝ) :
(∫ t in Icc (-pi) pi,
packetWave A K M t * packetRemainder ε (packetIntensity M t) /
(2 * pi)) =
2 * ∫ t in Icc (-pi) pi,
packetRemainder ε (packetIntensity M t) *
packetOscillation (A * K) (t - arccos (2 / M - 1)) /
(2 * pi) := by
have hi (θ : ℝ) : IntegrableOn (fun t =>
packetRemainder ε (packetIntensity M t) * packetOscillation (A * K) (t - θ) /
(2 * pi)) (Icc (-pi) pi) := by
apply ContinuousOn.integrableOn_compact isCompact_Icc
unfold packetRemainder packetIntensity packetOscillation packetKernel
apply ContinuousOn.div
· apply ContinuousOn.mul
· apply ContinuousOn.add
· fun_prop
· apply ContinuousOn.div
· fun_prop
· fun_prop
· intro t ht
have := (packetIntensity_mem_Icc hM t).1
unfold packetIntensity at this
linarith
· fun_prop (disch := exact
continuous_jackson _)
· fun_prop
· intro t ht; exact ne_of_gt (by positivity)
have he (t : ℝ) : packetWave A K M t * packetRemainder ε (packetIntensity M t) /
(2 * pi) =
packetRemainder ε (packetIntensity M t) *
packetOscillation (A * K) (t - arccos (2 / M - 1)) / (2 * pi) +
packetRemainder ε (packetIntensity M t) *
packetOscillation (A * K) (t + arccos (2 / M - 1)) / (2 * pi) := by
unfold packetWave
ring
simp_rw [he]
rw [integral_add (hi _) (by simpa only [IntegrableOn, sub_neg_eq_add] using hi (-arccos (2 / M - 1))),
packetRemainder_reflected_response]
ring
Fourier annihilation removes the affine part of phi, retaining the exact sum of the two nonlinear kink responses in roadmap (29). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
A K :
ℕ
hA :
1 ≤ A
hK :
2 ≤ K
M :
ℝ
hM :
0 ≤ M
ε :
ℝ
conclusion
= 2 * ∫ t in Icc (-pi) pi, packetRemainder ε (packetIntensity M t) * packetOscillation (A * K) (t - arccos (2 / M - 1)) / (2 * pi)
Proof (Lean source)
lemma packetWave_phiEps_response {A K : ℕ} (hA : 1 ≤ A) (hK : 2 ≤ K)
{M : ℝ} (hM : 0 ≤ M) (ε : ℝ) :
(∫ t in Icc (-pi) pi,
packetWave A K M t * phiEpsFormula ε (packetIntensity M t) / (2 * pi)) =
2 * ∫ t in Icc (-pi) pi,
packetRemainder ε (packetIntensity M t) *
packetOscillation (A * K) (t - arccos (2 / M - 1)) /
(2 * pi) := by
let p : Polynomial ℝ := C ε * (X - 1)
have hp : p.natDegree ≤ K := by
calc
_ ≤ (C ε).natDegree + (X - 1 : Polynomial ℝ).natDegree :=
Polynomial.natDegree_mul_le
_ ≤ 1 := by
simp only [Polynomial.natDegree_C, zero_add]
exact (Polynomial.natDegree_sub_le _ _).trans (by simp)
_ ≤ K := by omega
have hz := packetWave_polynomial_annihilation hA hK M p hp
have hpe (z : ℝ) : p.eval z = ε * (z - 1) := by simp [p]
simp_rw [hpe] at hz
have hsplit (t : ℝ) :
packetWave A K M t * phiEpsFormula ε (packetIntensity M t) / (2 * pi) =
packetWave A K M t * (ε * (packetIntensity M t - 1)) / (2 * pi) +
packetWave A K M t * packetRemainder ε (packetIntensity M t) / (2 * pi) := by
rw [phiEps_eq_packetRemainder (packetIntensity_mem_Icc hM t).1]
ring
have hi1 : IntegrableOn (fun t => packetWave A K M t *
(ε * (packetIntensity M t - 1)) / (2 * pi)) (Icc (-pi) pi) := by
apply ContinuousOn.integrableOn_compact isCompact_Icc
unfold packetWave packetOscillation packetKernel
fun_prop (disch := exact
continuous_jackson _)
have hi2 : IntegrableOn (fun t => packetWave A K M t *
packetRemainder ε (packetIntensity M t) / (2 * pi))
(Icc (-pi) pi) := by
apply ContinuousOn.integrableOn_compact isCompact_Icc
unfold packetRemainder
apply ContinuousOn.div
· apply ContinuousOn.mul
· unfold packetWave packetOscillation packetKernel
fun_prop (disch := exact
continuous_jackson _)
· apply ContinuousOn.add
· fun_prop
· apply ContinuousOn.div
· fun_prop
· fun_prop
· intro t ht
linarith [(packetIntensity_mem_Icc hM t).1]
· fun_prop
· intro t ht; exact ne_of_gt (by positivity)
simp_rw [hsplit]
rw [integral_add hi1 hi2, integral_div, hz, zero_div, zero_add,
packetWave_remainder_response A K hM ε]
The full unnormalized packet has a uniform response of order sqrt(M)/K, including zero overlap and the endpoint M = K². In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ (c : ℝ) (A : ℕ),
0 < c
1 ≤ A
Proof (Lean source)
lemma packetWave_response_lower :
∃ (c : ℝ) (A : ℕ), 0 < c ∧ 1 ≤ A ∧
∀ (K : ℕ) (M ε : ℝ), 2 ≤ K → 2 ≤ M → M ≤ (K : ℝ) ^ 2 →
0 ≤ ε → ε ≤ 1 / 2 →
c * sqrt M / K ≤
|∫ t in Icc (-pi) pi,
packetWave A K M t * phiEpsFormula ε (packetIntensity M t) / (2 * pi)| := by
obtain ⟨c, A, hc, hA, hresponse⟩ := packetRemainder_response_lower
refine ⟨2 * c, A, by positivity, hA, ?_⟩
intro K M ε hK hM hMhi hε hεhi
rw [packetWave_phiEps_response hA hK (by linarith) ε, abs_mul,
abs_of_pos (by norm_num : (0 : ℝ) < 2)]
have h := mul_le_mul_of_nonneg_left (hresponse K M ε hK hM hMhi hε hεhi)
(by norm_num : (0 : ℝ) ≤ 2)
simpa only [mul_div_assoc, mul_assoc] using h
Normalization costs at most the universal weighted-variation bound 36. In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ (c : ℝ) (A : ℕ),
0 < c
1 ≤ A
conclusion 1
K :
ℕ
M ε :
ℝ
2 ≤ K
2 ≤ M
M ≤ (K : ℝ) ^ 2
0 ≤ ε
ε ≤ 1 / 2
c * sqrt M / K
≤ |∫ t in Icc (-pi) pi, (packetWave A K M t / packetWeight A K M / (2 * pi)) * phiEpsFormula ε (packetIntensity M t)|
Proof (Lean source)
lemma packetNormalized_response_lower :
∃ (c : ℝ) (A : ℕ), 0 < c ∧ 1 ≤ A ∧
∀ (K : ℕ) (M ε : ℝ), 2 ≤ K → 2 ≤ M → M ≤ (K : ℝ) ^ 2 →
0 ≤ ε → ε ≤ 1 / 2 →
c * sqrt M / K ≤
|∫ t in Icc (-pi) pi,
(packetWave A K M t / packetWeight A K M / (2 * pi)) *
phiEpsFormula ε (packetIntensity M t)| := by
obtain ⟨c, A, hc, hA, hresponse⟩ := packetWave_response_lower
refine ⟨c / 36, A, by positivity, hA, ?_⟩
intro K M ε hK hM hMhi hε hεhi
have hW := packetWeight_pos hA hK hM
have hWle := packetWeight_le hA hK hM hMhi
have he (t : ℝ) :
(packetWave A K M t / packetWeight A K M / (2 * pi)) *
phiEpsFormula ε (packetIntensity M t) =
(packetWave A K M t * phiEpsFormula ε (packetIntensity M t) / (2 * pi)) /
packetWeight A K M := by ring
simp_rw [he]
rw [integral_div, abs_div, abs_of_pos hW]
have hl := hresponse K M ε hK hM hMhi hε hεhi
have hnum : 0 ≤ c * sqrt M / K := by positivity
calc
c / 36 * sqrt M / K = (c * sqrt M / K) / 36 := by ring
_ ≤ (c * sqrt M / K) / packetWeight A K M :=
div_le_div_of_nonneg_left hnum hW hWle
_ ≤ _ := div_le_div_of_nonneg_right hl hW.le
The explicit common-atom completion attains the uniform weighted lower separation, with support, mean one, and all K moments supplied by the constructed constrained-prior pair, as required by roadmap (30). In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ (c : ℝ) (A : ℕ),
0 < c
1 ≤ A
conclusion 1
K :
ℕ
M ε :
ℝ
2 ≤ K
2 ≤ M
M ≤ (K : ℝ) ^ 2
0 ≤ ε
ε ≤ 1 / 2
∃ (p : ConstrainedPriorPair K M),
(1 / 2 : ℝ)
Proof (Lean source)
lemma packetCompletion_separation_lower :
∃ (c : ℝ) (A : ℕ), 0 < c ∧ 1 ≤ A ∧
∀ (K : ℕ) (M ε : ℝ), 2 ≤ K → 2 ≤ M → M ≤ (K : ℝ) ^ 2 →
0 ≤ ε → ε ≤ 1 / 2 →
∃ (p : ConstrainedPriorPair K M),
∃ hdom : CommonAtomDomain (packetPositive A K M) (packetNegative A K M),
(packetPositive A K M univ).toReal ≤ 1 / 2 ∧
(1 / 2 : ℝ) ≤ (1 - ∫ z, z ∂packetPositive A K M) /
(1 - (packetPositive A K M univ).toReal) ∧
(1 - ∫ z, z ∂packetPositive A K M) /
(1 - (packetPositive A K M univ).toReal) ≤ 2 ∧
commonAtomCompletion (packetPositive A K M) (packetNegative A K M) hdom =
(p.ν₀, p.ν₁) ∧
c * sqrt M / K ≤
|(∫ z, phiEpsFormula ε z ∂p.ν₁) - (∫ z, phiEpsFormula ε z ∂p.ν₀)| := by
obtain ⟨c, A, hc, hA, hresponse⟩ := packetNormalized_response_lower
refine ⟨c, A, hc, hA, ?_⟩
intro K M ε hK hM hMhi hε hεhi
obtain ⟨p, hdom, ht, hzlo, hzhi, hcompletion, hgap⟩ :=
packetCompletion_witness hA hK hM ε
refine ⟨p, hdom, ht, hzlo, hzhi, hcompletion, ?_⟩
rw [hgap]
exact hresponse K M ε hK hM hMhi hε hεhi
theorem packetWeightedApproxError_lower
The attained packet gap forces the claimed order of the weighted best approximation error: every admissible polynomial has the same moment pairing, so the proved residual envelope bounds the gap before taking the infimum. In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ c : ℝ,
0 < c
conclusion 1
K :
ℕ
M ε :
ℝ
2 ≤ K
2 ≤ M
M ≤ (K : ℝ) ^ 2
0 ≤ ε
ε ≤ 1 / 2
c * sqrt M / K ≤ weightedApproxError K M ε
Proof (Lean source)
lemma packetWeightedApproxError_lower :
∃ c : ℝ, 0 < c ∧
∀ (K : ℕ) (M ε : ℝ), 2 ≤ K → 2 ≤ M → M ≤ (K : ℝ) ^ 2 →
0 ≤ ε → ε ≤ 1 / 2 →
c * sqrt M / K ≤ weightedApproxError K M ε := by
obtain ⟨c, A, hc, hA, hprior⟩ := packetCompletion_separation_lower
refine ⟨c / 4, by positivity, ?_⟩
intro K M ε hK hM hMhi hε hεhi
obtain ⟨p, _, _, _, _, _, hgap⟩ := hprior K M ε hK hM hMhi hε hεhi
have hne : {e : ℝ | ∃ q : Polynomial ℝ, q.natDegree ≤ K ∧
e = sSup ((fun z : ℝ => |phiEpsFormula ε z - q.eval z| / (1 + z)) ''
Icc 0 M)}.Nonempty := by
refine ⟨_, 0, ?_, rfl⟩
simp
apply le_csInf hne
rintro e ⟨q, hq, rfl⟩
have hbound := constrainedPriorPair_gap_le_polynomial_error p ε
(by linarith) q hq
have hscale : c / 4 * sqrt M / K = (c * sqrt M / K) / 4 := by ring
rw [hscale]
exact (div_le_div_of_nonneg_right hgap (by norm_num)).trans
((div_le_iff₀ (by norm_num : (0 : ℝ) < 4)).mpr (by linarith only [hbound]))
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.PacketSmoothRemainder 17 declarations Roadmap (22) and (25): the absolutely continuous curvature of the scalar kink composition is uniformly bounded by a constant times M.
Smooth remainder in the packet response
Roadmap (22) and (25): the absolutely continuous curvature of the scalar kink composition is uniformly bounded by a constant times M. Its pairing with the localized twice antiderivative is therefore of order M/N².
The nonlinear remainder after subtracting the affine term in roadmap (10). For the displayed parameters, packetRemainder is the object specified by this definition.
The curvature on the upper smooth arc, by the chain rule. For the displayed parameters, packetUpperCurvature is the object specified by this definition.
Definition (Lean source)
parameters
M ε t :
ℝ
def
packetUpperCurvature M ε t :
ℝ
given by
-4 * (1 - 2 * ε) * (-(M / 2) * sin t) ^ 2 / (1 + packetIntensity M t) ^ 3
+ 2 * (1 - 2 * ε) * (-(M / 2) * cos t) / (1 + packetIntensity M t) ^ 2
The density of the smooth second derivative; values at the two kinks are set to zero, since the derivative jumps are handled separately as atoms. For the displayed parameters, packetSmoothCurvature is the object specified by this definition.
Definition (Lean source)
parameters
M ε t :
ℝ
def
packetSmoothCurvature M ε t :
ℝ
given by
if packetIntensity M t < 1 then ε * (M / 2) * cos t else if 1 < packetIntensity M t then packetUpperCurvature M ε t else 0
The affine term and the nonlinear remainder recover the original functional. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma phiEps_eq_packetRemainder {z : ℝ} (hz : 0 ≤ z) (ε : ℝ) :
phiEpsFormula ε z = ε * (z - 1) + packetRemainder ε z := by
have hd : 1 + z ≠ 0 := by linarith
unfold phiEpsFormula packetRemainder
by_cases h : 1 ≤ z
· rw [max_eq_left (by linarith : 0 ≤ z - 1),
max_eq_right (by linarith : 1 - z ≤ 0)]
field_simp
ring
· rw [max_eq_right (by linarith : z - 1 ≤ 0),
max_eq_left (by linarith : 0 ≤ 1 - z)]
ring
The cosine coordinate has the derivative used in roadmap (11). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
Proof (Lean source)
lemma packetIntensity_hasDerivAt (M t : ℝ) :
HasDerivAt (packetIntensity M) (-(M / 2) * sin t) t := by
convert ((Real.hasDerivAt_cos t).const_add 1).const_mul (M / 2) using 1 <;>
first | rfl | ring
The squared coordinate speed is z(M-z), keeping the curvature bound linear in M rather than quadratic. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
M t :
ℝ
conclusion
Proof (Lean source)
lemma packetIntensity_speed_sq (M t : ℝ) :
(-(M / 2) * sin t) ^ 2 =
packetIntensity M t * (M - packetIntensity M t) := by
have h := Real.sin_sq_add_cos_sq t
unfold packetIntensity
nlinarith [sq_nonneg M]
The upper smooth branch has the rational slope stated in roadmap (22). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
M :
ℝ
hM :
0 ≤ M
ε t :
ℝ
conclusion
HasDerivAt (fun s => (1 - 2 * ε) * (packetIntensity M s - 1) / (1 + packetIntensity M s)) (2 * (1 - 2 * ε) * (-(M / 2) * sin t) / (1 + packetIntensity M t) ^ 2) t
Proof (Lean source)
lemma packetUpperBranch_hasDerivAt {M : ℝ} (hM : 0 ≤ M) (ε t : ℝ) :
HasDerivAt (fun s => (1 - 2 * ε) *
(packetIntensity M s - 1) / (1 + packetIntensity M s))
(2 * (1 - 2 * ε) * (-(M / 2) * sin t) /
(1 + packetIntensity M t) ^ 2) t := by
have hd : 1 + packetIntensity M t ≠ 0 := by
have := (packetIntensity_mem_Icc hM t).1
linarith
convert (((packetIntensity_hasDerivAt M t).sub_const 1).const_mul
(1 - 2 * ε)).div ((packetIntensity_hasDerivAt M t).const_add 1) hd using 1 <;>
first | rfl | (field_simp; ring)
theorem packetUpperSlope_hasDerivAt
Differentiating the upper slope gives the actual chain-rule curvature. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
M :
ℝ
hM :
0 ≤ M
ε t :
ℝ
conclusion
HasDerivAt (fun s => 2 * (1 - 2 * ε) * (-(M / 2) * sin s) / (1 + packetIntensity M s) ^ 2) (packetUpperCurvature M ε t) t
Proof (Lean source)
lemma packetUpperSlope_hasDerivAt {M : ℝ} (hM : 0 ≤ M) (ε t : ℝ) :
HasDerivAt (fun s => 2 * (1 - 2 * ε) * (-(M / 2) * sin s) /
(1 + packetIntensity M s) ^ 2) (packetUpperCurvature M ε t) t := by
have hd : 1 + packetIntensity M t ≠ 0 := by
have := (packetIntensity_mem_Icc hM t).1
linarith
convert ((((Real.hasDerivAt_sin t).const_mul (-(M / 2))).const_mul
(2 * (1 - 2 * ε))).div
(((packetIntensity_hasDerivAt M t).const_add 1).pow 2)
(pow_ne_zero 2 hd)) using 1 <;>
first | rfl | (dsimp [packetUpperCurvature]; field_simp; ring)
theorem packetSmoothCurvature_eq_second_deriv
On either open smooth arc the curvature density is exactly the ordinary second derivative of the nonlinear remainder composed with z(t). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
= packetSmoothCurvature M ε t
Proof (Lean source)
lemma packetSmoothCurvature_eq_second_deriv {M : ℝ} (hM : 0 ≤ M)
(ε t : ℝ) (ht : packetIntensity M t ≠ 1) :
deriv (deriv (fun s => packetRemainder ε (packetIntensity M s))) t =
packetSmoothCurvature M ε t := by
rcases lt_or_gt_of_ne ht with h | h
· have hevent : ∀ᶠ s in nhds t, packetIntensity M s < 1 :=
(continuous_packetIntensity M).continuousAt.eventually_lt continuousAt_const h
have heq : (fun s => packetRemainder ε (packetIntensity M s)) =ᶠ[nhds t]
(fun s => ε * (1 - packetIntensity M s)) := by
filter_upwards [hevent] with s hs
simp only [packetRemainder, max_eq_left (by linarith : 0 ≤ 1 - packetIntensity M s),
max_eq_right (by linarith : packetIntensity M s - 1 ≤ 0), mul_zero, zero_div, add_zero]
rw [heq.deriv.deriv_eq]
have hfirst : deriv (fun s => ε * (1 - packetIntensity M s)) =
(fun s => ε * (M / 2) * sin s) := by
funext s
convert (((packetIntensity_hasDerivAt M s).const_sub 1).const_mul ε).deriv using 1 <;> ring
rw [hfirst]
rw [show packetSmoothCurvature M ε t = ε * (M / 2) * cos t by
simp [packetSmoothCurvature, h]]
exact ((Real.hasDerivAt_sin t).const_mul (ε * (M / 2))).deriv
· have hevent : ∀ᶠ s in nhds t, 1 < packetIntensity M s :=
continuousAt_const.eventually_lt (continuous_packetIntensity M).continuousAt h
have heq : (fun s => packetRemainder ε (packetIntensity M s)) =ᶠ[nhds t]
(fun s => (1 - 2 * ε) * (packetIntensity M s - 1) /
(1 + packetIntensity M s)) := by
filter_upwards [hevent] with s hs
simp only [packetRemainder, max_eq_right (by linarith : 1 - packetIntensity M s ≤ 0),
max_eq_left (by linarith : 0 ≤ packetIntensity M s - 1), mul_zero, zero_add]
rw [heq.deriv.deriv_eq]
have hfirst : deriv (fun s => (1 - 2 * ε) * (packetIntensity M s - 1) /
(1 + packetIntensity M s)) =
(fun s => 2 * (1 - 2 * ε) * (-(M / 2) * sin s) /
(1 + packetIntensity M s) ^ 2) := by
funext s
exact (packetUpperBranch_hasDerivAt hM ε s).deriv
rw [hfirst, (packetUpperSlope_hasDerivAt hM ε t).deriv]
simp [packetSmoothCurvature, h, not_lt.mpr h.le]
theorem packetUpperCurvature_bound
The smooth upper-arc curvature is bounded by 5M, uniformly in ε. The speed identity is essential here: estimating the sine factor alone would give the wrong power of M. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
M ε :
ℝ
hM :
0 ≤ M
hε :
0 ≤ ε
hεhi :
ε ≤ 1 / 2
t :
ℝ
conclusion
|packetUpperCurvature M ε t| ≤ 5 * M
Proof (Lean source)
lemma packetUpperCurvature_bound {M ε : ℝ} (hM : 0 ≤ M)
(hε : 0 ≤ ε) (hεhi : ε ≤ 1 / 2) (t : ℝ) :
|packetUpperCurvature M ε t| ≤ 5 * M := by
let z := packetIntensity M t
have hz : 0 ≤ z := (packetIntensity_mem_Icc hM t).1
have hzM : z ≤ M := (packetIntensity_mem_Icc hM t).2
have ha : 0 ≤ 1 - 2 * ε := by linarith
have ha1 : 1 - 2 * ε ≤ 1 := by linarith
have hd : 0 < 1 + z := by linarith
have hd2 : 1 ≤ (1 + z) ^ 2 := by nlinarith
have hd3 : z ≤ (1 + z) ^ 3 := by nlinarith [sq_nonneg z, mul_nonneg hz (sq_nonneg z)]
have hv : (-(M / 2) * sin t) ^ 2 ≤ M * (1 + z) ^ 3 := by
rw [packetIntensity_speed_sq]
change z * (M - z) ≤ _
calc
_ ≤ M * z := by nlinarith
_ ≤ _ := mul_le_mul_of_nonneg_left hd3 hM
have hfirst : |-4 * (1 - 2 * ε) * (-(M / 2) * sin t) ^ 2 /
(1 + z) ^ 3| ≤ 4 * M := by
rw [abs_div, abs_mul, abs_mul, abs_of_nonneg ha,
abs_of_nonneg (sq_nonneg (-(M / 2) * sin t)), abs_of_pos (pow_pos hd 3)]
norm_num
apply (div_le_iff₀ (pow_pos hd 3)).2
have hv' := mul_le_mul_of_nonneg_left hv (by norm_num : (0 : ℝ) ≤ 4)
have hcoef := mul_le_mul_of_nonneg_right ha1
(sq_nonneg (-(M / 2) * sin t))
nlinarith
have hcos : |cos t| ≤ 1 := abs_le.mpr ⟨Real.neg_one_le_cos t, Real.cos_le_one t⟩
have hsecond : |2 * (1 - 2 * ε) * (-(M / 2) * cos t) /
(1 + z) ^ 2| ≤ M := by
rw [abs_div, abs_mul, abs_mul, abs_mul, abs_neg,
abs_of_nonneg ha, abs_of_nonneg (by positivity : 0 ≤ M / 2),
abs_of_pos (pow_pos hd 2)]
norm_num
apply (div_le_iff₀ (pow_pos hd 2)).2
calc
_ ≤ 2 * 1 * (M / 2 * 1) :=
mul_le_mul (mul_le_mul_of_nonneg_left ha1 (by norm_num))
(mul_le_mul_of_nonneg_left hcos (by positivity)) (by positivity) (by norm_num)
_ = M := by ring
_ ≤ M * (1 + z) ^ 2 := le_mul_of_one_le_right hM hd2
exact (abs_add_le _ _).trans (by
change |-4 * (1 - 2 * ε) * (-(M / 2) * sin t) ^ 2 / (1 + z) ^ 3| +
|2 * (1 - 2 * ε) * (-(M / 2) * cos t) / (1 + z) ^ 2| ≤ _
linarith)
theorem packetSmoothCurvature_bound
Both smooth arcs obey the same linear-in-M bound from roadmap (22). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
M ε :
ℝ
hM :
0 ≤ M
hε :
0 ≤ ε
hεhi :
ε ≤ 1 / 2
t :
ℝ
conclusion
|packetSmoothCurvature M ε t| ≤ 5 * M
Proof (Lean source)
lemma packetSmoothCurvature_bound {M ε : ℝ} (hM : 0 ≤ M)
(hε : 0 ≤ ε) (hεhi : ε ≤ 1 / 2) (t : ℝ) :
|packetSmoothCurvature M ε t| ≤ 5 * M := by
unfold packetSmoothCurvature
split_ifs
· rw [abs_mul, abs_mul, abs_of_nonneg hε,
abs_of_nonneg (by positivity : 0 ≤ M / 2)]
have hc : |cos t| ≤ 1 := abs_le.mpr ⟨Real.neg_one_le_cos t, Real.cos_le_one t⟩
calc
_ ≤ (1 / 2) * (M / 2) * 1 :=
mul_le_mul (mul_le_mul_of_nonneg_right hεhi (by positivity)) hc
(abs_nonneg _) (by positivity)
_ ≤ _ := by nlinarith
· exact packetUpperCurvature_bound hM hε hεhi t
· simp only [abs_zero]
positivity
theorem packetRemainder_second_deriv_bound
Roadmap (22), stated directly for the original composed remainder on each smooth arc. No regularity or curvature hypothesis is assumed. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma packetRemainder_second_deriv_bound {M ε : ℝ} (hM : 0 ≤ M)
(hε : 0 ≤ ε) (hεhi : ε ≤ 1 / 2) (t : ℝ)
(ht : packetIntensity M t ≠ 1) :
|deriv (deriv (fun s => packetRemainder ε (packetIntensity M s))) t| ≤ 5 * M := by
rw [packetSmoothCurvature_eq_second_deriv hM ε t ht]
exact packetSmoothCurvature_bound hM hε hεhi t
The finite Fourier antiderivative is continuous. The stated conclusion holds.
Formal statement
Proof (Lean source)
-- @node: continuous_packetAntideriv
@[fun_prop] lemma continuous_packetAntideriv (N : ℕ) :
Continuous (packetAntideriv N) := by
unfold packetAntideriv
fun_prop
Every integer Fourier frequency of the antiderivative has period 2π. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Proof (Lean source)
lemma packetAntideriv_periodic (N : ℕ) :
Periodic (packetAntideriv N) (2 * pi) := by
intro t
unfold packetAntideriv
congr 1
apply Finset.sum_congr rfl
intro j hj
exact congrArg₂ (fun x y => packetCoeffPlus N j * x + packetCoeffMinus N j * y)
(packet_cos_periodic (4 * (N : ℤ) + j) t)
(packet_cos_periodic (4 * (N : ℤ) - j) t)
theorem measurable_packetSmoothCurvature
The smooth curvature density is measurable, despite the two jumps. The stated conclusion holds.
Formal statement
Proof (Lean source)
-- @node: measurable_packetSmoothCurvature
@[fun_prop]
lemma measurable_packetSmoothCurvature (M ε : ℝ) :
Measurable (packetSmoothCurvature M ε) := by
unfold packetSmoothCurvature
apply Measurable.ite (measurableSet_lt (by fun_prop) measurable_const) (by fun_prop)
apply Measurable.ite (measurableSet_lt measurable_const (by fun_prop))
(by unfold packetUpperCurvature; fun_prop) measurable_const
theorem packetSmoothCurvature_integrable
The smooth density is integrable on the circle, without additional regularity premises. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
M ε :
ℝ
hM :
0 ≤ M
hε :
0 ≤ ε
hεhi :
ε ≤ 1 / 2
conclusion
Proof (Lean source)
lemma packetSmoothCurvature_integrable {M ε : ℝ} (hM : 0 ≤ M)
(hε : 0 ≤ ε) (hεhi : ε ≤ 1 / 2) :
Integrable (packetSmoothCurvature M ε)
(volume.restrict (Icc (-pi) pi)) := by
apply Integrable.of_bound (by fun_prop) (5 * M)
filter_upwards with t
simpa only [Real.norm_eq_abs] using packetSmoothCurvature_bound hM hε hεhi t
theorem packetSmoothRemainder_bound
Circle translation preserves the Jackson L¹ bound, so the smooth term in roadmap (25) has absolute value at most CM/N² for every center. In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ C : ℝ,
0 < C
conclusion 1
N :
ℕ
M ε θ :
ℝ
2 ≤ N
0 ≤ M
0 ≤ ε
ε ≤ 1 / 2
≤ C * M / (N : ℝ) ^ 2
Proof (Lean source)
lemma packetSmoothRemainder_bound :
∃ C : ℝ, 0 < C ∧ ∀ (N : ℕ) (M ε θ : ℝ),
2 ≤ N → 0 ≤ M → 0 ≤ ε → ε ≤ 1 / 2 →
|∫ t in Icc (-pi) pi,
packetSmoothCurvature M ε t * packetAntideriv N (t - θ) /
(2 * pi)| ≤ C * M / (N : ℝ) ^ 2 := by
obtain ⟨c, C, hc, hcC, hpacket⟩ := jackson_packet_localization
refine ⟨5 * C, by linarith, ?_⟩
intro N M ε θ hN hM hε hεhi
have hG : Integrable (fun t => |packetAntideriv N (t - θ)| / (2 * pi))
(volume.restrict (Icc (-pi) pi)) := by
apply ContinuousOn.integrableOn_compact isCompact_Icc
fun_prop
have hperiod : Periodic (fun t => |packetAntideriv N t| / (2 * pi))
(2 * pi) := by
intro t
exact congrArg (fun x => |x| / (2 * pi)) (packetAntideriv_periodic N t)
have hshift := packet_periodic_integral_translate
(fun t => |packetAntideriv N t| / (2 * pi)) hperiod θ
have hl1 := (hpacket N hN).2.2.2.2.2.2.2.2.2.2.2
have hbound : |∫ t in Icc (-pi) pi,
packetSmoothCurvature M ε t * packetAntideriv N (t - θ) /
(2 * pi)| ≤
∫ t in Icc (-pi) pi,
(5 * M) * (|packetAntideriv N (t - θ)| / (2 * pi)) := by
apply (show ‖∫ t in Icc (-pi) pi,
packetSmoothCurvature M ε t * packetAntideriv N (t - θ) / (2 * pi)‖ ≤ _ from
norm_integral_le_of_norm_le (hG.const_mul (5 * M)) ?_)
filter_upwards with t
rw [Real.norm_eq_abs, abs_div, abs_mul, abs_of_pos (by positivity : 0 < 2 * pi)]
calc
_ ≤ (5 * M) * |packetAntideriv N (t - θ)| / (2 * pi) :=
div_le_div_of_nonneg_right
(mul_le_mul_of_nonneg_right (packetSmoothCurvature_bound hM hε hεhi t)
(abs_nonneg _)) (by positivity)
_ = _ := by ring
calc
_ ≤ _ := hbound
_ = (5 * M) * ∫ t in Icc (-pi) pi,
|packetAntideriv N (t - θ)| / (2 * pi) := integral_const_mul _ _
_ = (5 * M) * ∫ t in Icc (-pi) pi,
|packetAntideriv N t| / (2 * pi) := by rw [hshift]
_ ≤ (5 * M) * (C / (N : ℝ) ^ 2) :=
mul_le_mul_of_nonneg_left hl1 (by positivity)
_ = _ := by ring
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.PacketWeightBound 5 declarations Roadmap (18)--(19): translating the two positive kernel envelopes together cancels their sine terms.
Uniform weighted variation of the Jackson packet
Roadmap (18)--(19): translating the two positive kernel envelopes together cancels their sine terms. The second Jackson moment then bounds the weight uniformly, including the endpoint M = K².
Fourier reconstruction gives the kernel period needed for circle translation. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma packetKernel_periodic {N : ℕ} (hN : 0 < N) :
Periodic (packetKernel N) (2 * pi) := by
intro t
change kernel N
(t + 2 * pi) = kernel N t
rw [kernel_fourier N hN,
kernel_fourier N hN]
apply Finset.sum_congr rfl
intro j hj
exact congrArg (fun x =>
normalizedCoeff N j * x)
(packet_cos_periodic j t)
The sum of the translated weights has no sine term; both centers are at z = 1. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
M :
ℝ
hM :
2 ≤ M
t :
ℝ
conclusion
packetIntensity M (t + arccos (2 / M - 1))
+ packetIntensity M (t - arccos (2 / M - 1))
= 2 + (M - 2) * (1 - cos t)
Proof (Lean source)
lemma packetIntensity_translate_sum {M : ℝ} (hM : 2 ≤ M) (t : ℝ) :
packetIntensity M (t + arccos (2 / M - 1)) +
packetIntensity M (t - arccos (2 / M - 1)) =
2 + (M - 2) * (1 - cos t) := by
have hM0 : 0 < M := by linarith
have hr : 2 / M ≤ 1 := (div_le_one hM0).2 hM
have hc := Real.cos_arccos (x := 2 / M - 1)
(by linarith [div_pos (by norm_num : (0 : ℝ) < 2) hM0]) (by linarith)
simp only [packetIntensity, Real.cos_add, Real.cos_sub, hc]
field_simp
<;> ring
Positive kernel envelopes dominate the absolute modulated packet pointwise. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
A₀ K :
ℕ
hN :
0 < A₀ * K
M t :
ℝ
conclusion
|packetWave A₀ K M t|
≤ packetKernel (A₀ * K) (t - arccos (2 / M - 1))
+ packetKernel (A₀ * K) (t + arccos (2 / M - 1))
Proof (Lean source)
lemma packetWave_abs_le_kernels {A₀ K : ℕ} (hN : 0 < A₀ * K) (M t : ℝ) :
|packetWave A₀ K M t| ≤
packetKernel (A₀ * K) (t - arccos (2 / M - 1)) +
packetKernel (A₀ * K) (t + arccos (2 / M - 1)) := by
have he (u : ℝ) : |packetOscillation (A₀ * K) u| ≤ packetKernel (A₀ * K) u := by
have hn : 0 ≤ packetKernel (A₀ * K) u :=
mul_nonneg (by positivity)
(jackson_nonneg _ hN u)
rw [packetOscillation, abs_mul, abs_of_nonneg hn]
exact mul_le_of_le_one_right hn (Real.abs_cos_le_one _)
exact (abs_add_le _ _).trans (add_le_add (he _) (he _))
A translated weighted kernel integral can be moved back to its center. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
N :
ℕ
hN :
0 < N
M θ :
ℝ
conclusion
Proof (Lean source)
lemma packetKernel_weight_translate {N : ℕ} (hN : 0 < N) (M θ : ℝ) :
(∫ t in Icc (-pi) pi,
(1 + packetIntensity M t) * packetKernel N (t - θ) / (2 * pi)) =
∫ t in Icc (-pi) pi,
(1 + packetIntensity M (t + θ)) * packetKernel N t / (2 * pi) := by
let f := fun t => (1 + packetIntensity M (t + θ)) * packetKernel N t / (2 * pi)
have hp : Periodic f (2 * pi) := by
intro t
dsimp [f]
rw [packetKernel_periodic hN t]
unfold packetIntensity
rw [show t + 2 * pi + θ = (t + θ) + 2 * pi by ring,
Real.cos_add_two_pi]
convert packet_periodic_integral_translate f hp θ using 1 <;>
simp [f]
The normalizing weighted variation is universally bounded throughout the paper's full range. This is roadmap (19), with the explicit constant 36. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
A₀ K :
ℕ
hA :
1 ≤ A₀
hK :
2 ≤ K
M :
ℝ
hM :
2 ≤ M
hMhi :
M ≤ (K : ℝ) ^ 2
conclusion
packetWeight A₀ K M ≤ 36
Proof (Lean source)
lemma packetWeight_le {A₀ K : ℕ} (hA : 1 ≤ A₀) (hK : 2 ≤ K)
{M : ℝ} (hM : 2 ≤ M) (hMhi : M ≤ (K : ℝ) ^ 2) :
packetWeight A₀ K M ≤ 36 := by
let N := A₀ * K
let θ := arccos (2 / M - 1)
have hN : 0 < N := by dsimp [N]; nlinarith
have hc : Continuous (packetKernel N) := by unfold packetKernel; fun_prop
have hi (s : ℝ) : IntegrableOn (fun t =>
(1 + packetIntensity M t) * packetKernel N (t - s) / (2 * Real.pi))
(Icc (-Real.pi) Real.pi) :=
(by fun_prop : Continuous (fun t =>
(1 + packetIntensity M t) * packetKernel N (t - s) / (2 * Real.pi))).continuousOn.integrableOn_compact isCompact_Icc
have hj (s : ℝ) : IntegrableOn (fun t =>
(1 + packetIntensity M (t + s)) * packetKernel N t / (2 * Real.pi))
(Icc (-Real.pi) Real.pi) :=
(by fun_prop : Continuous (fun t =>
(1 + packetIntensity M (t + s)) * packetKernel N t / (2 * Real.pi))).continuousOn.integrableOn_compact isCompact_Icc
have hf : IntegrableOn (fun t =>
(1 + packetIntensity M t) * |packetWave A₀ K M t| / (2 * Real.pi))
(Icc (-Real.pi) Real.pi) :=
(by fun_prop : Continuous (fun t =>
(1 + packetIntensity M t) * |packetWave A₀ K M t| / (2 * Real.pi))).continuousOn.integrableOn_compact isCompact_Icc
have he : packetWeight A₀ K M ≤
∫ t in Icc (-Real.pi) Real.pi,
(4 + (M - 2) * (1 - cos t)) *
jackson N t := by
calc
_ ≤ ∫ t in Icc (-Real.pi) Real.pi,
((1 + packetIntensity M t) * packetKernel N (t - θ) / (2 * Real.pi) +
(1 + packetIntensity M t) * packetKernel N (t - (-θ)) / (2 * Real.pi)) := by
apply integral_mono hf ((hi θ).add (hi (-θ)))
intro t
have hz := (packetIntensity_mem_Icc (by linarith : 0 ≤ M) t).1
have hw := packetWave_abs_le_kernels hN M t
dsimp [N, θ] at *
simpa only [sub_neg_eq_add, ← add_div, ← mul_add] using
div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_left hw (by linarith))
(by positivity : (0 : ℝ) ≤ 2 * Real.pi)
_ = _ := by
rw [integral_add (hi θ) (hi (-θ)),
packetKernel_weight_translate hN M θ, packetKernel_weight_translate hN M (-θ),
← integral_add (hj θ) (hj (-θ))]
apply integral_congr_ae
filter_upwards with t
have hs := packetIntensity_translate_sum hM t
dsimp [θ] at *
simp only [← sub_eq_add_neg] at *
unfold packetKernel
have hp : (2 * Real.pi : ℝ) ≠ 0 := by positivity
have halg (a b j : ℝ) : a * (2 * Real.pi * j) / (2 * Real.pi) +
b * (2 * Real.pi * j) / (2 * Real.pi) = (a + b) * j := by
field_simp
<;> ring
rw [halg]
congr 1
linarith [hs]
have hm := jackson_second_moment N hN
have hu := Causalean.Mathlib.Analysis.JacksonApproximation.jackson_integral_eq_one N hN
have hbound : (∫ t in Icc (-Real.pi) Real.pi,
(4 + (M - 2) * (1 - cos t)) *
jackson N t) ≤
4 + 32 * (M - 2) / (N : ℝ) ^ 2 := by
calc
_ ≤ ∫ t in Icc (-Real.pi) Real.pi,
(4 + (M - 2) * (t ^ 2 / 2)) *
jackson N t := by
apply integral_mono
((by fun_prop : Continuous (fun t => (4 + (M - 2) * (1 - cos t)) *
jackson N t)).continuousOn.integrableOn_compact isCompact_Icc)
((by fun_prop : Continuous (fun t => (4 + (M - 2) * (t ^ 2 / 2)) *
jackson N t)).continuousOn.integrableOn_compact isCompact_Icc)
intro t
apply mul_le_mul_of_nonneg_right ?_
(jackson_nonneg N hN t)
have hcos := Real.one_sub_sq_div_two_le_cos (x := t)
nlinarith
_ = 4 + ((M - 2) / 2) * (∫ t in Icc (-Real.pi) Real.pi,
t ^ 2 * jackson N t) := by
simp_rw [show ∀ t : ℝ, (4 + (M - 2) * (t ^ 2 / 2)) *
jackson N t =
4 * jackson N t +
((M - 2) / 2) * (t ^ 2 * jackson N t) by intro t; ring]
rw [integral_add, integral_const_mul, integral_const_mul, hu]
· ring
· exact ((by fun_prop : Continuous (fun t : ℝ =>
4 * jackson N t)).continuousOn.integrableOn_compact isCompact_Icc)
· exact ((by fun_prop : Continuous (fun t : ℝ => ((M - 2) / 2) *
(t ^ 2 * jackson N t))).continuousOn.integrableOn_compact isCompact_Icc)
_ ≤ _ := by
have hh := mul_le_mul_of_nonneg_left hm (show 0 ≤ (M - 2) / 2 by positivity)
convert add_le_add_left hh 4 using 1 <;> first | rfl | ring
have hNK : (K : ℝ) ≤ N := by dsimp [N]; exact_mod_cast (show K ≤ A₀ * K by nlinarith)
have hNR : (0 : ℝ) < N := by exact_mod_cast hN
have hratio : (M - 2) / (N : ℝ) ^ 2 ≤ 1 := by
apply (div_le_one (by positivity : (0 : ℝ) < (N : ℝ) ^ 2)).2
nlinarith [sq_nonneg ((N : ℝ) - K)]
have hh := mul_le_mul_of_nonneg_left hratio (by norm_num : (0 : ℝ) ≤ 32)
have := he.trans hbound
rw [mul_div_assoc] at this
linarith
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.SparseFamily 15 declarations Normalized sparse family and its full four-coordinate Poisson likelihood
Normalized sparse family and its full four-coordinate Poisson likelihood
def sparseNormalizerFormula
For the displayed parameters, sparseNormalizerFormula is the object specified by this definition.
def sparseNormalizer
Normalizing mass on the public positive-overlap domain. For the displayed parameters, sparseNormalizer is the object specified by this definition.
Definition (Lean source)
def sparseUnnormalizedMass
For the displayed parameters, sparseUnnormalizedMass is the object specified by this definition.
Definition (Lean source)
parameters
def
sparseUnnormalizedMass ε z o :
ℝ
given by
x :
o.1
a :
o.2.1
y :
o.2.2
if a then if y then ε * (1 - ε) * z x / d else ε * (1 - ε) / d else ((1 - ε) ^ 2 + ε ^ 2 * z x) / (2 * d)
def sparseMass
For the displayed parameters, sparseMass is the object specified by this definition.
Definition (Lean source)
parameters
def
sparseMass ε M z hz o :
ℝ
given by
sparseUnnormalizedMass ε z o / sparseNormalizerFormula d ε M z hz
In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
(∑ o : Obs d, ofReal (sparseMass ε M z hz o)) = 1
Proof (Lean source)
lemma sparseMass_pmf_sum {d : ℕ} (hd : 0 < d) (ε M : ℝ)
(hε : 0 < ε ∧ ε ≤ 1 / 2) (z : Fin d → ℝ)
(hz : ∀ x, z x ∈ Icc 0 M) :
(∑ o : Obs d, ofReal (sparseMass ε M z hz o)) = 1 := by
have hdR : (0 : ℝ) < d := Nat.cast_pos.mpr hd
have hz0 (x : Fin d) : 0 ≤ z x := (hz x).1
have hε0 : 0 ≤ ε := le_of_lt hε.1
have hε1 : 0 ≤ 1 - ε := by linarith [hε.2]
have hS : 0 < sparseNormalizerFormula d ε M z hz := by
unfold sparseNormalizerFormula
have hs : 0 ≤ ∑ x : Fin d, z x := Finset.sum_nonneg (fun x _ => hz0 x)
have : 0 ≤ ε / (d : ℝ) * ∑ x : Fin d, z x :=
mul_nonneg (div_nonneg hε0 (le_of_lt hdR)) hs
linarith [hε.2]
have hm (o : Obs d) : 0 ≤ sparseMass ε M z hz o := by
unfold sparseMass sparseUnnormalizedMass
rcases o with ⟨x, a, y⟩
have hzx : 0 ≤ z x := hz0 x
cases a <;> cases y <;> dsimp <;> positivity
rw [← ENNReal.ofReal_sum_of_nonneg (fun o _ => hm o)]
have hreal : (∑ o : Obs d, sparseMass ε M z hz o) = 1 := by
simp_rw [sparseMass, ← Finset.sum_div]
have htotal : (∑ o : Obs d, sparseUnnormalizedMass ε z o) =
sparseNormalizerFormula d ε M z hz := by
calc
_ = ∑ x : Fin d, (ε * z x + 1 - ε) / d := by
simp only [Fintype.sum_prod_type]
apply Finset.sum_congr rfl
intro x _
simp [sparseUnnormalizedMass]
ring
_ = sparseNormalizerFormula d ε M z hz := by
calc
_ = ∑ x : Fin d, ((ε / d) * z x + (1 - ε) / d) := by
apply Finset.sum_congr rfl
intro x _
ring
_ = (ε / d) * ∑ x : Fin d, z x + (d : ℝ) * ((1 - ε) / d) := by
rw [Finset.sum_add_distrib, ← Finset.mul_sum]
simp
_ = sparseNormalizerFormula d ε M z hz := by
unfold sparseNormalizerFormula
field_simp
ring
rw [htotal]
exact div_self hS.ne'
rw [hreal]
simp
def sparseLaw
For the displayed parameters, sparseLaw is the object specified by this definition.
Definition (Lean source)
parameters
def
given by
⟨PMF.ofFintype (fun o => ofReal (sparseUnnormalizedMass ε z o / sparseNormalizer d ε M hε z hz)) (sparseMass_pmf_sum hd ε M hε z hz)⟩
def sparseReference
For the displayed parameters, sparseReference is the object specified by this definition.
Definition (Lean source)
parameters
M :
ℝ
_hMpaper :
2 ≤ M
def
sparseReference M _hMpaper :
ℝ
given by
M / 2
def poissonCellMean
For the displayed parameters, poissonCellMean is the object specified by this definition.
def poissonCellLaw
For the displayed parameters, poissonCellLaw is the object specified by this definition.
Definition (Lean source)
parameters
B :
ℝ
n d :
ℕ
ε z :
ℝ
given by
def oneCellLikelihood
For the displayed parameters, oneCellLikelihood is the object specified by this definition.
Definition (Lean source)
parameters
def
oneCellLikelihood B n d ε M z N hMpaper :
ℝ
given by
∏ j : Fin 4, let μ := poissonCellMean B n d ε z j let μstar := poissonCellMean B n d ε (sparseReference (_hMpaper := hMpaper) M) j exp (μstar - μ) * (μ / μstar) ^ (N j)
def sparseGramCoefficient
For the displayed parameters, sparseGramCoefficient is the object specified by this definition.
Definition (Lean source)
parameters
B :
ℝ
_hB :
B > 2
n d :
ℕ
ε M :
ℝ
_hε :
0 < ε ∧ ε ≤ 1 / 2
_hM :
2 ≤ M
def
sparseGramCoefficient B _hB n d ε M _hε _hM :
ℝ
given by
B * n / d * (ε * (1 - ε) / sparseReference (_hMpaper := _hM) M + ε ^ 4 / ((1 - ε) ^ 2 + ε ^ 2 * sparseReference (_hMpaper := _hM) M))
In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
B :
ℝ
hB :
B > 2
n d :
ℕ
hn :
1 ≤ n
hd :
2 ≤ d
ε M :
ℝ
hε :
0 < ε
hεhalf :
ε ≤ 1 / 2
hM :
2 ≤ M
conclusion
sparseGramCoefficient B hB n d ε M ⟨hε, hεhalf⟩ hM ≤ 2 * B * n * ε / (d * M)
Proof (Lean source)
lemma sparseGramCoefficient_bound (B : ℝ) (hB : B > 2) (n d : ℕ)
(hn : 1 ≤ n) (hd : 2 ≤ d) (ε M : ℝ)
(hε : 0 < ε) (hεhalf : ε ≤ 1 / 2) (hM : 2 ≤ M) :
sparseGramCoefficient B hB n d ε M ⟨hε, hεhalf⟩ hM ≤ 2 * B * n * ε / (d * M) := by
have hd0 : (0 : ℝ) < d := by exact_mod_cast (show 0 < d by omega)
have hn0 : (0 : ℝ) ≤ n := by positivity
have hM0 : 0 < M := by linarith
have hD : 0 < (1 - ε) ^ 2 + ε ^ 2 * (M / 2) := by positivity
have hinner :
ε * (1 - ε) / (M / 2) +
ε ^ 4 / ((1 - ε) ^ 2 + ε ^ 2 * (M / 2)) ≤ 2 * ε / M := by
field_simp
nlinarith [sq_nonneg (1 - ε)]
unfold sparseGramCoefficient sparseReference
calc
B * n / d * (ε * (1 - ε) / (M / 2) +
ε ^ 4 / ((1 - ε) ^ 2 + ε ^ 2 * (M / 2)))
≤ B * n / d * (2 * ε / M) := by gcongr
_ = 2 * B * n * ε / (d * M) := by ring
theorem sparseLaw_jointMass
In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma sparseLaw_jointMass {d : ℕ} (hd : 0 < d) (ε M : ℝ)
(hε : 0 < ε ∧ ε ≤ 1 / 2) (z : Fin d → ℝ)
(hz : ∀ x, z x ∈ Icc 0 M) (x : Fin d) (a y : Bool) :
jointMass (sparseLaw hd ε M hε z hz) x a y =
sparseMass ε M z hz (x, a, y) := by
have hm : 0 ≤ sparseMass ε M z hz (x, a, y) := by
unfold sparseMass sparseUnnormalizedMass
have hzx : 0 ≤ z x := (hz x).1
have hε0 : 0 ≤ ε := le_of_lt hε.1
have hε1 : 0 ≤ 1 - ε := by linarith [hε.2]
have hdR : (0 : ℝ) < d := Nat.cast_pos.mpr hd
have hden : 0 < sparseNormalizerFormula d ε M z hz := by
unfold sparseNormalizerFormula
have hdR : (0 : ℝ) < d := Nat.cast_pos.mpr hd
have hs : 0 ≤ ∑ x : Fin d, z x := Finset.sum_nonneg (fun x _ => (hz x).1)
have : 0 ≤ ε / (d : ℝ) * ∑ x : Fin d, z x :=
mul_nonneg (div_nonneg (le_of_lt hε.1) (le_of_lt hdR)) hs
linarith [hε.2]
cases a <;> cases y <;> dsimp <;> positivity
change (ofReal (sparseMass ε M z hz (x, a, y))).toReal = _
exact ENNReal.toReal_ofReal hm
In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma poisson_likelihood_gram_scalar (μstar μ ν : ℝ) (hstar : 0 < μstar) :
(∫ N : ℕ, (exp (μstar - μ) * (μ / μstar) ^ N) *
(exp (μstar - ν) * (ν / μstar) ^ N)
∂poissonMeasure (toNNReal μstar)) =
exp ((μ - μstar) * (ν - μstar) / μstar) := by
have hmgf (r : NNReal) (c : ℝ) :
(∫ N : ℕ, c ^ N ∂poissonMeasure r) =
exp ((r : ℝ) * (c - 1)) := by
rw [ProbabilityTheory.integral_poissonMeasure]
simp only [smul_eq_mul]
rw [show (fun n : ℕ => exp (-(r : ℝ)) * (r : ℝ) ^ n /
(n.factorial : ℝ) * c ^ n) =
fun n => exp (-(r : ℝ)) * (((r : ℝ) * c) ^ n /
(n.factorial : ℝ)) by
funext n
rw [mul_pow]
ring]
rw [tsum_mul_left]
rw [show (∑' n : ℕ, ((r : ℝ) * c) ^ n / (n.factorial : ℝ)) =
exp ((r : ℝ) * c) by
simpa only [Real.exp_eq_exp_ℝ] using
(NormedSpace.expSeries_div_hasSum_exp ((r : ℝ) * c)).tsum_eq]
rw [← Real.exp_add]
congr 1
ring
have heq (N : ℕ) :
(exp (μstar - μ) * (μ / μstar) ^ N) *
(exp (μstar - ν) * (ν / μstar) ^ N) =
exp (μstar - μ + (μstar - ν)) *
((μ / μstar) * (ν / μstar)) ^ N := by
rw [Real.exp_add, mul_pow]
ring
simp_rw [heq]
rw [integral_const_mul, hmgf]
rw [← Real.exp_add, Real.coe_toNNReal _ (le_of_lt hstar)]
congr 1
field_simp
ring
theorem sparse_likelihood
In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
n :
ℕ
d :
ℕ
ε :
ℝ
M :
ℝ
B :
ℝ
hn :
1 ≤ n
hd :
2 ≤ d
hε :
0 < ε
hεhalf :
ε ≤ 1 / 2
hM :
2 ≤ M
hB :
B > 2
n d :
ℕ
ε M B :
ℝ
zv :
Fin d → ℝ
hn :
1 ≤ n
hd :
2 ≤ d
hε :
0 < ε
hεhalf :
ε ≤ 1 / 2
hM :
2 ≤ M
hB :
B > 2
hz :
∀ x, zv x ∈ Icc 0 M
conclusion
P :
sparseLaw (lt_of_lt_of_le (by decide : 0 < 2) hd) ε M ⟨hε, hεhalf⟩ zv hz
ObservedClass ε P
x :
Fin d
cellMass P x = (1 - ε + ε * zv x) / (d * sparseNormalizerFormula d ε M zv hz)
propensity P x = ε * (1 - ε) * (1 + zv x) / (1 - ε + ε * zv x)
outcomeMean P false x = 1 / 2
outcomeMean P true x = zv x / (1 + zv x)
observedValue P
= 1 / 2
(∫ N : Fin 4 → ℕ, oneCellLikelihood (hMpaper := hM) B n d ε M z N * oneCellLikelihood (hMpaper := hM) B n d ε M w N ∂poissonCellLaw B n d ε (sparseReference (_hMpaper := hM) M))
= exp (sparseGramCoefficient B hB n d ε M ⟨hε, hεhalf⟩ hM * (z - sparseReference (_hMpaper := hM) M) * (w - sparseReference (_hMpaper := hM) M))
sparseGramCoefficient B hB n d ε M ⟨hε, hεhalf⟩ hM ≤ 2 * B * n * ε / (d * M)
Proof (Lean source)
lemma sparse_likelihood :
∀ (n d : ℕ) (ε M B : ℝ) (zv : Fin d → ℝ)
(hn : 1 ≤ n) (hd : 2 ≤ d) (hε : 0 < ε) (hεhalf : ε ≤ 1 / 2)
(hM : 2 ≤ M) (hB : B > 2) (hz : ∀ x, zv x ∈ Icc 0 M),
let P := sparseLaw (lt_of_lt_of_le (by decide : 0 < 2) hd)
ε M ⟨hε, hεhalf⟩ zv hz
ObservedClass ε P ∧
(∀ x : Fin d,
cellMass P x = (1 - ε + ε * zv x) / (d * sparseNormalizerFormula d ε M zv hz) ∧
propensity P x =
ε * (1 - ε) * (1 + zv x) / (1 - ε + ε * zv x) ∧
outcomeMean P false x = 1 / 2 ∧
outcomeMean P true x = zv x / (1 + zv x)) ∧
observedValue P =
1 / 2 + (1 / (2 * d * sparseNormalizerFormula d ε M zv hz)) *
∑ x : Fin d, phiEpsFormula ε (zv x) ∧
(∀ z w : ℝ, z ∈ Icc 0 M → w ∈ Icc 0 M →
(∫ N : Fin 4 → ℕ,
oneCellLikelihood (hMpaper := hM) B n d ε M z N *
oneCellLikelihood (hMpaper := hM) B n d ε M w N
∂poissonCellLaw B n d ε (sparseReference (_hMpaper := hM) M)) =
exp (sparseGramCoefficient B hB n d ε M ⟨hε, hεhalf⟩ hM *
(z - sparseReference (_hMpaper := hM) M) * (w - sparseReference (_hMpaper := hM) M))) ∧
sparseGramCoefficient B hB n d ε M ⟨hε, hεhalf⟩ hM ≤ 2 * B * n * ε / (d * M) := by
intro n d ε M B zv hn hd hε hεhalf hM hB hz
dsimp
refine ⟨?_, ?_, ?_, ?_, sparseGramCoefficient_bound B hB n d hn hd ε M hε hεhalf hM⟩
· refine ⟨?_⟩
intro x _
have hdR : (0 : ℝ) < d := by exact_mod_cast (show 0 < d by omega)
have hS : 0 < sparseNormalizerFormula d ε M zv hz := by
unfold sparseNormalizerFormula
have hs : 0 ≤ ∑ x : Fin d, zv x :=
Finset.sum_nonneg (fun x _ => (hz x).1)
have : 0 ≤ ε / (d : ℝ) * ∑ x : Fin d, zv x :=
mul_nonneg (div_nonneg (le_of_lt hε) (le_of_lt hdR)) hs
linarith [hεhalf]
have hzx : 0 ≤ zv x := (hz x).1
have hc : 0 < 1 - ε + ε * zv x := by
nlinarith [mul_nonneg (le_of_lt hε) hzx]
let P := sparseLaw (lt_of_lt_of_le (by decide : 0 < 2) hd)
ε M ⟨hε, hεhalf⟩ zv hz
have hcell : cellMass P x =
(1 - ε + ε * zv x) / (d * sparseNormalizerFormula d ε M zv hz) := by
simp [P, cellMass, sparseLaw_jointMass, sparseMass,
sparseUnnormalizedMass]
field_simp
ring
have htrue : armMass P true x =
ε * (1 - ε) * (1 + zv x) / (d * sparseNormalizerFormula d ε M zv hz) := by
simp [P, armMass, sparseLaw_jointMass, sparseMass,
sparseUnnormalizedMass]
field_simp
ring
have hπ : propensity P x =
ε * (1 - ε) * (1 + zv x) / (1 - ε + ε * zv x) := by
rw [propensity, htrue, hcell]
field_simp
dsimp [P] at hπ ⊢
rw [hπ]
constructor
· apply (le_div_iff₀ hc).2
nlinarith [mul_nonneg (by linarith : 0 ≤ 1 - 2 * ε) hzx]
· apply (div_le_iff₀ hc).2
nlinarith [mul_nonneg (by linarith : 0 ≤ 1 - ε)
(by linarith : 0 ≤ 1 - 2 * ε)]
· intro x
have hdR : (0 : ℝ) < d := by exact_mod_cast (show 0 < d by omega)
have hS : 0 < sparseNormalizerFormula d ε M zv hz := by
unfold sparseNormalizerFormula
have hs : 0 ≤ ∑ x : Fin d, zv x := Finset.sum_nonneg (fun x _ => (hz x).1)
have : 0 ≤ ε / (d : ℝ) * ∑ x : Fin d, zv x :=
mul_nonneg (div_nonneg (le_of_lt hε) (le_of_lt hdR)) hs
linarith [hεhalf]
have hzx : 0 ≤ zv x := (hz x).1
have hc : 0 < 1 - ε + ε * zv x := by nlinarith [mul_nonneg (le_of_lt hε) hzx]
have ht : 0 < 1 + zv x := by linarith
let P := sparseLaw (lt_of_lt_of_le (by decide : 0 < 2) hd)
ε M ⟨hε, hεhalf⟩ zv hz
have hcell : cellMass P x =
(1 - ε + ε * zv x) / (d * sparseNormalizerFormula d ε M zv hz) := by
simp [P, cellMass, sparseLaw_jointMass, sparseMass,
sparseUnnormalizedMass]
field_simp
ring
have htrue : armMass P true x =
ε * (1 - ε) * (1 + zv x) / (d * sparseNormalizerFormula d ε M zv hz) := by
simp [P, armMass, sparseLaw_jointMass, sparseMass,
sparseUnnormalizedMass]
field_simp
ring
have hfalse : armMass P false x =
((1 - ε) ^ 2 + ε ^ 2 * zv x) / (d * sparseNormalizerFormula d ε M zv hz) := by
simp [P, armMass, sparseLaw_jointMass, sparseMass,
sparseUnnormalizedMass]
field_simp
have hyt : jointMass P x true true =
ε * (1 - ε) * zv x / (d * sparseNormalizerFormula d ε M zv hz) := by
simp [P, sparseLaw_jointMass, sparseMass, sparseUnnormalizedMass]
ring
have hyf : jointMass P x false true =
((1 - ε) ^ 2 + ε ^ 2 * zv x) / (2 * d * sparseNormalizerFormula d ε M zv hz) := by
simp [P, sparseLaw_jointMass, sparseMass, sparseUnnormalizedMass]
ring
have hε1 : 0 < 1 - ε := by linarith
have hA : 0 < (1 - ε) ^ 2 + ε ^ 2 * zv x := by positivity
dsimp [P] at hcell htrue hfalse hyt hyf ⊢
refine ⟨hcell, ?_, ?_, ?_⟩
· rw [propensity, htrue, hcell]
field_simp
· rw [outcomeMean, hyf, hfalse]
field_simp
· rw [outcomeMean, hyt, htrue]
field_simp
· let P := sparseLaw (lt_of_lt_of_le (by decide : 0 < 2) hd)
ε M ⟨hε, hεhalf⟩ zv hz
have hdR : (0 : ℝ) < d := by exact_mod_cast (show 0 < d by omega)
have hS : 0 < sparseNormalizerFormula d ε M zv hz := by
unfold sparseNormalizerFormula
have hs : 0 ≤ ∑ x : Fin d, zv x :=
Finset.sum_nonneg (fun x _ => (hz x).1)
have : 0 ≤ ε / (d : ℝ) * ∑ x : Fin d, zv x :=
mul_nonneg (div_nonneg (le_of_lt hε) (le_of_lt hdR)) hs
linarith [hεhalf]
have hden : 0 < (d : ℝ) * sparseNormalizerFormula d ε M zv hz := mul_pos hdR hS
have hcell (x : Fin d) : cellMass P x =
(1 - ε + ε * zv x) / (d * sparseNormalizerFormula d ε M zv hz) := by
simp [P, cellMass, sparseLaw_jointMass, sparseMass, sparseUnnormalizedMass]
field_simp
ring
have hmeans (x : Fin d) : outcomeMean P false x = 1 / 2 ∧
outcomeMean P true x = zv x / (1 + zv x) := by
have hzx : 0 ≤ zv x := (hz x).1
have ht : 0 < 1 + zv x := by linarith
have htrue : armMass P true x =
ε * (1 - ε) * (1 + zv x) / (d * sparseNormalizerFormula d ε M zv hz) := by
simp [P, armMass, sparseLaw_jointMass, sparseMass, sparseUnnormalizedMass]
field_simp
ring
have hfalse : armMass P false x =
((1 - ε) ^ 2 + ε ^ 2 * zv x) / (d * sparseNormalizerFormula d ε M zv hz) := by
simp [P, armMass, sparseLaw_jointMass, sparseMass, sparseUnnormalizedMass]
field_simp
have hyt : jointMass P x true true =
ε * (1 - ε) * zv x / (d * sparseNormalizerFormula d ε M zv hz) := by
simp [P, sparseLaw_jointMass, sparseMass, sparseUnnormalizedMass]
ring
have hyf : jointMass P x false true =
((1 - ε) ^ 2 + ε ^ 2 * zv x) / (2 * d * sparseNormalizerFormula d ε M zv hz) := by
simp [P, sparseLaw_jointMass, sparseMass, sparseUnnormalizedMass]
ring
have hε1 : 0 < 1 - ε := by linarith
have hA : 0 < (1 - ε) ^ 2 + ε ^ 2 * zv x := by positivity
constructor
· rw [outcomeMean, hyf, hfalse]
field_simp
· rw [outcomeMean, hyt, htrue]
field_simp
have hmax (x : Fin d) :
max (1 / 2 : ℝ) (zv x / (1 + zv x)) =
1 / 2 + max (zv x - 1) 0 / (2 * (1 + zv x)) := by
have ht : 0 < 1 + zv x := by linarith [(hz x).1]
rcases le_total (zv x) 1 with hle | hge
· have hfrac : zv x / (1 + zv x) ≤ (1 / 2 : ℝ) := by
apply (div_le_iff₀ ht).2
linarith
rw [max_eq_left hfrac, max_eq_right (by linarith : zv x - 1 ≤ 0)]
ring
· have hfrac : (1 / 2 : ℝ) ≤ zv x / (1 + zv x) := by
apply (le_div_iff₀ ht).2
linarith
rw [max_eq_right hfrac, max_eq_left (by linarith : 0 ≤ zv x - 1)]
field_simp
ring
have hpoint (x : Fin d) :
cellMass P x * max (outcomeMean P false x) (outcomeMean P true x) =
cellMass P x / 2 + phiEpsFormula ε (zv x) /
(2 * d * sparseNormalizerFormula d ε M zv hz) := by
rw [(hmeans x).1, (hmeans x).2, hmax, hcell]
unfold phiEpsFormula
have ht : 0 < 1 + zv x := by linarith [(hz x).1]
field_simp
have hsum : (∑ x : Fin d, cellMass P x) = 1 := by
simp_rw [hcell]
rw [← Finset.sum_div]
have hnum : (∑ x : Fin d, (1 - ε + ε * zv x)) =
(d : ℝ) * sparseNormalizerFormula d ε M zv hz := by
simp [sparseNormalizerFormula, Finset.sum_add_distrib, ← Finset.mul_sum]
field_simp
rw [hnum]
exact div_self hden.ne'
dsimp [P] at hpoint hsum ⊢
unfold observedValue
simp_rw [hpoint]
rw [Finset.sum_add_distrib, ← Finset.sum_div, hsum]
rw [← Finset.sum_div]
ring
· intro z w _ _
have hstar (j : Fin 4) :
0 < poissonCellMean B n d ε (sparseReference (_hMpaper := hM) M) j := by
have hdR : (0 : ℝ) < d := by exact_mod_cast (show 0 < d by omega)
have hnR : (0 : ℝ) < n := by exact_mod_cast (show 0 < n by omega)
have hMstar : 0 < sparseReference (_hMpaper := hM) M := by unfold sparseReference; linarith
have hε1 : 0 < 1 - ε := by linarith
fin_cases j <;> simp [poissonCellMean, cellIdx] <;> positivity
have hfun (N : Fin 4 → ℕ) :
oneCellLikelihood (hMpaper := hM) B n d ε M z N *
oneCellLikelihood (hMpaper := hM) B n d ε M w N =
∏ j : Fin 4,
(exp (poissonCellMean B n d ε (sparseReference (_hMpaper := hM) M) j -
poissonCellMean B n d ε z j) *
(poissonCellMean B n d ε z j /
poissonCellMean B n d ε (sparseReference (_hMpaper := hM) M) j) ^ N j) *
(exp (poissonCellMean B n d ε (sparseReference (_hMpaper := hM) M) j -
poissonCellMean B n d ε w j) *
(poissonCellMean B n d ε w j /
poissonCellMean B n d ε (sparseReference (_hMpaper := hM) M) j) ^ N j) := by
unfold oneCellLikelihood
rw [← Finset.prod_mul_distrib]
simp_rw [hfun]
rw [show poissonCellLaw B n d ε (sparseReference (_hMpaper := hM) M) =
Measure.pi (fun j : Fin 4 =>
poissonMeasure (toNNReal
(poissonCellMean B n d ε (sparseReference (_hMpaper := hM) M) j))) by rfl]
rw [integral_fintype_prod_eq_prod (fun j : Fin 4 => fun k : ℕ =>
(exp (poissonCellMean B n d ε (sparseReference (_hMpaper := hM) M) j -
poissonCellMean B n d ε z j) *
(poissonCellMean B n d ε z j /
poissonCellMean B n d ε (sparseReference (_hMpaper := hM) M) j) ^ k) *
(exp (poissonCellMean B n d ε (sparseReference (_hMpaper := hM) M) j -
poissonCellMean B n d ε w j) *
(poissonCellMean B n d ε w j /
poissonCellMean B n d ε (sparseReference (_hMpaper := hM) M) j) ^ k))]
simp_rw [poisson_likelihood_gram_scalar _ _ _ (hstar _)]
rw [← Real.exp_sum]
congr 1
simp [Fin.sum_univ_four, poissonCellMean, cellIdx,
sparseGramCoefficient, sparseReference]
field_simp
ring
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.UpperAggregation 4 declarations Roadmap equations (30) and (32) aggregate cellwise pilot second moments and square-root approximation biases.
Finite-sum bounds for upper-risk aggregation
Roadmap equations (30) and (32) aggregate cellwise pilot second moments and square-root approximation biases. These lemmas retain null cells and use only the probability-mass normalization and the active-branch rate bound.
theorem cellMass_sum_sqrt_sq_le
Equation (30): probability masses control the squared sum of their square roots. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
Proof (Lean source)
lemma cellMass_sum_sqrt_sq_le {d : ℕ} (P : DiscreteLaw d) :
(∑ x : Fin d, sqrt (cellMass P x)) ^ 2 ≤ (d : ℝ) := by
have hp (x : Fin d) : 0 ≤ cellMass P x :=
Finset.sum_nonneg fun a _ => Finset.sum_nonneg fun y _ => toReal_nonneg
have h := Finset.sum_mul_sq_le_sq_mul_sq (Finset.univ : Finset (Fin d))
(fun _ => (1 : ℝ)) (fun x => sqrt (cellMass P x))
simpa only [one_mul, one_pow, Finset.sum_const, Finset.card_univ,
Fintype.card_fin, nsmul_eq_mul, mul_one, Real.sq_sqrt (hp _),
cellMass_sum_eq_one] using h
theorem cellMass_sum_scaled_sqrt_sq_le
The square-root part of the bias in (32) has squared size at most R. The positive scale is m ε L in the active branch. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Proof (Lean source)
lemma cellMass_sum_scaled_sqrt_sq_le {d : ℕ} (P : DiscreteLaw d)
(a : ℝ) (ha : 0 < a) :
(∑ x : Fin d, sqrt (cellMass P x / a)) ^ 2 ≤ (d : ℝ) / a := by
have hp (x : Fin d) : 0 ≤ cellMass P x / a := by
apply div_nonneg _ ha.le
exact Finset.sum_nonneg fun b _ => Finset.sum_nonneg fun y _ => toReal_nonneg
have h := Finset.sum_mul_sq_le_sq_mul_sq (Finset.univ : Finset (Fin d))
(fun _ => (1 : ℝ)) (fun x => sqrt (cellMass P x / a))
simp only [one_mul, one_pow, Finset.sum_const, Finset.card_univ,
Fintype.card_fin, nsmul_eq_mul, mul_one, Real.sq_sqrt (hp _)] at h
rw [← Finset.sum_div, cellMass_sum_eq_one] at h
simpa only [mul_one_div] using h
theorem activeBranch_total_bias_sq_le
Equation (32) with an explicit universal constant on the active branch R ≤ 8. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Proof (Lean source)
lemma activeBranch_total_bias_sq_le {d : ℕ} (P : DiscreteLaw d)
(a : ℝ) (ha : 0 < a) (hactive : (d : ℝ) / a ≤ 8) :
((∑ x : Fin d, sqrt (cellMass P x / a)) + (d : ℝ) / a) ^ 2 ≤
18 * ((d : ℝ) / a) := by
have hs := cellMass_sum_scaled_sqrt_sq_le P a ha
have hR : 0 ≤ (d : ℝ) / a := div_nonneg (Nat.cast_nonneg _) ha.le
have hR2 : ((d : ℝ) / a) ^ 2 ≤ 8 * ((d : ℝ) / a) := by
nlinarith
nlinarith [sq_nonneg ((∑ x : Fin d, sqrt (cellMass P x / a)) - (d : ℝ) / a)]
theorem clippingScale_secondMoment_sum_le
Equation (30): summing a cellwise radius-second-moment estimate uses only total mass one. Here v is the second moment already established by the pilot calculation (19). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma clippingScale_secondMoment_sum_le {d : ℕ} (P : DiscreteLaw d)
(v : Fin d → ℝ) (C L m ε : ℝ)
(hv : ∀ x, v x ≤ C * (cellMass P x * L / (m * ε) + L ^ 2 / (m ^ 2 * ε ^ 2))) :
(∑ x : Fin d, v x) ≤
C * (L / (m * ε) + (d : ℝ) * L ^ 2 / (m ^ 2 * ε ^ 2)) := by
calc
_ ≤ ∑ x : Fin d,
C * (cellMass P x * L / (m * ε) + L ^ 2 / (m ^ 2 * ε ^ 2)) :=
Finset.sum_le_sum fun x _ => hv x
_ = _ := by
rw [← Finset.mul_sum, Finset.sum_add_distrib, ← Finset.sum_div,
← Finset.sum_mul, cellMass_sum_eq_one]
simp only [one_mul, Finset.sum_const, Finset.card_univ,
Fintype.card_fin, nsmul_eq_mul]
ring
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.UpperArmMean 13 declarations Roadmap equation (5) combines the centered selected-arm variance with the empty-arm contribution.
Bernoulli arm-mean second moments
Roadmap equation (5) combines the centered selected-arm variance with the empty-arm contribution. The finite observed law allows boundary means and null cells throughout the residual calculation.
theorem armMass_mul_outcomeMean
Multiplying the totalized arm mean by its arm mass recovers the success mass, including a null arm. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma armMass_mul_outcomeMean {d : ℕ} (P : DiscreteLaw d) (a : Bool) (x : Fin d) :
armMass P a x * outcomeMean P a x = jointMass P x a true := by
have hsum : armMass P a x = jointMass P x a false + jointMass P x a true := by
simp [armMass]; ring
by_cases h : armMass P a x = 0
· have hz : jointMass P x a true = 0 := by
have h0 : 0 ≤ jointMass P x a false := ENNReal.toReal_nonneg
have h1 : 0 ≤ jointMass P x a true := ENNReal.toReal_nonneg
rw [h] at hsum
linarith
simp [h, hz]
· rw [outcomeMean]
field_simp [h]
theorem integral_observedArm_eq
Integrals restricted to one observed arm reduce to its two outcome atoms. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
Proof (Lean source)
lemma integral_observedArm_eq {d : ℕ} (P : DiscreteLaw d) (a : Bool) (x : Fin d)
(f : Bool → ℝ) :
(∫ z in {z : Obs d | z.1 = x ∧ z.2.1 = a}, f z.2.2 ∂P.pmf.toMeasure) =
jointMass P x a false * f false + jointMass P x a true * f true := by
classical
rw [← integral_indicator (Set.toFinite _).measurableSet, PMF.integral_eq_sum]
cases a <;> simp [indicator, Fintype.sum_prod_type, jointMass, smul_eq_mul] <;> ring
theorem observedArm_residual_centered
The binary outcome residual is centered on every arm, including null arms. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
(∫ z in {z : Obs d | z.1 = x ∧ z.2.1 = a}, ((if z.2.2 then (1 : ℝ) else 0) - outcomeMean P a x) ∂P.pmf.toMeasure)
= 0
Proof (Lean source)
lemma observedArm_residual_centered {d : ℕ} (P : DiscreteLaw d) (a : Bool) (x : Fin d) :
(∫ z in {z : Obs d | z.1 = x ∧ z.2.1 = a},
((if z.2.2 then (1 : ℝ) else 0) - outcomeMean P a x) ∂P.pmf.toMeasure) = 0 := by
rw [integral_observedArm_eq P a x (fun y => (if y then (1 : ℝ) else 0) - outcomeMean P a x)]
have hm := armMass_mul_outcomeMean P a x
have hs : armMass P a x = jointMass P x a false + jointMass P x a true := by
simp [armMass]; ring
simp only [Bool.false_eq_true, ↓reduceIte]
rw [hs] at hm
nlinarith
theorem observedArm_residual_secondMoment_le
The binary residual second moment is at most one quarter of the arm mass. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
(∫ z in {z : Obs d | z.1 = x ∧ z.2.1 = a}, ((if z.2.2 then (1 : ℝ) else 0) - outcomeMean P a x) ^ 2 ∂P.pmf.toMeasure)
≤ armMass P a x * (1 / 2 : ℝ) ^ 2
Proof (Lean source)
lemma observedArm_residual_secondMoment_le {d : ℕ} (P : DiscreteLaw d)
(a : Bool) (x : Fin d) :
(∫ z in {z : Obs d | z.1 = x ∧ z.2.1 = a},
((if z.2.2 then (1 : ℝ) else 0) - outcomeMean P a x) ^ 2 ∂P.pmf.toMeasure) ≤
armMass P a x * (1 / 2 : ℝ) ^ 2 := by
rw [integral_observedArm_eq P a x (fun y => ((if y then (1 : ℝ) else 0) - outcomeMean P a x) ^ 2)]
have hm := armMass_mul_outcomeMean P a x
have hs : armMass P a x = jointMass P x a false + jointMass P x a true := by
simp [armMass]; ring
have hp : 0 ≤ armMass P a x := Finset.sum_nonneg fun _ _ => ENNReal.toReal_nonneg
have hvar : outcomeMean P a x * (1 - outcomeMean P a x) ≤ (1 / 2 : ℝ) ^ 2 := by
nlinarith [sq_nonneg (outcomeMean P a x - 1 / 2)]
have heq : jointMass P x a false * (0 - outcomeMean P a x) ^ 2 +
jointMass P x a true * (1 - outcomeMean P a x) ^ 2 =
armMass P a x * (outcomeMean P a x * (1 - outcomeMean P a x)) := by
calc
_ = armMass P a x * outcomeMean P a x ^ 2 + jointMass P x a true *
(1 - 2 * outcomeMean P a x) := by rw [hs]; ring
_ = _ := by rw [← hm]; ring
simp only [Bool.false_eq_true, ↓reduceIte]
rw [heq]
exact mul_le_mul_of_nonneg_left hvar hp
The sum of squares of selected-arm averaging coefficients is the totalized inverse count. This is the finite-design variance factor in equation (5). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
Proof (Lean source)
lemma selectedArm_coeff_sq_sum {n d : ℕ} (D : Fin n → Fin d × Bool)
(x : Fin d) (a : Bool) :
(∑ i, (if (D i).1 = x ∧ (D i).2 = a then
((∑ j : Fin n, if (D j).1 = x ∧ (D j).2 = a then 1 else 0 : ℕ) : ℝ)⁻¹
else 0) ^ 2) =
if 0 < (∑ j : Fin n, if (D j).1 = x ∧ (D j).2 = a then 1 else 0 : ℕ) then
((∑ j : Fin n, if (D j).1 = x ∧ (D j).2 = a then 1 else 0 : ℕ) : ℝ)⁻¹
else 0 := by
classical
let N : ℕ := ∑ j, if (D j).1 = x ∧ (D j).2 = a then 1 else 0
have heq : (∑ i, (if (D i).1 = x ∧ (D i).2 = a then (N : ℝ)⁻¹ else 0) ^ 2) =
(N : ℝ) * (N : ℝ)⁻¹ ^ 2 := by
rw [show (N : ℝ) = ∑ i : Fin n, if (D i).1 = x ∧ (D i).2 = a then
(1 : ℝ) else 0 by simp [N, Nat.cast_sum], Finset.sum_mul]
apply Finset.sum_congr rfl
intro i hi
split_ifs <;> simp
change (∑ i, (if (D i).1 = x ∧ (D i).2 = a then (N : ℝ)⁻¹ else 0) ^ 2) =
if 0 < N then (N : ℝ)⁻¹ else 0
rw [heq]
by_cases hN : 0 < N
· rw [if_pos hN]
have h : (N : ℝ) ≠ 0 := by exact_mod_cast (Nat.ne_of_gt hN)
field_simp
· have h : N = 0 := Nat.eq_zero_of_not_pos hN
simp [h]
theorem selectedArm_residual_eq
The selected design-weighted residual is the centered empirical numerator scaled by its arm count; at an empty arm it is zero. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
designWeightedResidual fst (fun z : Obs d => z.2.1) (fun z : Obs d => if z.2.2 then 1 else 0) (outcomeMean P) (fun D i => if (D i).1 = x ∧ (D i).2 = a then ((∑ j : Fin n, if (D j).1 = x ∧ (D j).2 = a then 1 else 0 : ℕ) : ℝ)⁻¹ else 0) o
= ((sampleSuccessCount o x a : ℝ) - (sampleArmCount o x a : ℝ) * outcomeMean P a x) / sampleArmCount o x a
Proof (Lean source)
lemma selectedArm_residual_eq {n d : ℕ} (P : DiscreteLaw d) (o : Fin n → Obs d)
(x : Fin d) (a : Bool) :
designWeightedResidual
fst (fun z : Obs d => z.2.1) (fun z : Obs d => if z.2.2 then 1 else 0)
(outcomeMean P)
(fun D i => if (D i).1 = x ∧ (D i).2 = a then
((∑ j : Fin n, if (D j).1 = x ∧ (D j).2 = a then 1 else 0 : ℕ) : ℝ)⁻¹
else 0) o =
((sampleSuccessCount o x a : ℝ) -
(sampleArmCount o x a : ℝ) * outcomeMean P a x) / sampleArmCount o x a := by
classical
simp only [designWeightedResidual,
sampleDesign]
change (∑ i, (if (o i).1 = x ∧ (o i).2.1 = a then
(sampleArmCount o x a : ℝ)⁻¹ else 0) *
((if (o i).2.2 then 1 else 0) - outcomeMean P (o i).2.1 (o i).1)) = _
have hterm (i : Fin n) : (if (o i).1 = x ∧ (o i).2.1 = a then
(sampleArmCount o x a : ℝ)⁻¹ else 0) *
((if (o i).2.2 then 1 else 0) - outcomeMean P (o i).2.1 (o i).1) =
(sampleArmCount o x a : ℝ)⁻¹ *
((if (o i).1 = x ∧ (o i).2.1 = a ∧ (o i).2.2 = true then 1 else 0) -
(if (o i).1 = x ∧ (o i).2.1 = a then 1 else 0) * outcomeMean P a x) := by
by_cases h : (o i).1 = x ∧ (o i).2.1 = a
· simp only [h, if_pos, h.1, h.2, true_and, one_mul]
· have hn : ¬ ((o i).1 = x ∧ (o i).2.1 = a ∧ (o i).2.2 = true) :=
fun hh => h ⟨hh.1, hh.2.1⟩
simp [h, hn]
simp_rw [hterm]
rw [← Finset.mul_sum, Finset.sum_sub_distrib, ← Finset.sum_mul]
simp only [sampleSuccessCount, sampleArmCount, Nat.cast_sum, Nat.cast_ite,
Nat.cast_one, Nat.cast_zero, div_eq_mul_inv, mul_comm]
theorem empiricalArmCentered_mse_le_inverse
Conditional Bernoulli centering removes cross-coordinate terms even though the averaging denominator depends on the entire observed design. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
(∫ o, (((sampleSuccessCount o x a : ℝ) - (sampleArmCount o x a : ℝ) * outcomeMean P a x) / sampleArmCount o x a) ^ 2 ∂productLaw P n)
≤ (1 / 4 : ℝ) * ∫ o, if 0 < sampleArmCount o x a then (sampleArmCount o x a : ℝ)⁻¹ else 0 ∂productLaw P n
Proof (Lean source)
lemma empiricalArmCentered_mse_le_inverse {n d : ℕ} (P : DiscreteLaw d)
(x : Fin d) (a : Bool) :
(∫ o, (((sampleSuccessCount o x a : ℝ) -
(sampleArmCount o x a : ℝ) * outcomeMean P a x) / sampleArmCount o x a) ^ 2
∂productLaw P n) ≤
(1 / 4 : ℝ) * ∫ o,
if 0 < sampleArmCount o x a then (sampleArmCount o x a : ℝ)⁻¹ else 0
∂productLaw P n := by
have hmem : ∀ b y, MemLp
(supportedArmGroupResidual fst
(fun z : Obs d => z.2.1) (fun z : Obs d => if z.2.2 then 1 else 0)
(outcomeMean P) b y) 2 P.pmf.toMeasure := by
intro b y
apply (memLp_two_iff_integrable_sq ?_).2 Integrable.of_finite
fun_prop
have hc : ∀ b y, (∫ z in armGroupEvent fst
(fun z : Obs d => z.2.1) b y,
armGroupResidual (fun z : Obs d => if z.2.2 then 1 else 0)
(outcomeMean P) b y z ∂P.pmf.toMeasure) = 0 :=
fun b y => observedArm_residual_centered P b y
have hs : ∀ b y, (∫ z in armGroupEvent fst
(fun z : Obs d => z.2.1) b y,
(armGroupResidual (fun z : Obs d => if z.2.2 then 1 else 0)
(outcomeMean P) b y z) ^ 2 ∂P.pmf.toMeasure) ≤
(P.pmf.toMeasure (armGroupEvent fst
(fun z : Obs d => z.2.1) b y)).toReal * (1 / 2 : ℝ) ^ 2 := by
intro b y
change _ ≤ P.pmf.toMeasure.real {z : Obs d | z.1 = y ∧ z.2.1 = b} * _
rw [armEvent_mass_eq_armMass]
exact observedArm_residual_secondMoment_le P b y
classical
have h := integral_designWeightedResidual_sq_le
n P.pmf.toMeasure fst (fun z : Obs d => z.2.1)
(fun z : Obs d => if z.2.2 then 1 else 0) (outcomeMean P) (1 / 2)
(by fun_prop) (by fun_prop) hmem hc hs
(fun D i => if (D i).1 = x ∧ (D i).2 = a then
((∑ j : Fin n, if (D j).1 = x ∧ (D j).2 = a then 1 else 0 : ℕ) : ℝ)⁻¹
else 0) (fun _ => 1) (fun _ => by norm_num)
simp only [one_mul, selectedArm_residual_eq, selectedArm_coeff_sq_sum] at h
convert h using 1
· rfl
· norm_num only [show (1 / 2 : ℝ) ^ 2 = 1 / 4 by norm_num]
congr 1
theorem empiricalArm_error_sq_eq
Separating the empty arm from the centered numerator gives the exact pointwise variance-plus-empty-arm decomposition. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
((sampleSuccessCount o x a : ℝ) / sampleArmCount o x a - outcomeMean P a x) ^ 2
= (((sampleSuccessCount o x a : ℝ) - (sampleArmCount o x a : ℝ) * outcomeMean P a x) / sampleArmCount o x a) ^ 2
+ (if sampleArmCount o x a = 0 then (1 : ℝ) else 0) * outcomeMean P a x ^ 2
Proof (Lean source)
lemma empiricalArm_error_sq_eq {n d : ℕ} (P : DiscreteLaw d) (o : Fin n → Obs d)
(x : Fin d) (a : Bool) :
((sampleSuccessCount o x a : ℝ) / sampleArmCount o x a - outcomeMean P a x) ^ 2 =
(((sampleSuccessCount o x a : ℝ) -
(sampleArmCount o x a : ℝ) * outcomeMean P a x) / sampleArmCount o x a) ^ 2 +
(if sampleArmCount o x a = 0 then (1 : ℝ) else 0) * outcomeMean P a x ^ 2 := by
by_cases h : sampleArmCount o x a = 0
· simp [h]
· have hR : (sampleArmCount o x a : ℝ) ≠ 0 := by exact_mod_cast h
rw [if_neg h, zero_mul, add_zero, sub_div, mul_div_cancel_left₀ _ hR]
theorem empiricalOutcomeMean_mse_le
Equation (5) for an actual observed arm. The inverse occupied-count term and the empty-arm term are bounded separately using equation (4). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
(∫ o, ((sampleSuccessCount o x a : ℝ) / sampleArmCount o x a - outcomeMean P a x) ^ 2 ∂productLaw P n)
≤ 3 / (2 * ((n + 1 : ℝ) * armMass P a x))
Proof (Lean source)
lemma empiricalOutcomeMean_mse_le {n d : ℕ} (P : DiscreteLaw d) (x : Fin d)
(a : Bool) (hp : 0 < armMass P a x) :
(∫ o, ((sampleSuccessCount o x a : ℝ) / sampleArmCount o x a -
outcomeMean P a x) ^ 2 ∂productLaw P n) ≤
3 / (2 * ((n + 1 : ℝ) * armMass P a x)) := by
let : IsProbabilityMeasure (productLaw P n) := by
dsimp [productLaw]
infer_instance
have hm := outcomeMean_mem_unitInterval P a x
have hm2 : outcomeMean P a x ^ 2 ≤ 1 := by nlinarith [hm.1, hm.2]
have hc := empiricalArmCentered_mse_le_inverse (n := n) P x a
have hi := integral_sampleArmCount_inverse_le (n := n) P x a hp
have he := integral_sampleArmCount_empty_le (n := n) P x a hp
have hden : 0 < (n + 1 : ℝ) * armMass P a x := mul_pos (by positivity) hp
calc
_ = (∫ o, (((sampleSuccessCount o x a : ℝ) -
(sampleArmCount o x a : ℝ) * outcomeMean P a x) / sampleArmCount o x a) ^ 2
∂productLaw P n) +
(∫ o, if sampleArmCount o x a = 0 then (1 : ℝ) else 0 ∂productLaw P n) *
outcomeMean P a x ^ 2 := by
simp_rw [empiricalArm_error_sq_eq]
rw [integral_add Integrable.of_finite Integrable.of_finite, integral_mul_const]
_ ≤ (1 / 4 : ℝ) * (2 / ((n + 1 : ℝ) * armMass P a x)) +
1 / ((n + 1 : ℝ) * armMass P a x) := by
apply add_le_add
· exact hc.trans (mul_le_mul_of_nonneg_left hi (by norm_num))
· exact (mul_le_mul_of_nonneg_right he (sq_nonneg _)).trans
(mul_le_of_le_one_right (by positivity) hm2)
_ = _ := by
field_simp [ne_of_gt hden]
<;> ring
theorem observedClass_armMass_lower
Occupied-cell propensity overlap gives the same lower mass bound in each arm. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
ε * cellMass P x ≤ armMass P a x
Proof (Lean source)
lemma observedClass_armMass_lower {d : ℕ} {ε : ℝ} (P : DiscreteLaw d)
(hP : ObservedClass ε P) (x : Fin d) (a : Bool) (hp : 0 < cellMass P x) :
ε * cellMass P x ≤ armMass P a x := by
obtain ⟨hlo, hhi⟩ := hP.overlap x hp
have hs : cellMass P x = armMass P false x + armMass P true x := by
simp [cellMass, armMass]; ring
rw [propensity] at hlo hhi
have hl := (le_div_iff₀ hp).mp hlo
have hh := (div_le_iff₀ hp).mp hhi
cases a
· nlinarith
· exact hl
theorem cellMass_mul_empiricalOutcomeMean_mse_le
Multiplying equation (5) by cell mass removes the unknown cell mass from its bound. Null cells vanish before any division. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
cellMass P x * (∫ o, ((sampleSuccessCount o x a : ℝ) / sampleArmCount o x a - outcomeMean P a x) ^ 2 ∂productLaw P n)
≤ 3 / (2 * (n + 1 : ℝ) * ε)
Proof (Lean source)
lemma cellMass_mul_empiricalOutcomeMean_mse_le {n d : ℕ} {ε : ℝ}
(P : DiscreteLaw d) (hP : ObservedClass ε P) (hε : 0 < ε)
(x : Fin d) (a : Bool) :
cellMass P x * (∫ o, ((sampleSuccessCount o x a : ℝ) / sampleArmCount o x a -
outcomeMean P a x) ^ 2 ∂productLaw P n) ≤ 3 / (2 * (n + 1 : ℝ) * ε) := by
have hp0 : 0 ≤ cellMass P x :=
Finset.sum_nonneg fun _ _ => Finset.sum_nonneg fun _ _ => ENNReal.toReal_nonneg
by_cases hp : cellMass P x = 0
· rw [hp, zero_mul]
positivity
· have hp1 : 0 < cellMass P x := lt_of_le_of_ne hp0 (Ne.symm hp)
have hl := observedClass_armMass_lower P hP x a hp1
have harm : 0 < armMass P a x := lt_of_lt_of_le (mul_pos hε hp1) hl
have hN : 0 < (n + 1 : ℝ) := by positivity
calc
_ ≤ cellMass P x * (3 / (2 * ((n + 1 : ℝ) * armMass P a x))) :=
mul_le_mul_of_nonneg_left (empiricalOutcomeMean_mse_le P x a harm) hp0
_ = 3 * cellMass P x / (2 * (n + 1 : ℝ) * armMass P a x) := by ring
_ ≤ 3 * cellMass P x / (2 * (n + 1 : ℝ) * (ε * cellMass P x)) := by
gcongr
_ = _ := by field_simp
theorem empiricalArmMax_mse_le
Equation (6): the squared mass-weighted maximum arm error is at most 3d/((n+1)ε), using the two Bernoulli arm-mean bounds. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
(∫ o, (∑ x : Fin d, cellMass P x * max |(sampleSuccessCount o x false : ℝ) / sampleArmCount o x false - outcomeMean P false x| |(sampleSuccessCount o x true : ℝ) / sampleArmCount o x true - outcomeMean P true x|) ^ 2 ∂productLaw P n)
≤ 3 * d / ((n + 1 : ℝ) * ε)
Proof (Lean source)
lemma empiricalArmMax_mse_le {n d : ℕ} {ε : ℝ} (P : DiscreteLaw d)
(hP : ObservedClass ε P) (hε : 0 < ε) :
(∫ o, (∑ x : Fin d, cellMass P x *
max |(sampleSuccessCount o x false : ℝ) / sampleArmCount o x false - outcomeMean P false x|
|(sampleSuccessCount o x true : ℝ) / sampleArmCount o x true - outcomeMean P true x|) ^ 2
∂productLaw P n) ≤ 3 * d / ((n + 1 : ℝ) * ε) := by
let : IsProbabilityMeasure (productLaw P n) := by dsimp [productLaw]; infer_instance
calc
_ ≤ ∫ o, ∑ x : Fin d, cellMass P x * ∑ a : Bool,
((sampleSuccessCount o x a : ℝ) / sampleArmCount o x a - outcomeMean P a x) ^ 2
∂productLaw P n :=
integral_mono Integrable.of_finite Integrable.of_finite
(fun o => cellMass_weighted_armMax_sq_le P (fun a x =>
(sampleSuccessCount o x a : ℝ) / sampleArmCount o x a - outcomeMean P a x))
_ = ∑ x : Fin d, ∑ a : Bool, cellMass P x * ∫ o,
((sampleSuccessCount o x a : ℝ) / sampleArmCount o x a - outcomeMean P a x) ^ 2
∂productLaw P n := by
rw [integral_finsetSum _ (fun _ _ => Integrable.of_finite)]
apply Finset.sum_congr rfl
intro x hx
rw [integral_const_mul, integral_finsetSum _ (fun _ _ => Integrable.of_finite),
Finset.mul_sum]
_ ≤ ∑ x : Fin d, ∑ a : Bool, 3 / (2 * (n + 1 : ℝ) * ε) := by
apply Finset.sum_le_sum
intro x hx
exact Finset.sum_le_sum fun a _ => cellMass_mul_empiricalOutcomeMean_mse_le P hP hε x a
_ = _ := by
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_bool, Fintype.card_fin, nsmul_eq_mul]
field_simp
<;> ring
theorem empiricalValue_sqRisk_le
The projected empirical-ratio estimator has the parametric upper bound needed on the bounded-alphabet branch; both terms in equation (6) are derived under the actual iid law. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
sqRisk (productLaw P n) empiricalValue (observedValue P)
≤ 8 * d / ((n : ℝ) * ε)
Proof (Lean source)
lemma empiricalValue_sqRisk_le {n d : ℕ} {ε : ℝ} (hn : 0 < n)
(P : DiscreteLaw d) (hP : ObservedClass ε P) (hε : 0 < ε) (hε1 : ε ≤ 1 / 2) :
sqRisk (productLaw P n) empiricalValue (observedValue P) ≤
8 * d / ((n : ℝ) * ε) := by
let : IsProbabilityMeasure (productLaw P n) := by dsimp [productLaw]; infer_instance
have hN : (0 : ℝ) < n := by exact_mod_cast hn
have hmass := empiricalMassError_mse_le hn P
have harm := empiricalArmMax_mse_le (n := n) P hP hε
calc
_ ≤ 2 * (∫ o, (∑ x : Fin d,
|(sampleCellCount o x : ℝ) / n - cellMass P x|) ^ 2 ∂productLaw P n) +
2 * (∫ o, (∑ x : Fin d, cellMass P x *
max |(sampleSuccessCount o x false : ℝ) / sampleArmCount o x false - outcomeMean P false x|
|(sampleSuccessCount o x true : ℝ) / sampleArmCount o x true - outcomeMean P true x|) ^ 2
∂productLaw P n) := by
unfold sqRisk
rw [← integral_const_mul, ← integral_const_mul,
← integral_add Integrable.of_finite Integrable.of_finite]
exact integral_mono Integrable.of_finite Integrable.of_finite
(fun o => empiricalValue_sq_error_le_components o P)
_ ≤ 2 * ((d : ℝ) / n) + 2 * (3 * d / ((n + 1 : ℝ) * ε)) := by linarith
_ ≤ 2 * ((d : ℝ) / ((n : ℝ) * ε)) + 2 * (3 * d / ((n : ℝ) * ε)) := by
apply add_le_add
· gcongr
nlinarith
· gcongr
linarith
_ = _ := by ring
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.UpperBadPilotAggregation 6 declarations The exponential estimates in (22) absorb the clipping inflation and yield an alphabet-decaying cell loss.
Aggregating actual bad-pilot losses
The exponential estimates in (22) absorb the clipping inflation and yield an alphabet-decaying cell loss. Summing these actual losses on the active branch gives the variance-scale rate in (31), including null cells.
theorem logAlphabet_exp_decay_mul_rpow_le
An exponential tail at the logarithmic alphabet scale absorbs any smaller prescribed polynomial inflation and decay. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
1 ≤ d
α β δ :
ℝ
hβ :
0 ≤ β
hgap :
α + δ ≤ β
conclusion
(d : ℝ) ^ α * exp (-β * logAlphabet d) ≤ (d : ℝ) ^ (-δ)
Proof (Lean source)
lemma logAlphabet_exp_decay_mul_rpow_le {d : ℕ} (hd : 1 ≤ d)
(α β δ : ℝ) (hβ : 0 ≤ β) (hgap : α + δ ≤ β) :
(d : ℝ) ^ α * exp (-β * logAlphabet d) ≤ (d : ℝ) ^ (-δ) := by
have hd1 : (1 : ℝ) ≤ d := by exact_mod_cast hd
have hd0 : (0 : ℝ) < d := by linarith
have hlog : 0 ≤ log (d : ℝ) := Real.log_nonneg hd1
have hL : logAlphabet d = 1 + log (d : ℝ) := by
rw [logAlphabet, Real.log_mul (Real.exp_pos 1).ne' hd0.ne', Real.log_exp]
rw [Real.rpow_def_of_pos hd0, Real.rpow_def_of_pos hd0, ← Real.exp_add, hL]
apply Real.exp_le_exp.mpr
nlinarith [mul_nonneg (sub_nonneg.mpr hgap) hlog]
theorem badPilot_squared_coefficient_decay_le
The precise squared clipping inflation in (22) is absorbed by the exponential bad-pilot tail, retaining the paper's d⁻⁴ factor. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
1 ≤ d
C D :
ℝ
hC :
0 ≤ C
hD :
0 ≤ D
conclusion
2 * (((d : ℝ) ^ (1 / 4 : ℝ)) ^ 2 * C + D) * exp (-20 * logAlphabet d)
≤ 2 * (C + D) * (d : ℝ) ^ (-4 : ℝ)
Proof (Lean source)
lemma badPilot_squared_coefficient_decay_le {d : ℕ} (hd : 1 ≤ d)
(C D : ℝ) (hC : 0 ≤ C) (hD : 0 ≤ D) :
2 * (((d : ℝ) ^ (1 / 4 : ℝ)) ^ 2 * C + D) *
exp (-20 * logAlphabet d) ≤ 2 * (C + D) * (d : ℝ) ^ (-4 : ℝ) := by
have hd0 : (0 : ℝ) ≤ d := Nat.cast_nonneg _
have hpow : ((d : ℝ) ^ (1 / 4 : ℝ)) ^ 2 = (d : ℝ) ^ (1 / 2 : ℝ) := by
rw [← Real.rpow_mul_natCast hd0]
norm_num
have h₁ := logAlphabet_exp_decay_mul_rpow_le hd (1 / 2) 20 4
(by norm_num) (by norm_num)
have h₂ := logAlphabet_exp_decay_mul_rpow_le hd 0 20 4
(by norm_num) (by norm_num)
simp only [Real.rpow_zero, one_mul] at h₂
rw [hpow]
nlinarith [mul_le_mul_of_nonneg_left h₁ hC, mul_le_mul_of_nonneg_left h₂ hD]
theorem pilotRectangle_bad_clippedCellValue_alphabet_sq_le Lemma pilotRectangle_bad_clippedCellValue_alphabet_sq_le in the paper ↗
The actual bad-pilot cell loss has the d⁻⁴ bound in (22), without an occupied-cell premise: null cells are localized almost surely. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
d :
ℕ
hd :
1 ≤ d
ε :
ℝ
P :
hε :
0 < ε
hε1 :
ε ≤ 1
hP :
ObservedClass ε P
x :
Fin d
W :
Fin 4 → Ω → ℕ
Ne :
Ω → Fin 4 → ℕ
m :
ℝ≥0
hm :
0 < m
H :
ℝ
hHpos :
0 < H
hH :
universalH ≤ H
hW :
∀ j, Measurable (W j)
hlaw :
∀ j, HasLaw (W j) (poissonMeasure (m * ⟨cellVector P x j, ENNReal.toReal_nonneg⟩)) μ
hind :
iIndepFun W μ
K :
ℕ
conclusion
C :
512 * H ^ 2 * (8 + 2 * (H / universalH) * productMomentConstant 4 1)
D :
(∫ ω, if cellVector P x ∉ pilotRectangle (pilotLower H hHpos m (logAlphabet d) (fun j => W j ω)) (pilotUpperFormula H hHpos m (logAlphabet d) (fun j => W j ω)) then (clippedCellValueFormula ε K d m (pilotMidpoint H hHpos m (logAlphabet d) (fun j => W j ω)) (pilotRadiusFormula H hHpos m (logAlphabet d) (fun j => W j ω)) (Ne ω) - armwiseExtensionFormula ε (cellVector P x)) ^ 2 else 0 ∂μ)
≤ 2 * (C + D) * (d : ℝ) ^ (-4 : ℝ) * (cellMass P x * logAlphabet d / (m * ε) + logAlphabet d ^ 2 / ((m : ℝ) ^ 2 * ε ^ 2))
Proof (Lean source)
lemma pilotRectangle_bad_clippedCellValue_alphabet_sq_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ]
{d : ℕ} (hd : 1 ≤ d) (ε : ℝ) (P : DiscreteLaw d)
(hε : 0 < ε) (hε1 : ε ≤ 1) (hP : ObservedClass ε P) (x : Fin d)
(W : Fin 4 → Ω → ℕ) (Ne : Ω → Fin 4 → ℕ) (m : ℝ≥0) (hm : 0 < m)
(H : ℝ) (hHpos : 0 < H) (hH : universalH ≤ H)
(hW : ∀ j, Measurable (W j))
(hlaw : ∀ j, HasLaw (W j)
(poissonMeasure (m * ⟨cellVector P x j, ENNReal.toReal_nonneg⟩)) μ)
(hind : iIndepFun W μ) (K : ℕ) :
let C := 512 * H ^ 2 * (8 + 2 * (H / universalH) * productMomentConstant 4 1)
let D := 1024 * sqrt (8 * scalarMomentConstant 4) * (H / universalH) ^ 2
(∫ ω, if cellVector P x ∉ pilotRectangle
(pilotLower H hHpos m (logAlphabet d) (fun j => W j ω))
(pilotUpperFormula H hHpos m (logAlphabet d) (fun j => W j ω)) then
(clippedCellValueFormula ε K d m
(pilotMidpoint H hHpos m (logAlphabet d) (fun j => W j ω))
(pilotRadiusFormula H hHpos m (logAlphabet d) (fun j => W j ω)) (Ne ω) -
armwiseExtensionFormula ε (cellVector P x)) ^ 2 else 0 ∂μ) ≤
2 * (C + D) * (d : ℝ) ^ (-4 : ℝ) *
(cellMass P x * logAlphabet d / (m * ε) +
logAlphabet d ^ 2 / ((m : ℝ) ^ 2 * ε ^ 2)) := by
classical
dsimp only
have hC : 0 ≤ 512 * H ^ 2 *
(8 + 2 * (H / universalH) * productMomentConstant 4 1) := by
have hu := universalH_pos
have hc := productMomentConstant_pos 4 1
positivity
have hD : 0 ≤ 1024 * sqrt (8 * scalarMomentConstant 4) * (H / universalH) ^ 2 :=
by positivity
have hp0 : 0 ≤ cellMass P x :=
Finset.sum_nonneg fun a _ => Finset.sum_nonneg fun y _ => ENNReal.toReal_nonneg
have hs : 0 ≤ cellMass P x * logAlphabet d / (m * ε) +
logAlphabet d ^ 2 / ((m : ℝ) ^ 2 * ε ^ 2) := by
have hL := logAlphabet_pos hd
positivity
by_cases hp : cellMass P x = 0
· have hnull := nullCell_pilotRectangle_ae_contains μ P x hp H hHpos m
(logAlphabet d) (NNReal.coe_pos.mpr hm) (logAlphabet_pos hd).le W (by
intro j
convert hlaw j using 1
congr 1)
have heq : (∫ ω, if cellVector P x ∉ pilotRectangle
(pilotLower H hHpos m (logAlphabet d) (fun j => W j ω))
(pilotUpperFormula H hHpos m (logAlphabet d) (fun j => W j ω)) then
(clippedCellValueFormula ε K d m
(pilotMidpoint H hHpos m (logAlphabet d) (fun j => W j ω))
(pilotRadiusFormula H hHpos m (logAlphabet d) (fun j => W j ω)) (Ne ω) -
armwiseExtensionFormula ε (cellVector P x)) ^ 2 else 0 ∂μ) = 0 := by
calc
_ = ∫ _, (0 : ℝ) ∂μ := integral_congr_ae (by
filter_upwards [hnull] with ω hω
simp only [if_neg (not_not.mpr hω)])
_ = 0 := integral_zero _ _
rw [heq]
exact mul_nonneg (mul_nonneg (by linarith)
(Real.rpow_nonneg (Nat.cast_nonneg d) _)) hs
· have hL : 1 ≤ logAlphabet d := by
have hd1 : (1 : ℝ) ≤ d := by exact_mod_cast hd
rw [logAlphabet, Real.log_mul (Real.exp_pos 1).ne' (by positivity), Real.log_exp]
linarith [Real.log_nonneg hd1]
exact (pilotRectangle_bad_clippedCellValue_integral_sq_le μ ε P hε hε1 hP x
(lt_of_le_of_ne hp0 (ne_comm.mp hp)) W Ne m hm H hHpos (logAlphabet d) hH hL
hW hlaw hind K).trans
(mul_le_mul_of_nonneg_right (badPilot_squared_coefficient_decay_le hd _ _ hC hD) hs)
theorem activeBranch_badPilot_clippedCellValue_sum_sq_rate_le Lemma activeBranch_badPilot_clippedCellValue_sum_sq_rate_le in the paper ↗
On the active branch, d times the summed actual bad-pilot squared loss is at most a universal multiple of d/(m ε L). This controls both its variance contribution and the Cauchy–Schwarz bound for its total bias. No cross-cell or evaluation-count independence is required. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
∃ A : ℝ,
0 < A
conclusion 1
Ω :
Type*
μ :
Measure Ω
d :
ℕ
1 ≤ d
ε :
ℝ
P :
0 < ε
ε ≤ 1
ObservedClass ε P
m :
ℝ≥0
0 < m
(d : ℝ) ≤ 8 * (m * ε) * logAlphabet d
(∀ x j, Measurable (W x j))
(∀ x j, HasLaw (W x j) (poissonMeasure (m * ⟨cellVector P x j, ENNReal.toReal_nonneg⟩)) μ)
(∀ x, iIndepFun (W x) μ)
K :
ℕ
(d : ℝ) * (∑ x : Fin d, ∫ ω, if cellVector P x ∉ pilotRectangle (pilotLower H hHpos m (logAlphabet d) (fun j => W x j ω)) (pilotUpperFormula H hHpos m (logAlphabet d) (fun j => W x j ω)) then (clippedCellValueFormula ε K d m (pilotMidpoint H hHpos m (logAlphabet d) (fun j => W x j ω)) (pilotRadiusFormula H hHpos m (logAlphabet d) (fun j => W x j ω)) (Ne x ω) - armwiseExtensionFormula ε (cellVector P x)) ^ 2 else 0 ∂μ)
≤ A * ((d : ℝ) / (m * ε * logAlphabet d))
Proof (Lean source)
lemma activeBranch_badPilot_clippedCellValue_sum_sq_rate_le
(H : ℝ) (hHpos : 0 < H) (hH : universalH ≤ H) :
∃ A : ℝ, 0 < A ∧ ∀ {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] {d : ℕ}, 1 ≤ d →
∀ (ε : ℝ) (P : DiscreteLaw d), 0 < ε → ε ≤ 1 → ObservedClass ε P →
∀ (W : Fin d → Fin 4 → Ω → ℕ) (Ne : Fin d → Ω → Fin 4 → ℕ)
(m : ℝ≥0), 0 < m → (d : ℝ) ≤ 8 * (m * ε) * logAlphabet d →
(∀ x j, Measurable (W x j)) →
(∀ x j, HasLaw (W x j)
(poissonMeasure (m * ⟨cellVector P x j, ENNReal.toReal_nonneg⟩)) μ) →
(∀ x, iIndepFun (W x) μ) → ∀ K : ℕ,
(d : ℝ) * (∑ x : Fin d, ∫ ω, if cellVector P x ∉ pilotRectangle
(pilotLower H hHpos m (logAlphabet d) (fun j => W x j ω))
(pilotUpperFormula H hHpos m (logAlphabet d) (fun j => W x j ω)) then
(clippedCellValueFormula ε K d m
(pilotMidpoint H hHpos m (logAlphabet d) (fun j => W x j ω))
(pilotRadiusFormula H hHpos m (logAlphabet d) (fun j => W x j ω)) (Ne x ω) -
armwiseExtensionFormula ε (cellVector P x)) ^ 2 else 0 ∂μ) ≤
A * ((d : ℝ) / (m * ε * logAlphabet d)) := by
let C := 512 * H ^ 2 * (8 + 2 * (H / universalH) * productMomentConstant 4 1)
let D := 1024 * sqrt (8 * scalarMomentConstant 4) * (H / universalH) ^ 2
have hC : 0 < C := by
have hu := universalH_pos
have hc := productMomentConstant_pos 4 1
dsimp [C]
positivity
have hD : 0 ≤ D := by dsimp [D]; positivity
obtain ⟨A₀, hA₀, hrate⟩ := activeBranch_pilot_moment_sum_rate_le 1 (by norm_num)
refine ⟨2 * (C + D) * A₀, by positivity, ?_⟩
intro Ω _ μ _ d hd ε P hε hε1 hP W Ne m hm hactive hW hlaw hind K
let v : Fin d → ℝ := fun x => ∫ ω, if cellVector P x ∉ pilotRectangle
(pilotLower H hHpos m (logAlphabet d) (fun j => W x j ω))
(pilotUpperFormula H hHpos m (logAlphabet d) (fun j => W x j ω)) then
(clippedCellValueFormula ε K d m
(pilotMidpoint H hHpos m (logAlphabet d) (fun j => W x j ω))
(pilotRadiusFormula H hHpos m (logAlphabet d) (fun j => W x j ω)) (Ne x ω) -
armwiseExtensionFormula ε (cellVector P x)) ^ 2 else 0 ∂μ
have hd1 : (1 : ℝ) ≤ d := by exact_mod_cast hd
have hd0 : (0 : ℝ) < d := by linarith
have hdecay : (d : ℝ) * (d : ℝ) ^ (-4 : ℝ) ≤ 1 := by
nth_rw 1 [← Real.rpow_one (d : ℝ)]
rw [← Real.rpow_add hd0]
exact Real.rpow_le_one_of_one_le_of_nonpos hd1 (by norm_num)
have hv (x : Fin d) : (d : ℝ) * v x ≤
(2 * (C + D)) * (cellMass P x * logAlphabet d / (m * ε) +
logAlphabet d ^ 2 / ((m : ℝ) * ε) ^ 2) := by
have ht := pilotRectangle_bad_clippedCellValue_alphabet_sq_le μ hd ε P hε hε1 hP x
(W x) (Ne x) m hm H hHpos hH (hW x) (hlaw x) (hind x) K
change v x ≤ 2 * (C + D) * (d : ℝ) ^ (-4 : ℝ) * _ at ht
have hp0 : 0 ≤ cellMass P x :=
Finset.sum_nonneg fun a _ => Finset.sum_nonneg fun y _ => toReal_nonneg
have hL := logAlphabet_pos hd
have hscale : 0 ≤ cellMass P x * logAlphabet d / (m * ε) +
logAlphabet d ^ 2 / ((m : ℝ) ^ 2 * ε ^ 2) := by positivity
have hs := mul_le_mul_of_nonneg_right hdecay
(mul_nonneg (show 0 ≤ 2 * (C + D) by positivity) hscale)
have ht' := mul_le_mul_of_nonneg_left ht hd0.le
simpa only [mul_pow] using (ht'.trans (by nlinarith [hs]))
have hr := hrate hd P (fun x => (d : ℝ) * v x) (2 * (C + D)) (m * ε)
(by positivity) (by positivity) hactive hv
norm_num only [sub_self, Real.rpow_zero, one_mul] at hr
rw [← Finset.mul_sum] at hr
exact hr
theorem pilotRectangle_bad_clippedCellValue_alphabet_abs_le Lemma pilotRectangle_bad_clippedCellValue_alphabet_abs_le in the paper ↗
The actual absolute bad-pilot error retains the d⁻² decay in (22), including null cells and arbitrary evaluation counts. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
d :
ℕ
hd :
1 ≤ d
ε :
ℝ
P :
hε :
0 < ε
hε1 :
ε ≤ 1
hP :
ObservedClass ε P
x :
Fin d
W :
Fin 4 → Ω → ℕ
Ne :
Ω → Fin 4 → ℕ
m :
ℝ≥0
hm :
0 < m
H :
ℝ
hHpos :
0 < H
hH :
universalH ≤ H
hW :
∀ j, Measurable (W j)
hlaw :
∀ j, HasLaw (W j) (poissonMeasure (m * ⟨cellVector P x j, ENNReal.toReal_nonneg⟩)) μ
hind :
iIndepFun W μ
K :
ℕ
conclusion
C :
D :
(∫ ω, if cellVector P x ∉ pilotRectangle (pilotLower H hHpos m (logAlphabet d) (fun j => W j ω)) (pilotUpperFormula H hHpos m (logAlphabet d) (fun j => W j ω)) then |clippedCellValueFormula ε K d m (pilotMidpoint H hHpos m (logAlphabet d) (fun j => W j ω)) (pilotRadiusFormula H hHpos m (logAlphabet d) (fun j => W j ω)) (Ne ω) - armwiseExtensionFormula ε (cellVector P x)| else 0 ∂μ)
≤ (C + D) * (d : ℝ) ^ (-2 : ℝ) * (sqrt (cellMass P x * logAlphabet d / (m * ε)) + logAlphabet d / (m * ε))
Proof (Lean source)
lemma pilotRectangle_bad_clippedCellValue_alphabet_abs_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ]
{d : ℕ} (hd : 1 ≤ d) (ε : ℝ) (P : DiscreteLaw d)
(hε : 0 < ε) (hε1 : ε ≤ 1) (hP : ObservedClass ε P) (x : Fin d)
(W : Fin 4 → Ω → ℕ) (Ne : Ω → Fin 4 → ℕ) (m : ℝ≥0) (hm : 0 < m)
(H : ℝ) (hHpos : 0 < H) (hH : universalH ≤ H)
(hW : ∀ j, Measurable (W j))
(hlaw : ∀ j, HasLaw (W j)
(poissonMeasure (m * ⟨cellVector P x j, ENNReal.toReal_nonneg⟩)) μ)
(hind : iIndepFun W μ) (K : ℕ) :
let C := sqrt (8 * (512 * H ^ 2 * (8 + 2 * (H / universalH) * productMomentConstant 4 1)))
let D := sqrt (8 * (1024 * sqrt (8 * scalarMomentConstant 4) * (H / universalH) ^ 2))
(∫ ω, if cellVector P x ∉ pilotRectangle
(pilotLower H hHpos m (logAlphabet d) (fun j => W j ω))
(pilotUpperFormula H hHpos m (logAlphabet d) (fun j => W j ω)) then
|clippedCellValueFormula ε K d m
(pilotMidpoint H hHpos m (logAlphabet d) (fun j => W j ω))
(pilotRadiusFormula H hHpos m (logAlphabet d) (fun j => W j ω)) (Ne ω) -
armwiseExtensionFormula ε (cellVector P x)| else 0 ∂μ) ≤
(C + D) * (d : ℝ) ^ (-2 : ℝ) *
(sqrt (cellMass P x * logAlphabet d / (m * ε)) +
logAlphabet d / (m * ε)) := by
classical
dsimp only
have hC : 0 ≤ sqrt (8 * (512 * H ^ 2 *
(8 + 2 * (H / universalH) * productMomentConstant 4 1))) := Real.sqrt_nonneg _
have hD : 0 ≤ sqrt (8 * (1024 * sqrt (8 * scalarMomentConstant 4) *
(H / universalH) ^ 2)) := Real.sqrt_nonneg _
have hL := logAlphabet_pos hd
have hs : 0 ≤ sqrt (cellMass P x * logAlphabet d / (m * ε)) +
logAlphabet d / (m * ε) := by positivity
have hp0 : 0 ≤ cellMass P x :=
Finset.sum_nonneg fun a _ => Finset.sum_nonneg fun y _ => ENNReal.toReal_nonneg
by_cases hp : cellMass P x = 0
· have hnull := nullCell_pilotRectangle_ae_contains μ P x hp H hHpos m
(logAlphabet d) (NNReal.coe_pos.mpr hm) hL.le W (by
intro j
convert hlaw j using 1
congr 1)
have heq : (∫ ω, if cellVector P x ∉ pilotRectangle
(pilotLower H hHpos m (logAlphabet d) (fun j => W j ω))
(pilotUpperFormula H hHpos m (logAlphabet d) (fun j => W j ω)) then
|clippedCellValueFormula ε K d m
(pilotMidpoint H hHpos m (logAlphabet d) (fun j => W j ω))
(pilotRadiusFormula H hHpos m (logAlphabet d) (fun j => W j ω)) (Ne ω) -
armwiseExtensionFormula ε (cellVector P x)| else 0 ∂μ) = 0 := by
calc
_ = ∫ _, (0 : ℝ) ∂μ := integral_congr_ae (by
filter_upwards [hnull] with ω hω
simp only [if_neg (not_not.mpr hω)])
_ = 0 := integral_zero _ _
rw [heq]
positivity
· have hL1 : 1 ≤ logAlphabet d := by
have hd1 : (1 : ℝ) ≤ d := by exact_mod_cast hd
rw [logAlphabet, Real.log_mul (Real.exp_pos 1).ne' (by positivity), Real.log_exp]
linarith [Real.log_nonneg hd1]
have ht := pilotRectangle_bad_clippedCellValue_integral_le μ ε P hε hε1 hP x
(lt_of_le_of_ne hp0 (ne_comm.mp hp)) W Ne m hm H hHpos (logAlphabet d) hH hL1
hW hlaw hind K
have h₁ := logAlphabet_exp_decay_mul_rpow_le hd (1 / 4) 30 2
(by norm_num) (by norm_num)
have h₂ := logAlphabet_exp_decay_mul_rpow_le hd 0 30 2
(by norm_num) (by norm_num)
simp only [Real.rpow_zero, one_mul] at h₂
apply ht.trans
apply mul_le_mul_of_nonneg_right _ hs
nlinarith [mul_le_mul_of_nonneg_left h₁ hC, mul_le_mul_of_nonneg_left h₂ hD]
theorem activeBranch_badPilot_clippedCellValue_total_abs_sq_rate_le Lemma activeBranch_badPilot_clippedCellValue_total_abs_sq_rate_le in the paper ↗
The total actual bad-pilot absolute bias, squared after summing cells, is controlled on the active branch. No cross-cell independence is needed. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
∃ A : ℝ,
0 < A
conclusion 1
Ω :
Type*
μ :
Measure Ω
d :
ℕ
1 ≤ d
ε :
ℝ
P :
0 < ε
ε ≤ 1
ObservedClass ε P
m :
ℝ≥0
0 < m
(d : ℝ) ≤ 8 * (m * ε) * logAlphabet d
(∀ x j, Measurable (W x j))
(∀ x j, HasLaw (W x j) (poissonMeasure (m * ⟨cellVector P x j, ENNReal.toReal_nonneg⟩)) μ)
(∀ x, iIndepFun (W x) μ)
K :
ℕ
(∑ x : Fin d, ∫ ω, if cellVector P x ∉ pilotRectangle (pilotLower H hHpos m (logAlphabet d) (fun j => W x j ω)) (pilotUpperFormula H hHpos m (logAlphabet d) (fun j => W x j ω)) then |clippedCellValueFormula ε K d m (pilotMidpoint H hHpos m (logAlphabet d) (fun j => W x j ω)) (pilotRadiusFormula H hHpos m (logAlphabet d) (fun j => W x j ω)) (Ne x ω) - armwiseExtensionFormula ε (cellVector P x)| else 0 ∂μ) ^ 2
≤ A * ((d : ℝ) / (m * ε * logAlphabet d))
Proof (Lean source)
lemma activeBranch_badPilot_clippedCellValue_total_abs_sq_rate_le
(H : ℝ) (hHpos : 0 < H) (hH : universalH ≤ H) :
∃ A : ℝ, 0 < A ∧ ∀ {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] {d : ℕ}, 1 ≤ d →
∀ (ε : ℝ) (P : DiscreteLaw d), 0 < ε → ε ≤ 1 → ObservedClass ε P →
∀ (W : Fin d → Fin 4 → Ω → ℕ) (Ne : Fin d → Ω → Fin 4 → ℕ)
(m : ℝ≥0), 0 < m → (d : ℝ) ≤ 8 * (m * ε) * logAlphabet d →
(∀ x j, Measurable (W x j)) →
(∀ x j, HasLaw (W x j)
(poissonMeasure (m * ⟨cellVector P x j, ENNReal.toReal_nonneg⟩)) μ) →
(∀ x, iIndepFun (W x) μ) → ∀ K : ℕ,
(∑ x : Fin d, ∫ ω, if cellVector P x ∉ pilotRectangle
(pilotLower H hHpos m (logAlphabet d) (fun j => W x j ω))
(pilotUpperFormula H hHpos m (logAlphabet d) (fun j => W x j ω)) then
|clippedCellValueFormula ε K d m
(pilotMidpoint H hHpos m (logAlphabet d) (fun j => W x j ω))
(pilotRadiusFormula H hHpos m (logAlphabet d) (fun j => W x j ω)) (Ne x ω) -
armwiseExtensionFormula ε (cellVector P x)| else 0 ∂μ) ^ 2 ≤
A * ((d : ℝ) / (m * ε * logAlphabet d)) := by
let C := sqrt (8 * (512 * H ^ 2 *
(8 + 2 * (H / universalH) * productMomentConstant 4 1))) +
sqrt (8 * (1024 * sqrt (8 * scalarMomentConstant 4) * (H / universalH) ^ 2))
have hC : 0 < C := by
have hu := universalH_pos
have hc := productMomentConstant_pos 4 1
dsimp [C]
positivity
obtain ⟨A₀, hA₀, hrate⟩ := activeBranch_pilot_moment_sum_rate_le 1 (by norm_num)
refine ⟨2 * C ^ 2 * A₀, by positivity, ?_⟩
intro Ω _ μ _ d hd ε P hε hε1 hP W Ne m hm hactive hW hlaw hind K
let v : Fin d → ℝ := fun x => ∫ ω, if cellVector P x ∉ pilotRectangle
(pilotLower H hHpos m (logAlphabet d) (fun j => W x j ω))
(pilotUpperFormula H hHpos m (logAlphabet d) (fun j => W x j ω)) then
|clippedCellValueFormula ε K d m
(pilotMidpoint H hHpos m (logAlphabet d) (fun j => W x j ω))
(pilotRadiusFormula H hHpos m (logAlphabet d) (fun j => W x j ω)) (Ne x ω) -
armwiseExtensionFormula ε (cellVector P x)| else 0 ∂μ
have hv0 (x : Fin d) : 0 ≤ v x := integral_nonneg (fun ω => by
dsimp only
split_ifs <;> positivity)
have hd1 : (1 : ℝ) ≤ d := by exact_mod_cast hd
have hd0 : (0 : ℝ) < d := by linarith
have hdecay : (d : ℝ) * ((d : ℝ) ^ (-2 : ℝ)) ^ 2 ≤ 1 := by
rw [← Real.rpow_mul_natCast hd0.le]
nth_rw 1 [← Real.rpow_one (d : ℝ)]
rw [← Real.rpow_add hd0]
exact Real.rpow_le_one_of_one_le_of_nonpos hd1 (by norm_num)
have hv (x : Fin d) : (d : ℝ) * (v x) ^ 2 ≤
(2 * C ^ 2) * (cellMass P x * logAlphabet d / (m * ε) +
logAlphabet d ^ 2 / ((m : ℝ) * ε) ^ 2) := by
have ht := pilotRectangle_bad_clippedCellValue_alphabet_abs_le μ hd ε P hε hε1 hP x
(W x) (Ne x) m hm H hHpos hH (hW x) (hlaw x) (hind x) K
change v x ≤ C * (d : ℝ) ^ (-2 : ℝ) *
(sqrt (cellMass P x * logAlphabet d / (m * ε)) +
logAlphabet d / (m * ε)) at ht
have hp0 : 0 ≤ cellMass P x :=
Finset.sum_nonneg fun a _ => Finset.sum_nonneg fun y _ => toReal_nonneg
have hL := logAlphabet_pos hd
have hscale : 0 ≤ cellMass P x * logAlphabet d / (m * ε) := by positivity
have hsqrt := Real.sq_sqrt hscale
have htwo : (sqrt (cellMass P x * logAlphabet d / (m * ε)) +
logAlphabet d / (m * ε)) ^ 2 ≤
2 * (cellMass P x * logAlphabet d / (m * ε) +
logAlphabet d ^ 2 / ((m : ℝ) * ε) ^ 2) := by
rw [← div_pow]
nlinarith [sq_nonneg (sqrt (cellMass P x * logAlphabet d / (m * ε)) -
logAlphabet d / (m * ε))]
calc
_ ≤ (d : ℝ) * (C * (d : ℝ) ^ (-2 : ℝ) *
(sqrt (cellMass P x * logAlphabet d / (m * ε)) +
logAlphabet d / (m * ε))) ^ 2 := by
gcongr
_ = C ^ 2 * ((d : ℝ) * ((d : ℝ) ^ (-2 : ℝ)) ^ 2) *
(sqrt (cellMass P x * logAlphabet d / (m * ε)) +
logAlphabet d / (m * ε)) ^ 2 := by ring
_ ≤ C ^ 2 * 1 * (2 * (cellMass P x * logAlphabet d / (m * ε) +
logAlphabet d ^ 2 / ((m : ℝ) * ε) ^ 2)) := by gcongr
_ = _ := by ring
have hr := hrate hd P (fun x => (d : ℝ) * (v x) ^ 2) (2 * C ^ 2) (m * ε)
(by positivity) (by positivity) hactive hv
norm_num only [sub_self, Real.rpow_zero, one_mul] at hr
rw [← Finset.mul_sum] at hr
have hsum := Finset.sum_mul_sq_le_sq_mul_sq (Finset.univ : Finset (Fin d))
(fun _ => (1 : ℝ)) v
simp only [one_mul, one_pow, Finset.sum_const, Finset.card_univ,
Fintype.card_fin, nsmul_eq_mul, mul_one] at hsum
exact hsum.trans hr
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.UpperBadPilotCenter 7 declarations The fourth marginal score moment and the rectangle failure probability give an exponentially small second moment for each coordinate on any pilot failure.
Coordinate moments on failed pilot rectangles
The fourth marginal score moment and the rectangle failure probability give an exponentially small second moment for each coordinate on any pilot failure. Unlike an aggregate mass bound, this retains each coordinate's true mass for the armwise inverse-propensity weighting in roadmap (22b). The actual pilot-center functional inherits these moments through the armwise modulus; summing outcome masses retains one inverse-overlap power in its mass term.
theorem pilotCoordinate_score_integral_pow_le Lemma pilotCoordinate_score_integral_pow_le in the paper ↗
Every coordinate score moment through degree four has its own true-mass scale, obtained from the marginal Poisson law and the raw-score comparison. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
Measure Ω
W :
Fin 4 → Ω → ℕ
q :
Fin 4 → ℝ≥0
m :
ℝ≥0
hm :
0 < m
H L :
ℝ
hH :
universalH ≤ H
hL :
1 ≤ L
hlaw :
∀ j, HasLaw (W j) (poissonMeasure (m * q j)) μ
j :
Fin 4
t :
ℕ
ht :
t ≤ 4
conclusion
(∫ ω, (normalizedDeviation m (q j) (W j ω) + empiricalRadius H m (L / m) (W j ω)) ^ t ∂μ)
Proof (Lean source)
lemma pilotCoordinate_score_integral_pow_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) (W : Fin 4 → Ω → ℕ) (q : Fin 4 → ℝ≥0)
(m : ℝ≥0) (hm : 0 < m) (H L : ℝ) (hH : universalH ≤ H) (hL : 1 ≤ L)
(hlaw : ∀ j, HasLaw (W j) (poissonMeasure (m * q j)) μ)
(j : Fin 4) (t : ℕ) (ht : t ≤ 4) :
(∫ ω, (normalizedDeviation m (q j) (W j ω) +
empiricalRadius H m (L / m) (W j ω)) ^ t ∂μ) ≤
(H / universalH) ^ t * scalarMomentConstant t *
(sqrt ((q j : ℝ) * L / m) + L / m) ^ t := by
let c : ℝ := H / universalH / (m : ℝ)
have hc : 0 ≤ c := by
dsimp [c]
exact div_nonneg (div_nonneg (universalH_pos.le.trans hH) universalH_pos.le) m.coe_nonneg
have hi : Integrable (fun ω => (score universalH L (m * q j) (W j ω)) ^ t) μ := by
have h := integrable_score_pow (m * q j) hL (t := t)
rw [← (hlaw j).map_eq] at h
exact h.comp_aemeasurable (hlaw j).aemeasurable
have he := (hlaw j).integral_comp
(Measurable.of_discrete.aestronglyMeasurable (f := fun w : ℕ =>
(score universalH L (m * q j) w) ^ t))
simp only [Function.comp_def] at he
calc
_ ≤ ∫ ω, c ^ t * (score universalH L (m * q j) (W j ω)) ^ t ∂μ := by
apply integral_mono_of_nonneg
· apply ae_of_all
intro ω
have := universalH_pos.le.trans hH
unfold normalizedDeviation empiricalRadius
positivity
· exact hi.const_mul _
· apply ae_of_all
intro ω
have hs := empiricalRadius_score_le_raw (q j) m hm H L hH hL (W j ω)
have hn : 0 ≤ normalizedDeviation m (q j) (W j ω) +
empiricalRadius H m (L / m) (W j ω) := by
have := universalH_pos.le.trans hH
unfold normalizedDeviation empiricalRadius
positivity
convert pow_le_pow_left₀ hn hs t using 1
dsimp [c]
ring
_ = c ^ t * ∫ w : ℕ, (score universalH L (m * q j) w) ^ t
∂poissonMeasure (m * q j) := by
rw [integral_const_mul, he]
_ ≤ c ^ t * (scalarMomentConstant t * (localScale (m * q j) L) ^ t) :=
mul_le_mul_of_nonneg_left (poisson_score_moment (m * q j) hL ht) (pow_nonneg hc t)
_ = _ := by
rw [show c ^ t * (scalarMomentConstant t * (localScale (m * q j) L) ^ t) =
(H / universalH) ^ t * scalarMomentConstant t *
(localScale (m * q j) L / (m : ℝ)) ^ t by
dsimp [c]; rw [div_pow, div_pow]; ring]
rw [empiricalRadius_localScale_div (q j) m hm L hL]
theorem pilotRectangle_bad_coordinate_integral_sq_le Lemma pilotRectangle_bad_coordinate_integral_sq_le in the paper ↗
Even when a different coordinate fails localization, a coordinate's second score moment is exponentially small with its own mass scale. Hölder uses the marginal fourth moment and the full rectangle failure probability; no coordinate independence is needed. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
W :
Fin 4 → Ω → ℕ
q :
Fin 4 → ℝ≥0
m :
ℝ≥0
hm :
0 < m
H :
ℝ
hHpos :
0 < H
L :
ℝ
hH :
universalH ≤ H
hL :
1 ≤ L
hW :
∀ j, Measurable (W j)
hlaw :
∀ j, HasLaw (W j) (poissonMeasure (m * q j)) μ
j :
Fin 4
conclusion
(∫ ω, if (fun i => (q i : ℝ)) ∉ pilotRectangle (pilotLower H hHpos m L (fun i => W i ω)) (pilotUpperFormula H hHpos m L (fun i => W i ω)) then (normalizedDeviation m (q j) (W j ω) + empiricalRadius H m (L / m) (W j ω)) ^ 2 else 0 ∂μ)
≤ sqrt (8 * scalarMomentConstant 4) * (H / universalH) ^ 2 * exp (-20 * L) * (sqrt ((q j : ℝ) * L / m) + L / m) ^ 2
Proof (Lean source)
lemma pilotRectangle_bad_coordinate_integral_sq_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ]
(W : Fin 4 → Ω → ℕ) (q : Fin 4 → ℝ≥0) (m : ℝ≥0) (hm : 0 < m)
(H : ℝ) (hHpos : 0 < H) (L : ℝ) (hH : universalH ≤ H) (hL : 1 ≤ L)
(hW : ∀ j, Measurable (W j))
(hlaw : ∀ j, HasLaw (W j) (poissonMeasure (m * q j)) μ) (j : Fin 4) :
(∫ ω, if (fun i => (q i : ℝ)) ∉ pilotRectangle
(pilotLower H hHpos m L (fun i => W i ω))
(pilotUpperFormula H hHpos m L (fun i => W i ω)) then
(normalizedDeviation m (q j) (W j ω) +
empiricalRadius H m (L / m) (W j ω)) ^ 2 else 0 ∂μ) ≤
sqrt (8 * scalarMomentConstant 4) * (H / universalH) ^ 2 *
exp (-20 * L) *
(sqrt ((q j : ℝ) * L / m) + L / m) ^ 2 := by
classical
let B : Set Ω := {ω | (fun i => (q i : ℝ)) ∉ pilotRectangle
(pilotLower H hHpos m L (fun i => W i ω))
(pilotUpperFormula H hHpos m L (fun i => W i ω))}
let S : Ω → ℝ := fun ω => normalizedDeviation m (q j) (W j ω) +
empiricalRadius H m (L / m) (W j ω)
let a : ℝ := sqrt ((q j : ℝ) * L / m) + L / m
have hB : MeasurableSet B := by
dsimp [B]
unfold pilotRectangle pilotLower pilotUpperFormula pilotHalfWidthFormula pilotCenterFormula
measurability
have hi4 : Integrable (fun ω => S ω ^ 4) μ := by
have h := pilotCoordinate_score_pow_integrable (q j) m hm H L hH hL 4 (by omega)
rw [← (hlaw j).map_eq] at h
exact h.comp_aemeasurable (hlaw j).aemeasurable
have hS : MemLp (fun ω => S ω ^ 2) 2 μ := by
apply (memLp_two_iff_integrable_sq (by dsimp [S]; fun_prop)).2
simpa only [← pow_mul] using hi4
have hp : μ.real B ≤ 8 * exp (-40 * L) := by
have h := pilotRectangle_bad_probability_le μ W q m hm H hHpos L hH hL hlaw
exact (ENNReal.toReal_mono ENNReal.ofReal_ne_top h).trans_eq
(ENNReal.toReal_ofReal (by positivity))
have hs := pilotCoordinate_score_integral_pow_le μ W q m hm H L hH hL hlaw j 4 (by omega)
have hh := integral_mul_le_Lp_mul_Lq_of_nonneg Real.HolderConjugate.two_two
(f := fun ω => S ω ^ 2) (g := B.indicator (fun _ => (1 : ℝ)))
(ae_of_all _ fun ω => sq_nonneg _)
(ae_of_all _ fun ω => Set.indicator_nonneg (fun _ _ => zero_le_one) ω)
(by simpa using hS)
(by simpa using (memLp_const (1 : ℝ) (p := (2 : ENNReal)) (μ := μ)).indicator hB)
have hmul (ω : Ω) : S ω ^ 2 * B.indicator (fun _ => (1 : ℝ)) ω =
B.indicator (fun ω => S ω ^ 2) ω := by by_cases h : ω ∈ B <;> simp [h]
have hone (ω : Ω) : (B.indicator (fun _ => (1 : ℝ)) ω) ^ (2 : ℝ) =
B.indicator (fun _ => (1 : ℝ)) ω := by by_cases h : ω ∈ B <;> simp [h]
have hpow (ω : Ω) : (S ω ^ 2) ^ (2 : ℝ) = S ω ^ 4 := by
rw [Real.rpow_two, ← pow_mul]
simp_rw [hmul, hone, hpow, ← Real.sqrt_eq_rpow] at hh
rw [integral_indicator_const _ hB] at hh
simp only [smul_eq_mul, mul_one] at hh
have hb := mul_le_mul (Real.sqrt_le_sqrt hs) (Real.sqrt_le_sqrt hp)
(Real.sqrt_nonneg _) (Real.sqrt_nonneg _)
have hc : 0 ≤ H / universalH := div_nonneg hHpos.le universalH_pos.le
have ha : 0 ≤ a := by dsimp [a]; positivity
have hconstant := (scalarMomentConstant_pos 4).le
have he : sqrt ((H / universalH) ^ 4 * scalarMomentConstant 4 * a ^ 4) *
sqrt (8 * exp (-40 * L)) =
sqrt (8 * scalarMomentConstant 4) * (H / universalH) ^ 2 *
exp (-20 * L) * a ^ 2 := by
rw [← Real.sqrt_mul (by positivity)]
have hexp : exp (-40 * L) = exp (-20 * L) ^ 2 := by
rw [pow_two, ← Real.exp_add]; congr 1; ring
have heq : (H / universalH) ^ 4 * scalarMomentConstant 4 * a ^ 4 *
(8 * exp (-40 * L)) =
(sqrt (8 * scalarMomentConstant 4) * (H / universalH) ^ 2 *
exp (-20 * L) * a ^ 2) ^ 2 := by
rw [hexp, mul_pow, mul_pow, mul_pow, Real.sq_sqrt (by
have := (scalarMomentConstant_pos 4).le; positivity)]
ring
rw [heq, Real.sqrt_sq (by positivity)]
simpa only [B, S, a, indicator, Set.mem_ofPred_eq] using
(hh.trans hb).trans_eq he
theorem pilotRectangle_bad_weightedScore_integral_sq_le Lemma pilotRectangle_bad_weightedScore_integral_sq_le in the paper ↗
A weighted coordinate score retains each weight next to its own true-mass scale on rectangle failure. This is the finite Cauchy–Schwarz assembly needed before substituting inverse arm propensities in roadmap (22b). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
W :
Fin 4 → Ω → ℕ
q :
Fin 4 → ℝ≥0
m :
ℝ≥0
hm :
0 < m
H :
ℝ
hHpos :
0 < H
L :
ℝ
hH :
universalH ≤ H
hL :
1 ≤ L
hW :
∀ j, Measurable (W j)
hlaw :
∀ j, HasLaw (W j) (poissonMeasure (m * q j)) μ
w :
Fin 4 → ℝ
hw :
∀ j, 0 ≤ w j
conclusion
(∫ ω, if (fun j => (q j : ℝ)) ∉ pilotRectangle (pilotLower H hHpos m L (fun j => W j ω)) (pilotUpperFormula H hHpos m L (fun j => W j ω)) then (∑ j, w j * (normalizedDeviation m (q j) (W j ω) + empiricalRadius H m (L / m) (W j ω))) ^ 2 else 0 ∂μ)
≤ 4 * (sqrt (8 * scalarMomentConstant 4) * (H / universalH) ^ 2 * exp (-20 * L)) * ∑ j, w j ^ 2 * (sqrt ((q j : ℝ) * L / m) + L / m) ^ 2
Proof (Lean source)
lemma pilotRectangle_bad_weightedScore_integral_sq_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ]
(W : Fin 4 → Ω → ℕ) (q : Fin 4 → ℝ≥0) (m : ℝ≥0) (hm : 0 < m)
(H : ℝ) (hHpos : 0 < H) (L : ℝ) (hH : universalH ≤ H) (hL : 1 ≤ L)
(hW : ∀ j, Measurable (W j))
(hlaw : ∀ j, HasLaw (W j) (poissonMeasure (m * q j)) μ)
(w : Fin 4 → ℝ) (hw : ∀ j, 0 ≤ w j) :
(∫ ω, if (fun j => (q j : ℝ)) ∉ pilotRectangle
(pilotLower H hHpos m L (fun j => W j ω))
(pilotUpperFormula H hHpos m L (fun j => W j ω)) then
(∑ j, w j * (normalizedDeviation m (q j) (W j ω) +
empiricalRadius H m (L / m) (W j ω))) ^ 2 else 0 ∂μ) ≤
4 * (sqrt (8 * scalarMomentConstant 4) * (H / universalH) ^ 2 *
exp (-20 * L)) *
∑ j, w j ^ 2 * (sqrt ((q j : ℝ) * L / m) + L / m) ^ 2 := by
classical
let B : Set Ω := {ω | (fun j => (q j : ℝ)) ∉ pilotRectangle
(pilotLower H hHpos m L (fun j => W j ω))
(pilotUpperFormula H hHpos m L (fun j => W j ω))}
let S : Fin 4 → Ω → ℝ := fun j ω => normalizedDeviation m (q j) (W j ω) +
empiricalRadius H m (L / m) (W j ω)
let C : ℝ := sqrt (8 * scalarMomentConstant 4) * (H / universalH) ^ 2 *
exp (-20 * L)
have hB : MeasurableSet B := by
dsimp [B]
unfold pilotRectangle pilotLower pilotUpperFormula pilotHalfWidthFormula pilotCenterFormula
measurability
have hi (j : Fin 4) : Integrable (B.indicator (fun ω => S j ω ^ 2)) μ := by
have h := pilotCoordinate_score_pow_integrable (q j) m hm H L hH hL 2 (by omega)
rw [← (hlaw j).map_eq] at h
exact (h.comp_aemeasurable (hlaw j).aemeasurable).indicator hB
have hn (j : Fin 4) (ω : Ω) : 0 ≤ S j ω := by
have := universalH_pos.le.trans hH
dsimp [S]; unfold normalizedDeviation empiricalRadius; positivity
have hs (j : Fin 4) : (∫ ω, B.indicator (fun ω => S j ω ^ 2) ω ∂μ) ≤
C * (sqrt ((q j : ℝ) * L / m) + L / m) ^ 2 := by
simpa only [B, S, C, indicator, Set.mem_ofPred_eq] using
pilotRectangle_bad_coordinate_integral_sq_le μ W q m hm H hHpos L hH hL hW hlaw j
calc
_ ≤ ∫ ω, 4 * ∑ j, w j ^ 2 * B.indicator (fun ω => S j ω ^ 2) ω ∂μ := by
apply integral_mono_of_nonneg
· exact ae_of_all _ fun ω => by dsimp only; split_ifs <;> positivity
· exact (integrable_finsetSum Finset.univ (fun j _ => (hi j).const_mul _)).const_mul 4
· apply ae_of_all
intro ω
by_cases hb : ω ∈ B
· change (if ω ∈ B then (∑ j, w j * S j ω) ^ 2 else 0) ≤ _
rw [if_pos hb]
simp only [Set.indicator_of_mem hb]
simpa only [Finset.card_univ, Fintype.card_fin, Nat.cast_ofNat, pow_one, mul_pow] using
pow_sum_le_card_mul_sum_pow (s := (Finset.univ : Finset (Fin 4)))
(fun j _ => mul_nonneg (hw j) (hn j ω)) 1
· change (if ω ∈ B then _ else 0) ≤ _
simp [hb]
_ = 4 * ∑ j, w j ^ 2 * ∫ ω, B.indicator (fun ω => S j ω ^ 2) ω ∂μ := by
rw [integral_const_mul, integral_finsetSum Finset.univ
(fun j _ => (hi j).const_mul _)]
simp_rw [integral_const_mul]
_ ≤ 4 * ∑ j, w j ^ 2 * (C *
(sqrt ((q j : ℝ) * L / m) + L / m) ^ 2) := by
gcongr with j
exact hs j
_ = _ := by
change 4 * ∑ j, w j ^ 2 * (C * _) = (4 * C) * ∑ j, w j ^ 2 * _
simp_rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro j _
ring
theorem pilotRectangle_bad_cellValue_integral_sq_le Lemma pilotRectangle_bad_cellValue_integral_sq_le in the paper ↗
The actual pilot-center functional has an exponentially small second moment on rectangle failure. Each inverse propensity remains attached to its own coordinate mass, before any overlap-floor simplification. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
d :
ℕ
ε :
ℝ
P :
hε :
0 < ε
hP :
ObservedClass ε P
x :
Fin d
hp :
0 < cellMass P x
W :
Fin 4 → Ω → ℕ
m :
ℝ≥0
hm :
0 < m
H :
ℝ
hHpos :
0 < H
L :
ℝ
hH :
universalH ≤ H
hL :
1 ≤ L
hW :
∀ j, Measurable (W j)
hlaw :
∀ j, HasLaw (W j) (poissonMeasure (m * ⟨cellVector P x j, ENNReal.toReal_nonneg⟩)) μ
conclusion
(∫ ω, if cellVector P x ∉ pilotRectangle (pilotLower H hHpos m L (fun j => W j ω)) (pilotUpperFormula H hHpos m L (fun j => W j ω)) then (armwiseExtensionFormula ε (pilotMidpoint H hHpos m L (fun j => W j ω)) - armwiseExtensionFormula ε (cellVector P x)) ^ 2 else 0 ∂μ)
≤ 64 * (sqrt (8 * scalarMomentConstant 4) * (H / universalH) ^ 2 * exp (-20 * L)) * ∑ a : Bool, (cellMass P x / armMass P a x) ^ 2 * ∑ y : Bool, (sqrt (jointMass P x a y * L / m) + L / m) ^ 2
Proof (Lean source)
lemma pilotRectangle_bad_cellValue_integral_sq_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ]
{d : ℕ} (ε : ℝ) (P : DiscreteLaw d) (hε : 0 < ε)
(hP : ObservedClass ε P) (x : Fin d) (hp : 0 < cellMass P x)
(W : Fin 4 → Ω → ℕ) (m : ℝ≥0) (hm : 0 < m)
(H : ℝ) (hHpos : 0 < H) (L : ℝ) (hH : universalH ≤ H) (hL : 1 ≤ L)
(hW : ∀ j, Measurable (W j))
(hlaw : ∀ j, HasLaw (W j)
(poissonMeasure (m * ⟨cellVector P x j, ENNReal.toReal_nonneg⟩)) μ) :
(∫ ω, if cellVector P x ∉ pilotRectangle
(pilotLower H hHpos m L (fun j => W j ω))
(pilotUpperFormula H hHpos m L (fun j => W j ω)) then
(armwiseExtensionFormula ε (pilotMidpoint H hHpos m L (fun j => W j ω)) -
armwiseExtensionFormula ε (cellVector P x)) ^ 2 else 0 ∂μ) ≤
64 * (sqrt (8 * scalarMomentConstant 4) * (H / universalH) ^ 2 *
exp (-20 * L)) *
∑ a : Bool, (cellMass P x / armMass P a x) ^ 2 *
∑ y : Bool, (sqrt (jointMass P x a y * L / m) + L / m) ^ 2 := by
classical
let q : Fin 4 → ℝ≥0 := fun j => ⟨cellVector P x j, ENNReal.toReal_nonneg⟩
have hqcoe (j : Fin 4) : (q j : ℝ) = cellVector P x j := rfl
have hqfun : (fun j => (q j : ℝ)) = cellVector P x := funext hqcoe
let w : Fin 4 → ℝ := fun j => cellMass P x / armMass P (decide (2 ≤ j.val)) x
let B : Set Ω := {ω | cellVector P x ∉ pilotRectangle
(pilotLower H hHpos m L (fun j => W j ω))
(pilotUpperFormula H hHpos m L (fun j => W j ω))}
let S : Ω → ℝ := fun ω => ∑ j, w j * (normalizedDeviation m (q j) (W j ω) +
empiricalRadius H m (L / m) (W j ω))
have hw (j : Fin 4) : 0 ≤ w j := by
dsimp [w]
exact div_nonneg hp.le (Finset.sum_nonneg fun _ _ => toReal_nonneg)
have hB : MeasurableSet B := by
dsimp [B]
unfold pilotRectangle pilotLower pilotUpperFormula pilotHalfWidthFormula pilotCenterFormula
measurability
have hi (j : Fin 4) : Integrable (fun ω =>
(normalizedDeviation m (q j) (W j ω) +
empiricalRadius H m (L / m) (W j ω)) ^ 2) μ := by
have h := pilotCoordinate_score_pow_integrable (q j) m hm H L hH hL 2 (by omega)
rw [← (hlaw j).map_eq] at h
exact h.comp_aemeasurable (hlaw j).aemeasurable
have hS : MemLp S 2 μ := by
apply memLp_finsetSum
intro j _
exact ((memLp_two_iff_integrable_sq (by fun_prop)).2 (hi j)).const_mul (w j)
have hpoint (ω : Ω) : |armwiseExtensionFormula ε
(pilotMidpoint H hHpos m L (fun j => W j ω)) -
armwiseExtensionFormula ε (cellVector P x)| ≤ 4 * S ω := by
have h := pilotCellValue_abs_error_le ε P hε hP x hp H hHpos m L
(NNReal.coe_pos.mpr hm) (by linarith) (fun j => W j ω)
convert h using 1 <;>
simp [S, w, hqcoe, Fin.sum_univ_succ, cellIdx,
normalizedDeviation, pilotCenterFormula, pilotHalfWidth_eq_empiricalRadius] <;> ring
have hb := pilotRectangle_bad_weightedScore_integral_sq_le μ W q m hm H hHpos L
hH hL hW hlaw w hw
calc
_ ≤ ∫ ω, 16 * B.indicator (fun ω => S ω ^ 2) ω ∂μ := by
apply integral_mono_of_nonneg
· exact ae_of_all _ fun ω => by dsimp only; split_ifs <;> positivity
· exact ((memLp_two_iff_integrable_sq hS.aestronglyMeasurable).1 hS).indicator hB
|>.const_mul 16
· apply ae_of_all
intro ω
by_cases h : ω ∈ B
· change (if ω ∈ B then _ else 0) ≤ 16 * B.indicator (fun ω => S ω ^ 2) ω
rw [if_pos h, Set.indicator_of_mem h]
have hs : 0 ≤ S ω := Finset.sum_nonneg fun j _ =>
mul_nonneg (hw j) (by
have := universalH_pos.le.trans hH
unfold normalizedDeviation empiricalRadius
positivity)
have hpow := pow_le_pow_left₀ (abs_nonneg _) (hpoint ω) 2
simpa only [sq_abs, mul_pow, show (4 : ℝ) ^ 2 = 16 by norm_num] using hpow
· change (if ω ∈ B then _ else 0) ≤ _
simp [h]
_ = 16 * ∫ ω, B.indicator (fun ω => S ω ^ 2) ω ∂μ := integral_const_mul _ _
_ ≤ 16 * (4 * (sqrt (8 * scalarMomentConstant 4) * (H / universalH) ^ 2 *
exp (-20 * L)) *
∑ j, w j ^ 2 * (sqrt ((q j : ℝ) * L / m) + L / m) ^ 2) := by
apply mul_le_mul_of_nonneg_left _ (by norm_num)
simpa only [B, S, hqfun, indicator, Set.mem_ofPred_eq] using hb
_ = _ := by
simp only [w, hqcoe]
simp [Fin.sum_univ_succ, cellVector, jointMass]
ring
theorem inverseArm_coordinate_scale_sq_sum_le Lemma inverseArm_coordinate_scale_sq_sum_le in the paper ↗
Summing the two outcome masses before applying overlap preserves one inverse-overlap power in the mass term of the bad-pilot second moment. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma inverseArm_coordinate_scale_sq_sum_le {p s q₀ q₁ δ ε : ℝ}
(hp : 0 < p) (hε : 0 < ε) (hδ : 0 ≤ δ)
(hq₀ : 0 ≤ q₀) (hq₁ : 0 ≤ q₁) (hsum : q₀ + q₁ = s)
(hfloor : ε * p ≤ s) :
(p / s) ^ 2 * ((sqrt (q₀ * δ) + δ) ^ 2 +
(sqrt (q₁ * δ) + δ) ^ 2) ≤
8 * (p * δ / ε + (δ / ε) ^ 2) := by
have hs : 0 < s := lt_of_lt_of_le (mul_pos hε hp) hfloor
have hw : p / s ≤ 1 / ε := by
apply (div_le_div_iff₀ hs hε).2
simpa only [one_mul, mul_one, mul_comm] using hfloor
have hsumSq : (sqrt (q₀ * δ) + δ) ^ 2 +
(sqrt (q₁ * δ) + δ) ^ 2 ≤ 2 * (s * δ + 2 * δ ^ 2) := by
have h₀ := add_sq_le (a := sqrt (q₀ * δ)) (b := δ)
have h₁ := add_sq_le (a := sqrt (q₁ * δ)) (b := δ)
rw [Real.sq_sqrt (mul_nonneg hq₀ hδ)] at h₀
rw [Real.sq_sqrt (mul_nonneg hq₁ hδ)] at h₁
nlinarith [hsum]
have hmass : (p / s) ^ 2 * (s * δ) ≤ p * δ / ε := by
calc
_ = (p * δ) * (p / s) := by field_simp
_ ≤ (p * δ) * (1 / ε) := mul_le_mul_of_nonneg_left hw (mul_nonneg hp.le hδ)
_ = _ := by ring
have hlinear : (p / s) ^ 2 * δ ^ 2 ≤ (δ / ε) ^ 2 := by
have h := pow_le_pow_left₀ (div_nonneg hp.le hs.le) hw 2
have h' := mul_le_mul_of_nonneg_right h (sq_nonneg δ)
convert h' using 1 <;> first | rfl | ring
have h := mul_le_mul_of_nonneg_left hsumSq (sq_nonneg (p / s))
nlinarith [div_nonneg (mul_nonneg hp.le hδ) hε.le]
theorem pilotRectangle_bad_cellValue_normalized_le Lemma pilotRectangle_bad_cellValue_normalized_le in the paper ↗
The center-value component of roadmap (22) has the required normalized mass and overlap scales on failed rectangles, without coordinate independence. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
d :
ℕ
ε :
ℝ
P :
hε :
0 < ε
hP :
ObservedClass ε P
x :
Fin d
hp :
0 < cellMass P x
W :
Fin 4 → Ω → ℕ
m :
ℝ≥0
hm :
0 < m
H :
ℝ
hHpos :
0 < H
L :
ℝ
hH :
universalH ≤ H
hL :
1 ≤ L
hW :
∀ j, Measurable (W j)
hlaw :
∀ j, HasLaw (W j) (poissonMeasure (m * ⟨cellVector P x j, ENNReal.toReal_nonneg⟩)) μ
conclusion
(∫ ω, if cellVector P x ∉ pilotRectangle (pilotLower H hHpos m L (fun j => W j ω)) (pilotUpperFormula H hHpos m L (fun j => W j ω)) then (armwiseExtensionFormula ε (pilotMidpoint H hHpos m L (fun j => W j ω)) - armwiseExtensionFormula ε (cellVector P x)) ^ 2 else 0 ∂μ)
≤ 1024 * (sqrt (8 * scalarMomentConstant 4) * (H / universalH) ^ 2 * exp (-20 * L)) * (cellMass P x * L / (m * ε) + L ^ 2 / ((m : ℝ) ^ 2 * ε ^ 2))
Proof (Lean source)
lemma pilotRectangle_bad_cellValue_normalized_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ]
{d : ℕ} (ε : ℝ) (P : DiscreteLaw d) (hε : 0 < ε)
(hP : ObservedClass ε P) (x : Fin d) (hp : 0 < cellMass P x)
(W : Fin 4 → Ω → ℕ) (m : ℝ≥0) (hm : 0 < m)
(H : ℝ) (hHpos : 0 < H) (L : ℝ) (hH : universalH ≤ H) (hL : 1 ≤ L)
(hW : ∀ j, Measurable (W j))
(hlaw : ∀ j, HasLaw (W j)
(poissonMeasure (m * ⟨cellVector P x j, ENNReal.toReal_nonneg⟩)) μ) :
(∫ ω, if cellVector P x ∉ pilotRectangle
(pilotLower H hHpos m L (fun j => W j ω))
(pilotUpperFormula H hHpos m L (fun j => W j ω)) then
(armwiseExtensionFormula ε (pilotMidpoint H hHpos m L (fun j => W j ω)) -
armwiseExtensionFormula ε (cellVector P x)) ^ 2 else 0 ∂μ) ≤
1024 * (sqrt (8 * scalarMomentConstant 4) * (H / universalH) ^ 2 *
exp (-20 * L)) *
(cellMass P x * L / (m * ε) + L ^ 2 / ((m : ℝ) ^ 2 * ε ^ 2)) := by
have hb := pilotRectangle_bad_cellValue_integral_sq_le μ ε P hε hP x hp W m hm
H hHpos L hH hL hW hlaw
let C : ℝ := sqrt (8 * scalarMomentConstant 4) * (H / universalH) ^ 2 *
exp (-20 * L)
have hscale (a : Bool) : (cellMass P x / armMass P a x) ^ 2 *
∑ y : Bool, (sqrt (jointMass P x a y * L / m) + L / m) ^ 2 ≤
8 * (cellMass P x * L / (m * ε) + L ^ 2 / ((m : ℝ) ^ 2 * ε ^ 2)) := by
have hmass : totalMassVecFormula (cellVector P x) = cellMass P x := by
simp [totalMassVecFormula, armMassVecFormula, cellVector, cellMass, cellIdx]; ring
have hfloor : ε * cellMass P x ≤ armMass P a x := by
rw [← anchoredDenom_cellVector_eq_armMass ε P hP x hp a, ← hmass]
exact le_max_right _ _
have h := inverseArm_coordinate_scale_sq_sum_le hp hε
(div_nonneg (by linarith : 0 ≤ L) m.coe_nonneg)
(show 0 ≤ jointMass P x a false from toReal_nonneg)
(show 0 ≤ jointMass P x a true from toReal_nonneg)
(show jointMass P x a false + jointMass P x a true = armMass P a x by
simp [armMass, jointMass]; ring) hfloor
convert h using 1 <;> simp [mul_div_assoc] <;> first | rfl | (ring <;> simp)
have hsum := Finset.sum_le_sum (fun a (_ : a ∈ (Finset.univ : Finset Bool)) => hscale a)
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_bool, nsmul_eq_mul,
Nat.cast_ofNat] at hsum
apply hb.trans
have h := mul_le_mul_of_nonneg_left hsum (show 0 ≤ 64 * C by dsimp [C]; positivity)
convert h using 1 <;> dsimp only [C] <;> first | rfl | ring
theorem pilotRectangle_bad_cellValue_integral_le Lemma pilotRectangle_bad_cellValue_integral_le in the paper ↗
The center-error first moment on failed rectangles follows from its normalized second moment and the exponentially small failure probability. This completes the center component of roadmap (22), retaining the square root of one inverse-overlap power in the mass term. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
d :
ℕ
ε :
ℝ
P :
hε :
0 < ε
hP :
ObservedClass ε P
x :
Fin d
hp :
0 < cellMass P x
W :
Fin 4 → Ω → ℕ
m :
ℝ≥0
hm :
0 < m
H :
ℝ
hHpos :
0 < H
L :
ℝ
hH :
universalH ≤ H
hL :
1 ≤ L
hW :
∀ j, Measurable (W j)
hlaw :
∀ j, HasLaw (W j) (poissonMeasure (m * ⟨cellVector P x j, ENNReal.toReal_nonneg⟩)) μ
conclusion
(∫ ω, if cellVector P x ∉ pilotRectangle (pilotLower H hHpos m L (fun j => W j ω)) (pilotUpperFormula H hHpos m L (fun j => W j ω)) then |armwiseExtensionFormula ε (pilotMidpoint H hHpos m L (fun j => W j ω)) - armwiseExtensionFormula ε (cellVector P x)| else 0 ∂μ)
≤ sqrt (8 * (1024 * sqrt (8 * scalarMomentConstant 4) * (H / universalH) ^ 2)) * exp (-30 * L) * (sqrt (cellMass P x * L / (m * ε)) + L / (m * ε))
Proof (Lean source)
lemma pilotRectangle_bad_cellValue_integral_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ]
{d : ℕ} (ε : ℝ) (P : DiscreteLaw d) (hε : 0 < ε)
(hP : ObservedClass ε P) (x : Fin d) (hp : 0 < cellMass P x)
(W : Fin 4 → Ω → ℕ) (m : ℝ≥0) (hm : 0 < m)
(H : ℝ) (hHpos : 0 < H) (L : ℝ) (hH : universalH ≤ H) (hL : 1 ≤ L)
(hW : ∀ j, Measurable (W j))
(hlaw : ∀ j, HasLaw (W j)
(poissonMeasure (m * ⟨cellVector P x j, ENNReal.toReal_nonneg⟩)) μ) :
(∫ ω, if cellVector P x ∉ pilotRectangle
(pilotLower H hHpos m L (fun j => W j ω))
(pilotUpperFormula H hHpos m L (fun j => W j ω)) then
|armwiseExtensionFormula ε (pilotMidpoint H hHpos m L (fun j => W j ω)) -
armwiseExtensionFormula ε (cellVector P x)| else 0 ∂μ) ≤
sqrt (8 * (1024 * sqrt (8 * scalarMomentConstant 4) *
(H / universalH) ^ 2)) * exp (-30 * L) *
(sqrt (cellMass P x * L / (m * ε)) + L / (m * ε)) := by
classical
let q : Fin 4 → ℝ≥0 := fun j => ⟨cellVector P x j, ENNReal.toReal_nonneg⟩
let B : Set Ω := {ω | cellVector P x ∉ pilotRectangle
(pilotLower H hHpos m L (fun j => W j ω))
(pilotUpperFormula H hHpos m L (fun j => W j ω))}
let S : Ω → ℝ := fun ω => |armwiseExtensionFormula ε
(pilotMidpoint H hHpos m L (fun j => W j ω)) -
armwiseExtensionFormula ε (cellVector P x)|
let C : ℝ := 1024 * sqrt (8 * scalarMomentConstant 4) * (H / universalH) ^ 2
let a : ℝ := cellMass P x * L / (m * ε)
let b : ℝ := L / (m * ε)
let s : ℝ := sqrt a + b
have hB : MeasurableSet B := by
dsimp [B]
unfold pilotRectangle pilotLower pilotUpperFormula pilotHalfWidthFormula pilotCenterFormula
measurability
have hcenter : MemLp (fun ω => armwiseExtensionFormula ε
(pilotMidpoint H hHpos m L (fun j => W j ω))) 2 μ := by
have hmass := pilotTotalMass_memLp_two μ W q m hm H hHpos L hH hL hW hlaw
apply hmass.of_le
· first | fun_prop | exact ((Measurable.of_discrete (f := fun z : Fin 4 → ℕ =>
armwiseExtensionFormula ε (pilotMidpoint H hHpos m L z))).comp
(measurable_pi_lambda _ hW)).aestronglyMeasurable
· apply ae_of_all
intro ω
have hb := pilotMidpoint_nonneg H hHpos m L (NNReal.coe_pos.mpr hm) (fun j => W j ω)
have hf := armwiseExtension_nonneg_le_totalMassVec hε hb
rw [Real.norm_eq_abs, Real.norm_eq_abs, abs_of_nonneg hf.1]
exact hf.2.trans (le_abs_self _)
have hS : MemLp S 2 μ := (hcenter.sub (memLp_const _)).abs
have hSn (ω : Ω) : 0 ≤ S ω := abs_nonneg _
have ha : 0 ≤ a := by dsimp [a]; positivity
have hb : 0 ≤ b := by dsimp [b]; positivity
have hs : 0 ≤ s := add_nonneg (Real.sqrt_nonneg _) hb
have hC : 0 ≤ C := by dsimp [C]; positivity
have hprob : μ.real B ≤ 8 * exp (-40 * L) := by
have hp := pilotRectangle_bad_probability_le μ W q m hm H hHpos L hH hL hlaw
exact (ENNReal.toReal_mono ENNReal.ofReal_ne_top hp).trans_eq
(ENNReal.toReal_ofReal (by positivity))
have hsecond : (∫ ω, B.indicator (fun ω => S ω ^ 2) ω ∂μ) ≤
C * exp (-20 * L) * s ^ 2 := by
have h := pilotRectangle_bad_cellValue_normalized_le μ ε P hε hP x hp W m hm
H hHpos L hH hL hW hlaw
have hab : a + b ^ 2 ≤ s ^ 2 := by
dsimp [s]
nlinarith [Real.sq_sqrt ha, Real.sqrt_nonneg a]
have he : L ^ 2 / ((m : ℝ) ^ 2 * ε ^ 2) = b ^ 2 := by dsimp [b]; ring
simp only [he, ← mul_assoc] at h
change _ ≤ C * exp (-20 * L) * (a + b ^ 2) at h
simpa only [indicator, B, S, Set.mem_ofPred_eq, sq_abs] using
h.trans (mul_le_mul_of_nonneg_left hab (mul_nonneg hC (Real.exp_nonneg _)))
have hholder := integral_mul_le_Lp_mul_Lq_of_nonneg Real.HolderConjugate.two_two
(f := B.indicator S) (g := B.indicator (fun _ => (1 : ℝ)))
(ae_of_all _ fun ω => Set.indicator_nonneg (fun ω _ => hSn ω) ω)
(ae_of_all _ fun ω => Set.indicator_nonneg (fun _ _ => zero_le_one) ω)
(by simpa using hS.indicator hB)
(by simpa using (memLp_const (1 : ℝ) (p := (2 : ENNReal)) (μ := μ)).indicator hB)
have hmul (ω : Ω) : B.indicator S ω * B.indicator (fun _ => (1 : ℝ)) ω =
B.indicator S ω := by by_cases h : ω ∈ B <;> simp [h]
have hsq (ω : Ω) : (B.indicator S ω) ^ (2 : ℝ) =
B.indicator (fun ω => S ω ^ 2) ω := by
by_cases h : ω ∈ B <;> simp [h]
have hone (ω : Ω) : (B.indicator (fun _ => (1 : ℝ)) ω) ^ (2 : ℝ) =
B.indicator (fun _ => (1 : ℝ)) ω := by by_cases h : ω ∈ B <;> simp [h]
simp_rw [hmul, hsq, hone, ← Real.sqrt_eq_rpow] at hholder
rw [integral_indicator_const _ hB] at hholder
simp only [smul_eq_mul, mul_one] at hholder
have hbound := mul_le_mul (Real.sqrt_le_sqrt hsecond) (Real.sqrt_le_sqrt hprob)
(Real.sqrt_nonneg _) (Real.sqrt_nonneg _)
have heq : sqrt (C * exp (-20 * L) * s ^ 2) *
sqrt (8 * exp (-40 * L)) =
sqrt (8 * C) * exp (-30 * L) * s := by
rw [← Real.sqrt_mul (by positivity)]
have hexp : exp (-20 * L) * exp (-40 * L) =
exp (-30 * L) ^ 2 := by
rw [pow_two, ← Real.exp_add, ← Real.exp_add]
congr 1; ring
have he : C * exp (-20 * L) * s ^ 2 * (8 * exp (-40 * L)) =
(sqrt (8 * C) * exp (-30 * L) * s) ^ 2 := by
rw [mul_pow, mul_pow, Real.sq_sqrt (by positivity)]
rw [← hexp]
ring
rw [he, Real.sqrt_sq (by positivity)]
simpa only [indicator, B, S, C, s, a, b, Set.mem_ofPred_eq] using
(hholder.trans hbound).trans_eq heq
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.UpperBadPilotClipped 3 declarations The width and center estimates in roadmap (22) control the actual cell statistic on bad pilots, as required after (28).
Actual clipped-cell moments on failed pilot rectangles
The width and center estimates in roadmap (22) control the actual cell statistic on bad pilots, as required after (28). Evaluation counts may be arbitrary: clipping supplies the pathwise bound without evaluation moments.
The pilot center functional is square-integrable by domination by total midpoint mass. This regularity holds on failed rectangles as well. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
W :
Fin 4 → Ω → ℕ
q :
Fin 4 → ℝ≥0
m :
ℝ≥0
hm :
0 < m
H :
ℝ
hHpos :
0 < H
L ε :
ℝ
hH :
universalH ≤ H
hL :
1 ≤ L
hε :
0 < ε
hW :
∀ j, Measurable (W j)
hlaw :
∀ j, HasLaw (W j) (poissonMeasure (m * q j)) μ
conclusion
Proof (Lean source)
lemma pilotCellValue_memLp_two {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ]
(W : Fin 4 → Ω → ℕ) (q : Fin 4 → ℝ≥0) (m : ℝ≥0) (hm : 0 < m)
(H : ℝ) (hHpos : 0 < H) (L ε : ℝ) (hH : universalH ≤ H) (hL : 1 ≤ L)
(hε : 0 < ε) (hW : ∀ j, Measurable (W j))
(hlaw : ∀ j, HasLaw (W j) (poissonMeasure (m * q j)) μ) :
MemLp (fun ω => armwiseExtensionFormula ε
(pilotMidpoint H hHpos m L (fun j => W j ω))) 2 μ := by
apply (pilotTotalMass_memLp_two μ W q m hm H hHpos L hH hL hW hlaw).of_le
· first | fun_prop | exact ((Measurable.of_discrete (f := fun z : Fin 4 → ℕ =>
armwiseExtensionFormula ε (pilotMidpoint H hHpos m L z))).comp
(measurable_pi_lambda _ hW)).aestronglyMeasurable
· apply ae_of_all
intro ω
have hb := pilotMidpoint_nonneg H hHpos m L (NNReal.coe_pos.mpr hm)
(fun j => W j ω)
have hf := armwiseExtension_nonneg_le_totalMassVec hε hb
rw [Real.norm_eq_abs, Real.norm_eq_abs, abs_of_nonneg hf.1]
exact hf.2.trans (le_abs_self _)
theorem pilotRectangle_bad_clippedCellValue_integral_le Lemma pilotRectangle_bad_clippedCellValue_integral_le in the paper ↗
On failed pilot rectangles, the actual clipped occupied-cell absolute error is exponentially small on the normalized mass and overlap scales of (22). No law or moment assumption on evaluation counts is needed. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
d :
ℕ
ε :
ℝ
P :
hε :
0 < ε
hε1 :
ε ≤ 1
hP :
ObservedClass ε P
x :
Fin d
hp :
0 < cellMass P x
W :
Fin 4 → Ω → ℕ
Ne :
Ω → Fin 4 → ℕ
m :
ℝ≥0
hm :
0 < m
H :
ℝ
hHpos :
0 < H
L :
ℝ
hH :
universalH ≤ H
hL :
1 ≤ L
hW :
∀ j, Measurable (W j)
hlaw :
∀ j, HasLaw (W j) (poissonMeasure (m * ⟨cellVector P x j, ENNReal.toReal_nonneg⟩)) μ
hind :
iIndepFun W μ
K :
ℕ
conclusion
(∫ ω, if cellVector P x ∉ pilotRectangle (pilotLower H hHpos m L (fun j => W j ω)) (pilotUpperFormula H hHpos m L (fun j => W j ω)) then |clippedCellValueFormula ε K d m (pilotMidpoint H hHpos m L (fun j => W j ω)) (pilotRadiusFormula H hHpos m L (fun j => W j ω)) (Ne ω) - armwiseExtensionFormula ε (cellVector P x)| else 0 ∂μ)
≤ ((d : ℝ) ^ (1 / 4 : ℝ) * sqrt (8 * (512 * H ^ 2 * (8 + 2 * (H / universalH) * productMomentConstant 4 1))) + sqrt (8 * (1024 * sqrt (8 * scalarMomentConstant 4) * (H / universalH) ^ 2))) * exp (-30 * L) * (sqrt (cellMass P x * L / (m * ε)) + L / (m * ε))
Proof (Lean source)
lemma pilotRectangle_bad_clippedCellValue_integral_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ]
{d : ℕ} (ε : ℝ) (P : DiscreteLaw d) (hε : 0 < ε) (hε1 : ε ≤ 1)
(hP : ObservedClass ε P) (x : Fin d) (hp : 0 < cellMass P x)
(W : Fin 4 → Ω → ℕ) (Ne : Ω → Fin 4 → ℕ) (m : ℝ≥0) (hm : 0 < m)
(H : ℝ) (hHpos : 0 < H) (L : ℝ) (hH : universalH ≤ H) (hL : 1 ≤ L)
(hW : ∀ j, Measurable (W j))
(hlaw : ∀ j, HasLaw (W j)
(poissonMeasure (m * ⟨cellVector P x j, ENNReal.toReal_nonneg⟩)) μ)
(hind : iIndepFun W μ) (K : ℕ) :
(∫ ω, if cellVector P x ∉ pilotRectangle
(pilotLower H hHpos m L (fun j => W j ω))
(pilotUpperFormula H hHpos m L (fun j => W j ω)) then
|clippedCellValueFormula ε K d m
(pilotMidpoint H hHpos m L (fun j => W j ω))
(pilotRadiusFormula H hHpos m L (fun j => W j ω)) (Ne ω) -
armwiseExtensionFormula ε (cellVector P x)| else 0 ∂μ) ≤
((d : ℝ) ^ (1 / 4 : ℝ) * sqrt (8 * (512 * H ^ 2 *
(8 + 2 * (H / universalH) * productMomentConstant 4 1))) +
sqrt (8 * (1024 * sqrt (8 * scalarMomentConstant 4) * (H / universalH) ^ 2))) *
exp (-30 * L) *
(sqrt (cellMass P x * L / (m * ε)) + L / (m * ε)) := by
classical
let q : Fin 4 → ℝ≥0 := fun j => ⟨cellVector P x j, ENNReal.toReal_nonneg⟩
let B : Set Ω := {ω | cellVector P x ∉ pilotRectangle
(pilotLower H hHpos m L (fun j => W j ω))
(pilotUpperFormula H hHpos m L (fun j => W j ω))}
let b : Ω → Fin 4 → ℝ := fun ω => pilotMidpoint H hHpos m L (fun j => W j ω)
let r : Ω → Fin 4 → ℝ := fun ω => pilotRadiusFormula H hHpos m L (fun j => W j ω)
have hB : MeasurableSet B := by
dsimp [B]
unfold pilotRectangle pilotLower pilotUpperFormula pilotHalfWidthFormula pilotCenterFormula
measurability
have hS := pilotClippingScale_memLp_two μ W q m hm H hHpos L ε hH hL hε hε1 hW hlaw
have hF := (pilotCellValue_memLp_two μ W q m hm H hHpos L ε hH hL hε hW hlaw).sub
(memLp_const (armwiseExtensionFormula ε (cellVector P x)))
have hw (ω : Ω) (_ : ω ∈ B) :
0 ≤ (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε (b ω) (r ω) := by
apply mul_nonneg (Real.rpow_nonneg (Nat.cast_nonneg _) _)
exact clippingScale_nonneg ε _ _
(pilotMidpoint_nonneg H hHpos m L (NNReal.coe_pos.mpr hm) _)
(fun j => (pilotRadius_pos H hHpos m L (NNReal.coe_pos.mpr hm) (by linarith) _ j).le)
have hmass : (∑ j, (q j : ℝ)) = cellMass P x := by
change (∑ j, cellVector P x j) = cellMass P x
simp [cellVector, cellMass, Fin.sum_univ_succ]
ring
have ht := clippedCellValue_integral_abs_on_le μ B b r Ne ε K d m
(armwiseExtensionFormula ε (cellVector P x)) hw
((hS.integrable (by norm_num)).const_mul _).restrict
(hF.abs.integrable (by norm_num)).restrict hB
have hs := pilotRectangle_bad_clippingScale_integral_le μ W q m hm H hHpos L ε
hH hL hε hε1 hW hlaw hind
have hf := pilotRectangle_bad_cellValue_integral_le μ ε P hε hP x hp W m hm
H hHpos L hH hL hW hlaw
rw [hmass] at hs
have hqfun : (fun j => (q j : ℝ)) = cellVector P x := rfl
rw [hqfun] at hs
rw [← integral_indicator hB] at ht
rw [← integral_indicator hB, ← integral_indicator hB] at ht
have hscale (ω : Ω) : B.indicator (fun ω =>
(d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε (b ω) (r ω)) ω =
(d : ℝ) ^ (1 / 4 : ℝ) *
B.indicator (fun ω => clippingScaleFormula ε (b ω) (r ω)) ω := by
by_cases h : ω ∈ B <;> simp [h]
simp_rw [hscale] at ht
rw [integral_const_mul] at ht
simp only [indicator, B, b, r, Set.mem_ofPred_eq] at ht
have hsum := add_le_add (mul_le_mul_of_nonneg_left hs
(Real.rpow_nonneg (Nat.cast_nonneg d) (1 / 4 : ℝ))) hf
exact (ht.trans hsum).trans_eq (by ring)
theorem pilotRectangle_bad_clippedCellValue_integral_sq_le Lemma pilotRectangle_bad_clippedCellValue_integral_sq_le in the paper ↗
On failed pilot rectangles, the actual clipped occupied-cell loss is exponentially small on the normalized mass and overlap scales of (22). No law or moment assumption on evaluation counts is needed. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
d :
ℕ
ε :
ℝ
P :
hε :
0 < ε
hε1 :
ε ≤ 1
hP :
ObservedClass ε P
x :
Fin d
hp :
0 < cellMass P x
W :
Fin 4 → Ω → ℕ
Ne :
Ω → Fin 4 → ℕ
m :
ℝ≥0
hm :
0 < m
H :
ℝ
hHpos :
0 < H
L :
ℝ
hH :
universalH ≤ H
hL :
1 ≤ L
hW :
∀ j, Measurable (W j)
hlaw :
∀ j, HasLaw (W j) (poissonMeasure (m * ⟨cellVector P x j, ENNReal.toReal_nonneg⟩)) μ
hind :
iIndepFun W μ
K :
ℕ
conclusion
(∫ ω, if cellVector P x ∉ pilotRectangle (pilotLower H hHpos m L (fun j => W j ω)) (pilotUpperFormula H hHpos m L (fun j => W j ω)) then (clippedCellValueFormula ε K d m (pilotMidpoint H hHpos m L (fun j => W j ω)) (pilotRadiusFormula H hHpos m L (fun j => W j ω)) (Ne ω) - armwiseExtensionFormula ε (cellVector P x)) ^ 2 else 0 ∂μ)
≤ 2 * (((d : ℝ) ^ (1 / 4 : ℝ)) ^ 2 * (512 * H ^ 2 * (8 + 2 * (H / universalH) * productMomentConstant 4 1)) + (1024 * sqrt (8 * scalarMomentConstant 4) * (H / universalH) ^ 2)) * exp (-20 * L) * (cellMass P x * L / (m * ε) + L ^ 2 / ((m : ℝ) ^ 2 * ε ^ 2))
Proof (Lean source)
lemma pilotRectangle_bad_clippedCellValue_integral_sq_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ]
{d : ℕ} (ε : ℝ) (P : DiscreteLaw d) (hε : 0 < ε) (hε1 : ε ≤ 1)
(hP : ObservedClass ε P) (x : Fin d) (hp : 0 < cellMass P x)
(W : Fin 4 → Ω → ℕ) (Ne : Ω → Fin 4 → ℕ) (m : ℝ≥0) (hm : 0 < m)
(H : ℝ) (hHpos : 0 < H) (L : ℝ) (hH : universalH ≤ H) (hL : 1 ≤ L)
(hW : ∀ j, Measurable (W j))
(hlaw : ∀ j, HasLaw (W j)
(poissonMeasure (m * ⟨cellVector P x j, ENNReal.toReal_nonneg⟩)) μ)
(hind : iIndepFun W μ) (K : ℕ) :
(∫ ω, if cellVector P x ∉ pilotRectangle
(pilotLower H hHpos m L (fun j => W j ω))
(pilotUpperFormula H hHpos m L (fun j => W j ω)) then
(clippedCellValueFormula ε K d m
(pilotMidpoint H hHpos m L (fun j => W j ω))
(pilotRadiusFormula H hHpos m L (fun j => W j ω)) (Ne ω) -
armwiseExtensionFormula ε (cellVector P x)) ^ 2 else 0 ∂μ) ≤
2 * (((d : ℝ) ^ (1 / 4 : ℝ)) ^ 2 * (512 * H ^ 2 *
(8 + 2 * (H / universalH) * productMomentConstant 4 1)) +
(1024 * sqrt (8 * scalarMomentConstant 4) * (H / universalH) ^ 2)) *
exp (-20 * L) *
(cellMass P x * L / (m * ε) + L ^ 2 / ((m : ℝ) ^ 2 * ε ^ 2)) := by
classical
let q : Fin 4 → ℝ≥0 := fun j => ⟨cellVector P x j, ENNReal.toReal_nonneg⟩
let B : Set Ω := {ω | cellVector P x ∉ pilotRectangle
(pilotLower H hHpos m L (fun j => W j ω))
(pilotUpperFormula H hHpos m L (fun j => W j ω))}
let b : Ω → Fin 4 → ℝ := fun ω => pilotMidpoint H hHpos m L (fun j => W j ω)
let r : Ω → Fin 4 → ℝ := fun ω => pilotRadiusFormula H hHpos m L (fun j => W j ω)
have hB : MeasurableSet B := by
dsimp [B]
unfold pilotRectangle pilotLower pilotUpperFormula pilotHalfWidthFormula pilotCenterFormula
measurability
have hS := pilotClippingScale_memLp_two μ W q m hm H hHpos L ε hH hL hε hε1 hW hlaw
have hF := (pilotCellValue_memLp_two μ W q m hm H hHpos L ε hH hL hε hW hlaw).sub
(memLp_const (armwiseExtensionFormula ε (cellVector P x)))
have hw (ω : Ω) (_ : ω ∈ B) :
0 ≤ (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε (b ω) (r ω) := by
apply mul_nonneg (Real.rpow_nonneg (Nat.cast_nonneg _) _)
exact clippingScale_nonneg ε _ _
(pilotMidpoint_nonneg H hHpos m L (NNReal.coe_pos.mpr hm) _)
(fun j => (pilotRadius_pos H hHpos m L (NNReal.coe_pos.mpr hm) (by linarith) _ j).le)
have hmass : (∑ j, (q j : ℝ)) = cellMass P x := by
change (∑ j, cellVector P x j) = cellMass P x
simp [cellVector, cellMass, Fin.sum_univ_succ]
ring
have hiS := (memLp_two_iff_integrable_sq hS.aestronglyMeasurable).1 hS
have hiF := (memLp_two_iff_integrable_sq hF.aestronglyMeasurable).1 hF
have ht := clippedCellValue_integral_sq_on_le μ B b r Ne ε K d m
(armwiseExtensionFormula ε (cellVector P x)) hw
(by simpa only [mul_pow] using (hiS.const_mul (((d : ℝ) ^ (1 / 4 : ℝ)) ^ 2)).restrict)
hiF.restrict hB
have hs := pilotRectangle_bad_clippingScale_normalized_le μ W q m hm H hHpos L ε
hH hL hε hε1 hW hlaw hind
have hf := pilotRectangle_bad_cellValue_normalized_le μ ε P hε hP x hp W m hm
H hHpos L hH hL hW hlaw
rw [hmass] at hs
have hqfun : (fun j => (q j : ℝ)) = cellVector P x := rfl
rw [hqfun] at hs
rw [← integral_indicator hB] at ht
rw [← integral_indicator hB, ← integral_indicator hB] at ht
have hscale (ω : Ω) : B.indicator (fun ω =>
((d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε (b ω) (r ω)) ^ 2) ω =
((d : ℝ) ^ (1 / 4 : ℝ)) ^ 2 *
B.indicator (fun ω => clippingScaleFormula ε (b ω) (r ω) ^ 2) ω := by
by_cases h : ω ∈ B <;> simp [h, mul_pow]
simp_rw [hscale] at ht
rw [integral_const_mul] at ht
simp only [indicator, B, b, r, Set.mem_ofPred_eq] at ht
have hsum := add_le_add (mul_le_mul_of_nonneg_left hs (sq_nonneg ((d : ℝ) ^ (1 / 4 : ℝ)))) hf
have hb := mul_le_mul_of_nonneg_left hsum (show (0 : ℝ) ≤ 2 by norm_num)
rw [← mul_add] at ht
exact (ht.trans hb).trans_eq (by ring)
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.UpperBadPilotScale 4 declarations The clipping-scale component of roadmap (22) follows from its affine square envelope and the joint first score moment on rectangle failure.
Clipping-scale moments on failed pilot rectangles
The clipping-scale component of roadmap (22) follows from its affine square envelope and the joint first score moment on rectangle failure. The midpoint and failure event remain coupled throughout the argument.
theorem pilotTotalMass_le_true_mass_add_score Lemma pilotTotalMass_le_true_mass_add_score in the paper ↗
The total midpoint mass is at most the true mass plus the aggregate deviation-and-width score, including failed pilots. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
W :
Fin 4 → Ω → ℕ
q :
Fin 4 → ℝ≥0
m :
ℝ≥0
hm :
0 < m
H :
ℝ
hH :
0 < H
L :
ℝ
hL :
0 ≤ L
ω :
Ω
conclusion
totalMassVecFormula (pilotMidpoint H hH m L (fun j => W j ω))
≤ (∑ j, (q j : ℝ)) + empiricalAggregateScore W H L m q ω
Proof (Lean source)
lemma pilotTotalMass_le_true_mass_add_score {Ω : Type*}
(W : Fin 4 → Ω → ℕ) (q : Fin 4 → ℝ≥0) (m : ℝ≥0) (hm : 0 < m)
(H : ℝ) (hH : 0 < H) (L : ℝ) (hL : 0 ≤ L) (ω : Ω) :
totalMassVecFormula (pilotMidpoint H hH m L (fun j => W j ω)) ≤
(∑ j, (q j : ℝ)) + empiricalAggregateScore W H L m q ω := by
have hs := pilotMidpoint_total_abs_error_le_score W q m hm H hH L hL ω
have hd : totalMassVecFormula (pilotMidpoint H hH m L (fun j => W j ω)) -
∑ j, (q j : ℝ) ≤ ∑ j, |pilotMidpoint H hH m L (fun i => W i ω) j - q j| := by
rw [totalMassVec_eq_sum_coordinates, ← Finset.sum_sub_distrib]
exact Finset.sum_le_sum fun j _ => le_abs_self _
linarith
theorem pilotRectangle_bad_clippingScale_integral_sq_le Lemma pilotRectangle_bad_clippingScale_integral_sq_le in the paper ↗
The bad-pilot second moment of the clipping scale has an exponential factor multiplying only mass-normalized deviations. The probability term and joint score term are both derived from the independent Poisson laws. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
W :
Fin 4 → Ω → ℕ
q :
Fin 4 → ℝ≥0
m :
ℝ≥0
hm :
0 < m
H :
ℝ
hHpos :
0 < H
L ε :
ℝ
hH :
universalH ≤ H
hL :
1 ≤ L
hε :
0 < ε
hε1 :
ε ≤ 1
hW :
∀ j, Measurable (W j)
hlaw :
∀ j, HasLaw (W j) (poissonMeasure (m * q j)) μ
hind :
iIndepFun W μ
conclusion
(∫ ω, if (fun j => (q j : ℝ)) ∉ pilotRectangle (pilotLower H hHpos m L (fun j => W j ω)) (pilotUpperFormula H hHpos m L (fun j => W j ω)) then clippingScaleFormula ε (pilotMidpoint H hHpos m L (fun j => W j ω)) (pilotRadiusFormula H hHpos m L (fun j => W j ω)) ^ 2 else 0 ∂μ)
≤ 512 * H ^ 2 * (((∑ j, (q j : ℝ)) * (L / m) / ε + ((L / m) / ε) ^ 2) * (8 * exp (-40 * L)) + ((L / m) / ε) * ((H / universalH) * productMomentConstant 4 1 * exp (-20 * L) * (sqrt ((∑ j, (q j : ℝ)) * L / m) + L / m)))
Proof (Lean source)
lemma pilotRectangle_bad_clippingScale_integral_sq_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ]
(W : Fin 4 → Ω → ℕ) (q : Fin 4 → ℝ≥0) (m : ℝ≥0) (hm : 0 < m)
(H : ℝ) (hHpos : 0 < H) (L ε : ℝ) (hH : universalH ≤ H) (hL : 1 ≤ L)
(hε : 0 < ε) (hε1 : ε ≤ 1)
(hW : ∀ j, Measurable (W j))
(hlaw : ∀ j, HasLaw (W j) (poissonMeasure (m * q j)) μ)
(hind : iIndepFun W μ) :
(∫ ω, if (fun j => (q j : ℝ)) ∉ pilotRectangle
(pilotLower H hHpos m L (fun j => W j ω))
(pilotUpperFormula H hHpos m L (fun j => W j ω)) then
clippingScaleFormula ε (pilotMidpoint H hHpos m L (fun j => W j ω))
(pilotRadiusFormula H hHpos m L (fun j => W j ω)) ^ 2 else 0 ∂μ) ≤
512 * H ^ 2 *
(((∑ j, (q j : ℝ)) * (L / m) / ε + ((L / m) / ε) ^ 2) *
(8 * exp (-40 * L)) +
((L / m) / ε) * ((H / universalH) * productMomentConstant 4 1 *
exp (-20 * L) *
(sqrt ((∑ j, (q j : ℝ)) * L / m) + L / m))) := by
classical
let B : Set Ω := {ω | (fun j => (q j : ℝ)) ∉ pilotRectangle
(pilotLower H hHpos m L (fun j => W j ω))
(pilotUpperFormula H hHpos m L (fun j => W j ω))}
let Q : ℝ := ∑ j, (q j : ℝ)
let δ : ℝ := L / m
let E := empiricalAggregateScore W H L m q
let a : ℝ := Q * δ / ε + (δ / ε) ^ 2
have hδ : 0 ≤ δ := div_nonneg (by linarith) m.coe_nonneg
have ha : 0 ≤ a := by dsimp [a, Q]; positivity
have hB : MeasurableSet B := by
dsimp [B]
unfold pilotRectangle pilotLower pilotUpperFormula pilotHalfWidthFormula pilotCenterFormula
measurability
have hiE : Integrable E μ := by
simpa only [pow_one] using
pilotAggregateScore_pow_integrable μ W q m hm H L hH hL hW hlaw 1 (inl rfl)
have hprob : μ.real B ≤ 8 * exp (-40 * L) := by
have hp := pilotRectangle_bad_probability_le μ W q m hm H hHpos L hH hL hlaw
exact (ENNReal.toReal_mono ENNReal.ofReal_ne_top hp).trans_eq
(ENNReal.toReal_ofReal (by positivity))
have hscore : (∫ ω, B.indicator E ω ∂μ) ≤
(H / universalH) * productMomentConstant 4 1 * exp (-20 * L) *
(sqrt (Q * L / m) + L / m) := by
have hs := pilotRectangle_bad_aggregate_moment_le μ W q m hm H hHpos L hH hL
hW hlaw hind 1 (inl rfl)
simpa only [indicator, B, E, empiricalAggregateScore, normalizedDeviation,
pilotCenterFormula, pilotHalfWidth_eq_empiricalRadius, pow_one, Q, Set.mem_setOf_eq] using hs
have hi : Integrable (fun ω => 512 * H ^ 2 *
(B.indicator (fun _ => a) ω + (δ / ε) * B.indicator E ω)) μ :=
(((integrable_const a).indicator hB).add
((hiE.indicator hB).const_mul (δ / ε))).const_mul _
have hpoint (ω : Ω) :
(if ω ∈ B then clippingScaleFormula ε
(pilotMidpoint H hHpos m L (fun j => W j ω))
(pilotRadiusFormula H hHpos m L (fun j => W j ω)) ^ 2 else 0) ≤
512 * H ^ 2 * (B.indicator (fun _ => a) ω + (δ / ε) * B.indicator E ω) := by
by_cases hb : ω ∈ B
· rw [if_pos hb, Set.indicator_of_mem hb, Set.indicator_of_mem hb]
have hs := pilotClippingScale_sq_le H hHpos m L ε (NNReal.coe_pos.mpr hm)
(by linarith) hε hε1 (fun j => W j ω)
apply hs.trans
have ht := pilotTotalMass_le_true_mass_add_score W q m hm H hHpos L (by linarith) ω
have hm := mul_le_mul_of_nonneg_right ht (div_nonneg hδ hε.le)
apply mul_le_mul_of_nonneg_left _ (by positivity : 0 ≤ 512 * H ^ 2)
dsimp [a, δ, Q, E] at *
convert add_le_add_right hm ((L / m / ε) ^ 2) using 1 <;> ring
· simp [hb, Set.indicator_of_notMem hb]
calc
_ ≤ ∫ ω, 512 * H ^ 2 *
(B.indicator (fun _ => a) ω + (δ / ε) * B.indicator E ω) ∂μ :=
integral_mono_of_nonneg (ae_of_all _ fun ω => by
change 0 ≤ if ω ∈ B then _ else 0
split_ifs <;> positivity) hi (ae_of_all _ hpoint)
_ = 512 * H ^ 2 * (a * μ.real B + (δ / ε) * ∫ ω, B.indicator E ω ∂μ) := by
rw [integral_const_mul, integral_add ((integrable_const a).indicator hB)
((hiE.indicator hB).const_mul _), integral_indicator_const a hB, integral_const_mul]
simp only [smul_eq_mul, mul_comm a]
_ ≤ _ := by
apply mul_le_mul_of_nonneg_left _ (by positivity : 0 ≤ 512 * H ^ 2)
exact add_le_add (mul_le_mul_of_nonneg_left hprob ha)
(mul_le_mul_of_nonneg_left hscore (div_nonneg hδ hε.le))
theorem pilotRectangle_bad_clippingScale_normalized_le Lemma pilotRectangle_bad_clippingScale_normalized_le in the paper ↗
The clipping-scale component of (22) has the same mass and overlap scales as (19), with an exponentially small multiplier. In particular the mass term has only one inverse-overlap power even on failed pilot rectangles. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
W :
Fin 4 → Ω → ℕ
q :
Fin 4 → ℝ≥0
m :
ℝ≥0
hm :
0 < m
H :
ℝ
hHpos :
0 < H
L ε :
ℝ
hH :
universalH ≤ H
hL :
1 ≤ L
hε :
0 < ε
hε1 :
ε ≤ 1
hW :
∀ j, Measurable (W j)
hlaw :
∀ j, HasLaw (W j) (poissonMeasure (m * q j)) μ
hind :
iIndepFun W μ
conclusion
(∫ ω, if (fun j => (q j : ℝ)) ∉ pilotRectangle (pilotLower H hHpos m L (fun j => W j ω)) (pilotUpperFormula H hHpos m L (fun j => W j ω)) then clippingScaleFormula ε (pilotMidpoint H hHpos m L (fun j => W j ω)) (pilotRadiusFormula H hHpos m L (fun j => W j ω)) ^ 2 else 0 ∂μ)
≤ 512 * H ^ 2 * (8 + 2 * (H / universalH) * productMomentConstant 4 1) * exp (-20 * L) * ((∑ j, (q j : ℝ)) * L / (m * ε) + L ^ 2 / ((m : ℝ) ^ 2 * ε ^ 2))
Proof (Lean source)
lemma pilotRectangle_bad_clippingScale_normalized_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ]
(W : Fin 4 → Ω → ℕ) (q : Fin 4 → ℝ≥0) (m : ℝ≥0) (hm : 0 < m)
(H : ℝ) (hHpos : 0 < H) (L ε : ℝ) (hH : universalH ≤ H) (hL : 1 ≤ L)
(hε : 0 < ε) (hε1 : ε ≤ 1)
(hW : ∀ j, Measurable (W j))
(hlaw : ∀ j, HasLaw (W j) (poissonMeasure (m * q j)) μ)
(hind : iIndepFun W μ) :
(∫ ω, if (fun j => (q j : ℝ)) ∉ pilotRectangle
(pilotLower H hHpos m L (fun j => W j ω))
(pilotUpperFormula H hHpos m L (fun j => W j ω)) then
clippingScaleFormula ε (pilotMidpoint H hHpos m L (fun j => W j ω))
(pilotRadiusFormula H hHpos m L (fun j => W j ω)) ^ 2 else 0 ∂μ) ≤
512 * H ^ 2 * (8 + 2 * (H / universalH) * productMomentConstant 4 1) *
exp (-20 * L) *
((∑ j, (q j : ℝ)) * L / (m * ε) + L ^ 2 / ((m : ℝ) ^ 2 * ε ^ 2)) := by
have hb := pilotRectangle_bad_clippingScale_integral_sq_le μ W q m hm H hHpos L ε
hH hL hε hε1 hW hlaw hind
let Q : ℝ := ∑ j, (q j : ℝ)
let δ : ℝ := L / m
let a : ℝ := Q * δ / ε + (δ / ε) ^ 2
let C : ℝ := (H / universalH) * productMomentConstant 4 1
have hQ : 0 ≤ Q := Finset.sum_nonneg fun j _ => (q j).coe_nonneg
have hδ : 0 ≤ δ := div_nonneg (by linarith) m.coe_nonneg
have ha : 0 ≤ a := by dsimp [a]; positivity
have hC : 0 ≤ C := mul_nonneg (div_nonneg hHpos.le universalH_pos.le)
(productMomentConstant_pos 4 1).le
have hsqrt : sqrt (Q * δ) ≤ Q + δ := by
nlinarith [Real.sq_sqrt (mul_nonneg hQ hδ), Real.sqrt_nonneg (Q * δ),
sq_nonneg (Q - δ)]
have hδsq : δ * δ / ε ≤ (δ / ε) ^ 2 := by
apply (div_le_iff₀ hε).2
have h := mul_le_of_le_one_right (sq_nonneg (δ / ε)) hε1
have he : δ * δ = (δ / ε) ^ 2 * ε * ε := by field_simp
rw [he]
exact mul_le_mul_of_nonneg_right h hε.le
have hscale : (δ / ε) * (sqrt (Q * δ) + δ) ≤ 2 * a := by
have h := mul_le_mul_of_nonneg_left hsqrt (div_nonneg hδ hε.le)
dsimp [a]
have he : δ / ε * (Q + δ) = Q * δ / ε + δ * δ / ε := by ring
rw [he] at h
have he' : δ / ε * (sqrt (Q * δ) + δ) =
δ / ε * sqrt (Q * δ) + δ * δ / ε := by ring
rw [he']
nlinarith [div_nonneg (mul_nonneg hQ hδ) hε.le]
have hexp : exp (-40 * L) ≤ exp (-20 * L) := by
apply Real.exp_le_exp.mpr
linarith
have h1 := mul_le_mul_of_nonneg_left hexp (show 0 ≤ 8 * a by positivity)
have h2 := mul_le_mul_of_nonneg_left hscale
(mul_nonneg hC (Real.exp_nonneg (-20 * L)))
apply hb.trans
have he : (∑ j, (q j : ℝ)) * L / (m * ε) + L ^ 2 / ((m : ℝ) ^ 2 * ε ^ 2) = a := by
dsimp [a, Q, δ]
ring
rw [he]
apply (show _ ≤ 512 * H ^ 2 * ((8 + 2 * C) * exp (-20 * L) * a) from ?_).trans_eq
(by dsimp [C]; ring)
apply mul_le_mul_of_nonneg_left _ (by positivity : 0 ≤ 512 * H ^ 2)
change a * (8 * exp (-40 * L)) +
(δ / ε) * (C * exp (-20 * L) * (sqrt (Q * L / m) + δ)) ≤ _
have heq : Q * L / m = Q * δ := by dsimp [δ]; ring
rw [heq]
nlinarith
theorem pilotRectangle_bad_clippingScale_integral_le Lemma pilotRectangle_bad_clippingScale_integral_le in the paper ↗
Cauchy–Schwarz couples the failed-rectangle probability to its clipping-scale second moment, giving the first-moment component of roadmap (22). The mass term retains a square root of one inverse-overlap power. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
W :
Fin 4 → Ω → ℕ
q :
Fin 4 → ℝ≥0
m :
ℝ≥0
hm :
0 < m
H :
ℝ
hHpos :
0 < H
L ε :
ℝ
hH :
universalH ≤ H
hL :
1 ≤ L
hε :
0 < ε
hε1 :
ε ≤ 1
hW :
∀ j, Measurable (W j)
hlaw :
∀ j, HasLaw (W j) (poissonMeasure (m * q j)) μ
hind :
iIndepFun W μ
conclusion
(∫ ω, if (fun j => (q j : ℝ)) ∉ pilotRectangle (pilotLower H hHpos m L (fun j => W j ω)) (pilotUpperFormula H hHpos m L (fun j => W j ω)) then clippingScaleFormula ε (pilotMidpoint H hHpos m L (fun j => W j ω)) (pilotRadiusFormula H hHpos m L (fun j => W j ω)) else 0 ∂μ)
≤ sqrt (8 * (512 * H ^ 2 * (8 + 2 * (H / universalH) * productMomentConstant 4 1))) * exp (-30 * L) * (sqrt ((∑ j, (q j : ℝ)) * L / (m * ε)) + L / (m * ε))
Proof (Lean source)
lemma pilotRectangle_bad_clippingScale_integral_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ]
(W : Fin 4 → Ω → ℕ) (q : Fin 4 → ℝ≥0) (m : ℝ≥0) (hm : 0 < m)
(H : ℝ) (hHpos : 0 < H) (L ε : ℝ) (hH : universalH ≤ H) (hL : 1 ≤ L)
(hε : 0 < ε) (hε1 : ε ≤ 1)
(hW : ∀ j, Measurable (W j))
(hlaw : ∀ j, HasLaw (W j) (poissonMeasure (m * q j)) μ)
(hind : iIndepFun W μ) :
(∫ ω, if (fun j => (q j : ℝ)) ∉ pilotRectangle
(pilotLower H hHpos m L (fun j => W j ω))
(pilotUpperFormula H hHpos m L (fun j => W j ω)) then
clippingScaleFormula ε (pilotMidpoint H hHpos m L (fun j => W j ω))
(pilotRadiusFormula H hHpos m L (fun j => W j ω)) else 0 ∂μ) ≤
sqrt (8 * (512 * H ^ 2 *
(8 + 2 * (H / universalH) * productMomentConstant 4 1))) *
exp (-30 * L) *
(sqrt ((∑ j, (q j : ℝ)) * L / (m * ε)) + L / (m * ε)) := by
classical
let B : Set Ω := {ω | (fun j => (q j : ℝ)) ∉ pilotRectangle
(pilotLower H hHpos m L (fun j => W j ω))
(pilotUpperFormula H hHpos m L (fun j => W j ω))}
let S : Ω → ℝ := fun ω => clippingScaleFormula ε
(pilotMidpoint H hHpos m L (fun j => W j ω))
(pilotRadiusFormula H hHpos m L (fun j => W j ω))
let C : ℝ := 512 * H ^ 2 *
(8 + 2 * (H / universalH) * productMomentConstant 4 1)
let a : ℝ := (∑ j, (q j : ℝ)) * L / (m * ε)
let b : ℝ := L / (m * ε)
let s : ℝ := sqrt a + b
have hB : MeasurableSet B := by
dsimp [B]
unfold pilotRectangle pilotLower pilotUpperFormula pilotHalfWidthFormula pilotCenterFormula
measurability
have hS := pilotClippingScale_memLp_two μ W q m hm H hHpos L ε hH hL hε hε1 hW hlaw
have hSn (ω : Ω) : 0 ≤ S ω := clippingScale_nonneg ε _ _
(pilotMidpoint_nonneg H hHpos m L (NNReal.coe_pos.mpr hm) _)
(fun j => (pilotRadius_pos H hHpos m L (NNReal.coe_pos.mpr hm) (by linarith) _ j).le)
have ha : 0 ≤ a := by dsimp [a]; positivity
have hb : 0 ≤ b := by dsimp [b]; positivity
have hs : 0 ≤ s := add_nonneg (Real.sqrt_nonneg _) hb
have hC : 0 ≤ C := by
dsimp [C]
have := (productMomentConstant_pos 4 1).le
have := universalH_pos
positivity
have hprob : μ.real B ≤ 8 * exp (-40 * L) := by
have hp := pilotRectangle_bad_probability_le μ W q m hm H hHpos L hH hL hlaw
exact (ENNReal.toReal_mono ENNReal.ofReal_ne_top hp).trans_eq
(ENNReal.toReal_ofReal (by positivity))
have hsecond : (∫ ω, B.indicator (fun ω => S ω ^ 2) ω ∂μ) ≤
C * exp (-20 * L) * s ^ 2 := by
have h := pilotRectangle_bad_clippingScale_normalized_le μ W q m hm H hHpos L ε
hH hL hε hε1 hW hlaw hind
have hab : a + b ^ 2 ≤ s ^ 2 := by
dsimp [s]
nlinarith [Real.sq_sqrt ha, Real.sqrt_nonneg a]
have he : L ^ 2 / ((m : ℝ) ^ 2 * ε ^ 2) = b ^ 2 := by dsimp [b]; ring
simp only [he] at h
simpa only [indicator, B, S, Set.mem_ofPred_eq] using
h.trans (mul_le_mul_of_nonneg_left hab (mul_nonneg hC (Real.exp_nonneg _)))
have hholder := integral_mul_le_Lp_mul_Lq_of_nonneg Real.HolderConjugate.two_two
(f := B.indicator S) (g := B.indicator (fun _ => (1 : ℝ)))
(ae_of_all _ fun ω => Set.indicator_nonneg (fun ω _ => hSn ω) ω)
(ae_of_all _ fun ω => Set.indicator_nonneg (fun _ _ => zero_le_one) ω)
(by simpa using hS.indicator hB)
(by simpa using (memLp_const (1 : ℝ) (p := (2 : ENNReal)) (μ := μ)).indicator hB)
have hmul (ω : Ω) : B.indicator S ω * B.indicator (fun _ => (1 : ℝ)) ω =
B.indicator S ω := by by_cases h : ω ∈ B <;> simp [h]
have hsq (ω : Ω) : (B.indicator S ω) ^ (2 : ℝ) =
B.indicator (fun ω => S ω ^ 2) ω := by
by_cases h : ω ∈ B <;> simp [h]
have hone (ω : Ω) : (B.indicator (fun _ => (1 : ℝ)) ω) ^ (2 : ℝ) =
B.indicator (fun _ => (1 : ℝ)) ω := by by_cases h : ω ∈ B <;> simp [h]
simp_rw [hmul, hsq, hone, ← Real.sqrt_eq_rpow] at hholder
rw [integral_indicator_const _ hB] at hholder
simp only [smul_eq_mul, mul_one] at hholder
have hbound := mul_le_mul (Real.sqrt_le_sqrt hsecond) (Real.sqrt_le_sqrt hprob)
(Real.sqrt_nonneg _) (Real.sqrt_nonneg _)
have heq : sqrt (C * exp (-20 * L) * s ^ 2) *
sqrt (8 * exp (-40 * L)) =
sqrt (8 * C) * exp (-30 * L) * s := by
rw [← Real.sqrt_mul (by positivity)]
have hexp : exp (-20 * L) * exp (-40 * L) =
exp (-30 * L) ^ 2 := by
rw [pow_two, ← Real.exp_add, ← Real.exp_add]
congr 1; ring
have he : C * exp (-20 * L) * s ^ 2 * (8 * exp (-40 * L)) =
(sqrt (8 * C) * exp (-30 * L) * s) ^ 2 := by
rw [mul_pow, mul_pow, Real.sq_sqrt (by positivity)]
rw [← hexp]
ring
rw [he, Real.sqrt_sq (by positivity)]
simpa only [indicator, B, S, C, s, a, b, Set.mem_ofPred_eq] using
(hholder.trans hbound).trans_eq heq
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.UpperBias 9 declarations On the localization event, the empirical half width is controlled by the true coordinate's square-root scale.
Localized pilot geometry for Jackson bias
On the localization event, the empirical half width is controlled by the true coordinate's square-root scale. The boundary-distance factor loses its additive level term, including at zero coordinates. These are the deterministic pilot inputs to roadmap equations (11) and (20).
Solving the localization inequality replaces the empirical square-root scale by the true coordinate scale plus a level-sized remainder. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma localized_pilot_sqrt_le {c q δ H : ℝ}
(hc : 0 ≤ c) (hq : 0 ≤ q) (hδ : 0 ≤ δ) (hH : 0 ≤ H)
(hloc : c - H * (sqrt (c * δ) + δ) ≤ q) :
sqrt (c * δ) ≤ sqrt (q * δ) + (H + 1) * δ := by
have hs := Real.sq_sqrt (mul_nonneg hc hδ)
have ht := Real.sq_sqrt (mul_nonneg hq hδ)
have hlocδ := mul_le_mul_of_nonneg_right hloc hδ
by_contra h
have hgap : 0 < sqrt (c * δ) - sqrt (q * δ) - (H + 1) * δ := by
linarith
have hsum : 0 < sqrt (c * δ) + sqrt (q * δ) + δ := by
have := Real.sqrt_nonneg (q * δ)
nlinarith
have hprod := mul_pos hgap hsum
have hcross : 0 ≤ (H + 2) * sqrt (q * δ) * δ := by positivity
nlinarith [sq_nonneg δ]
The localized half width has a true-mass envelope with a constant depending only on the universal pilot multiplier. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma localized_pilot_halfWidth_le {c q δ H : ℝ}
(hc : 0 ≤ c) (hq : 0 ≤ q) (hδ : 0 ≤ δ) (hH : 1 ≤ H)
(hloc : c - H * (sqrt (c * δ) + δ) ≤ q) :
H * (sqrt (c * δ) + δ) ≤
H * (H + 2) * (sqrt (q * δ) + δ) := by
have hs := localized_pilot_sqrt_le hc hq hδ (by linarith) hloc
have hmul := mul_le_mul_of_nonneg_left hs (show 0 ≤ H by linarith)
have hcross : 0 ≤ H * (H + 1) * sqrt (q * δ) := by positivity
nlinarith
For a localized nonnegative interval, its boundary-distance product is controlled by true mass times level, with no level-squared remainder. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma localized_interval_boundary_le {c q h δ B : ℝ}
(hq : 0 ≤ q) (hh : 0 ≤ h) (hδ : 0 ≤ δ) (hB : 1 ≤ B)
(hlo : max 0 (c - h) ≤ q) (hhi : q ≤ c + h)
(hwidth : h ≤ B * (sqrt (q * δ) + δ)) :
sqrt ((q - max 0 (c - h)) * (c + h - q)) ≤
2 * B * sqrt (q * δ) := by
have hleft : 0 ≤ q - max 0 (c - h) := sub_nonneg.mpr hlo
have hright : 0 ≤ c + h - q := sub_nonneg.mpr hhi
have hleftq : q - max 0 (c - h) ≤ q := by linarith [le_max_left 0 (c - h)]
have hright2 : c + h - q ≤ 2 * h := by linarith [le_max_right 0 (c - h)]
have hboundary : (q - max 0 (c - h)) * (c + h - q) ≤
(2 * B * sqrt (q * δ)) ^ 2 := by
by_cases hsmall : q ≤ δ
· have hs : sqrt (q * δ) ≤ δ := by
nlinarith [Real.sq_sqrt (mul_nonneg hq hδ), Real.sqrt_nonneg (q * δ),
mul_le_mul_of_nonneg_right hsmall hδ]
have hw : h ≤ 2 * B * δ := by nlinarith
have hprod := mul_le_mul hleftq (hright2.trans (by nlinarith : 2 * h ≤ 4 * B * δ))
hright hq
have hBs : B ≤ B ^ 2 := by nlinarith
have hb := mul_le_mul_of_nonneg_right hBs (mul_nonneg hq hδ)
rw [mul_pow, mul_pow, Real.sq_sqrt (mul_nonneg hq hδ)]
nlinarith
· have hs : δ ≤ sqrt (q * δ) := by
nlinarith [Real.sq_sqrt (mul_nonneg hq hδ), Real.sqrt_nonneg (q * δ),
mul_le_mul_of_nonneg_right (le_of_not_ge hsmall) hδ]
have hw : h ≤ 2 * B * sqrt (q * δ) := by nlinarith
have hprod : (q - max 0 (c - h)) * (c + h - q) ≤ h ^ 2 := by
have hl := le_max_right 0 (c - h)
nlinarith [sq_nonneg (q - c)]
exact hprod.trans (pow_le_pow_left₀ hh hw 2)
have hs := Real.sq_sqrt (mul_nonneg hleft hright)
have ht : 0 ≤ 2 * B * sqrt (q * δ) := by positivity
nlinarith [Real.sqrt_nonneg ((q - max 0 (c - h)) * (c + h - q))]
On the actual pilot interval, the coordinate half width obeys the true-mass envelope used before equation (20). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
H :
ℝ
hH :
0 < H
hHlarge :
1 ≤ H
m L :
ℝ
hm :
0 < m
hL :
0 ≤ L
Np :
Fin 4 → ℕ
j :
Fin 4
q :
ℝ
hq :
0 ≤ q
hlo :
pilotLower H hH m L Np j ≤ q
conclusion
pilotHalfWidthFormula H hH m L Np j ≤ H * (H + 2) * (sqrt (q * L / m) + L / m)
Proof (Lean source)
lemma pilotHalfWidth_le_true_scale (H : ℝ) (hH : 0 < H) (hHlarge : 1 ≤ H)
(m L : ℝ) (hm : 0 < m) (hL : 0 ≤ L) (Np : Fin 4 → ℕ)
(j : Fin 4) (q : ℝ) (hq : 0 ≤ q)
(hlo : pilotLower H hH m L Np j ≤ q) :
pilotHalfWidthFormula H hH m L Np j ≤
H * (H + 2) * (sqrt (q * L / m) + L / m) := by
have hc : 0 ≤ pilotCenterFormula m Np j := by dsimp [pilotCenterFormula]; positivity
have hloc := (le_max_right 0
(pilotCenterFormula m Np j - pilotHalfWidthFormula H hH m L Np j)).trans hlo
simp only [pilotHalfWidthFormula, mul_div_assoc] at hloc ⊢
simpa only [pilotHalfWidthFormula, mul_div_assoc] using
localized_pilot_halfWidth_le hc hq (div_nonneg hL hm.le) hHlarge hloc
The actual pilot face-distance factor retains the true coordinate's square-root scale, and vanishes at a zero coordinate. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma pilotBoundary_le_true_scale (H : ℝ) (hH : 0 < H) (hHlarge : 1 ≤ H)
(m L : ℝ) (hm : 0 < m) (hL : 0 ≤ L) (Np : Fin 4 → ℕ)
(j : Fin 4) (q : ℝ) (hq : 0 ≤ q)
(hloc : q ∈ Icc (pilotLower H hH m L Np j) (pilotUpperFormula H hH m L Np j)) :
sqrt ((q - pilotLower H hH m L Np j) *
(pilotUpperFormula H hH m L Np j - q)) ≤
2 * (H * (H + 2)) * sqrt (q * L / m) := by
have hh : 0 ≤ pilotHalfWidthFormula H hH m L Np j := by dsimp [pilotHalfWidthFormula]; positivity
have hB : 1 ≤ H * (H + 2) := by nlinarith
have hw := pilotHalfWidth_le_true_scale H hH hHlarge m L hm hL Np j q hq hloc.1
simp only [mul_div_assoc] at hw ⊢
simpa only [pilotLower, pilotUpperFormula, mul_div_assoc] using localized_interval_boundary_le
hq hh (div_nonneg hL hm.le) hB hloc.1 hloc.2 hw
The coordinate radius is bounded by the same true-mass scale as its half width whenever the true coordinate lies above the pilot lower endpoint. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
H :
ℝ
hH :
0 < H
hHlarge :
1 ≤ H
m L :
ℝ
hm :
0 < m
hL :
0 ≤ L
Np :
Fin 4 → ℕ
j :
Fin 4
q :
ℝ
hq :
0 ≤ q
hlo :
pilotLower H hH m L Np j ≤ q
conclusion
pilotRadiusFormula H hH m L Np j ≤ H * (H + 2) * (sqrt (q * L / m) + L / m)
Proof (Lean source)
lemma pilotRadius_le_true_scale (H : ℝ) (hH : 0 < H) (hHlarge : 1 ≤ H)
(m L : ℝ) (hm : 0 < m) (hL : 0 ≤ L) (Np : Fin 4 → ℕ)
(j : Fin 4) (q : ℝ) (hq : 0 ≤ q)
(hlo : pilotLower H hH m L Np j ≤ q) :
pilotRadiusFormula H hH m L Np j ≤
H * (H + 2) * (sqrt (q * L / m) + L / m) := by
have hr : pilotRadiusFormula H hH m L Np j ≤ pilotHalfWidthFormula H hH m L Np j := by
have h := le_max_right 0 (pilotCenterFormula m Np j - pilotHalfWidthFormula H hH m L Np j)
dsimp [pilotRadiusFormula, pilotUpperFormula, pilotLower]
linarith
exact hr.trans (pilotHalfWidth_le_true_scale H hH hHlarge m L hm hL Np j q hq hlo)
On a localized rectangle, summing the coordinate radius bounds gives the true arm-mass scale used in equation (20). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
H :
ℝ
hH :
0 < H
hHlarge :
1 ≤ H
m L :
ℝ
hm :
0 < m
hL :
0 ≤ L
Np :
Fin 4 → ℕ
q :
Fin 4 → ℝ
hq :
∀ j, 0 ≤ q j
hlo :
∀ j, pilotLower H hH m L Np j ≤ q j
a :
conclusion
armRadiusFormula (pilotRadiusFormula H hH m L Np) a
≤ 2 * (H * (H + 2)) * (sqrt (armMassVecFormula q a * L / m) + L / m)
Proof (Lean source)
lemma pilotArmRadius_le_true_scale (H : ℝ) (hH : 0 < H) (hHlarge : 1 ≤ H)
(m L : ℝ) (hm : 0 < m) (hL : 0 ≤ L) (Np : Fin 4 → ℕ)
(q : Fin 4 → ℝ) (hq : ∀ j, 0 ≤ q j)
(hlo : ∀ j, pilotLower H hH m L Np j ≤ q j) (a : Bool) :
armRadiusFormula (pilotRadiusFormula H hH m L Np) a ≤
2 * (H * (H + 2)) * (sqrt (armMassVecFormula q a * L / m) + L / m) := by
have h₀ := pilotRadius_le_true_scale H hH hHlarge m L hm hL Np
(cellIdx a false) _ (hq _) (hlo _)
have h₁ := pilotRadius_le_true_scale H hH hHlarge m L hm hL Np
(cellIdx a true) _ (hq _) (hlo _)
have hs₀ : sqrt (q (cellIdx a false) * L / m) ≤
sqrt (armMassVecFormula q a * L / m) := by
apply Real.sqrt_le_sqrt
dsimp [armMassVecFormula]
gcongr
exact le_add_of_nonneg_right (hq _)
have hs₁ : sqrt (q (cellIdx a true) * L / m) ≤
sqrt (armMassVecFormula q a * L / m) := by
apply Real.sqrt_le_sqrt
dsimp [armMassVecFormula]
gcongr
exact le_add_of_nonneg_left (hq _)
have ht₀ := mul_le_mul_of_nonneg_left hs₀ (show 0 ≤ H * (H + 2) by positivity)
have ht₁ := mul_le_mul_of_nonneg_left hs₁ (show 0 ≤ H * (H + 2) by positivity)
dsimp only [armRadiusFormula]
linarith
The arm's two boundary-distance factors have no additive level remainder, including arms with zero true mass. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
H :
ℝ
hH :
0 < H
hHlarge :
1 ≤ H
m L :
ℝ
hm :
0 < m
hL :
0 ≤ L
Np :
Fin 4 → ℕ
q :
Fin 4 → ℝ
hq :
∀ j, 0 ≤ q j
hloc :
a :
conclusion
(∑ y : Bool, sqrt ((q (cellIdx a y) - pilotLower H hH m L Np (cellIdx a y)) * (pilotUpperFormula H hH m L Np (cellIdx a y) - q (cellIdx a y))))
≤ 4 * (H * (H + 2)) * sqrt (armMassVecFormula q a * L / m)
Proof (Lean source)
lemma pilotArmBoundary_le_true_scale (H : ℝ) (hH : 0 < H) (hHlarge : 1 ≤ H)
(m L : ℝ) (hm : 0 < m) (hL : 0 ≤ L) (Np : Fin 4 → ℕ)
(q : Fin 4 → ℝ) (hq : ∀ j, 0 ≤ q j)
(hloc : q ∈ pilotRectangle (pilotLower H hH m L Np) (pilotUpperFormula H hH m L Np))
(a : Bool) :
(∑ y : Bool, sqrt ((q (cellIdx a y) - pilotLower H hH m L Np (cellIdx a y)) *
(pilotUpperFormula H hH m L Np (cellIdx a y) - q (cellIdx a y)))) ≤
4 * (H * (H + 2)) * sqrt (armMassVecFormula q a * L / m) := by
have hterm (y : Bool) := pilotBoundary_le_true_scale H hH hHlarge m L hm hL Np
(cellIdx a y) _ (hq _) (hloc _)
have hs (y : Bool) : sqrt (q (cellIdx a y) * L / m) ≤
sqrt (armMassVecFormula q a * L / m) := by
apply Real.sqrt_le_sqrt
apply div_le_div_of_nonneg_right _ hm.le
apply mul_le_mul_of_nonneg_right _ hL
cases y
· exact le_add_of_nonneg_right (hq _)
· exact le_add_of_nonneg_left (hq _)
have hmul (y : Bool) := mul_le_mul_of_nonneg_left (hs y)
(show 0 ≤ 2 * (H * (H + 2)) by positivity)
simp only [Fintype.sum_bool]
linarith [hterm false, hterm true, hmul false, hmul true]
The localized boundary factors keep the inverse arm masses attached to their coordinates. Applying the observed armwise square-root bound (21) produces one square-root inverse-overlap factor, as needed for (20). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
ε :
ℝ
P :
hε :
0 < ε
hP :
ObservedClass ε P
x :
Fin d
hp :
0 < cellMass P x
H :
ℝ
hH :
0 < H
hHlarge :
1 ≤ H
m L :
ℝ
hm :
0 < m
hL :
0 ≤ L
Np :
Fin 4 → ℕ
hloc :
conclusion
(∑ a : Bool, cellMass P x / armMass P a x * ∑ y : Bool, sqrt (((cellVector P x) (cellIdx a y) - pilotLower H hH m L Np (cellIdx a y)) * (pilotUpperFormula H hH m L Np (cellIdx a y) - (cellVector P x) (cellIdx a y))))
Proof (Lean source)
lemma observed_pilot_weightedBoundary_le {d : ℕ} (ε : ℝ) (P : DiscreteLaw d)
(hε : 0 < ε) (hP : ObservedClass ε P) (x : Fin d) (hp : 0 < cellMass P x)
(H : ℝ) (hH : 0 < H) (hHlarge : 1 ≤ H) (m L : ℝ)
(hm : 0 < m) (hL : 0 ≤ L) (Np : Fin 4 → ℕ)
(hloc : cellVector P x ∈
pilotRectangle (pilotLower H hH m L Np) (pilotUpperFormula H hH m L Np)) :
(∑ a : Bool, cellMass P x / armMass P a x * ∑ y : Bool,
sqrt (((cellVector P x) (cellIdx a y) - pilotLower H hH m L Np (cellIdx a y)) *
(pilotUpperFormula H hH m L Np (cellIdx a y) - (cellVector P x) (cellIdx a y)))) ≤
4 * (H * (H + 2)) * sqrt 2 * sqrt (cellMass P x * L / (m * ε)) := by
have hq (j : Fin 4) : 0 ≤ (cellVector P x) j := ENNReal.toReal_nonneg
have hscale : 0 ≤ 2 * (H * (H + 2)) * sqrt (L / m) := by positivity
have hs := observed_armwise_sqrt_bound d ε P hε hP x hp
calc
_ ≤ ∑ a : Bool, cellMass P x / armMass P a x * ∑ y : Bool,
(2 * (H * (H + 2)) * sqrt (L / m)) * sqrt (jointMass P x a y) := by
apply Finset.sum_le_sum
intro a _
apply mul_le_mul_of_nonneg_left _ (div_nonneg hp.le
(Finset.sum_nonneg fun _ _ => ENNReal.toReal_nonneg))
apply Finset.sum_le_sum
intro y _
have ht := pilotBoundary_le_true_scale H hH hHlarge m L hm hL Np
(cellIdx a y) _ (hq _) (hloc _)
rw [mul_div_assoc, Real.sqrt_mul (hq _)] at ht
have heq : (cellVector P x) (cellIdx a y) = jointMass P x a y := by
cases a <;> cases y <;> simp [cellVector, cellIdx]
simpa only [heq, mul_assoc, mul_comm, mul_left_comm] using ht
_ = (2 * (H * (H + 2)) * sqrt (L / m)) *
(∑ a : Bool, cellMass P x / armMass P a x * ∑ y : Bool,
sqrt (jointMass P x a y)) := by
simp only [Fintype.sum_bool]
ring
_ ≤ (2 * (H * (H + 2)) * sqrt (L / m)) *
(2 * sqrt 2 * sqrt (cellMass P x / ε)) :=
mul_le_mul_of_nonneg_left hs hscale
_ = _ := by
have he : cellMass P x * L / (m * ε) = (L / m) * (cellMass P x / ε) := by ring
rw [he, Real.sqrt_mul (div_nonneg hL hm.le)]
ring
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.UpperBinomialCounts 12 declarations Roadmap equation (4) controls the inverse occupied-arm count and the empty-arm probability.
Binomial count bounds for the empirical-ratio branch
Roadmap equation (4) controls the inverse occupied-arm count and the empty-arm probability. These bounds retain the extra trial in the denominator and allow success probability one.
A function of the success count in iid Boolean trials has the binomial expectation. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
n :
ℕ
p :
ℝ
hp0 :
0 ≤ p
hp1 :
p ≤ 1
F :
ℕ → ℝ
conclusion
(∫ b : Fin n → Bool, F (Finset.univ.filter fun i => b i = true).card ∂Measure.pi (fun _ => bernoulliBool p))
= ∑ k ∈ range (n + 1), binomialWeight n p k * F k
Proof (Lean source)
lemma integral_boolCount_eq_binomial (n : ℕ) (p : ℝ)
(hp0 : 0 ≤ p) (hp1 : p ≤ 1) (F : ℕ → ℝ) :
(∫ b : Fin n → Bool, F (Finset.univ.filter fun i => b i = true).card
∂Measure.pi (fun _ => bernoulliBool p)) =
∑ k ∈ range (n + 1), binomialWeight n p k * F k := by
let : IsProbabilityMeasure (bernoulliBool p) := bernoulliBool_isProbabilityMeasure hp0 hp1
rw [integral_fintype Integrable.of_finite]
have hmass (b : Fin n → Bool) :
(Measure.pi (fun _ : Fin n => bernoulliBool p)).real {b} =
∏ i, if b i then p else 1 - p := by
simp only [measureReal_def, Measure.pi_singleton, ENNReal.toReal_prod]
apply Finset.prod_congr rfl
intro i hi
cases b i <;> simp [bernoulliBool, hp0, sub_nonneg.mpr hp1]
simp_rw [hmass, smul_eq_mul]
simpa using sum_bernoulli_eq_binomial (ι := Fin n) p F
Equation (4)'s reciprocal bound for iid Boolean success counts. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
n :
ℕ
p :
ℝ
hp :
0 < p
hp1 :
p ≤ 1
conclusion
(∫ b : Fin n → Bool, if 0 < (Finset.univ.filter fun i => b i = true).card then ((Finset.univ.filter fun i => b i = true).card : ℝ)⁻¹ else 0 ∂Measure.pi (fun _ => bernoulliBool p))
≤ 2 / ((n + 1 : ℝ) * p)
Proof (Lean source)
lemma integral_boolCount_inverse_le (n : ℕ) (p : ℝ)
(hp : 0 < p) (hp1 : p ≤ 1) :
(∫ b : Fin n → Bool,
if 0 < (Finset.univ.filter fun i => b i = true).card then
((Finset.univ.filter fun i => b i = true).card : ℝ)⁻¹ else 0
∂Measure.pi (fun _ => bernoulliBool p)) ≤ 2 / ((n + 1 : ℝ) * p) := by
rw [integral_boolCount_eq_binomial n p hp.le hp1 (fun k => if 0 < k then (k : ℝ)⁻¹ else 0)]
simpa only [Nat.cast_add, Nat.cast_one] using binomial_totalized_inverse_count_le n p hp hp1
Empty iid Boolean counts have the exact binomial zero probability. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
n :
ℕ
p :
ℝ
hp0 :
0 ≤ p
hp1 :
p ≤ 1
conclusion
(∫ b : Fin n → Bool, if (Finset.univ.filter fun i => b i = true).card = 0 then (1 : ℝ) else 0 ∂Measure.pi (fun _ => bernoulliBool p))
= (1 - p) ^ n
Proof (Lean source)
lemma integral_boolCount_empty_eq (n : ℕ) (p : ℝ)
(hp0 : 0 ≤ p) (hp1 : p ≤ 1) :
(∫ b : Fin n → Bool,
if (Finset.univ.filter fun i => b i = true).card = 0 then (1 : ℝ) else 0
∂Measure.pi (fun _ => bernoulliBool p)) = (1 - p) ^ n := by
rw [integral_boolCount_eq_binomial n p hp0 hp1 (fun k => if k = 0 then 1 else 0)]
simp [binomialWeight]
The probability of no binomial successes is bounded by the reciprocal expected count with one extra trial, as in equation (4). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
n :
ℕ
p :
ℝ
hp :
0 < p
hp1 :
p ≤ 1
conclusion
(1 - p) ^ n ≤ 1 / ((n + 1 : ℝ) * p)
Proof (Lean source)
lemma binomial_empty_probability_le (n : ℕ) (p : ℝ)
(hp : 0 < p) (hp1 : p ≤ 1) :
(1 - p) ^ n ≤ 1 / ((n + 1 : ℝ) * p) := by
have hq : 0 ≤ 1 - p := sub_nonneg.mpr hp1
have hsum : (n + 1 : ℝ) * (1 - p) ^ n ≤
∑ k ∈ range (n + 1), (1 - p) ^ k := by
calc
_ = ∑ k ∈ range (n + 1), (1 - p) ^ n := by simp
_ ≤ _ := by
apply Finset.sum_le_sum
intro k hk
exact pow_le_pow_of_le_one hq (by linarith) (by simpa using Finset.mem_range.mp hk)
have hid := geom_sum_mul_neg (1 - p) (n + 1)
have hbound : (n + 1 : ℝ) * (1 - p) ^ n * p ≤ 1 := by
have h := mul_le_mul_of_nonneg_right hsum hp.le
have hpow := pow_nonneg hq (n + 1)
nlinarith
apply (le_div_iff₀ (mul_pos (by positivity) hp)).2
nlinarith
The empty-count expectation obeys the second inequality in equation (4). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
n :
ℕ
p :
ℝ
hp :
0 < p
hp1 :
p ≤ 1
conclusion
(∫ b : Fin n → Bool, if (Finset.univ.filter fun i => b i = true).card = 0 then (1 : ℝ) else 0 ∂Measure.pi (fun _ => bernoulliBool p))
≤ 1 / ((n + 1 : ℝ) * p)
Proof (Lean source)
lemma integral_boolCount_empty_le (n : ℕ) (p : ℝ)
(hp : 0 < p) (hp1 : p ≤ 1) :
(∫ b : Fin n → Bool,
if (Finset.univ.filter fun i => b i = true).card = 0 then (1 : ℝ) else 0
∂Measure.pi (fun _ => bernoulliBool p)) ≤ 1 / ((n + 1 : ℝ) * p) := by
rw [integral_boolCount_empty_eq n p hp.le hp1]
exact binomial_empty_probability_le n p hp hp1
A measurable event indicator pushes its probability law to the Boolean Bernoulli law. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma eventIndicator_measurePreserving {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (A : Set Ω) (hA : MeasurableSet A) :
MeasurePreserving (fun z => decide (z ∈ A)) μ (bernoulliBool (μ.real A)) := by
classical
have hf : Measurable (fun z => decide (z ∈ A)) := by
have h : Measurable (fun z => if z ∈ A then true else false) :=
Measurable.ite hA measurable_const measurable_const
simpa using h
have hp0 : 0 ≤ μ.real A := ENNReal.toReal_nonneg
have hp1 : μ.real A ≤ 1 := measureReal_le_one
let : IsProbabilityMeasure (bernoulliBool (μ.real A)) :=
bernoulliBool_isProbabilityMeasure hp0 hp1
refine ⟨hf, Measure.ext_of_singleton fun b => ?_⟩
apply (ENNReal.toReal_eq_toReal_iff' (measure_ne_top _ _) (measure_ne_top _ _)).mp
change (μ.map (fun z => decide (z ∈ A))).real {b} = _
rw [map_measureReal_apply hf (measurableSet_singleton b)]
cases b
· have hcompl : μ.real {z | z ∉ A} = 1 - μ.real A := by
change μ.real Aᶜ = 1 - μ.real A
simpa using measureReal_compl (μ := μ) hA
simpa [bernoulliBool, sub_nonneg.mpr hp1, preimage] using hcompl
· simp [bernoulliBool, preimage]
rfl
Under iid sampling, any function of an event count has the binomial expectation. This transports equation (4) from Boolean trials to actual observed-category counts. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
(∫ z : Fin n → Ω, F (Finset.univ.filter fun i => z i ∈ A).card ∂Measure.pi (fun _ => μ))
= ∑ k ∈ range (n + 1), binomialWeight n (μ.real A) k * F k
Proof (Lean source)
lemma integral_eventCount_eq_binomial {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (A : Set Ω) (hA : MeasurableSet A)
(n : ℕ) (F : ℕ → ℝ) :
(∫ z : Fin n → Ω, F (Finset.univ.filter fun i => z i ∈ A).card
∂Measure.pi (fun _ => μ)) =
∑ k ∈ range (n + 1), binomialWeight n (μ.real A) k * F k := by
classical
let : IsProbabilityMeasure (bernoulliBool (μ.real A)) :=
bernoulliBool_isProbabilityMeasure toReal_nonneg measureReal_le_one
have hmp := measurePreserving_pi (fun _ : Fin n => μ)
(fun _ => bernoulliBool (μ.real A))
(fun _ => eventIndicator_measurePreserving μ A hA)
rw [← integral_boolCount_eq_binomial n (μ.real A) toReal_nonneg measureReal_le_one F]
rw [← hmp.map_eq, integral_map_of_stronglyMeasurable hmp.measurable
(Measurable.of_discrete.stronglyMeasurable)]
simp
theorem armEvent_mass_eq_armMass
The event selecting one covariate cell and treatment arm has mass equal to its atom sum. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
P.pmf.toMeasure.real {z : Obs d | z.1 = x ∧ z.2.1 = a} = armMass P a x
Proof (Lean source)
lemma armEvent_mass_eq_armMass {d : ℕ} (P : DiscreteLaw d) (x : Fin d) (a : Bool) :
P.pmf.toMeasure.real {z : Obs d | z.1 = x ∧ z.2.1 = a} = armMass P a x := by
classical
rw [← integral_indicator_one (Set.toFinite _).measurableSet, PMF.integral_eq_sum]
cases a <;>
simp [indicator, Fintype.sum_prod_type, armMass, jointMass, smul_eq_mul,
Finset.sum_add_distrib]
theorem integral_sampleArmCount_eq_binomial
The observed arm count has the binomial expectation formula under the actual iid law. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
(∫ o, F (sampleArmCount o x a) ∂productLaw P n)
Proof (Lean source)
lemma integral_sampleArmCount_eq_binomial {n d : ℕ}
(P : DiscreteLaw d) (x : Fin d) (a : Bool) (F : ℕ → ℝ) :
(∫ o, F (sampleArmCount o x a) ∂productLaw P n) =
∑ k ∈ range (n + 1), binomialWeight n (armMass P a x) k * F k := by
classical
have h := integral_eventCount_eq_binomial P.pmf.toMeasure
{z : Obs d | z.1 = x ∧ z.2.1 = a} (Set.toFinite _).measurableSet n F
rw [armEvent_mass_eq_armMass] at h
have hc (o : Fin n → Obs d) :
(Finset.univ.filter fun i => o i ∈ {z : Obs d | z.1 = x ∧ z.2.1 = a}).card =
sampleArmCount o x a := by
rw [Finset.card_eq_sum_ones, Finset.sum_filter]
rfl
calc
_ = ∫ o : Fin n → Obs d,
F (Finset.univ.filter fun i => o i ∈ {z : Obs d | z.1 = x ∧ z.2.1 = a}).card
∂productLaw P n := by
apply integral_congr_ae
filter_upwards [] with o
exact congrArg F (hc o).symm
_ = _ := by
convert h using 1
simp only [productLaw]
apply integral_congr_ae
filter_upwards [] with o
congr 2
ext i
simp
theorem integral_sampleArmCount_inverse_le
Equation (4)'s occupied-count reciprocal bound for an observed cell and arm. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
(∫ o, if 0 < sampleArmCount o x a then (sampleArmCount o x a : ℝ)⁻¹ else 0 ∂productLaw P n)
≤ 2 / ((n + 1 : ℝ) * armMass P a x)
Proof (Lean source)
lemma integral_sampleArmCount_inverse_le {n d : ℕ}
(P : DiscreteLaw d) (x : Fin d) (a : Bool) (hp : 0 < armMass P a x) :
(∫ o, if 0 < sampleArmCount o x a then (sampleArmCount o x a : ℝ)⁻¹ else 0
∂productLaw P n) ≤ 2 / ((n + 1 : ℝ) * armMass P a x) := by
have hp1 : armMass P a x ≤ 1 := by
rw [← armEvent_mass_eq_armMass P x a]
exact measureReal_le_one
rw [integral_sampleArmCount_eq_binomial P x a (fun k => if 0 < k then (k : ℝ)⁻¹ else 0)]
simpa only [Nat.cast_add, Nat.cast_one] using
binomial_totalized_inverse_count_le n (armMass P a x) hp hp1
theorem integral_sampleArmCount_empty_eq
Empty observed arms have the exact binomial zero probability, including null arms. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
(∫ o, if sampleArmCount o x a = 0 then (1 : ℝ) else 0 ∂productLaw P n)
= (1 - armMass P a x) ^ n
Proof (Lean source)
lemma integral_sampleArmCount_empty_eq {n d : ℕ}
(P : DiscreteLaw d) (x : Fin d) (a : Bool) :
(∫ o, if sampleArmCount o x a = 0 then (1 : ℝ) else 0 ∂productLaw P n) =
(1 - armMass P a x) ^ n := by
rw [integral_sampleArmCount_eq_binomial P x a (fun k => if k = 0 then 1 else 0)]
simp [binomialWeight]
theorem integral_sampleArmCount_empty_le
Equation (4)'s empty-arm probability bound holds for every occupied observed arm. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
(∫ o, if sampleArmCount o x a = 0 then (1 : ℝ) else 0 ∂productLaw P n)
≤ 1 / ((n + 1 : ℝ) * armMass P a x)
Proof (Lean source)
lemma integral_sampleArmCount_empty_le {n d : ℕ}
(P : DiscreteLaw d) (x : Fin d) (a : Bool) (hp : 0 < armMass P a x) :
(∫ o, if sampleArmCount o x a = 0 then (1 : ℝ) else 0 ∂productLaw P n) ≤
1 / ((n + 1 : ℝ) * armMass P a x) := by
rw [integral_sampleArmCount_empty_eq]
apply binomial_empty_probability_le n (armMass P a x) hp
rw [← armEvent_mass_eq_armMass P x a]
exact measureReal_le_one
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.UpperBoundedAlphabet 2 declarations The empirical-ratio risk bound from roadmap equations (4)–(6), together with unit-range loss, gives the headline minimum-rate bound whenever d is below the fixed universal cutoff.
Completed bounded-alphabet upper branch
The empirical-ratio risk bound from roadmap equations (4)–(6), together with unit-range loss, gives the headline minimum-rate bound whenever d is below the fixed universal cutoff. No unknown cell mass enters the final constant.
theorem logAlphabet_le_of_le
Below a fixed alphabet cutoff, the logarithmic rate scale is uniformly bounded. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d D :
ℕ
hd :
1 ≤ d
hdD :
d ≤ D
conclusion
logAlphabet d ≤ logAlphabet D
Proof (Lean source)
lemma logAlphabet_le_of_le {d D : ℕ} (hd : 1 ≤ d) (hdD : d ≤ D) :
logAlphabet d ≤ logAlphabet D := by
apply Real.log_le_log
· have hd0 : (0 : ℝ) < d := by exact_mod_cast (by omega : 0 < d)
positivity
· exact mul_le_mul_of_nonneg_left (by exact_mod_cast hdD) (Real.exp_pos _).le
theorem armwiseEstimator_boundedAlphabet_sqRisk_le Lemma armwiseEstimator_boundedAlphabet_sqRisk_le in the paper ↗
On the bounded-alphabet branch the exact headline estimator satisfies the minimum-form rate, with a constant depending only on the fixed cutoff. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
H₀ κ :
ℝ
D₀ :
ℕ
hH₀ :
0 < H₀
hκ :
0 < κ
hD₀ :
2 ≤ D₀
n d :
ℕ
ε :
ℝ
hn :
1 ≤ n
hd :
2 ≤ d
hdD :
d < D₀
hε :
0 < ε
hε1 :
ε ≤ 1 / 2
P :
hP :
ObservedClass ε P
hiid :
IidSampling P μ
conclusion
sqRisk μ (armwiseEstimatorFormula H₀ κ D₀ hH₀ hκ hD₀ n d ε) (observedValue P)
≤ (1 + 8 * logAlphabet D₀) * rateScale n d ε
Proof (Lean source)
lemma armwiseEstimator_boundedAlphabet_sqRisk_le
(H₀ κ : ℝ) (D₀ : ℕ) (hH₀ : 0 < H₀) (hκ : 0 < κ) (hD₀ : 2 ≤ D₀)
{n d : ℕ} {ε : ℝ} (hn : 1 ≤ n) (hd : 2 ≤ d) (hdD : d < D₀)
(hε : 0 < ε) (hε1 : ε ≤ 1 / 2) (P : DiscreteLaw d) (hP : ObservedClass ε P)
(μ : Measure (Fin n → Obs d)) (hiid : IidSampling P μ) :
sqRisk μ (armwiseEstimatorFormula H₀ κ D₀ hH₀ hκ hD₀ n d ε)
(observedValue P) ≤ (1 + 8 * logAlphabet D₀) * rateScale n d ε := by
have hn0 : 0 < n := by omega
have hN : (0 : ℝ) < n := by exact_mod_cast hn0
have hL : 0 < logAlphabet d := logAlphabet_pos (by omega)
have hL₀ : 0 < logAlphabet D₀ := logAlphabet_pos (by omega)
have hLL := logAlphabet_le_of_le (by omega : 1 ≤ d) (le_of_lt hdD)
have hparam := empiricalValue_sqRisk_le hn0 P hP hε hε1
let : IsProbabilityMeasure μ := by
rw [hiid]
dsimp [productLaw]
infer_instance
have hi := armwiseEstimator_sqRisk_le_one H₀ κ D₀ hH₀ hκ hD₀ ε P μ
have heq : armwiseEstimatorFormula H₀ κ D₀ hH₀ hκ hD₀ n d ε = empiricalValue := by
funext o
simp only [armwiseEstimatorFormula, hdD, if_pos]
rw [heq] at hi ⊢
rw [hiid] at hi ⊢
have hR : 0 ≤ (d : ℝ) / ((n : ℝ) * ε * logAlphabet d) := by positivity
have hscale : 8 * d / ((n : ℝ) * ε) =
8 * logAlphabet d * ((d : ℝ) / ((n : ℝ) * ε * logAlphabet d)) := by
field_simp
rw [hscale] at hparam
unfold rateScale
rcases le_total (1 : ℝ) ((d : ℝ) / ((n : ℝ) * ε * logAlphabet d)) with hsat | hsmall
· rw [min_eq_left hsat, mul_one]
exact hi.trans (by linarith)
· rw [min_eq_right hsmall]
exact hparam.trans (mul_le_mul_of_nonneg_right (by linarith) hR)
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.UpperCap 6 declarations Roadmap equations (35)–(36) compare coupled unit-range scalar statistics, transfer the quarter-mean Poisson tail, and absorb its penalty into the observable rate.
Restoring the Poisson cap
Roadmap equations (35)–(36) compare coupled unit-range scalar statistics, transfer the quarter-mean Poisson tail, and absorb its penalty into the observable rate. These estimates do not impose a relation between log n and d.
theorem logAlphabet_le_card
The alphabet logarithm is bounded by the alphabet size. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
1 ≤ d
conclusion
logAlphabet d ≤ d
Proof (Lean source)
lemma logAlphabet_le_card {d : ℕ} (hd : 1 ≤ d) : logAlphabet d ≤ d := by
have hd0 : (0 : ℝ) < d := by exact_mod_cast (by omega : 0 < d)
have hlog := Real.log_le_sub_one_of_pos hd0
unfold logAlphabet
rw [Real.log_mul (Real.exp_ne_zero _) hd0.ne', Real.log_exp]
linarith
The cap's exponential penalty is at most twice the inverse sample size. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma capExponential_le_inverse_sample {n : ℕ} (hn : 1 ≤ n) :
exp (-(n : ℝ) / 2) ≤ 2 / n := by
have hn0 : (0 : ℝ) < n := by exact_mod_cast (by omega : 0 < n)
have h := Real.mul_exp_neg_le_exp_neg_one ((n : ℝ) / 2)
have he : exp (-1) ≤ (1 : ℝ) := by
exact Real.exp_le_one_iff.mpr (by norm_num)
apply (le_div_iff₀ hn0).2
have h' : (n : ℝ) / 2 * exp (-(n : ℝ) / 2) ≤ 1 := by
simpa only [neg_div] using h.trans he
nlinarith
theorem capExponential_le_observable_rate
Equation (36): the cap penalty has the unsaturated observable rate scale. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
n d :
ℕ
ε :
ℝ
hn :
1 ≤ n
hd :
2 ≤ d
hε :
0 < ε
hε1 :
ε ≤ 1 / 2
conclusion
exp (-(n : ℝ) / 2) ≤ 2 * ((d : ℝ) / ((n : ℝ) * ε * logAlphabet d))
Proof (Lean source)
lemma capExponential_le_observable_rate {n d : ℕ} {ε : ℝ}
(hn : 1 ≤ n) (hd : 2 ≤ d) (hε : 0 < ε) (hε1 : ε ≤ 1 / 2) :
exp (-(n : ℝ) / 2) ≤
2 * ((d : ℝ) / ((n : ℝ) * ε * logAlphabet d)) := by
have hn0 : (0 : ℝ) < n := by exact_mod_cast (by omega : 0 < n)
have hd0 : (0 : ℝ) < d := by exact_mod_cast (by omega : 0 < d)
have hL := logAlphabet_pos (by omega : 1 ≤ d)
have hLd := logAlphabet_le_card (by omega : 1 ≤ d)
have hεL : ε * logAlphabet d ≤ d := by nlinarith
have hinv : 1 / (n : ℝ) ≤ (d : ℝ) / ((n : ℝ) * ε * logAlphabet d) := by
apply (div_le_div_iff₀ hn0 (by positivity)).2
nlinarith
apply (capExponential_le_inverse_sample hn).trans
calc
2 / (n : ℝ) = 2 * (1 / (n : ℝ)) := by ring
_ ≤ _ := mul_le_mul_of_nonneg_left hinv (by norm_num)
Equation (35) in real probability for any count with the quarter-mean Poisson law. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
μ.real {ω | n < M ω} ≤ exp (-(n : ℝ) / 2)
Proof (Lean source)
lemma quarterPoisson_overflow_prob_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (n : ℕ) (M : Ω → ℕ)
(hM : HasLaw M (poissonMeasure ((n : ℝ≥0) / 4)) μ) :
μ.real {ω | n < M ω} ≤ exp (-(n : ℝ) / 2) := by
have hs : MeasurableSet {w : ℕ | n < w} := measurableSet_Ioi
rw [hM.measureReal_eq hs]
exact (ENNReal.le_ofReal_iff_toReal_le (measure_ne_top _ _)
(Real.exp_nonneg _)).mp
(poisson_quarter_mean_cap_tail n)
Coupled unit-range statistics agreeing below the cap differ in risk by at most one unit of loss times the overflow probability. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
n :
ℕ
M :
Ω → ℕ
hM :
HasLaw M (poissonMeasure ((n : ℝ≥0) / 4)) μ
Vcap Vinf :
Ω → ℝ
t :
ℝ
ht :
0 ≤ t ∧ t ≤ 1
hcap :
∀ ω, 0 ≤ Vcap ω ∧ Vcap ω ≤ 1
hagree :
∀ ω
if
M ω ≤ n
then
Vcap ω = Vinf ω
hcapInt :
Integrable (fun ω => (Vcap ω - t) ^ 2) μ
hinfInt :
Integrable (fun ω => (Vinf ω - t) ^ 2) μ
Proof (Lean source)
lemma coupled_cap_sqRisk_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (n : ℕ) (M : Ω → ℕ)
(hM : HasLaw M (poissonMeasure ((n : ℝ≥0) / 4)) μ)
(Vcap Vinf : Ω → ℝ) (t : ℝ) (ht : 0 ≤ t ∧ t ≤ 1)
(hcap : ∀ ω, 0 ≤ Vcap ω ∧ Vcap ω ≤ 1)
(hagree : ∀ ω, M ω ≤ n → Vcap ω = Vinf ω)
(hcapInt : Integrable (fun ω => (Vcap ω - t) ^ 2) μ)
(hinfInt : Integrable (fun ω => (Vinf ω - t) ^ 2) μ) :
sqRisk μ Vcap t ≤
sqRisk μ Vinf t + exp (-(n : ℝ) / 2) := by
let s : Set Ω := {ω | n < M ω}
have hs : NullMeasurableSet s μ :=
hM.aemeasurable.nullMeasurableSet_preimage
(show MeasurableSet {w : ℕ | n < w} from measurableSet_Ioi)
have hi : Integrable (s.indicator (fun _ => (1 : ℝ))) μ :=
(integrable_const 1).indicator₀ hs
have hp (ω : Ω) : (Vcap ω - t) ^ 2 ≤
(Vinf ω - t) ^ 2 + s.indicator (fun _ => (1 : ℝ)) ω := by
by_cases h : M ω ≤ n
· rw [hagree ω h]
simp [s, Nat.not_lt.mpr h]
· have hv := hcap ω
have hb : |Vcap ω - t| ≤ 1 := abs_le.mpr ⟨by linarith [ht.2], by linarith [ht.1]⟩
have hb₂ : (Vcap ω - t) ^ 2 ≤ 1 := by
simpa only [sq_abs, one_pow] using pow_le_pow_left₀ (abs_nonneg _) hb 2
rw [Set.indicator_of_mem (show ω ∈ s from Nat.lt_of_not_ge h)]
linarith [sq_nonneg (Vinf ω - t)]
have hr := integral_mono hcapInt (hinfInt.add hi) hp
simp only [Pi.add_apply] at hr
rw [integral_add hinfInt hi, integral_indicator₀ hs, setIntegral_const] at hr
simp only [smul_eq_mul, mul_one] at hr
exact hr.trans (add_le_add_right (quarterPoisson_overflow_prob_le μ n M hM) _)
theorem coupled_cap_observable_rate_le
Restoring the cap converts the uncapped m=n/8 risk to the fixed-n scale, with an explicit additive universal constant for overflow. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
n d :
ℕ
ε :
ℝ
hn :
1 ≤ n
hd :
2 ≤ d
hε :
0 < ε
hε1 :
ε ≤ 1 / 2
M :
Ω → ℕ
hM :
HasLaw M (poissonMeasure ((n : ℝ≥0) / 4)) μ
Vcap Vinf :
Ω → ℝ
t C :
ℝ
ht :
0 ≤ t ∧ t ≤ 1
hcap :
∀ ω, 0 ≤ Vcap ω ∧ Vcap ω ≤ 1
hagree :
∀ ω
if
M ω ≤ n
then
Vcap ω = Vinf ω
hcapInt :
Integrable (fun ω => (Vcap ω - t) ^ 2) μ
hinfInt :
Integrable (fun ω => (Vinf ω - t) ^ 2) μ
huncapped :
sqRisk μ Vinf t ≤ C * ((d : ℝ) / (((n : ℝ) / 8) * ε * logAlphabet d))
conclusion
sqRisk μ Vcap t ≤ (8 * C + 2) * ((d : ℝ) / ((n : ℝ) * ε * logAlphabet d))
Proof (Lean source)
lemma coupled_cap_observable_rate_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] {n d : ℕ} {ε : ℝ}
(hn : 1 ≤ n) (hd : 2 ≤ d) (hε : 0 < ε) (hε1 : ε ≤ 1 / 2)
(M : Ω → ℕ) (hM : HasLaw M (poissonMeasure ((n : ℝ≥0) / 4)) μ)
(Vcap Vinf : Ω → ℝ) (t C : ℝ) (ht : 0 ≤ t ∧ t ≤ 1)
(hcap : ∀ ω, 0 ≤ Vcap ω ∧ Vcap ω ≤ 1)
(hagree : ∀ ω, M ω ≤ n → Vcap ω = Vinf ω)
(hcapInt : Integrable (fun ω => (Vcap ω - t) ^ 2) μ)
(hinfInt : Integrable (fun ω => (Vinf ω - t) ^ 2) μ)
(huncapped : sqRisk μ Vinf t ≤
C * ((d : ℝ) / (((n : ℝ) / 8) * ε * logAlphabet d))) :
sqRisk μ Vcap t ≤
(8 * C + 2) * ((d : ℝ) / ((n : ℝ) * ε * logAlphabet d)) := by
have hr := coupled_cap_sqRisk_le μ n M hM Vcap Vinf t ht hcap hagree hcapInt hinfInt
have he := capExponential_le_observable_rate hn hd hε hε1
have hid : (d : ℝ) / (((n : ℝ) / 8) * ε * logAlphabet d) =
8 * ((d : ℝ) / ((n : ℝ) * ε * logAlphabet d)) := by ring
rw [hid] at huncapped
nlinarith
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.UpperClipping 14 declarations The clipping step in roadmap equation (28) contracts squared error around an in-interval polynomial mean.
Clipping bias and squared-error contraction
The clipping step in roadmap equation (28) contracts squared error around an in-interval polynomial mean. Its change in expectation is controlled by the second moment around the clipping center divided by the clipping width.
Translating symmetric clipping gives the interval used by the cell estimator. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Proof (Lean source)
lemma centeredClip_eq (c A z : ℝ) :
max (c - A) (min (c + A) z) = c + clip A (z - c) := by
simp only [clip, add_max, add_min]
congr 1
congr 1
ring
The paper's cell clipping contracts squared error at every target in its interval. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
ε :
ℝ
K d :
ℕ
m :
ℝ
c r :
Fin 4 → ℝ
Ne :
Fin 4 → ℕ
p :
ℝ
hwidth :
0 ≤ (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r
hp :
|p - armwiseExtensionFormula ε c| ≤ (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r
conclusion
(clippedCellValueFormula ε K d m c r Ne - p) ^ 2
≤ (factorialCellValue ε K m c r Ne - p) ^ 2
Proof (Lean source)
lemma clippedCellValue_sq_error_le (ε : ℝ) (K d : ℕ) (m : ℝ)
(c r : Fin 4 → ℝ) (Ne : Fin 4 → ℕ) (p : ℝ)
(hwidth : 0 ≤ (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r)
(hp : |p - armwiseExtensionFormula ε c| ≤
(d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r) :
(clippedCellValueFormula ε K d m c r Ne - p) ^ 2 ≤
(factorialCellValue ε K m c r Ne - p) ^ 2 := by
rw [clippedCellValueFormula, centeredClip_eq]
have h := clip_sq_error_le
((d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r)
(factorialCellValue ε K m c r Ne - armwiseExtensionFormula ε c)
(p - armwiseExtensionFormula ε c) hwidth (abs_le.mp hp)
convert h using 1 <;> congr 1 <;> ring
A positive clipping width controls clipping displacement by a quadratic tail scale. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
A z :
ℝ
hA :
0 < A
conclusion
|clip A z - z| ≤ z ^ 2 / A
Proof (Lean source)
lemma clip_displacement_le_sq_div {A z : ℝ} (hA : 0 < A) :
|clip A z - z| ≤ z ^ 2 / A := by
apply (le_div_iff₀ hA).2
by_cases hlo : z < -A
· have hmin : min A z = z := min_eq_right (by linarith)
rw [clip, hmin, max_eq_left hlo.le, abs_of_nonneg (by linarith)]
nlinarith [sq_nonneg (z + A)]
· by_cases hhi : A < z
· rw [clip, min_eq_left hhi.le, max_eq_right (by linarith),
abs_of_nonpos (by linarith)]
nlinarith [sq_nonneg (z - A)]
· rw [clip, min_eq_right (le_of_not_gt hhi), max_eq_right (le_of_not_gt hlo)]
simp only [sub_self, abs_zero, zero_mul]
exact sq_nonneg z
The exact cell statistic obeys the quadratic clipping-displacement bound. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
ε :
ℝ
K d :
ℕ
m :
ℝ
c r :
Fin 4 → ℝ
Ne :
Fin 4 → ℕ
hwidth :
0 < (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r
conclusion
|clippedCellValueFormula ε K d m c r Ne - factorialCellValue ε K m c r Ne|
≤ (factorialCellValue ε K m c r Ne - armwiseExtensionFormula ε c) ^ 2 / ((d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r)
Proof (Lean source)
lemma clippedCellValue_displacement_le (ε : ℝ) (K d : ℕ) (m : ℝ)
(c r : Fin 4 → ℝ) (Ne : Fin 4 → ℕ)
(hwidth : 0 < (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r) :
|clippedCellValueFormula ε K d m c r Ne - factorialCellValue ε K m c r Ne| ≤
(factorialCellValue ε K m c r Ne - armwiseExtensionFormula ε c) ^ 2 /
((d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r) := by
rw [clippedCellValueFormula, centeredClip_eq]
convert clip_displacement_le_sq_div
(z := factorialCellValue ε K m c r Ne - armwiseExtensionFormula ε c) hwidth using 1
congr 1
ring
Integrating the displacement bound gives the clipping bias in equation (28). The second moment is an input to this transfer lemma, rather than an assumed risk rate. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
Z :
Ω → ℝ
c A :
ℝ
hA :
0 < A
hZ :
Integrable Z μ
hsecond :
Integrable (fun ω => (Z ω - c) ^ 2) μ
Proof (Lean source)
lemma centeredClip_integral_bias_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsFiniteMeasure μ] (Z : Ω → ℝ) (c A : ℝ) (hA : 0 < A)
(hZ : Integrable Z μ)
(hsecond : Integrable (fun ω => (Z ω - c) ^ 2) μ) :
|(∫ ω, max (c - A) (min (c + A) (Z ω)) ∂μ) - ∫ ω, Z ω ∂μ| ≤
(∫ ω, (Z ω - c) ^ 2 ∂μ) / A := by
have hT : Integrable (fun ω => max (c - A) (min (c + A) (Z ω))) μ :=
(integrable_const (c - A)).sup ((integrable_const (c + A)).inf hZ)
rw [← integral_sub hT hZ]
calc
_ ≤ ∫ ω, |max (c - A) (min (c + A) (Z ω)) - Z ω| ∂μ :=
abs_integral_le_integral_abs
_ ≤ ∫ ω, (Z ω - c) ^ 2 / A ∂μ := by
apply integral_mono (hT.sub hZ).abs (hsecond.div_const A)
intro ω
change |max (c - A) (min (c + A) (Z ω)) - Z ω| ≤ (Z ω - c) ^ 2 / A
rw [centeredClip_eq]
convert clip_displacement_le_sq_div (z := Z ω - c) hA using 1
congr 1
ring
_ = _ := by simp only [integral_div]
Projection to the unit interval contracts the loss relative to the observed target. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma projectUnit_sq_error_le {z p : ℝ} (hp : p ∈ Icc (0 : ℝ) 1) :
(projectUnit z - p) ^ 2 ≤ (z - p) ^ 2 := by
have heq : projectUnit z = (1 / 2 : ℝ) + clip (1 / 2) (z - 1 / 2) := by
convert centeredClip_eq (1 / 2) (1 / 2) z using 1
norm_num [projectUnit]
rw [heq]
have h := clip_sq_error_le (1 / 2) (z - 1 / 2) (p - 1 / 2)
(by norm_num) (by constructor <;> linarith [hp.1, hp.2])
convert h using 1 <;> congr 1 <;> ring
theorem clippedCellValue_integral_sq_error_le Lemma clippedCellValue_integral_sq_error_le in the paper ↗
Integrating the pointwise contraction proves the second-moment part of (28). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
Measure Ω
Ne :
Ω → Fin 4 → ℕ
ε :
ℝ
K d :
ℕ
m :
ℝ
c r :
Fin 4 → ℝ
p :
ℝ
hwidth :
0 ≤ (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r
hp :
|p - armwiseExtensionFormula ε c| ≤ (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r
hsecond :
Integrable (fun ω => (factorialCellValue ε K m c r (Ne ω) - p) ^ 2) μ
conclusion
(∫ ω, (clippedCellValueFormula ε K d m c r (Ne ω) - p) ^ 2 ∂μ)
≤ ∫ ω, (factorialCellValue ε K m c r (Ne ω) - p) ^ 2 ∂μ
Proof (Lean source)
lemma clippedCellValue_integral_sq_error_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) (Ne : Ω → Fin 4 → ℕ)
(ε : ℝ) (K d : ℕ) (m : ℝ) (c r : Fin 4 → ℝ) (p : ℝ)
(hwidth : 0 ≤ (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r)
(hp : |p - armwiseExtensionFormula ε c| ≤
(d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r)
(hsecond : Integrable (fun ω =>
(factorialCellValue ε K m c r (Ne ω) - p) ^ 2) μ) :
(∫ ω, (clippedCellValueFormula ε K d m c r (Ne ω) - p) ^ 2 ∂μ) ≤
∫ ω, (factorialCellValue ε K m c r (Ne ω) - p) ^ 2 ∂μ := by
apply integral_mono_of_nonneg (Filter.Eventually.of_forall (fun _ => sq_nonneg _))
hsecond
exact Filter.Eventually.of_forall fun ω =>
clippedCellValue_sq_error_le ε K d m c r (Ne ω) p hwidth hp
An unbiased factorial lift transfers its center second moment to clipping bias. This is the first part of (28) before substituting the factorial moment bound (27). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
Ne :
Ω → Fin 4 → ℕ
ε :
ℝ
K d :
ℕ
m :
ℝ
c r :
Fin 4 → ℝ
p :
ℝ
hwidth :
0 < (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r
hZ :
Integrable (fun ω => factorialCellValue ε K m c r (Ne ω)) μ
hmean :
(∫ ω, factorialCellValue ε K m c r (Ne ω) ∂μ) = p
hsecond :
conclusion
|(∫ ω, clippedCellValueFormula ε K d m c r (Ne ω) ∂μ) - p|
≤ (∫ ω, (factorialCellValue ε K m c r (Ne ω) - armwiseExtensionFormula ε c) ^ 2 ∂μ) / ((d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r)
Proof (Lean source)
lemma clippedCellValue_integral_bias_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsFiniteMeasure μ] (Ne : Ω → Fin 4 → ℕ)
(ε : ℝ) (K d : ℕ) (m : ℝ) (c r : Fin 4 → ℝ) (p : ℝ)
(hwidth : 0 < (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r)
(hZ : Integrable (fun ω => factorialCellValue ε K m c r (Ne ω)) μ)
(hmean : (∫ ω, factorialCellValue ε K m c r (Ne ω) ∂μ) = p)
(hsecond : Integrable (fun ω =>
(factorialCellValue ε K m c r (Ne ω) - armwiseExtensionFormula ε c) ^ 2) μ) :
|(∫ ω, clippedCellValueFormula ε K d m c r (Ne ω) ∂μ) - p| ≤
(∫ ω, (factorialCellValue ε K m c r (Ne ω) - armwiseExtensionFormula ε c) ^ 2 ∂μ) /
((d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r) := by
simpa only [clippedCellValueFormula, hmean] using centeredClip_integral_bias_le μ
(fun ω => factorialCellValue ε K m c r (Ne ω)) (armwiseExtensionFormula ε c)
((d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r) hwidth hZ hsecond
theorem clippedCellValue_eq_center_of_zero_scale Lemma clippedCellValue_eq_center_of_zero_scale in the paper ↗
Zero oscillation scale makes clipping exactly equal to its center. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
clippedCellValueFormula ε K d m c r Ne = armwiseExtensionFormula ε c
Proof (Lean source)
lemma clippedCellValue_eq_center_of_zero_scale (ε : ℝ) (K d : ℕ) (m : ℝ)
(c r : Fin 4 → ℝ) (Ne : Fin 4 → ℕ) (hscale : clippingScaleFormula ε c r = 0) :
clippedCellValueFormula ε K d m c r Ne = armwiseExtensionFormula ε c := by
simp only [clippedCellValueFormula, hscale, mul_zero, sub_zero, add_zero]
exact max_eq_left (min_le_left _ _)
The clipped cell statistic stays within its chosen width of the center, regardless of the evaluation counts or pilot localization. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
ε :
ℝ
K d :
ℕ
m :
ℝ
c r :
Fin 4 → ℝ
Ne :
Fin 4 → ℕ
hwidth :
0 ≤ (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r
conclusion
|clippedCellValueFormula ε K d m c r Ne - armwiseExtensionFormula ε c|
≤ (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r
Proof (Lean source)
lemma clippedCellValue_abs_sub_center_le (ε : ℝ) (K d : ℕ) (m : ℝ)
(c r : Fin 4 → ℝ) (Ne : Fin 4 → ℕ)
(hwidth : 0 ≤ (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r) :
|clippedCellValueFormula ε K d m c r Ne - armwiseExtensionFormula ε c| ≤
(d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r := by
apply abs_le.mpr
constructor
· have h := le_max_left
(armwiseExtensionFormula ε c - (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r)
(min (armwiseExtensionFormula ε c + (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r)
(factorialCellValue ε K m c r Ne))
change armwiseExtensionFormula ε c - _ ≤ clippedCellValueFormula ε K d m c r Ne at h
linarith
· have h : clippedCellValueFormula ε K d m c r Ne ≤
armwiseExtensionFormula ε c + (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r := by
exact max_le (by linarith) (min_le_left _ _)
linarith
On bad pilots the cell error is bounded by width plus center error. This is the pathwise inequality following roadmap (28), retaining the local scales rather than substituting unit loss. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
ε :
ℝ
K d :
ℕ
m :
ℝ
c r :
Fin 4 → ℝ
Ne :
Fin 4 → ℕ
p :
ℝ
hwidth :
0 ≤ (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r
conclusion
|clippedCellValueFormula ε K d m c r Ne - p|
≤ (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r + |armwiseExtensionFormula ε c - p|
Proof (Lean source)
lemma clippedCellValue_abs_error_le (ε : ℝ) (K d : ℕ) (m : ℝ)
(c r : Fin 4 → ℝ) (Ne : Fin 4 → ℕ) (p : ℝ)
(hwidth : 0 ≤ (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r) :
|clippedCellValueFormula ε K d m c r Ne - p| ≤
(d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r + |armwiseExtensionFormula ε c - p| := by
calc
_ ≤ |clippedCellValueFormula ε K d m c r Ne - armwiseExtensionFormula ε c| +
|armwiseExtensionFormula ε c - p| := abs_sub_le _ _ _
_ ≤ _ := add_le_add
(clippedCellValue_abs_sub_center_le ε K d m c r Ne hwidth) le_rfl
theorem clippedCellValue_sq_error_le_width_center Lemma clippedCellValue_sq_error_le_width_center in the paper ↗
The bad-pilot squared error is controlled by twice the width square and twice the center-error square, precisely the two quantities in roadmap (22). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
ε :
ℝ
K d :
ℕ
m :
ℝ
c r :
Fin 4 → ℝ
Ne :
Fin 4 → ℕ
p :
ℝ
hwidth :
0 ≤ (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r
conclusion
(clippedCellValueFormula ε K d m c r Ne - p) ^ 2
≤ 2 * ((d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r) ^ 2
+ 2 * (armwiseExtensionFormula ε c - p) ^ 2
Proof (Lean source)
lemma clippedCellValue_sq_error_le_width_center (ε : ℝ) (K d : ℕ) (m : ℝ)
(c r : Fin 4 → ℝ) (Ne : Fin 4 → ℕ) (p : ℝ)
(hwidth : 0 ≤ (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r) :
(clippedCellValueFormula ε K d m c r Ne - p) ^ 2 ≤
2 * ((d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r) ^ 2 +
2 * (armwiseExtensionFormula ε c - p) ^ 2 := by
have h := pow_le_pow_left₀ (abs_nonneg _)
(clippedCellValue_abs_error_le ε K d m c r Ne p hwidth) 2
rw [sq_abs] at h
have ht :
((d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r + |armwiseExtensionFormula ε c - p|) ^ 2 ≤
2 * ((d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r) ^ 2 +
2 * (armwiseExtensionFormula ε c - p) ^ 2 := by
nlinarith [sq_nonneg ((d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r -
|armwiseExtensionFormula ε c - p|), sq_abs (armwiseExtensionFormula ε c - p)]
exact h.trans ht
theorem clippedCellValue_integral_abs_on_le Lemma clippedCellValue_integral_abs_on_le in the paper ↗
On any pilot event, the integrated absolute cell error is controlled by the local width and center deviation. This is the first-moment use of roadmap (22) and does not require an integrability premise on the cell statistic. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
Measure Ω
B :
Set Ω
c r :
Ω → Fin 4 → ℝ
Ne :
Ω → Fin 4 → ℕ
ε :
ℝ
K d :
ℕ
m p :
ℝ
hwidth :
∀ ω ∈ B, 0 ≤ (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε (c ω) (r ω)
hA :
Integrable (fun ω => (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε (c ω) (r ω)) (μ.restrict B)
hF :
Integrable (fun ω => |armwiseExtensionFormula ε (c ω) - p|) (μ.restrict B)
hB :
conclusion
(∫ ω in B, |clippedCellValueFormula ε K d m (c ω) (r ω) (Ne ω) - p| ∂μ)
≤ (∫ ω in B, (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε (c ω) (r ω) ∂μ)
+ (∫ ω in B, |armwiseExtensionFormula ε (c ω) - p| ∂μ)
Proof (Lean source)
lemma clippedCellValue_integral_abs_on_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) (B : Set Ω) (c r : Ω → Fin 4 → ℝ) (Ne : Ω → Fin 4 → ℕ)
(ε : ℝ) (K d : ℕ) (m p : ℝ)
(hwidth : ∀ ω ∈ B, 0 ≤ (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε (c ω) (r ω))
(hA : Integrable (fun ω =>
(d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε (c ω) (r ω)) (μ.restrict B))
(hF : Integrable (fun ω => |armwiseExtensionFormula ε (c ω) - p|) (μ.restrict B))
(hB : MeasurableSet B) :
(∫ ω in B, |clippedCellValueFormula ε K d m (c ω) (r ω) (Ne ω) - p| ∂μ) ≤
(∫ ω in B, (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε (c ω) (r ω) ∂μ) +
(∫ ω in B, |armwiseExtensionFormula ε (c ω) - p| ∂μ) := by
calc
_ ≤ ∫ ω in B,
(d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε (c ω) (r ω) +
|armwiseExtensionFormula ε (c ω) - p| ∂μ := by
apply integral_mono_of_nonneg (ae_of_all _ fun _ => abs_nonneg _) (hA.add hF)
exact (ae_restrict_iff' hB).mpr (ae_of_all _ fun ω hω =>
clippedCellValue_abs_error_le ε K d m (c ω) (r ω) (Ne ω) p (hwidth ω hω))
_ = _ := integral_add hA hF
Integrating over any pilot event transfers the tail width and center second moments to the actual clipped cell loss. Applying this to the bad-pilot event is the second-moment use of roadmap (22) after (28). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
Measure Ω
B :
Set Ω
c r :
Ω → Fin 4 → ℝ
Ne :
Ω → Fin 4 → ℕ
ε :
ℝ
K d :
ℕ
m p :
ℝ
hwidth :
∀ ω ∈ B, 0 ≤ (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε (c ω) (r ω)
hA :
Integrable (fun ω => ((d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε (c ω) (r ω)) ^ 2) (μ.restrict B)
hF :
Integrable (fun ω => (armwiseExtensionFormula ε (c ω) - p) ^ 2) (μ.restrict B)
hB :
conclusion
(∫ ω in B, (clippedCellValueFormula ε K d m (c ω) (r ω) (Ne ω) - p) ^ 2 ∂μ)
≤ 2 * (∫ ω in B, ((d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε (c ω) (r ω)) ^ 2 ∂μ)
+ 2 * (∫ ω in B, (armwiseExtensionFormula ε (c ω) - p) ^ 2 ∂μ)
Proof (Lean source)
lemma clippedCellValue_integral_sq_on_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) (B : Set Ω) (c r : Ω → Fin 4 → ℝ) (Ne : Ω → Fin 4 → ℕ)
(ε : ℝ) (K d : ℕ) (m p : ℝ)
(hwidth : ∀ ω ∈ B, 0 ≤ (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε (c ω) (r ω))
(hA : Integrable (fun ω =>
((d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε (c ω) (r ω)) ^ 2) (μ.restrict B))
(hF : Integrable (fun ω => (armwiseExtensionFormula ε (c ω) - p) ^ 2) (μ.restrict B))
(hB : MeasurableSet B) :
(∫ ω in B, (clippedCellValueFormula ε K d m (c ω) (r ω) (Ne ω) - p) ^ 2 ∂μ) ≤
2 * (∫ ω in B, ((d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε (c ω) (r ω)) ^ 2 ∂μ) +
2 * (∫ ω in B, (armwiseExtensionFormula ε (c ω) - p) ^ 2 ∂μ) := by
calc
_ ≤ ∫ ω in B,
2 * ((d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε (c ω) (r ω)) ^ 2 +
2 * (armwiseExtensionFormula ε (c ω) - p) ^ 2 ∂μ := by
apply integral_mono_of_nonneg (ae_of_all _ fun _ => sq_nonneg _)
((hA.const_mul 2).add (hF.const_mul 2))
exact (ae_restrict_iff' hB).mpr (ae_of_all _ fun ω hω =>
clippedCellValue_sq_error_le_width_center ε K d m (c ω) (r ω) (Ne ω) p (hwidth ω hω))
_ = _ := by
rw [integral_add (hA.const_mul 2) (hF.const_mul 2)]
simp only [integral_const_mul]
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.UpperDegreeCalibration 5 declarations Roadmap equation (27) chooses a universal small degree coefficient so the linear and quadratic exponential growth in (26) costs only d^(1/16).
Universal Jackson degree calibration
Roadmap equation (27) chooses a universal small degree coefficient so the linear and quadratic exponential growth in (26) costs only d^(1/16). The additive degree floor contributes a universal multiplicative constant.
theorem jacksonDegree_le_two_add
The degree floor costs at most two in addition to the tuned logarithmic degree. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
κ :
ℝ
d :
ℕ
hκ :
0 ≤ κ
hL :
0 ≤ logAlphabet d
conclusion
(jacksonDegree κ d : ℝ) ≤ 2 + κ * logAlphabet d
Proof (Lean source)
lemma jacksonDegree_le_two_add {κ : ℝ} {d : ℕ} (hκ : 0 ≤ κ)
(hL : 0 ≤ logAlphabet d) :
(jacksonDegree κ d : ℝ) ≤ 2 + κ * logAlphabet d := by
have hf := Nat.floor_le (mul_nonneg hκ hL)
rw [jacksonDegree, Nat.cast_max]
exact max_le (by norm_num; exact mul_nonneg hκ hL) (by linarith)
theorem jacksonDegree_ge_half_log
The degree floor and minimum degree two retain at least half the tuned logarithmic degree, uniformly without a large-alphabet cutoff. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
κ :
ℝ
d :
ℕ
conclusion
κ / 2 * logAlphabet d ≤ (jacksonDegree κ d : ℝ)
Proof (Lean source)
lemma jacksonDegree_ge_half_log (κ : ℝ) (d : ℕ) :
κ / 2 * logAlphabet d ≤ (jacksonDegree κ d : ℝ) := by
have hf := Nat.lt_floor_add_one (κ * logAlphabet d)
have htwo : (2 : ℝ) ≤ jacksonDegree κ d := by
exact_mod_cast (le_max_left 2 ⌊κ * logAlphabet d⌋₊)
have hfloor : (⌊κ * logAlphabet d⌋₊ : ℝ) ≤ jacksonDegree κ d := by
exact_mod_cast (le_max_right 2 ⌊κ * logAlphabet d⌋₊)
by_cases hsmall : κ * logAlphabet d ≤ 4
· linarith
· linarith
A logarithmic degree bound controls the quadratic exponent without a log n term. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
C₁ C₂ κ L K :
ℝ
hC₁ :
0 ≤ C₁
hC₂ :
0 ≤ C₂
hL :
1 ≤ L
hK :
0 ≤ K
hdegree :
K ≤ 2 + κ * L
htune :
C₁ * κ + 2 * C₂ * κ ^ 2 ≤ 1 / 16
conclusion
C₁ * K + C₂ * K ^ 2 / L ≤ 2 * C₁ + 8 * C₂ + L / 16
Proof (Lean source)
lemma degree_exponent_le {C₁ C₂ κ L K : ℝ}
(hC₁ : 0 ≤ C₁) (hC₂ : 0 ≤ C₂) (hL : 1 ≤ L)
(hK : 0 ≤ K) (hdegree : K ≤ 2 + κ * L)
(htune : C₁ * κ + 2 * C₂ * κ ^ 2 ≤ 1 / 16) :
C₁ * K + C₂ * K ^ 2 / L ≤ 2 * C₁ + 8 * C₂ + L / 16 := by
have hLp : 0 < L := by linarith
have hsq : K ^ 2 ≤ 8 + 2 * κ ^ 2 * L ^ 2 := by
have h := pow_le_pow_left₀ hK hdegree 2
nlinarith [sq_nonneg (2 - κ * L)]
have hlin := mul_le_mul_of_nonneg_left hdegree hC₁
have hquad := mul_le_mul_of_nonneg_left hsq hC₂
have ht := mul_le_mul_of_nonneg_right htune hLp.le
apply (mul_le_mul_iff_right₀ hLp).mp
have he : (C₁ * K + C₂ * K ^ 2 / L) * L = C₁ * K * L + C₂ * K ^ 2 := by
field_simp
rw [mul_comm L (C₁ * K + C₂ * K ^ 2 / L), he]
have hl := mul_le_mul_of_nonneg_right hlin hLp.le
have ht₂ := mul_le_mul_of_nonneg_right ht hLp.le
have hc₂ := mul_le_mul_of_nonneg_left hL (show 0 ≤ 8 * C₂ by positivity)
nlinarith
One universal positive tuning coefficient suffices for any fixed nonnegative linear and quadratic moment-growth constants. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
C₁ C₂ :
ℝ
hC₁ :
0 ≤ C₁
hC₂ :
0 ≤ C₂
conclusion
∃ κ : ℝ,
conclusion 1
0 < κ
conclusion 2
C₁ * κ + 2 * C₂ * κ ^ 2 ≤ 1 / 16
Proof (Lean source)
lemma degree_tuning_exists (C₁ C₂ : ℝ) (hC₁ : 0 ≤ C₁) (hC₂ : 0 ≤ C₂) :
∃ κ : ℝ, 0 < κ ∧ C₁ * κ + 2 * C₂ * κ ^ 2 ≤ 1 / 16 := by
let B := C₁ + 2 * C₂ + 1
have hB : 0 < B := by dsimp [B]; linarith
let κ := 1 / (32 * B)
have hκ : 0 < κ := by dsimp [κ]; positivity
have hκ1 : κ ≤ 1 := by
dsimp [κ]
apply (div_le_iff₀ (by positivity : 0 < 32 * B)).2
dsimp [B]
linarith
have he : B * κ = 1 / 32 := by dsimp [κ]; field_simp
refine ⟨κ, hκ, ?_⟩
have hs : κ ^ 2 ≤ κ := by nlinarith
have hq := mul_le_mul_of_nonneg_left hs (show 0 ≤ 2 * C₂ by positivity)
dsimp [B] at he
nlinarith
theorem jacksonDegree_exponential_growth_le
The tuned Jackson degree has at most d^(1/16) exponential moment inflation, with a universal prefactor, as in roadmap (27). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
C₁ C₂ :
ℝ
hC₁ :
0 ≤ C₁
hC₂ :
0 ≤ C₂
conclusion
∃ κ A : ℝ,
0 < κ
0 < A
conclusion 1
d :
ℕ
1 ≤ d
exp (C₁ * jacksonDegree κ d + C₂ * (jacksonDegree κ d : ℝ) ^ 2 / logAlphabet d)
≤ A * (d : ℝ) ^ (1 / 16 : ℝ)
Proof (Lean source)
lemma jacksonDegree_exponential_growth_le (C₁ C₂ : ℝ)
(hC₁ : 0 ≤ C₁) (hC₂ : 0 ≤ C₂) :
∃ κ A : ℝ, 0 < κ ∧ 0 < A ∧ ∀ {d : ℕ}, 1 ≤ d →
exp (C₁ * jacksonDegree κ d + C₂ * (jacksonDegree κ d : ℝ) ^ 2 /
logAlphabet d) ≤ A * (d : ℝ) ^ (1 / 16 : ℝ) := by
obtain ⟨κ, hκ, htune⟩ := degree_tuning_exists C₁ C₂ hC₁ hC₂
refine ⟨κ, exp (2 * C₁ + 8 * C₂ + 1 / 16), hκ, Real.exp_pos _, ?_⟩
intro d hd
have hd0 : (0 : ℝ) < d := by exact_mod_cast (lt_of_lt_of_le Nat.zero_lt_one hd)
have hd1 : (1 : ℝ) ≤ d := by exact_mod_cast hd
have hlog : logAlphabet d = 1 + log d := by
rw [logAlphabet, Real.log_mul (ne_of_gt (Real.exp_pos _)) (ne_of_gt hd0), Real.log_exp]
have hL : 1 ≤ logAlphabet d := by rw [hlog]; linarith [Real.log_nonneg hd1]
have hex := degree_exponent_le hC₁ hC₂ hL (Nat.cast_nonneg _)
(jacksonDegree_le_two_add hκ.le (by linarith)) htune
calc
_ ≤ exp (2 * C₁ + 8 * C₂ + logAlphabet d / 16) := Real.exp_le_exp.mpr hex
_ = _ := by
rw [hlog, Real.rpow_def_of_pos hd0, ← Real.exp_add]
congr 1
ring
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.UpperEmpiricalRatio 20 declarations The observed value lies in the unit interval, and the constant branch has risk at most one quarter in the saturated regime.
Nonpolynomial upper-bound branches
The observed value lies in the unit interval, and the constant branch has risk
at most one quarter in the saturated regime. Empirical arm means stay in the
unit interval even for empty arms. The pathwise empirical-ratio error splits
into mass error and mass-weighted arm error, giving the two squared-loss terms
in roadmap equation (6) after projection. Weighted Cauchy–Schwarz reduces arm
error to individual second moments without excluding null cells. The iid
cell-indicator mean-square bound proves the mass term is at most d / n.
theorem cellMass_sum_eq_one
The covariate cell masses of a finite observed probability law sum to one. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
Proof (Lean source)
lemma cellMass_sum_eq_one {d : ℕ} (P : DiscreteLaw d) :
∑ x : Fin d, cellMass P x = 1 := by
have h := ENNReal.tsum_toReal_eq (fun z => P.pmf.apply_ne_top z)
rw [P.pmf.tsum_coe] at h
simpa [cellMass, jointMass, tsum_fintype, Fintype.sum_prod_type] using h.symm
theorem observedValue_mem_unitInterval
The oracle observed value is between zero and one for every finite observed law. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
P :
conclusion
conclusion 1
0 ≤ observedValue P
conclusion 2
observedValue P ≤ 1
Proof (Lean source)
lemma observedValue_mem_unitInterval {d : ℕ} (P : DiscreteLaw d) :
0 ≤ observedValue P ∧ observedValue P ≤ 1 := by
have hp (x : Fin d) : 0 ≤ cellMass P x := by
exact Finset.sum_nonneg fun a _ => Finset.sum_nonneg fun y _ => toReal_nonneg
have hm (x : Fin d) :
0 ≤ max (outcomeMean P false x) (outcomeMean P true x) ∧
max (outcomeMean P false x) (outcomeMean P true x) ≤ 1 :=
⟨le_trans (outcomeMean_mem_unitInterval P false x).1 (le_max_left _ _),
max_le (outcomeMean_mem_unitInterval P false x).2
(outcomeMean_mem_unitInterval P true x).2⟩
constructor
· exact Finset.sum_nonneg fun x _ => mul_nonneg (hp x) (hm x).1
· calc
observedValue P ≤ ∑ x : Fin d, cellMass P x := by
apply Finset.sum_le_sum
intro x _
simpa using mul_le_mul_of_nonneg_left (hm x).2 (hp x)
_ = 1 := cellMass_sum_eq_one P
theorem halfEstimator_sqRisk_le_quarter
The constant one-half estimator has squared risk at most one quarter. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
sqRisk μ (fun _ => 1 / 2) (observedValue P) ≤ 1 / 4
Proof (Lean source)
lemma halfEstimator_sqRisk_le_quarter {d n : ℕ} (P : DiscreteLaw d)
(μ : Measure (Fin n → Obs d)) [IsProbabilityMeasure μ] :
sqRisk μ (fun _ => 1 / 2) (observedValue P) ≤ 1 / 4 := by
obtain ⟨hlo, hhi⟩ := observedValue_mem_unitInterval P
have hprod : 0 ≤ observedValue P * (1 - observedValue P) :=
mul_nonneg hlo (by linarith)
simpa [sqRisk] using
(show (1 / 2 - observedValue P) ^ 2 ≤ (1 / 4 : ℝ) by nlinarith)
theorem logAlphabet_pos
The logarithmic alphabet scale is positive for a nonempty alphabet. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
1 ≤ d
conclusion
0 < logAlphabet d
Proof (Lean source)
lemma logAlphabet_pos {d : ℕ} (hd : 1 ≤ d) : 0 < logAlphabet d := by
apply Real.log_pos
have hd' : (1 : ℝ) ≤ d := by exact_mod_cast hd
have he : 1 < exp 1 := Real.one_lt_exp_iff.mpr (by norm_num)
nlinarith [Real.exp_pos (1 : ℝ)]
theorem rateScale_eq_one_of_saturated
On the constant branch the minimum-form rate scale is exactly one. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
n d :
ℕ
ε :
ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hε :
0 < ε
hsat :
(n : ℝ) * ε < (d : ℝ) / logAlphabet d
conclusion
rateScale n d ε = 1
Proof (Lean source)
lemma rateScale_eq_one_of_saturated {n d : ℕ} {ε : ℝ}
(hn : 1 ≤ n) (hd : 1 ≤ d) (hε : 0 < ε)
(hsat : (n : ℝ) * ε < (d : ℝ) / logAlphabet d) :
rateScale n d ε = 1 := by
have hL := logAlphabet_pos hd
have hn' : (0 : ℝ) < n := by exact_mod_cast (lt_of_lt_of_le Nat.zero_lt_one hn)
have hden : 0 < (n : ℝ) * ε * logAlphabet d := mul_pos (mul_pos hn' hε) hL
have hnum : (n : ℝ) * ε * logAlphabet d < d := (lt_div_iff₀ hL).mp hsat
exact min_eq_left ((lt_div_iff₀ hden).mpr (by simpa using hnum)).le
The large-alphabet constant branch is measurable and has risk at most one quarter. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
H₀ κ :
ℝ
D₀ :
ℕ
hH₀ :
0 < H₀
hκ :
0 < κ
hD₀ :
2 ≤ D₀
n d :
ℕ
ε :
ℝ
hd :
D₀ ≤ d
hsat :
(n : ℝ) * ε < (d : ℝ) / logAlphabet d
P :
hiid :
IidSampling P μ
conclusion
conclusion 1
Measurable (armwiseEstimatorFormula H₀ κ D₀ hH₀ hκ hD₀ n d ε)
conclusion 2
sqRisk μ (armwiseEstimatorFormula H₀ κ D₀ hH₀ hκ hD₀ n d ε) (observedValue P)
≤ 1 / 4
Proof (Lean source)
lemma armwiseEstimator_saturated_risk (H₀ κ : ℝ) (D₀ : ℕ)
(hH₀ : 0 < H₀) (hκ : 0 < κ) (hD₀ : 2 ≤ D₀)
{n d : ℕ} {ε : ℝ} (hd : D₀ ≤ d)
(hsat : (n : ℝ) * ε < (d : ℝ) / logAlphabet d)
(P : DiscreteLaw d) (μ : Measure (Fin n → Obs d)) (hiid : IidSampling P μ) :
Measurable (armwiseEstimatorFormula H₀ κ D₀ hH₀ hκ hD₀ n d ε) ∧
sqRisk μ
(armwiseEstimatorFormula H₀ κ D₀ hH₀ hκ hD₀ n d ε) (observedValue P) ≤ 1 / 4 := by
have heq : armwiseEstimatorFormula H₀ κ D₀ hH₀ hκ hD₀ n d ε = fun _ => 1 / 2 := by
funext o
simp [armwiseEstimatorFormula, not_lt.mpr hd, hsat]
rw [heq]
constructor
· fun_prop
· have hμ : μ = productLaw P n := hiid
subst μ
let : IsProbabilityMeasure (productLaw P n) := by
dsimp [productLaw]
infer_instance
exact halfEstimator_sqRisk_le_quarter P (productLaw P n)
theorem armwiseEstimator_saturated_rate_bound Lemma armwiseEstimator_saturated_rate_bound in the paper ↗
The saturated branch satisfies the desired minimum-form risk bound with constant one quarter. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
H₀ κ :
ℝ
D₀ :
ℕ
hH₀ :
0 < H₀
hκ :
0 < κ
hD₀ :
2 ≤ D₀
n d :
ℕ
ε :
ℝ
hn :
1 ≤ n
hd :
D₀ ≤ d
hε :
0 < ε
hsat :
(n : ℝ) * ε < (d : ℝ) / logAlphabet d
P :
hiid :
IidSampling P μ
conclusion
sqRisk μ (armwiseEstimatorFormula H₀ κ D₀ hH₀ hκ hD₀ n d ε) (observedValue P)
≤ (1 / 4 : ℝ) * rateScale n d ε
Proof (Lean source)
lemma armwiseEstimator_saturated_rate_bound (H₀ κ : ℝ) (D₀ : ℕ)
(hH₀ : 0 < H₀) (hκ : 0 < κ) (hD₀ : 2 ≤ D₀)
{n d : ℕ} {ε : ℝ} (hn : 1 ≤ n) (hd : D₀ ≤ d) (hε : 0 < ε)
(hsat : (n : ℝ) * ε < (d : ℝ) / logAlphabet d)
(P : DiscreteLaw d) (μ : Measure (Fin n → Obs d)) (hiid : IidSampling P μ) :
sqRisk μ
(armwiseEstimatorFormula H₀ κ D₀ hH₀ hκ hD₀ n d ε) (observedValue P) ≤
(1 / 4 : ℝ) * rateScale n d ε := by
rw [rateScale_eq_one_of_saturated hn (by omega) hε hsat, mul_one]
exact (armwiseEstimator_saturated_risk H₀ κ D₀ hH₀ hκ hD₀ hd hsat P μ hiid).2
theorem sampleSuccessCount_le_sampleArmCount
Successes form a subset of the observations in each treatment arm. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
sampleSuccessCount o x a ≤ sampleArmCount o x a
Proof (Lean source)
lemma sampleSuccessCount_le_sampleArmCount {n d : ℕ} (o : Fin n → Obs d)
(x : Fin d) (a : Bool) : sampleSuccessCount o x a ≤ sampleArmCount o x a := by
apply Finset.sum_le_sum
intro i hi
by_cases h : (o i).1 = x ∧ (o i).2.1 = a ∧ (o i).2.2 = true
· rw [if_pos h, if_pos ⟨h.1, h.2.1⟩]
· rw [if_neg h]
exact zero_le _
theorem empiricalOutcomeMean_mem_unitInterval
The empirical arm mean lies in the unit interval, including empty arms. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
conclusion 1
0 ≤ (sampleSuccessCount o x a : ℝ) / sampleArmCount o x a
conclusion 2
(sampleSuccessCount o x a : ℝ) / sampleArmCount o x a ≤ 1
Proof (Lean source)
lemma empiricalOutcomeMean_mem_unitInterval {n d : ℕ} (o : Fin n → Obs d)
(x : Fin d) (a : Bool) :
0 ≤ (sampleSuccessCount o x a : ℝ) / sampleArmCount o x a ∧
(sampleSuccessCount o x a : ℝ) / sampleArmCount o x a ≤ 1 := by
constructor
· positivity
· exact div_le_one_of_le₀ (by exact_mod_cast sampleSuccessCount_le_sampleArmCount o x a)
(Nat.cast_nonneg _)
theorem empiricalArmMax_mem_unitInterval
The two empirical arm means have a maximum in the unit interval. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
v :
max ((sampleSuccessCount o x false : ℝ) / sampleArmCount o x false) ((sampleSuccessCount o x true : ℝ) / sampleArmCount o x true)
0 ≤ v
v ≤ 1
Proof (Lean source)
lemma empiricalArmMax_mem_unitInterval {n d : ℕ} (o : Fin n → Obs d)
(x : Fin d) :
let v := max ((sampleSuccessCount o x false : ℝ) / sampleArmCount o x false)
((sampleSuccessCount o x true : ℝ) / sampleArmCount o x true)
0 ≤ v ∧ v ≤ 1 := by
exact ⟨le_trans (empiricalOutcomeMean_mem_unitInterval o x false).1 (le_max_left _ _),
max_le (empiricalOutcomeMean_mem_unitInterval o x false).2
(empiricalOutcomeMean_mem_unitInterval o x true).2⟩
theorem empiricalCell_abs_error_le
The cell error splits into empirical mass error and mass-weighted arm error. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
v :
max ((sampleSuccessCount o x false : ℝ) / sampleArmCount o x false) ((sampleSuccessCount o x true : ℝ) / sampleArmCount o x true)
|(sampleCellCount o x : ℝ) / n * v
≤ |(sampleCellCount o x : ℝ) / n
- cellMass P x|
+ cellMass P x * max |(sampleSuccessCount o x false : ℝ) / sampleArmCount o x false
- outcomeMean P false x| |(sampleSuccessCount o x true : ℝ) / sampleArmCount o x true
- outcomeMean P true x|
Proof (Lean source)
lemma empiricalCell_abs_error_le {n d : ℕ} (o : Fin n → Obs d)
(P : DiscreteLaw d) (x : Fin d) :
let v := max ((sampleSuccessCount o x false : ℝ) / sampleArmCount o x false)
((sampleSuccessCount o x true : ℝ) / sampleArmCount o x true)
|(sampleCellCount o x : ℝ) / n * v -
cellMass P x * max (outcomeMean P false x) (outcomeMean P true x)| ≤
|(sampleCellCount o x : ℝ) / n - cellMass P x| + cellMass P x *
max |(sampleSuccessCount o x false : ℝ) / sampleArmCount o x false - outcomeMean P false x|
|(sampleSuccessCount o x true : ℝ) / sampleArmCount o x true - outcomeMean P true x| := by
dsimp only
let v := max ((sampleSuccessCount o x false : ℝ) / sampleArmCount o x false)
((sampleSuccessCount o x true : ℝ) / sampleArmCount o x true)
let w := max (outcomeMean P false x) (outcomeMean P true x)
have hp : 0 ≤ cellMass P x :=
Finset.sum_nonneg fun a _ => Finset.sum_nonneg fun y _ => toReal_nonneg
have hv := empiricalArmMax_mem_unitInterval o x
change |(sampleCellCount o x : ℝ) / n * v - cellMass P x * w| ≤ _
calc
_ = |((sampleCellCount o x : ℝ) / n - cellMass P x) * v +
cellMass P x * (v - w)| := by congr 1; ring
_ ≤ |((sampleCellCount o x : ℝ) / n - cellMass P x) * v| +
|cellMass P x * (v - w)| := abs_add_le _ _
_ ≤ |(sampleCellCount o x : ℝ) / n - cellMass P x| + cellMass P x *
max |(sampleSuccessCount o x false : ℝ) / sampleArmCount o x false - outcomeMean P false x|
|(sampleSuccessCount o x true : ℝ) / sampleArmCount o x true - outcomeMean P true x| := by
rw [abs_mul, abs_mul, abs_of_nonneg hv.1, abs_of_nonneg hp]
apply add_le_add
· exact mul_le_of_le_one_right (abs_nonneg _) hv.2
· exact mul_le_mul_of_nonneg_left (abs_max_sub_max_le_max _ _ _ _) hp
theorem empiricalRawValue_abs_error_le
Summing the cell bounds gives the pathwise decomposition preceding equation (6). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
|(∑ x : Fin d, (sampleCellCount o x : ℝ) / n * max ((sampleSuccessCount o x false : ℝ) / sampleArmCount o x false) ((sampleSuccessCount o x true : ℝ) / sampleArmCount o x true))
- observedValue P|
≤ (∑ x : Fin d, |(sampleCellCount o x : ℝ) / n - cellMass P x|)
- outcomeMean P false x| |(sampleSuccessCount o x true : ℝ) / sampleArmCount o x true
- outcomeMean P true x|
Proof (Lean source)
lemma empiricalRawValue_abs_error_le {n d : ℕ} (o : Fin n → Obs d)
(P : DiscreteLaw d) :
|(∑ x : Fin d, (sampleCellCount o x : ℝ) / n *
max ((sampleSuccessCount o x false : ℝ) / sampleArmCount o x false)
((sampleSuccessCount o x true : ℝ) / sampleArmCount o x true)) - observedValue P| ≤
(∑ x : Fin d, |(sampleCellCount o x : ℝ) / n - cellMass P x|) +
∑ x : Fin d, cellMass P x *
max |(sampleSuccessCount o x false : ℝ) / sampleArmCount o x false - outcomeMean P false x|
|(sampleSuccessCount o x true : ℝ) / sampleArmCount o x true - outcomeMean P true x| := by
rw [observedValue, ← Finset.sum_sub_distrib, ← Finset.sum_add_distrib]
exact (Finset.abs_sum_le_sum_abs _ _).trans
(Finset.sum_le_sum fun x _ => empiricalCell_abs_error_le o P x)
theorem empiricalValue_sq_error_le_components
Projection and the pathwise decomposition reduce squared loss to the two terms in (6). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
A :
∑ x : Fin d, |(sampleCellCount o x : ℝ) / n - cellMass P x|
B :
- outcomeMean P false x| |(sampleSuccessCount o x true : ℝ) / sampleArmCount o x true
- outcomeMean P true x|
(empiricalValue o - observedValue P) ^ 2 ≤ 2 * A ^ 2 + 2 * B ^ 2
Proof (Lean source)
lemma empiricalValue_sq_error_le_components {n d : ℕ} (o : Fin n → Obs d)
(P : DiscreteLaw d) :
let A := ∑ x : Fin d, |(sampleCellCount o x : ℝ) / n - cellMass P x|
let B := ∑ x : Fin d, cellMass P x *
max |(sampleSuccessCount o x false : ℝ) / sampleArmCount o x false - outcomeMean P false x|
|(sampleSuccessCount o x true : ℝ) / sampleArmCount o x true - outcomeMean P true x|
(empiricalValue o - observedValue P) ^ 2 ≤ 2 * A ^ 2 + 2 * B ^ 2 := by
dsimp only
let T := ∑ x : Fin d, (sampleCellCount o x : ℝ) / n *
max ((sampleSuccessCount o x false : ℝ) / sampleArmCount o x false)
((sampleSuccessCount o x true : ℝ) / sampleArmCount o x true)
let A := ∑ x : Fin d, |(sampleCellCount o x : ℝ) / n - cellMass P x|
let B := ∑ x : Fin d, cellMass P x *
max |(sampleSuccessCount o x false : ℝ) / sampleArmCount o x false - outcomeMean P false x|
|(sampleSuccessCount o x true : ℝ) / sampleArmCount o x true - outcomeMean P true x|
have hA : 0 ≤ A := Finset.sum_nonneg fun x _ => abs_nonneg _
have hB : 0 ≤ B := by
apply Finset.sum_nonneg
intro x hx
apply mul_nonneg
· exact Finset.sum_nonneg fun a _ => Finset.sum_nonneg fun y _ => toReal_nonneg
· exact le_trans (abs_nonneg _) (le_max_left _ _)
have habs : |T - observedValue P| ≤ A + B := empiricalRawValue_abs_error_le o P
have hsq : (T - observedValue P) ^ 2 ≤ (A + B) ^ 2 := by
simpa only [← pow_two, sq_abs] using
mul_self_le_mul_self (abs_nonneg (T - observedValue P)) habs
have hp := observedValue_mem_unitInterval P
calc
(empiricalValue o - observedValue P) ^ 2 ≤ (T - observedValue P) ^ 2 :=
projectUnit_sq_error_le hp
_ ≤ 2 * A ^ 2 + 2 * B ^ 2 := by nlinarith [sq_nonneg (A - B)]
theorem empiricalMassError_sq_le
Cauchy–Schwarz bounds the squared total mass error by the sum of cellwise errors. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
Proof (Lean source)
lemma empiricalMassError_sq_le {n d : ℕ} (o : Fin n → Obs d)
(P : DiscreteLaw d) :
(∑ x : Fin d, |(sampleCellCount o x : ℝ) / n - cellMass P x|) ^ 2 ≤
(d : ℝ) * ∑ x : Fin d, ((sampleCellCount o x : ℝ) / n - cellMass P x) ^ 2 := by
have h := Finset.sum_mul_sq_le_sq_mul_sq (Finset.univ : Finset (Fin d))
(fun _ => (1 : ℝ)) (fun x => |(sampleCellCount o x : ℝ) / n - cellMass P x|)
simpa only [one_mul, one_pow, Finset.sum_const, Finset.card_univ,
Fintype.card_fin, nsmul_eq_mul, mul_one, sq_abs] using h
theorem cellMass_weighted_armMax_sq_le
Probability-weighted Cauchy–Schwarz controls the arm error without dividing by cell masses. In particular, null cells contribute zero. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
Proof (Lean source)
lemma cellMass_weighted_armMax_sq_le {d : ℕ} (P : DiscreteLaw d)
(e : Bool → Fin d → ℝ) :
(∑ x : Fin d, cellMass P x * max |e false x| |e true x|) ^ 2 ≤
∑ x : Fin d, cellMass P x * ∑ a : Bool, (e a x) ^ 2 := by
have hp (x : Fin d) : 0 ≤ cellMass P x :=
Finset.sum_nonneg fun a _ => Finset.sum_nonneg fun y _ => toReal_nonneg
have h := weighted_inner_sq_le univ
(cellMass P) (fun _ => 1) (fun x => max |e false x| |e true x|)
(fun x _ => hp x)
simp only [one_mul, one_pow, mul_one, cellMass_sum_eq_one] at h
apply h.trans
apply Finset.sum_le_sum
intro x hx
apply mul_le_mul_of_nonneg_left _ (hp x)
simp only [Fintype.sum_bool]
rcases le_total |e false x| |e true x| with he | he
· rw [max_eq_right he, sq_abs]
nlinarith [sq_nonneg (e false x)]
· rw [max_eq_left he, sq_abs]
nlinarith [sq_nonneg (e true x)]
theorem empiricalValue_sq_error_le_cellwise
The projected empirical statistic is controlled by individual mass and arm squared errors, as required for the expectation bounds in equation (6). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
(empiricalValue o - observedValue P) ^ 2
≤ 2 * ((d : ℝ) * ∑ x : Fin d, ((sampleCellCount o x : ℝ) / n - cellMass P x) ^ 2)
+ 2 * ∑ x : Fin d, cellMass P x * ∑ a : Bool, ((sampleSuccessCount o x a : ℝ) / sampleArmCount o x a - outcomeMean P a x) ^ 2
Proof (Lean source)
lemma empiricalValue_sq_error_le_cellwise {n d : ℕ} (o : Fin n → Obs d)
(P : DiscreteLaw d) :
(empiricalValue o - observedValue P) ^ 2 ≤
2 * ((d : ℝ) * ∑ x : Fin d,
((sampleCellCount o x : ℝ) / n - cellMass P x) ^ 2) +
2 * ∑ x : Fin d, cellMass P x * ∑ a : Bool,
((sampleSuccessCount o x a : ℝ) / sampleArmCount o x a - outcomeMean P a x) ^ 2 := by
apply (empiricalValue_sq_error_le_components o P).trans
apply add_le_add
· exact mul_le_mul_of_nonneg_left (empiricalMassError_sq_le o P) (by norm_num)
· exact mul_le_mul_of_nonneg_left
(cellMass_weighted_armMax_sq_le P (fun a x =>
(sampleSuccessCount o x a : ℝ) / sampleArmCount o x a - outcomeMean P a x))
(by norm_num)
theorem empiricalValue_sqRisk_le_cellwise
Integrating the cellwise inequality reduces empirical-ratio risk to marginal second moments. All integrability conditions are automatic on the finite observed sample space. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
sqRisk μ empiricalValue (observedValue P)
≤ 2 * ((d : ℝ) * ∑ x : Fin d, ∫ o, ((sampleCellCount o x : ℝ) / n - cellMass P x) ^ 2 ∂μ)
+ 2 * ∑ x : Fin d, cellMass P x * ∑ a : Bool, ∫ o, ((sampleSuccessCount o x a : ℝ) / sampleArmCount o x a - outcomeMean P a x) ^ 2 ∂μ
Proof (Lean source)
lemma empiricalValue_sqRisk_le_cellwise {n d : ℕ} (P : DiscreteLaw d)
(μ : Measure (Fin n → Obs d)) [IsFiniteMeasure μ] :
sqRisk μ empiricalValue (observedValue P) ≤
2 * ((d : ℝ) * ∑ x : Fin d, ∫ o,
((sampleCellCount o x : ℝ) / n - cellMass P x) ^ 2 ∂μ) +
2 * ∑ x : Fin d, cellMass P x * ∑ a : Bool, ∫ o,
((sampleSuccessCount o x a : ℝ) / sampleArmCount o x a - outcomeMean P a x) ^ 2 ∂μ := by
unfold sqRisk
calc
_ ≤ ∫ o, (2 * ((d : ℝ) * ∑ x : Fin d,
((sampleCellCount o x : ℝ) / n - cellMass P x) ^ 2) +
2 * ∑ x : Fin d, cellMass P x * ∑ a : Bool,
((sampleSuccessCount o x a : ℝ) / sampleArmCount o x a - outcomeMean P a x) ^ 2) ∂μ :=
integral_mono Integrable.of_finite Integrable.of_finite
(fun o => empiricalValue_sq_error_le_cellwise o P)
_ = _ := by
rw [integral_add Integrable.of_finite Integrable.of_finite]
simp only [integral_const_mul]
congr 1
· congr 1
congr 1
exact integral_finsetSum _ (fun _ _ => Integrable.of_finite)
· congr 1
rw [integral_finsetSum _ (fun _ _ => Integrable.of_finite)]
apply Finset.sum_congr rfl
intro x hx
rw [integral_const_mul, integral_finsetSum _ (fun _ _ => Integrable.of_finite)]
theorem integral_cellIndicator_eq_cellMass
The one-observation indicator of a covariate cell has mean equal to its cell mass. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
(∫ z : Obs d, (if z.1 = x then (1 : ℝ) else 0) ∂P.pmf.toMeasure) = cellMass P x
Proof (Lean source)
lemma integral_cellIndicator_eq_cellMass {d : ℕ} (P : DiscreteLaw d) (x : Fin d) :
(∫ z : Obs d, (if z.1 = x then (1 : ℝ) else 0) ∂P.pmf.toMeasure) =
cellMass P x := by
rw [PMF.integral_eq_sum]
simp [Fintype.sum_prod_type, cellMass, jointMass, smul_eq_mul, Finset.sum_add_distrib]
theorem empiricalCellMass_mse_le
Under iid sampling, each empirical cell mass has mean squared error at most its mass over n. The bound also holds for null cells, without a positivity assumption on their mass. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
(∫ o, ((sampleCellCount o x : ℝ) / n - cellMass P x) ^ 2 ∂productLaw P n)
≤ cellMass P x / n
Proof (Lean source)
lemma empiricalCellMass_mse_le {n d : ℕ} (hn : 0 < n)
(P : DiscreteLaw d) (x : Fin d) :
(∫ o, ((sampleCellCount o x : ℝ) / n - cellMass P x) ^ 2 ∂productLaw P n) ≤
cellMass P x / n := by
let f : Obs d → ℝ := fun z => if z.1 = x then 1 else 0
have hf : MemLp f 2 P.pmf.toMeasure := by
apply (memLp_two_iff_integrable_sq ?_).2 Integrable.of_finite
fun_prop
have hm : (∫ z, f z ∂P.pmf.toMeasure) = cellMass P x :=
integral_cellIndicator_eq_cellMass P x
have hs : (∫ z, (f z) ^ 2 ∂P.pmf.toMeasure) = cellMass P x := by
have heq : (fun z => (f z) ^ 2) = f := by
funext z
simp only [f]
split_ifs <;> norm_num
rw [heq, hm]
have hc (o : Fin n → Obs d) : (sampleCellCount o x : ℝ) = ∑ i : Fin n, f (o i) := by
simp [sampleCellCount, f]
have h := iid_mean_sq_le P.pmf.toMeasure hn f hf
rw [hm, hs] at h
simpa only [productLaw, hc, div_eq_mul_inv, mul_comm] using h
theorem empiricalMassError_mse_le
Equation (6)'s mass-error term is bounded by d/n under the actual iid observed law. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
(∫ o, (∑ x : Fin d, |(sampleCellCount o x : ℝ) / n - cellMass P x|) ^ 2 ∂productLaw P n)
≤ (d : ℝ) / n
Proof (Lean source)
lemma empiricalMassError_mse_le {n d : ℕ} (hn : 0 < n) (P : DiscreteLaw d) :
(∫ o, (∑ x : Fin d, |(sampleCellCount o x : ℝ) / n - cellMass P x|) ^ 2
∂productLaw P n) ≤ (d : ℝ) / n := by
let : IsProbabilityMeasure (productLaw P n) := by
dsimp [productLaw]
infer_instance
calc
_ ≤ ∫ o, (d : ℝ) * ∑ x : Fin d,
((sampleCellCount o x : ℝ) / n - cellMass P x) ^ 2 ∂productLaw P n :=
integral_mono Integrable.of_finite Integrable.of_finite
(fun o => empiricalMassError_sq_le o P)
_ = (d : ℝ) * ∑ x : Fin d, ∫ o,
((sampleCellCount o x : ℝ) / n - cellMass P x) ^ 2 ∂productLaw P n := by
rw [integral_const_mul, integral_finsetSum _ (fun _ _ => Integrable.of_finite)]
_ ≤ (d : ℝ) * ∑ x : Fin d, cellMass P x / n := by
gcongr with x
exact empiricalCellMass_mse_le hn P x
_ = (d : ℝ) / n := by rw [← Finset.sum_div, cellMass_sum_eq_one]; ring
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.UpperFactorial 15 declarations Roadmap equations (23)–(26) connect independent Poisson factorial moments to actual pilot rectangles and the centered Jackson factorial statistic.
Localized factorial moments
Roadmap equations (23)–(26) connect independent Poisson factorial moments to actual pilot rectangles and the centered Jackson factorial statistic.
Positive pilot level gives strictly positive coordinate radii, even at zero counts. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
0 < pilotRadiusFormula H hH m L Np j
Proof (Lean source)
lemma pilotRadius_pos (H : ℝ) (hH : 0 < H) (m L : ℝ)
(hm : 0 < m) (hL : 0 < L) (Np : Fin 4 → ℕ) (j : Fin 4) :
0 < pilotRadiusFormula H hH m L Np j := by
have hc : 0 ≤ pilotCenterFormula m Np j := by unfold pilotCenterFormula; positivity
have hh : 0 < pilotHalfWidthFormula H hH m L Np j := by
unfold pilotHalfWidthFormula
exact mul_pos hH (add_pos_of_nonneg_of_pos (Real.sqrt_nonneg _) (div_pos hL hm))
dsimp [pilotRadiusFormula, pilotLower, pilotUpperFormula]
rw [max_def]
split_ifs <;> linarith
Membership in the pilot interval is precisely the centering condition for the normalized factorial envelope, including a lower endpoint truncated at zero. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
|q j - pilotMidpoint H hH m L Np j| ≤ pilotRadiusFormula H hH m L Np j
Proof (Lean source)
lemma pilotRectangle_center_bound (H : ℝ) (hH : 0 < H) (m L : ℝ)
(Np : Fin 4 → ℕ) (q : Fin 4 → ℝ)
(hq : q ∈ pilotRectangle (pilotLower H hH m L Np) (pilotUpperFormula H hH m L Np))
(j : Fin 4) :
|q j - pilotMidpoint H hH m L Np j| ≤ pilotRadiusFormula H hH m L Np j := by
have hj := hq j
apply abs_le.mpr
dsimp [pilotMidpoint, pilotRadiusFormula]
constructor <;> linarith [hj.1, hj.2]
Equation (23) specializes the exact mixed factorial identity to intensity m q. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
m q b :
ℝ
hm :
0 < m
hq :
0 ≤ q
h t :
ℕ
conclusion
(∫ N : ℕ, factorialLift m b N h * factorialLift m b N t ∂poissonMeasure ⟨m * q, mul_nonneg hm.le hq⟩)
Proof (Lean source)
lemma cellFactorial_mixed_moment (m q b : ℝ) (hm : 0 < m) (hq : 0 ≤ q)
(h t : ℕ) :
(∫ N : ℕ, factorialLift m b N h * factorialLift m b N t
∂poissonMeasure ⟨m * q, mul_nonneg hm.le hq⟩) =
∑ l ∈ range (min h t + 1),
(h.choose l : ℝ) * (t.choose l : ℝ) * (Nat.factorial l : ℝ) *
(q / m) ^ l * (q - b) ^ (h + t - 2 * l) := by
have hmean : m * q / m = q := mul_div_cancel_left₀ q (ne_of_gt hm)
have hvar : m * q / m ^ 2 = q / m := by field_simp
have hx := poisson_factorialLift_mixed ⟨m * q, mul_nonneg hm.le hq⟩ m b (ne_of_gt hm) h t
change (∫ N : ℕ, factorialLift m b N h * factorialLift m b N t
∂poissonMeasure ⟨m * q, mul_nonneg hm.le hq⟩) =
∑ l ∈ range (min h t + 1),
(h.choose l : ℝ) * (t.choose l : ℝ) * (Nat.factorial l : ℝ) *
(m * q / m ^ 2) ^ l * (m * q / m - b) ^ (h + t - 2 * l) at hx
simpa only [hmean, hvar] using hx
theorem pilotFactorialProduct_square_envelope Lemma pilotFactorialProduct_square_envelope in the paper ↗
Equations (24)–(25) give the four-coordinate exponential envelope on the actual pilot rectangle. The factor eight comes from the pilot variance bound. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
W :
Fin 4 → Ω → ℕ
q :
Fin 4 → ℝ
hqnonneg :
∀ j, 0 ≤ q j
H :
ℝ
hH :
0 < H
hHlarge :
1 ≤ H
m L :
ℝ
hm :
0 < m
hL :
0 < L
Np :
Fin 4 → ℕ
hWlaw :
hind :
iIndepFun W μ
hq :
h :
Fin 4 → ℕ
D :
ℕ
hdegree :
∀ j, h j ≤ D
conclusion
(∫ ω, (∏ j, factorialLift m (pilotMidpoint H hH m L Np j) (W j ω) (h j) / pilotRadiusFormula H hH m L Np j ^ h j) ^ 2 ∂μ)
≤ exp (32 * (D : ℝ) ^ 2 / L)
Proof (Lean source)
lemma pilotFactorialProduct_square_envelope {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (W : Fin 4 → Ω → ℕ)
(q : Fin 4 → ℝ) (hqnonneg : ∀ j, 0 ≤ q j)
(H : ℝ) (hH : 0 < H) (hHlarge : 1 ≤ H) (m L : ℝ)
(hm : 0 < m) (hL : 0 < L) (Np : Fin 4 → ℕ)
(hWlaw : ∀ j, HasLaw (W j) (poissonMeasure ⟨m * q j, mul_nonneg hm.le (hqnonneg j)⟩) μ)
(hind : iIndepFun W μ)
(hq : q ∈ pilotRectangle (pilotLower H hH m L Np) (pilotUpperFormula H hH m L Np))
(h : Fin 4 → ℕ) (D : ℕ) (hdegree : ∀ j, h j ≤ D) :
(∫ ω, (∏ j, factorialLift m (pilotMidpoint H hH m L Np j) (W j ω) (h j) /
pilotRadiusFormula H hH m L Np j ^ h j) ^ 2 ∂μ) ≤
exp (32 * (D : ℝ) ^ 2 / L) := by
have hc (j : Fin 4) :
|(m * q j) / m - pilotMidpoint H hH m L Np j| ≤ pilotRadiusFormula H hH m L Np j := by
rw [mul_div_cancel_left₀ _ (ne_of_gt hm)]
exact pilotRectangle_center_bound H hH m L Np q hq j
have hv (j : Fin 4) :
(m * q j) / (m ^ 2 * pilotRadiusFormula H hH m L Np j ^ 2) ≤ 1 / (L / 8) := by
have he : (m * q j) / (m ^ 2 * pilotRadiusFormula H hH m L Np j ^ 2) =
q j / (m * pilotRadiusFormula H hH m L Np j ^ 2) := by field_simp
rw [he, show (1 : ℝ) / (L / 8) = 8 / L by ring]
exact pilotRadius_variance_bound H hH hHlarge m L hm hL Np j (q j) (hq j).2
have hb := factorialProduct_four_square_envelope μ W
(fun j => ⟨m * q j, mul_nonneg hm.le (hqnonneg j)⟩) hWlaw hind m (L / 8)
hm (by positivity) (pilotMidpoint H hH m L Np) (pilotRadiusFormula H hH m L Np)
(pilotRadius_pos H hH m L hm hL Np) hc hv h D hdegree
simpa only [factorialProduct, Finset.prod_div_distrib,
show 4 * (D : ℝ) ^ 2 / (L / 8) = 32 * (D : ℝ) ^ 2 / L by ring] using hb
Every scalar factorial lift has an integrable square under a Poisson law. This regularity is derived from raw factorial moments, rather than assumed. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma cellFactorial_square_integrable (rate : NNReal) (m b : ℝ) (h : ℕ) :
Integrable (fun N : ℕ => (factorialLift m b N h) ^ 2) (poissonMeasure rate) := by
have hraw (j k : ℕ) : Integrable (fun N : ℕ =>
(N.descFactorial j : ℝ) * (N.descFactorial k : ℝ)) (poissonMeasure rate) := by
have hi := integrable_finsetSum (range (min j k + 1))
(fun l _ => (poisson_descFactorial_integrable rate (j + k - l)).const_mul
((j.choose l : ℝ) * (k.choose l : ℝ) * (Nat.factorial l : ℝ)))
have he : (fun N : ℕ => (N.descFactorial j : ℝ) * (N.descFactorial k : ℝ)) =
(fun N : ℕ => ∑ l ∈ range (min j k + 1),
(j.choose l : ℝ) * (k.choose l : ℝ) * (Nat.factorial l : ℝ) *
(N.descFactorial (j + k - l) : ℝ)) := by
funext N
exact descFactorial_mul N j k
rw [he]
exact hi
have hi := integrable_finsetSum (range (h + 1)) (fun j _ =>
integrable_finsetSum (range (h + 1)) (fun k _ =>
(hraw j k).const_mul (((h.choose j : ℝ) * (-b) ^ (h - j) / m ^ j) *
((h.choose k : ℝ) * (-b) ^ (h - k) / m ^ k))))
have he : (fun N : ℕ => (factorialLift m b N h) ^ 2) =
(fun N : ℕ => ∑ j ∈ range (h + 1), ∑ k ∈ range (h + 1),
(((h.choose j : ℝ) * (-b) ^ (h - j) / m ^ j) *
((h.choose k : ℝ) * (-b) ^ (h - k) / m ^ k)) *
((N.descFactorial j : ℝ) * (N.descFactorial k : ℝ))) := by
funext N
simp only [factorialLift, sq, Finset.sum_mul_sum]
apply Finset.sum_congr rfl
intro j _
apply Finset.sum_congr rfl
intro k _
ring
rw [he]
exact hi
Independent Poisson coordinates make every normalized factorial monomial L². No integrability premise is needed for the subsequent polynomial assembly. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
W :
Fin 4 → Ω → ℕ
hWlaw :
∀ j, HasLaw (W j) (poissonMeasure (rate j)) μ
hind :
iIndepFun W μ
m :
ℝ
b r :
Fin 4 → ℝ
h :
Fin 4 → ℕ
conclusion
MemLp (fun ω => ∏ j, factorialLift m (b j) (W j ω) (h j) / r j ^ h j) 2 μ
Proof (Lean source)
lemma cellFactorialProduct_memLp_two {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (W : Fin 4 → Ω → ℕ)
(rate : Fin 4 → NNReal)
(hWlaw : ∀ j, HasLaw (W j) (poissonMeasure (rate j)) μ)
(hind : iIndepFun W μ) (m : ℝ) (b r : Fin 4 → ℝ) (h : Fin 4 → ℕ) :
MemLp (fun ω => ∏ j, factorialLift m (b j) (W j ω) (h j) / r j ^ h j) 2 μ := by
let f (j : Fin 4) (N : ℕ) : ℝ := (factorialLift m (b j) N (h j) / r j ^ h j) ^ 2
have hf (j : Fin 4) : Integrable (f j) (poissonMeasure (rate j)) := by
simpa only [f, div_pow] using
(cellFactorial_square_integrable (rate j) m (b j) (h j)).div_const ((r j ^ h j) ^ 2)
have hpi := Integrable.fintype_prod hf
have hmap : μ.map (fun ω j => W j ω) = Measure.pi (fun j => poissonMeasure (rate j)) := by
rw [hind.map_fun_eq_pi_map (fun j => (hWlaw j).aemeasurable)]
congr 1
funext j
exact (hWlaw j).map_eq
have hmeas : AEMeasurable (fun ω j => W j ω) μ :=
aemeasurable_pi_lambda _ (fun j => (hWlaw j).aemeasurable)
have hsq : Integrable (fun ω => (∏ j, factorialLift m (b j) (W j ω) (h j) /
r j ^ h j) ^ 2) μ := by
have hcomp := (hmap ▸ hpi).comp_aemeasurable hmeas
convert hcomp using 1
funext ω
simp only [Function.comp_apply, f, Finset.prod_pow]
have ha : AEStronglyMeasurable (fun ω => ∏ j,
factorialLift m (b j) (W j ω) (h j) / r j ^ h j) μ := by
exact ((Measurable.of_discrete : Measurable (fun v : Fin 4 → ℕ =>
∏ j, factorialLift m (b j) (v j) (h j) / r j ^ h j)).comp_aemeasurable
hmeas).aestronglyMeasurable
exact (memLp_two_iff_integrable_sq ha).2 hsq
The actual Jackson factorial statistic is square-integrable under the independent Poisson experiment. This derives all regularity needed for clipping. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
W :
Fin 4 → Ω → ℕ
hWlaw :
∀ j, HasLaw (W j) (poissonMeasure (rate j)) μ
hind :
iIndepFun W μ
ε :
ℝ
K :
ℕ
m :
ℝ
b r :
Fin 4 → ℝ
conclusion
MemLp (fun ω => factorialCellValue ε K m b r (fun j => W j ω)) 2 μ
Proof (Lean source)
lemma factorialCellValue_memLp_two {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (W : Fin 4 → Ω → ℕ)
(rate : Fin 4 → NNReal)
(hWlaw : ∀ j, HasLaw (W j) (poissonMeasure (rate j)) μ)
(hind : iIndepFun W μ) (ε : ℝ) (K : ℕ) (m : ℝ) (b r : Fin 4 → ℝ) :
MemLp (fun ω => factorialCellValue ε K m b r (fun j => W j ω)) 2 μ := by
have hmono (α : Fin 4 → Fin (2 * (K - 1) + 1)) :=
(cellFactorialProduct_memLp_two μ W rate hWlaw hind m b r
(fun j => (α j).val)).const_mul (jacksonCoeff ε K b r α)
exact (memLp_const (armwiseExtensionFormula ε b)).add
(memLp_finsetSum univ (fun α _ => hmono α))
theorem factorialCellValue_centered_square_integrable Lemma factorialCellValue_centered_square_integrable in the paper ↗
Centering the factorial cell statistic at any fixed value preserves L², so its second moments exist without extra integrability assumptions. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
W :
Fin 4 → Ω → ℕ
hWlaw :
∀ j, HasLaw (W j) (poissonMeasure (rate j)) μ
hind :
iIndepFun W μ
ε :
ℝ
K :
ℕ
m :
ℝ
b r :
Fin 4 → ℝ
p :
ℝ
conclusion
Integrable (fun ω => (factorialCellValue ε K m b r (fun j => W j ω) - p) ^ 2) μ
Proof (Lean source)
lemma factorialCellValue_centered_square_integrable {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (W : Fin 4 → Ω → ℕ)
(rate : Fin 4 → NNReal)
(hWlaw : ∀ j, HasLaw (W j) (poissonMeasure (rate j)) μ)
(hind : iIndepFun W μ) (ε : ℝ) (K : ℕ) (m : ℝ) (b r : Fin 4 → ℝ) (p : ℝ) :
Integrable (fun ω =>
(factorialCellValue ε K m b r (fun j => W j ω) - p) ^ 2) μ := by
exact ((factorialCellValue_memLp_two μ W rate hWlaw hind ε K m b r).sub
(memLp_const p)).integrable_sq
The actual factorial cell statistic is unbiased for evaluation of its centered Jackson monomial expansion at the Poisson mean (roadmap equation (23)). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
W :
Fin 4 → Ω → ℕ
hWlaw :
∀ j, HasLaw (W j) (poissonMeasure (rate j)) μ
hind :
iIndepFun W μ
ε :
ℝ
K :
ℕ
m :
ℝ
hm :
m ≠ 0
b r :
Fin 4 → ℝ
conclusion
(∫ ω, factorialCellValue ε K m b r (fun j => W j ω) ∂μ)
= armwiseExtensionFormula ε b
+ ∑ α : Fin 4 → Fin (2 * (K - 1) + 1), jacksonCoeff ε K b r α * ∏ j, (((rate j : ℝ) / m - b j) / r j) ^ (α j).val
Proof (Lean source)
lemma factorialCellValue_integral_eq {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (W : Fin 4 → Ω → ℕ)
(rate : Fin 4 → NNReal)
(hWlaw : ∀ j, HasLaw (W j) (poissonMeasure (rate j)) μ)
(hind : iIndepFun W μ) (ε : ℝ) (K : ℕ) (m : ℝ) (hm : m ≠ 0)
(b r : Fin 4 → ℝ) :
(∫ ω, factorialCellValue ε K m b r (fun j => W j ω) ∂μ) =
armwiseExtensionFormula ε b + ∑ α : Fin 4 → Fin (2 * (K - 1) + 1),
jacksonCoeff ε K b r α * ∏ j, (((rate j : ℝ) / m - b j) / r j) ^ (α j).val := by
have hX (α : Fin 4 → Fin (2 * (K - 1) + 1)) :=
(cellFactorialProduct_memLp_two μ W rate hWlaw hind m b r (fun j => (α j).val)).integrable
(by norm_num : (1 : ENNReal) ≤ 2)
simp only [factorialCellValue]
rw [integral_add (integrable_const _) (integrable_finsetSum _ (fun α _ =>
(hX α).const_mul _)), integral_const, probReal_univ, one_smul,
integral_finsetSum _ (fun α _ => (hX α).const_mul _)]
congr 1
apply Finset.sum_congr rfl
intro α _
rw [integral_const_mul]
congr 1
simp only [Finset.prod_div_distrib]
rw [integral_div]
change (∫ ω, factorialProduct W m b (fun j => (α j).val) ω ∂μ) / _ = _
rw [factorialProduct_mean μ W rate hWlaw hind m]
simp only [Finset.prod_div_distrib, div_pow]
Clipping around a fixed center contracts squared error at the unclipped mean when that mean lies in the clipping interval. Its risk is bounded by the original second moment around the fixed center, as in roadmap equation (28). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Proof (Lean source)
lemma centeredClip_integral_sq_mean_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (Z : Ω → ℝ) (c A : ℝ)
(hA : 0 ≤ A) (hZ : MemLp Z 2 μ)
(hmean : |(∫ ω, Z ω ∂μ) - c| ≤ A) :
(∫ ω, (max (c - A) (min (c + A) (Z ω)) - ∫ v, Z v ∂μ) ^ 2 ∂μ) ≤
∫ ω, (Z ω - c) ^ 2 ∂μ := by
have hpoint (ω : Ω) :
(max (c - A) (min (c + A) (Z ω)) - ∫ v, Z v ∂μ) ^ 2 ≤
(Z ω - ∫ v, Z v ∂μ) ^ 2 := by
rw [centeredClip_eq]
have h := clip_sq_error_le A (Z ω - c) ((∫ v, Z v ∂μ) - c)
hA (abs_le.mp hmean)
convert h using 1 <;> congr 1 <;> ring
calc
_ ≤ ∫ ω, (Z ω - ∫ v, Z v ∂μ) ^ 2 ∂μ :=
integral_mono_of_nonneg (ae_of_all _ fun _ => sq_nonneg _)
((hZ.sub (memLp_const _)).integrable_sq) (ae_of_all _ hpoint)
_ = variance Z μ :=
(ProbabilityTheory.variance_eq_integral hZ.aemeasurable).symm
_ = variance (fun ω => Z ω - c) μ :=
(ProbabilityTheory.variance_sub_const hZ.aestronglyMeasurable c).symm
_ ≤ _ := ProbabilityTheory.variance_le_expectation_sq
((hZ.sub (memLp_const c)).aestronglyMeasurable)
theorem factorialCellValue_integral_eq_scaled Lemma factorialCellValue_integral_eq_scaled in the paper ↗
At intensities m times q, the factorial statistic has exactly the centered Jackson polynomial as its expectation, with the intensity normalization canceled. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
W :
Fin 4 → Ω → ℕ
q :
Fin 4 → ℝ
hqnonneg :
∀ j, 0 ≤ q j
m :
ℝ
hm :
0 < m
hWlaw :
hind :
iIndepFun W μ
ε :
ℝ
K :
ℕ
b r :
Fin 4 → ℝ
conclusion
(∫ ω, factorialCellValue ε K m b r (fun j => W j ω) ∂μ)
= armwiseExtensionFormula ε b
+ ∑ α : Fin 4 → Fin (2 * (K - 1) + 1), jacksonCoeff ε K b r α * ∏ j, ((q j - b j) / r j) ^ (α j).val
Proof (Lean source)
lemma factorialCellValue_integral_eq_scaled {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (W : Fin 4 → Ω → ℕ)
(q : Fin 4 → ℝ) (hqnonneg : ∀ j, 0 ≤ q j) (m : ℝ) (hm : 0 < m)
(hWlaw : ∀ j, HasLaw (W j)
(poissonMeasure ⟨m * q j, mul_nonneg hm.le (hqnonneg j)⟩) μ)
(hind : iIndepFun W μ) (ε : ℝ) (K : ℕ) (b r : Fin 4 → ℝ) :
(∫ ω, factorialCellValue ε K m b r (fun j => W j ω) ∂μ) =
armwiseExtensionFormula ε b + ∑ α : Fin 4 → Fin (2 * (K - 1) + 1),
jacksonCoeff ε K b r α * ∏ j, ((q j - b j) / r j) ^ (α j).val := by
rw [factorialCellValue_integral_eq μ W _ hWlaw hind ε K m (ne_of_gt hm)]
congr 1
apply Finset.sum_congr rfl
intro α _
congr 1
apply Finset.prod_congr rfl
intro j _
change ((m * q j / m - b j) / r j) ^ (α j).val = _
rw [mul_div_cancel_left₀ _ (ne_of_gt hm)]
theorem factorialPolynomial_square_envelope Lemma factorialPolynomial_square_envelope in the paper ↗
Summing normalized monomials costs the square of the coefficient ℓ¹ norm, rather than an extra alphabet factor. This is the finite polynomial step of (26). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma factorialPolynomial_square_envelope {Ω ι : Type*} [MeasurableSpace Ω] [Fintype ι]
(μ : Measure Ω) (β : ι → ℝ) (X : ι → Ω → ℝ) (B : ℝ)
(hX : ∀ i, MemLp (X i) 2 μ)
(hmom : ∀ i, (∫ ω, (X i ω) ^ 2 ∂μ) ≤ B) :
(∫ ω, (∑ i, β i * X i ω) ^ 2 ∂μ) ≤ (∑ i, |β i|) ^ 2 * B := by
have hpoint (ω : Ω) : (∑ i, β i * X i ω) ^ 2 ≤
(∑ i, |β i|) * ∑ i, |β i| * (X i ω) ^ 2 := by
apply Finset.sum_sq_le_sum_mul_sum_of_sq_le_mul univ
(fun i _ => abs_nonneg (β i)) (fun i _ => mul_nonneg (abs_nonneg _) (sq_nonneg _))
intro i _
apply le_of_eq
rw [mul_pow, ← sq_abs (β i)]
ring
have hs : Integrable (fun ω => (∑ i, |β i|) * ∑ i, |β i| * (X i ω) ^ 2) μ :=
(integrable_finsetSum _ (fun i _ => (hX i).integrable_sq.const_mul _)).const_mul _
calc
_ ≤ ∫ ω, (∑ i, |β i|) * ∑ i, |β i| * (X i ω) ^ 2 ∂μ :=
integral_mono_of_nonneg (ae_of_all _ fun _ => sq_nonneg _) hs (ae_of_all _ hpoint)
_ = (∑ i, |β i|) * ∑ i, |β i| * (∫ ω, (X i ω) ^ 2 ∂μ) := by
rw [integral_const_mul, integral_finsetSum _ (fun i _ => (hX i).integrable_sq.const_mul _)]
simp only [integral_const_mul]
_ ≤ (∑ i, |β i|) * ∑ i, |β i| * B := by
apply mul_le_mul_of_nonneg_left _ (Finset.sum_nonneg fun i _ => abs_nonneg _)
exact Finset.sum_le_sum fun i _ => mul_le_mul_of_nonneg_left (hmom i) (abs_nonneg _)
_ = _ := by rw [← Finset.sum_mul]; ring
theorem pilotFactorialCellValue_square_envelope Lemma pilotFactorialCellValue_square_envelope in the paper ↗
On the localized pilot rectangle, equation (26) holds for the estimator's actual centered factorial cell statistic, with its explicit coefficient norm. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
W :
Fin 4 → Ω → ℕ
q :
Fin 4 → ℝ
hqnonneg :
∀ j, 0 ≤ q j
H :
ℝ
hH :
0 < H
hHlarge :
1 ≤ H
m L :
ℝ
hm :
0 < m
hL :
0 < L
Np :
Fin 4 → ℕ
hWlaw :
hind :
iIndepFun W μ
hq :
ε :
ℝ
K :
ℕ
conclusion
b :
pilotMidpoint H hH m L Np
r :
pilotRadiusFormula H hH m L Np
(∫ ω, (factorialCellValue ε K m b r (fun j => W j ω) - armwiseExtensionFormula ε b) ^ 2 ∂μ)
≤ (∑ α : Fin 4 → Fin (2 * (K - 1) + 1), |jacksonCoeff ε K b r α|) ^ 2 * exp (32 * (2 * (K - 1) : ℕ) ^ 2 / L)
Proof (Lean source)
lemma pilotFactorialCellValue_square_envelope {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (W : Fin 4 → Ω → ℕ)
(q : Fin 4 → ℝ) (hqnonneg : ∀ j, 0 ≤ q j)
(H : ℝ) (hH : 0 < H) (hHlarge : 1 ≤ H) (m L : ℝ)
(hm : 0 < m) (hL : 0 < L) (Np : Fin 4 → ℕ)
(hWlaw : ∀ j, HasLaw (W j) (poissonMeasure ⟨m * q j, mul_nonneg hm.le (hqnonneg j)⟩) μ)
(hind : iIndepFun W μ)
(hq : q ∈ pilotRectangle (pilotLower H hH m L Np) (pilotUpperFormula H hH m L Np))
(ε : ℝ) (K : ℕ) :
let b := pilotMidpoint H hH m L Np
let r := pilotRadiusFormula H hH m L Np
(∫ ω, (factorialCellValue ε K m b r (fun j => W j ω) - armwiseExtensionFormula ε b) ^ 2 ∂μ) ≤
(∑ α : Fin 4 → Fin (2 * (K - 1) + 1), |jacksonCoeff ε K b r α|) ^ 2 *
exp (32 * (2 * (K - 1) : ℕ) ^ 2 / L) := by
dsimp only
simp only [factorialCellValue, add_sub_cancel_left]
apply factorialPolynomial_square_envelope
· intro α
exact cellFactorialProduct_memLp_two μ W _ hWlaw hind m
(pilotMidpoint H hH m L Np) (pilotRadiusFormula H hH m L Np) (fun j => (α j).val)
· intro α
exact pilotFactorialProduct_square_envelope μ W q hqnonneg H hH hHlarge m L hm hL
Np hWlaw hind hq (fun j => (α j).val) (2 * (K - 1))
(fun j => le_of_lt_succ (α j).isLt)
theorem pilotClippedCellValue_bias_envelope Lemma pilotClippedCellValue_bias_envelope in the paper ↗
Combining the exact Poisson mean, derived L² regularity, the localized factorial envelope, and clipping displacement proves the bias transfer in (28) for the actual statistic. The coefficient norm remains explicit for (14). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
W :
Fin 4 → Ω → ℕ
q :
Fin 4 → ℝ
hqnonneg :
∀ j, 0 ≤ q j
H :
ℝ
hH :
0 < H
hHlarge :
1 ≤ H
m L :
ℝ
hm :
0 < m
hL :
0 < L
Np :
Fin 4 → ℕ
hWlaw :
hind :
iIndepFun W μ
hq :
ε :
ℝ
K d :
ℕ
hwidth :
0 < (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε (pilotMidpoint H hH m L Np) (pilotRadiusFormula H hH m L Np)
conclusion
b :
pilotMidpoint H hH m L Np
r :
pilotRadiusFormula H hH m L Np
|(∫ ω, clippedCellValueFormula ε K d m b r (fun j => W j ω) ∂μ)
- (armwiseExtensionFormula ε b + ∑ α : Fin 4 → Fin (2 * (K - 1) + 1), jacksonCoeff ε K b r α * ∏ j, ((q j - b j) / r j) ^ (α j).val)|
≤ ((∑ α : Fin 4 → Fin (2 * (K - 1) + 1), |jacksonCoeff ε K b r α|) ^ 2 * exp (32 * (2 * (K - 1) : ℕ) ^ 2 / L)) / ((d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε b r)
Proof (Lean source)
lemma pilotClippedCellValue_bias_envelope {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (W : Fin 4 → Ω → ℕ)
(q : Fin 4 → ℝ) (hqnonneg : ∀ j, 0 ≤ q j)
(H : ℝ) (hH : 0 < H) (hHlarge : 1 ≤ H) (m L : ℝ)
(hm : 0 < m) (hL : 0 < L) (Np : Fin 4 → ℕ)
(hWlaw : ∀ j, HasLaw (W j) (poissonMeasure ⟨m * q j, mul_nonneg hm.le (hqnonneg j)⟩) μ)
(hind : iIndepFun W μ)
(hq : q ∈ pilotRectangle (pilotLower H hH m L Np) (pilotUpperFormula H hH m L Np))
(ε : ℝ) (K d : ℕ)
(hwidth : 0 < (d : ℝ) ^ (1 / 4 : ℝ) *
clippingScaleFormula ε (pilotMidpoint H hH m L Np) (pilotRadiusFormula H hH m L Np)) :
let b := pilotMidpoint H hH m L Np
let r := pilotRadiusFormula H hH m L Np
|(∫ ω, clippedCellValueFormula ε K d m b r (fun j => W j ω) ∂μ) -
(armwiseExtensionFormula ε b + ∑ α : Fin 4 → Fin (2 * (K - 1) + 1),
jacksonCoeff ε K b r α * ∏ j, ((q j - b j) / r j) ^ (α j).val)| ≤
((∑ α : Fin 4 → Fin (2 * (K - 1) + 1), |jacksonCoeff ε K b r α|) ^ 2 *
exp (32 * (2 * (K - 1) : ℕ) ^ 2 / L)) /
((d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε b r) := by
dsimp only
have hmean' := factorialCellValue_integral_eq_scaled μ W q hqnonneg m hm
hWlaw hind ε K (pilotMidpoint H hH m L Np) (pilotRadiusFormula H hH m L Np)
have hreg := factorialCellValue_memLp_two μ W _ hWlaw hind ε K m
(pilotMidpoint H hH m L Np) (pilotRadiusFormula H hH m L Np)
have hsecond := factorialCellValue_centered_square_integrable μ W _ hWlaw hind ε K m
(pilotMidpoint H hH m L Np) (pilotRadiusFormula H hH m L Np)
(armwiseExtensionFormula ε (pilotMidpoint H hH m L Np))
have hb := clippedCellValue_integral_bias_le μ (fun ω j => W j ω) ε K d m
(pilotMidpoint H hH m L Np) (pilotRadiusFormula H hH m L Np) _ hwidth
(hreg.integrable (by norm_num)) hmean' hsecond
exact hb.trans (div_le_div_of_nonneg_right
(pilotFactorialCellValue_square_envelope μ W q hqnonneg H hH hHlarge m L hm hL
Np hWlaw hind hq ε K) hwidth.le)
theorem pilotClippedCellValue_square_envelope Lemma pilotClippedCellValue_square_envelope in the paper ↗
On a localized pilot rectangle, clipping the actual Poisson factorial statistic preserves the second-moment envelope around its exact polynomial mean provided that mean lies in the clipping interval. The coefficient norm remains explicit; this is the contraction step of the second assertion of equation (28). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
W :
Fin 4 → Ω → ℕ
q :
Fin 4 → ℝ
hqnonneg :
∀ j, 0 ≤ q j
H :
ℝ
hH :
0 < H
hHlarge :
1 ≤ H
m L :
ℝ
hm :
0 < m
hL :
0 < L
Np :
Fin 4 → ℕ
hWlaw :
hind :
iIndepFun W μ
hq :
ε :
ℝ
K d :
ℕ
conclusion
b :
pilotMidpoint H hH m L Np
r :
pilotRadiusFormula H hH m L Np
p :
+ ∑ α : Fin 4 → Fin (2 * (K - 1) + 1), jacksonCoeff ε K b r α * ∏ j, ((q j - b j) / r j) ^ (α j).val
0 ≤ (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε b r
|p - armwiseExtensionFormula ε b| ≤ (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε b r
(∫ ω, (clippedCellValueFormula ε K d m b r (fun j => W j ω) - p) ^ 2 ∂μ)
≤ (∑ α : Fin 4 → Fin (2 * (K - 1) + 1), |jacksonCoeff ε K b r α|) ^ 2 * exp (32 * (2 * (K - 1) : ℕ) ^ 2 / L)
Proof (Lean source)
lemma pilotClippedCellValue_square_envelope {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (W : Fin 4 → Ω → ℕ)
(q : Fin 4 → ℝ) (hqnonneg : ∀ j, 0 ≤ q j)
(H : ℝ) (hH : 0 < H) (hHlarge : 1 ≤ H) (m L : ℝ)
(hm : 0 < m) (hL : 0 < L) (Np : Fin 4 → ℕ)
(hWlaw : ∀ j, HasLaw (W j)
(poissonMeasure ⟨m * q j, mul_nonneg hm.le (hqnonneg j)⟩) μ)
(hind : iIndepFun W μ)
(hq : q ∈ pilotRectangle (pilotLower H hH m L Np) (pilotUpperFormula H hH m L Np))
(ε : ℝ) (K d : ℕ) :
let b := pilotMidpoint H hH m L Np
let r := pilotRadiusFormula H hH m L Np
let p := armwiseExtensionFormula ε b + ∑ α : Fin 4 → Fin (2 * (K - 1) + 1),
jacksonCoeff ε K b r α * ∏ j, ((q j - b j) / r j) ^ (α j).val
0 ≤ (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε b r →
|p - armwiseExtensionFormula ε b| ≤
(d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε b r →
(∫ ω, (clippedCellValueFormula ε K d m b r (fun j => W j ω) - p) ^ 2 ∂μ) ≤
(∑ α : Fin 4 → Fin (2 * (K - 1) + 1), |jacksonCoeff ε K b r α|) ^ 2 *
exp (32 * (2 * (K - 1) : ℕ) ^ 2 / L) := by
dsimp only
intro hwidth hp
have hmean := factorialCellValue_integral_eq_scaled μ W q hqnonneg m hm
hWlaw hind ε K (pilotMidpoint H hH m L Np) (pilotRadiusFormula H hH m L Np)
have hreg := factorialCellValue_memLp_two μ W _ hWlaw hind ε K m
(pilotMidpoint H hH m L Np) (pilotRadiusFormula H hH m L Np)
have hclip := centeredClip_integral_sq_mean_le μ
(fun ω => factorialCellValue ε K m
(pilotMidpoint H hH m L Np) (pilotRadiusFormula H hH m L Np) (fun j => W j ω))
(armwiseExtensionFormula ε (pilotMidpoint H hH m L Np))
((d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε
(pilotMidpoint H hH m L Np) (pilotRadiusFormula H hH m L Np))
hwidth hreg (by simpa only [hmean] using hp)
simp only [hmean] at hclip
exact hclip.trans (pilotFactorialCellValue_square_envelope μ W q hqnonneg
H hH hHlarge m L hm hL Np hWlaw hind hq ε K)
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.UpperGoodBadAssembly 7 declarations Roadmap (29) splits each actual cell error into its disjoint good- and bad-pilot parts.
Good- and bad-pilot risk assembly
Roadmap (29) splits each actual cell error into its disjoint good- and bad-pilot parts. Independent cell variances add; the good bias is summed before squaring, and the bad bias is controlled by its second moments.
Pairwise independence suffices for (29): only distinct-cell covariances enter the variance of the sum. This matches the implemented marked experiment. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma pairwiseCell_sum_sqRisk_eq {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] {d : ℕ}
(T : Fin d → Ω → ℝ) (t : Fin d → ℝ)
(hT : ∀ x, MemLp (T x) 2 μ)
(hind : Pairwise (fun x y => IndepFun (T x) (T y) μ)) :
sqRisk μ (fun ω => ∑ x, T x ω) (∑ x, t x) =
(∑ x, variance (T x) μ) + (∑ x, ((∫ ω, T x ω ∂μ) - t x)) ^ 2 := by
have hsum := memLp_finsetSum univ (fun x _ => hT x)
have hfun : (∑ x, T x) = (fun ω => ∑ x, T x ω) := by funext ω; simp
have hv := IndepFun.variance_sum (X := T) (s := univ)
(fun x _ => hT x) (fun x _ y _ hxy => hind hxy)
rw [hfun] at hv
rw [cellStatistic_sqRisk_eq_variance_add_bias μ _ _ hsum, hv,
integral_finsetSum _ (fun x _ => (hT x).integrable (by norm_num)),
Finset.sum_sub_distrib]
Disjoint good and bad errors yield the independent-cell risk estimate in (29). Only the full cell statistics need be independent. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
d :
ℕ
T G B :
Fin d → Ω → ℝ
t :
Fin d → ℝ
hT :
∀ x, MemLp (T x) 2 μ
hind :
Pairwise (fun x y => IndepFun (T x) (T y) μ)
hG :
∀ x, MemLp (G x) 2 μ
hB :
∀ x, MemLp (B x) 2 μ
hsplit :
∀ x ω, T x ω - t x = G x ω + B x ω
hdisjoint :
∀ x ω, G x ω * B x ω = 0
conclusion
sqRisk μ (fun ω => ∑ x, T x ω) (∑ x, t x)
≤ (∑ x, ∫ ω, (G x ω) ^ 2 ∂μ)
+ 2 * (∑ x, ∫ ω, G x ω ∂μ) ^ 2
+ (1 + 2 * (d : ℝ)) * (∑ x, ∫ ω, (B x ω) ^ 2 ∂μ)
Proof (Lean source)
lemma independentCell_good_bad_sqRisk_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] {d : ℕ}
(T G B : Fin d → Ω → ℝ) (t : Fin d → ℝ)
(hT : ∀ x, MemLp (T x) 2 μ) (hind : Pairwise (fun x y => IndepFun (T x) (T y) μ))
(hG : ∀ x, MemLp (G x) 2 μ) (hB : ∀ x, MemLp (B x) 2 μ)
(hsplit : ∀ x ω, T x ω - t x = G x ω + B x ω)
(hdisjoint : ∀ x ω, G x ω * B x ω = 0) :
sqRisk μ (fun ω => ∑ x, T x ω) (∑ x, t x) ≤
(∑ x, ∫ ω, (G x ω) ^ 2 ∂μ) +
2 * (∑ x, ∫ ω, G x ω ∂μ) ^ 2 +
(1 + 2 * (d : ℝ)) * (∑ x, ∫ ω, (B x ω) ^ 2 ∂μ) := by
have hv (x : Fin d) : variance (T x) μ ≤
(∫ ω, (G x ω) ^ 2 ∂μ) + (∫ ω, (B x ω) ^ 2 ∂μ) := by
have hr := cellStatistic_sqRisk_eq_variance_add_bias μ (T x) (t x) (hT x)
have he : sqRisk μ (T x) (t x) =
(∫ ω, (G x ω) ^ 2 ∂μ) + (∫ ω, (B x ω) ^ 2 ∂μ) := by
unfold sqRisk
calc
_ = ∫ ω, (G x ω) ^ 2 + (B x ω) ^ 2 ∂μ := integral_congr_ae (by
filter_upwards [] with ω
rw [hsplit x ω]
nlinarith [hdisjoint x ω])
_ = _ := integral_add (hG x).integrable_sq (hB x).integrable_sq
rw [he] at hr
linarith [sq_nonneg ((∫ ω, T x ω ∂μ) - t x)]
have hb (x : Fin d) : (∫ ω, T x ω ∂μ) - t x =
(∫ ω, G x ω ∂μ) + (∫ ω, B x ω ∂μ) := by
rw [← integral_add ((hG x).integrable (by norm_num))
((hB x).integrable (by norm_num))]
have he : (fun ω => G x ω + B x ω) = (fun ω => T x ω - t x) := by
funext ω
exact (hsplit x ω).symm
rw [he, integral_sub ((hT x).integrable (by norm_num)) (integrable_const _)]
simp
have hbad := clippingScale_integral_sum_sq_le μ B
(fun x => (hB x).integrable (by norm_num)) (fun x => (hB x).integrable_sq)
rw [pairwiseCell_sum_sqRisk_eq μ T t hT hind]
simp_rw [hb]
rw [Finset.sum_add_distrib]
have hvar := Finset.sum_le_sum (fun x (_ : x ∈ univ) => hv x)
rw [Finset.sum_add_distrib] at hvar
nlinarith [sq_nonneg ((∑ x, ∫ ω, G x ω ∂μ) - (∑ x, ∫ ω, B x ω ∂μ))]
theorem independentCell_pilotEvent_sqRisk_le Lemma independentCell_pilotEvent_sqRisk_le in the paper ↗
Measurable pilot events provide the disjoint error split automatically. Both pieces inherit L² regularity from the cell statistic, so this step introduces no extra moment assumption on the localized pieces. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
d :
ℕ
T :
Fin d → Ω → ℝ
t :
Fin d → ℝ
hT :
∀ x, MemLp (T x) 2 μ
hind :
Pairwise (fun x y => IndepFun (T x) (T y) μ)
hE :
∀ x, MeasurableSet (E x)
conclusion
sqRisk μ (fun ω => ∑ x, T x ω) (∑ x, t x)
≤ (∑ x, ∫ ω, if ω ∈ E x then (T x ω - t x) ^ 2 else 0 ∂μ)
+ 2 * (∑ x, ∫ ω, if ω ∈ E x then T x ω - t x else 0 ∂μ) ^ 2
+ (1 + 2 * (d : ℝ)) * (∑ x, ∫ ω, if ω ∉ E x then (T x ω - t x) ^ 2 else 0 ∂μ)
Proof (Lean source)
lemma independentCell_pilotEvent_sqRisk_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] {d : ℕ}
(T : Fin d → Ω → ℝ) (t : Fin d → ℝ) (E : Fin d → Set Ω)
(hT : ∀ x, MemLp (T x) 2 μ) (hind : Pairwise (fun x y => IndepFun (T x) (T y) μ))
(hE : ∀ x, MeasurableSet (E x)) :
sqRisk μ (fun ω => ∑ x, T x ω) (∑ x, t x) ≤
(∑ x, ∫ ω, if ω ∈ E x then (T x ω - t x) ^ 2 else 0 ∂μ) +
2 * (∑ x, ∫ ω, if ω ∈ E x then T x ω - t x else 0 ∂μ) ^ 2 +
(1 + 2 * (d : ℝ)) *
(∑ x, ∫ ω, if ω ∉ E x then (T x ω - t x) ^ 2 else 0 ∂μ) := by
classical
let G : Fin d → Ω → ℝ := fun x ω => if ω ∈ E x then T x ω - t x else 0
let B : Fin d → Ω → ℝ := fun x ω => if ω ∉ E x then T x ω - t x else 0
have hG (x : Fin d) : MemLp (G x) 2 μ := by
change MemLp ((E x).indicator (fun ω => T x ω - t x)) 2 μ
exact ((hT x).sub (memLp_const (t x))).indicator (hE x)
have hB (x : Fin d) : MemLp (B x) 2 μ := by
have he : B x = (E x)ᶜ.indicator (fun ω => T x ω - t x) := by
funext ω
simp only [B, Set.indicator_apply, Set.mem_compl_iff]
rw [he]
exact ((hT x).sub (memLp_const (t x))).indicator (hE x).compl
have hs (x : Fin d) (ω : Ω) : T x ω - t x = G x ω + B x ω := by
by_cases he : ω ∈ E x <;> simp [G, B, he]
have hdj (x : Fin d) (ω : Ω) : G x ω * B x ω = 0 := by
by_cases he : ω ∈ E x <;> simp [G, B, he]
have h := independentCell_good_bad_sqRisk_le μ T G B t hT hind hG hB hs hdj
simpa only [G, B, ite_pow, zero_pow (by norm_num : 2 ≠ 0)] using h
theorem activeBranch_good_bad_independentCell_sqRisk_rate_le
The good-pilot estimates (19), (20), (28) and the bad-pilot squared cell estimate (22) imply the full independent-cell rate (34). The bad error retains its actual second moment; no unit-loss replacement is used. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Cs Cv Cb Cd :
ℝ
hCs :
0 ≤ Cs
hCv :
0 ≤ Cv
hCd :
0 ≤ Cd
conclusion
∃ C : ℝ,
0 < C
conclusion 1
Ω :
Type
μ :
Measure Ω
d :
ℕ
1 ≤ d
P :
T G B S :
Fin d → Ω → ℝ
t :
Fin d → ℝ
a :
ℝ
0 < a
(d : ℝ) ≤ 8 * a * logAlphabet d
(∀ x, MemLp (T x) 2 μ)
Pairwise (fun x y => IndepFun (T x) (T y) μ)
(∀ x, MemLp (G x) 2 μ)
(∀ x, MemLp (B x) 2 μ)
(∀ x ω, T x ω - t x = G x ω + B x ω)
(∀ x ω, G x ω * B x ω = 0)
(∀ x, Integrable (S x) μ)
(∀ x, Integrable (fun ω => (S x ω) ^ 2) μ)
(∀ x, (∫ ω, (S x ω) ^ 2 ∂μ) ≤ Cs * (cellMass P x * logAlphabet d / a + logAlphabet d ^ 2 / a ^ 2))
(∀ x, (∫ ω, (G x ω) ^ 2 ∂μ) ≤ Cv * (d : ℝ) ^ (1 / 16 : ℝ) * (∫ ω, (S x ω) ^ 2 ∂μ))
(∀ x, |∫ ω, G x ω ∂μ| ≤ Cb * (sqrt (cellMass P x / (a * logAlphabet d)) + 1 / (a * logAlphabet d) + (d : ℝ) ^ (-(3 / 16 : ℝ)) * (∫ ω, S x ω ∂μ)))
(∀ x, (∫ ω, (B x ω) ^ 2 ∂μ) ≤ Cd * (d : ℝ) ^ (-4 : ℝ) * (cellMass P x * logAlphabet d / a + logAlphabet d ^ 2 / a ^ 2))
sqRisk μ (fun ω => ∑ x, T x ω) (∑ x, t x)
≤ C * ((d : ℝ) / (a * logAlphabet d))
Proof (Lean source)
lemma activeBranch_good_bad_independentCell_sqRisk_rate_le (Cs Cv Cb Cd : ℝ)
(hCs : 0 ≤ Cs) (hCv : 0 ≤ Cv) (hCd : 0 ≤ Cd) :
∃ C : ℝ, 0 < C ∧ ∀ {Ω : Type} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] {d : ℕ}, 1 ≤ d →
∀ (P : DiscreteLaw d) (T G B S : Fin d → Ω → ℝ) (t : Fin d → ℝ) (a : ℝ),
0 < a → (d : ℝ) ≤ 8 * a * logAlphabet d →
(∀ x, MemLp (T x) 2 μ) → Pairwise (fun x y => IndepFun (T x) (T y) μ) →
(∀ x, MemLp (G x) 2 μ) → (∀ x, MemLp (B x) 2 μ) →
(∀ x ω, T x ω - t x = G x ω + B x ω) →
(∀ x ω, G x ω * B x ω = 0) →
(∀ x, Integrable (S x) μ) →
(∀ x, Integrable (fun ω => (S x ω) ^ 2) μ) →
(∀ x, (∫ ω, (S x ω) ^ 2 ∂μ) ≤
Cs * (cellMass P x * logAlphabet d / a + logAlphabet d ^ 2 / a ^ 2)) →
(∀ x, (∫ ω, (G x ω) ^ 2 ∂μ) ≤
Cv * (d : ℝ) ^ (1 / 16 : ℝ) * (∫ ω, (S x ω) ^ 2 ∂μ)) →
(∀ x, |∫ ω, G x ω ∂μ| ≤
Cb * (sqrt (cellMass P x / (a * logAlphabet d)) +
1 / (a * logAlphabet d) + (d : ℝ) ^ (-(3 / 16 : ℝ)) * (∫ ω, S x ω ∂μ))) →
(∀ x, (∫ ω, (B x ω) ^ 2 ∂μ) ≤
Cd * (d : ℝ) ^ (-4 : ℝ) *
(cellMass P x * logAlphabet d / a + logAlphabet d ^ 2 / a ^ 2)) →
sqRisk μ (fun ω => ∑ x, T x ω) (∑ x, t x) ≤
C * ((d : ℝ) / (a * logAlphabet d)) := by
obtain ⟨Av, hAv, hvariance⟩ := activeBranch_variance_scale_rate_le
obtain ⟨Ab, hAb, hbias⟩ := activeBranch_combined_bias_rate_le
obtain ⟨Ad, hAd, hbadRate⟩ := activeBranch_pilot_moment_sum_rate_le 1 (by norm_num)
let C := Cv * Cs * Av + 4 * Cb ^ 2 * (18 + Cs * Ab) + 3 * Cd * Ad + 1
refine ⟨C, by dsimp [C]; positivity, ?_⟩
intro Ω _ μ _ d hd P T G B S t a ha hactive hT hind hG hB hsplit hdisjoint
hS hS₂ hmom hgood₂ hgood₁ hbad₂
have hd1 : (1 : ℝ) ≤ d := by exact_mod_cast hd
have hd0 : (0 : ℝ) < d := by linarith
have hR : 0 ≤ (d : ℝ) / (a * logAlphabet d) := by
have hL := logAlphabet_pos hd
positivity
have hvs := hvariance hd P (fun x => ∫ ω, (S x ω) ^ 2 ∂μ)
Cs a hCs ha hactive hmom
have hg := Finset.sum_le_sum (fun x (_ : x ∈ univ) => hgood₂ x)
rw [← Finset.mul_sum] at hg
have hvg : (∑ x, ∫ ω, (G x ω) ^ 2 ∂μ) ≤
Cv * Cs * Av * ((d : ℝ) / (a * logAlphabet d)) := hg.trans (by
have hh := mul_le_mul_of_nonneg_left hvs hCv
nlinarith [hh])
have hbs := hbias μ hd P S Cs Cb a hCs ha hactive hS hS₂ hmom
have hgb : |∑ x, ∫ ω, G x ω ∂μ| ≤
∑ x : Fin d, Cb * (sqrt (cellMass P x / (a * logAlphabet d)) +
1 / (a * logAlphabet d) + (d : ℝ) ^ (-(3 / 16 : ℝ)) * (∫ ω, S x ω ∂μ)) :=
(Finset.abs_sum_le_sum_abs _ _).trans
(Finset.sum_le_sum (fun x _ => hgood₁ x))
have hgb₂ := pow_le_pow_left₀ (abs_nonneg _) hgb 2
rw [sq_abs] at hgb₂
have hdecay : (d : ℝ) * (d : ℝ) ^ (-4 : ℝ) ≤ 1 := by
nth_rw 1 [← Real.rpow_one (d : ℝ)]
rw [← Real.rpow_add hd0]
exact Real.rpow_le_one_of_one_le_of_nonpos hd1 (by norm_num)
have hscaled (x : Fin d) : (d : ℝ) * (∫ ω, (B x ω) ^ 2 ∂μ) ≤
Cd * (cellMass P x * logAlphabet d / a + logAlphabet d ^ 2 / a ^ 2) := by
have hp : 0 ≤ cellMass P x :=
Finset.sum_nonneg fun a _ => Finset.sum_nonneg fun y _ => toReal_nonneg
have hL := logAlphabet_pos hd
have hs : 0 ≤ cellMass P x * logAlphabet d / a + logAlphabet d ^ 2 / a ^ 2 :=
by positivity
calc
_ ≤ (d : ℝ) * (Cd * (d : ℝ) ^ (-4 : ℝ) *
(cellMass P x * logAlphabet d / a + logAlphabet d ^ 2 / a ^ 2)) :=
mul_le_mul_of_nonneg_left (hbad₂ x) hd0.le
_ = Cd * ((d : ℝ) * (d : ℝ) ^ (-4 : ℝ)) *
(cellMass P x * logAlphabet d / a + logAlphabet d ^ 2 / a ^ 2) := by ring
_ ≤ _ := by
simpa using mul_le_mul_of_nonneg_right
(mul_le_mul_of_nonneg_left hdecay hCd) hs
have hbd := hbadRate hd P (fun x => (d : ℝ) * (∫ ω, (B x ω) ^ 2 ∂μ))
Cd a hCd ha hactive hscaled
norm_num only [sub_self, Real.rpow_zero, one_mul] at hbd
rw [← Finset.mul_sum] at hbd
have hb0 : 0 ≤ ∑ x, ∫ ω, (B x ω) ^ 2 ∂μ :=
Finset.sum_nonneg (fun x _ => integral_nonneg (fun ω => sq_nonneg _))
have htotal := independentCell_good_bad_sqRisk_le μ T G B t hT hind hG hB
hsplit hdisjoint
have hbadterm : (1 + 2 * (d : ℝ)) * (∑ x, ∫ ω, (B x ω) ^ 2 ∂μ) ≤
3 * (Cd * Ad * ((d : ℝ) / (a * logAlphabet d))) := by
nlinarith [mul_nonneg (sub_nonneg.mpr hd1) hb0]
dsimp [C]
nlinarith
theorem activeBranch_approximation_sum_sq_rate_le
The squared cellwise Jackson approximation envelopes sum to at most 18 times the active rate, including null cells. This retains the approximation term in (28) when passing from polynomial loss to loss around the cell target. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Proof (Lean source)
lemma activeBranch_approximation_sum_sq_rate_le {d : ℕ} (P : DiscreteLaw d)
(a : ℝ) (ha : 0 < a) (hactive : (d : ℝ) / a ≤ 8) :
(∑ x : Fin d, (sqrt (cellMass P x / a) + 1 / a) ^ 2) ≤
18 * ((d : ℝ) / a) := by
have hsum := Finset.sum_sq_le_sq_sum_of_nonneg
(s := univ) (f := fun x : Fin d => sqrt (cellMass P x / a) + 1 / a)
(fun x _ => add_nonneg (Real.sqrt_nonneg _) (by positivity))
have htotal := activeBranch_total_bias_sq_le P a ha hactive
have he : (∑ x : Fin d, (sqrt (cellMass P x / a) + 1 / a)) =
(∑ x : Fin d, sqrt (cellMass P x / a)) + (d : ℝ) / a := by
rw [Finset.sum_add_distrib]
simp [div_eq_mul_inv]
rw [he] at hsum
exact hsum.trans htotal
theorem activeBranch_good_bad_approximation_sqRisk_rate_le
The good-pilot target loss includes both the factorial fluctuation and the squared Jackson approximation error. Summing that extra error with (32) preserves (34); the bad-pilot contribution is still its actual second moment. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Cs Cv Cb Cd Ce :
ℝ
hCs :
0 ≤ Cs
hCv :
0 ≤ Cv
hCd :
0 ≤ Cd
hCe :
0 ≤ Ce
conclusion
∃ C : ℝ,
0 < C
conclusion 1
Ω :
Type
μ :
Measure Ω
d :
ℕ
1 ≤ d
P :
T G B S :
Fin d → Ω → ℝ
t :
Fin d → ℝ
a :
ℝ
0 < a
(d : ℝ) ≤ 8 * a * logAlphabet d
(∀ x, MemLp (T x) 2 μ)
Pairwise (fun x y => IndepFun (T x) (T y) μ)
(∀ x, MemLp (G x) 2 μ)
(∀ x, MemLp (B x) 2 μ)
(∀ x ω, T x ω - t x = G x ω + B x ω)
(∀ x ω, G x ω * B x ω = 0)
(∀ x, Integrable (S x) μ)
(∀ x, Integrable (fun ω => (S x ω) ^ 2) μ)
(∀ x, (∫ ω, (S x ω) ^ 2 ∂μ) ≤ Cs * (cellMass P x * logAlphabet d / a + logAlphabet d ^ 2 / a ^ 2))
(∀ x, (∫ ω, (G x ω) ^ 2 ∂μ) ≤ Cv * (d : ℝ) ^ (1 / 16 : ℝ) * (∫ ω, (S x ω) ^ 2 ∂μ) + Ce * (sqrt (cellMass P x / (a * logAlphabet d)) + 1 / (a * logAlphabet d)) ^ 2)
(∀ x, |∫ ω, G x ω ∂μ| ≤ Cb * (sqrt (cellMass P x / (a * logAlphabet d)) + 1 / (a * logAlphabet d) + (d : ℝ) ^ (-(3 / 16 : ℝ)) * (∫ ω, S x ω ∂μ)))
(∀ x, (∫ ω, (B x ω) ^ 2 ∂μ) ≤ Cd * (d : ℝ) ^ (-4 : ℝ) * (cellMass P x * logAlphabet d / a + logAlphabet d ^ 2 / a ^ 2))
sqRisk μ (fun ω => ∑ x, T x ω) (∑ x, t x)
≤ C * ((d : ℝ) / (a * logAlphabet d))
Proof (Lean source)
lemma activeBranch_good_bad_approximation_sqRisk_rate_le (Cs Cv Cb Cd Ce : ℝ)
(hCs : 0 ≤ Cs) (hCv : 0 ≤ Cv) (hCd : 0 ≤ Cd) (hCe : 0 ≤ Ce) :
∃ C : ℝ, 0 < C ∧ ∀ {Ω : Type} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] {d : ℕ}, 1 ≤ d →
∀ (P : DiscreteLaw d) (T G B S : Fin d → Ω → ℝ) (t : Fin d → ℝ) (a : ℝ),
0 < a → (d : ℝ) ≤ 8 * a * logAlphabet d →
(∀ x, MemLp (T x) 2 μ) → Pairwise (fun x y => IndepFun (T x) (T y) μ) →
(∀ x, MemLp (G x) 2 μ) → (∀ x, MemLp (B x) 2 μ) →
(∀ x ω, T x ω - t x = G x ω + B x ω) →
(∀ x ω, G x ω * B x ω = 0) →
(∀ x, Integrable (S x) μ) →
(∀ x, Integrable (fun ω => (S x ω) ^ 2) μ) →
(∀ x, (∫ ω, (S x ω) ^ 2 ∂μ) ≤
Cs * (cellMass P x * logAlphabet d / a + logAlphabet d ^ 2 / a ^ 2)) →
(∀ x, (∫ ω, (G x ω) ^ 2 ∂μ) ≤
Cv * (d : ℝ) ^ (1 / 16 : ℝ) * (∫ ω, (S x ω) ^ 2 ∂μ) +
Ce * (sqrt (cellMass P x / (a * logAlphabet d)) +
1 / (a * logAlphabet d)) ^ 2) →
(∀ x, |∫ ω, G x ω ∂μ| ≤
Cb * (sqrt (cellMass P x / (a * logAlphabet d)) +
1 / (a * logAlphabet d) + (d : ℝ) ^ (-(3 / 16 : ℝ)) * (∫ ω, S x ω ∂μ))) →
(∀ x, (∫ ω, (B x ω) ^ 2 ∂μ) ≤
Cd * (d : ℝ) ^ (-4 : ℝ) *
(cellMass P x * logAlphabet d / a + logAlphabet d ^ 2 / a ^ 2)) →
sqRisk μ (fun ω => ∑ x, T x ω) (∑ x, t x) ≤
C * ((d : ℝ) / (a * logAlphabet d)) := by
obtain ⟨Av, hAv, hvariance⟩ := activeBranch_variance_scale_rate_le
obtain ⟨Ab, hAb, hbias⟩ := activeBranch_combined_bias_rate_le
obtain ⟨Ad, hAd, hbadRate⟩ := activeBranch_pilot_moment_sum_rate_le 1 (by norm_num)
let C := Cv * Cs * Av + 4 * Cb ^ 2 * (18 + Cs * Ab) + 3 * Cd * Ad + 18 * Ce + 1
refine ⟨C, by dsimp [C]; positivity, ?_⟩
intro Ω _ μ _ d hd P T G B S t a ha hactive hT hind hG hB hsplit hdisjoint
hS hS₂ hmom hgood₂ hgood₁ hbad₂
have hd1 : (1 : ℝ) ≤ d := by exact_mod_cast hd
have hd0 : (0 : ℝ) < d := by linarith
have hR : 0 ≤ (d : ℝ) / (a * logAlphabet d) := by
have hL := logAlphabet_pos hd
positivity
have hvs := hvariance hd P (fun x => ∫ ω, (S x ω) ^ 2 ∂μ)
Cs a hCs ha hactive hmom
have hg := Finset.sum_le_sum (fun x (_ : x ∈ univ) => hgood₂ x)
rw [Finset.sum_add_distrib, ← Finset.mul_sum, ← Finset.mul_sum] at hg
have happrox := activeBranch_approximation_sum_sq_rate_le P
(a * logAlphabet d) (mul_pos ha (logAlphabet_pos hd))
((div_le_iff₀ (mul_pos ha (logAlphabet_pos hd))).2 (by nlinarith))
have hvg : (∑ x, ∫ ω, (G x ω) ^ 2 ∂μ) ≤
(Cv * Cs * Av + 18 * Ce) * ((d : ℝ) / (a * logAlphabet d)) := hg.trans (by
have hh := mul_le_mul_of_nonneg_left hvs hCv
nlinarith [hh, mul_le_mul_of_nonneg_left happrox hCe])
have hbs := hbias μ hd P S Cs Cb a hCs ha hactive hS hS₂ hmom
have hgb : |∑ x, ∫ ω, G x ω ∂μ| ≤
∑ x : Fin d, Cb * (sqrt (cellMass P x / (a * logAlphabet d)) +
1 / (a * logAlphabet d) + (d : ℝ) ^ (-(3 / 16 : ℝ)) * (∫ ω, S x ω ∂μ)) :=
(Finset.abs_sum_le_sum_abs _ _).trans
(Finset.sum_le_sum (fun x _ => hgood₁ x))
have hgb₂ := pow_le_pow_left₀ (abs_nonneg _) hgb 2
rw [sq_abs] at hgb₂
have hdecay : (d : ℝ) * (d : ℝ) ^ (-4 : ℝ) ≤ 1 := by
nth_rw 1 [← Real.rpow_one (d : ℝ)]
rw [← Real.rpow_add hd0]
exact Real.rpow_le_one_of_one_le_of_nonpos hd1 (by norm_num)
have hscaled (x : Fin d) : (d : ℝ) * (∫ ω, (B x ω) ^ 2 ∂μ) ≤
Cd * (cellMass P x * logAlphabet d / a + logAlphabet d ^ 2 / a ^ 2) := by
have hp : 0 ≤ cellMass P x :=
Finset.sum_nonneg fun a _ => Finset.sum_nonneg fun y _ => toReal_nonneg
have hL := logAlphabet_pos hd
have hs : 0 ≤ cellMass P x * logAlphabet d / a + logAlphabet d ^ 2 / a ^ 2 :=
by positivity
calc
_ ≤ (d : ℝ) * (Cd * (d : ℝ) ^ (-4 : ℝ) *
(cellMass P x * logAlphabet d / a + logAlphabet d ^ 2 / a ^ 2)) :=
mul_le_mul_of_nonneg_left (hbad₂ x) hd0.le
_ = Cd * ((d : ℝ) * (d : ℝ) ^ (-4 : ℝ)) *
(cellMass P x * logAlphabet d / a + logAlphabet d ^ 2 / a ^ 2) := by ring
_ ≤ _ := by
simpa using mul_le_mul_of_nonneg_right
(mul_le_mul_of_nonneg_left hdecay hCd) hs
have hbd := hbadRate hd P (fun x => (d : ℝ) * (∫ ω, (B x ω) ^ 2 ∂μ))
Cd a hCd ha hactive hscaled
norm_num only [sub_self, Real.rpow_zero, one_mul] at hbd
rw [← Finset.mul_sum] at hbd
have hb0 : 0 ≤ ∑ x, ∫ ω, (B x ω) ^ 2 ∂μ :=
Finset.sum_nonneg (fun x _ => integral_nonneg (fun ω => sq_nonneg _))
have htotal := independentCell_good_bad_sqRisk_le μ T G B t hT hind hG hB
hsplit hdisjoint
have hbadterm : (1 + 2 * (d : ℝ)) * (∑ x, ∫ ω, (B x ω) ^ 2 ∂μ) ≤
3 * (Cd * Ad * ((d : ℝ) / (a * logAlphabet d))) := by
nlinarith [mul_nonneg (sub_nonneg.mpr hd1) hb0]
dsimp [C]
nlinarith
theorem activeBranch_pilotEvent_approximation_sqRisk_rate_le
Measurable good-pilot events turn the target-loss estimates of (20), (22), and (28) into the independent-cell rate. The Jackson squared error is retained, and the localized L² properties follow from the full cell statistics. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Cs Cv Cb Cd Ce :
ℝ
hCs :
0 ≤ Cs
hCv :
0 ≤ Cv
hCd :
0 ≤ Cd
hCe :
0 ≤ Ce
conclusion
∃ C : ℝ,
0 < C
conclusion 1
Ω :
Type
μ :
Measure Ω
d :
ℕ
1 ≤ d
P :
T S :
Fin d → Ω → ℝ
t :
Fin d → ℝ
a :
ℝ
0 < a
(d : ℝ) ≤ 8 * a * logAlphabet d
(∀ x, MemLp (T x) 2 μ)
Pairwise (fun x y => IndepFun (T x) (T y) μ)
(∀ x, MeasurableSet (E x))
(∀ x, Integrable (S x) μ)
(∀ x, Integrable (fun ω => (S x ω) ^ 2) μ)
(∀ x, (∫ ω, (S x ω) ^ 2 ∂μ) ≤ Cs * (cellMass P x * logAlphabet d / a + logAlphabet d ^ 2 / a ^ 2))
(∀ x, (∫ ω, if ω ∈ E x then (T x ω - t x) ^ 2 else 0 ∂μ) ≤ Cv * (d : ℝ) ^ (1 / 16 : ℝ) * (∫ ω, (S x ω) ^ 2 ∂μ) + Ce * (sqrt (cellMass P x / (a * logAlphabet d)) + 1 / (a * logAlphabet d)) ^ 2)
(∀ x, |∫ ω, if ω ∈ E x then T x ω - t x else 0 ∂μ| ≤ Cb * (sqrt (cellMass P x / (a * logAlphabet d)) + 1 / (a * logAlphabet d) + (d : ℝ) ^ (-(3 / 16 : ℝ)) * (∫ ω, S x ω ∂μ)))
(∀ x, (∫ ω, if ω ∉ E x then (T x ω - t x) ^ 2 else 0 ∂μ) ≤ Cd * (d : ℝ) ^ (-4 : ℝ) * (cellMass P x * logAlphabet d / a + logAlphabet d ^ 2 / a ^ 2))
sqRisk μ (fun ω => ∑ x, T x ω) (∑ x, t x)
≤ C * ((d : ℝ) / (a * logAlphabet d))
Proof (Lean source)
lemma activeBranch_pilotEvent_approximation_sqRisk_rate_le (Cs Cv Cb Cd Ce : ℝ)
(hCs : 0 ≤ Cs) (hCv : 0 ≤ Cv) (hCd : 0 ≤ Cd) (hCe : 0 ≤ Ce) :
∃ C : ℝ, 0 < C ∧ ∀ {Ω : Type} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] {d : ℕ}, 1 ≤ d →
∀ (P : DiscreteLaw d) (T S : Fin d → Ω → ℝ) (t : Fin d → ℝ)
(E : Fin d → Set Ω) (a : ℝ),
0 < a → (d : ℝ) ≤ 8 * a * logAlphabet d →
(∀ x, MemLp (T x) 2 μ) → Pairwise (fun x y => IndepFun (T x) (T y) μ) →
(∀ x, MeasurableSet (E x)) →
(∀ x, Integrable (S x) μ) →
(∀ x, Integrable (fun ω => (S x ω) ^ 2) μ) →
(∀ x, (∫ ω, (S x ω) ^ 2 ∂μ) ≤
Cs * (cellMass P x * logAlphabet d / a + logAlphabet d ^ 2 / a ^ 2)) →
(∀ x, (∫ ω, if ω ∈ E x then (T x ω - t x) ^ 2 else 0 ∂μ) ≤
Cv * (d : ℝ) ^ (1 / 16 : ℝ) * (∫ ω, (S x ω) ^ 2 ∂μ) +
Ce * (sqrt (cellMass P x / (a * logAlphabet d)) +
1 / (a * logAlphabet d)) ^ 2) →
(∀ x, |∫ ω, if ω ∈ E x then T x ω - t x else 0 ∂μ| ≤
Cb * (sqrt (cellMass P x / (a * logAlphabet d)) +
1 / (a * logAlphabet d) + (d : ℝ) ^ (-(3 / 16 : ℝ)) * (∫ ω, S x ω ∂μ))) →
(∀ x, (∫ ω, if ω ∉ E x then (T x ω - t x) ^ 2 else 0 ∂μ) ≤
Cd * (d : ℝ) ^ (-4 : ℝ) *
(cellMass P x * logAlphabet d / a + logAlphabet d ^ 2 / a ^ 2)) →
sqRisk μ (fun ω => ∑ x, T x ω) (∑ x, t x) ≤
C * ((d : ℝ) / (a * logAlphabet d)) := by
classical
obtain ⟨C, hC, hrate⟩ := activeBranch_good_bad_approximation_sqRisk_rate_le
Cs Cv Cb Cd Ce hCs hCv hCd hCe
refine ⟨C, hC, ?_⟩
intro Ω _ μ _ d hd P T S t E a ha hactive hT hind hE hS hS₂ hmom hg₂ hg₁ hb₂
let G : Fin d → Ω → ℝ := fun x ω => if ω ∈ E x then T x ω - t x else 0
let B : Fin d → Ω → ℝ := fun x ω => if ω ∉ E x then T x ω - t x else 0
have hG (x : Fin d) : MemLp (G x) 2 μ := by
change MemLp ((E x).indicator (fun ω => T x ω - t x)) 2 μ
exact ((hT x).sub (memLp_const (t x))).indicator (hE x)
have hB (x : Fin d) : MemLp (B x) 2 μ := by
have he : B x = (E x)ᶜ.indicator (fun ω => T x ω - t x) := by
funext ω
simp only [B, Set.indicator_apply, Set.mem_compl_iff]
rw [he]
exact ((hT x).sub (memLp_const (t x))).indicator (hE x).compl
apply hrate μ hd P T G B S t a ha hactive hT hind hG hB
· intro x ω
by_cases he : ω ∈ E x <;> simp [G, B, he]
· intro x ω
by_cases he : ω ∈ E x <;> simp [G, B, he]
· exact hS
· exact hS₂
· exact hmom
· simpa only [G, ite_pow, zero_pow (by norm_num : 2 ≠ 0)] using hg₂
· exact hg₁
· simpa only [B, ite_pow, zero_pow (by norm_num : 2 ≠ 0)] using hb₂
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.UpperIndependentAggregation 7 declarations Roadmap equations (29)–(34) add the independent cell variances and square the sum of their biases.
Independent-cell risk assembly
Roadmap equations (29)–(34) add the independent cell variances and square the sum of their biases. The rate assembly below consumes cellwise estimates; it does not assume the risk bound for the estimator.
theorem cellStatistic_sqRisk_eq_variance_add_bias Lemma cellStatistic_sqRisk_eq_variance_add_bias in the paper ↗
Squared loss decomposes into variance and squared bias for an L² statistic. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Proof (Lean source)
lemma cellStatistic_sqRisk_eq_variance_add_bias {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (T : Ω → ℝ) (t : ℝ)
(hT : MemLp T 2 μ) :
sqRisk μ T t = variance T μ + ((∫ ω, T ω ∂μ) - t) ^ 2 := by
have hInt := hT.integrable (by norm_num : (1 : ENNReal) ≤ 2)
have hCross : Integrable (fun ω => (2 * t) * T ω) μ := hInt.const_mul (2 * t)
have hDiff : Integrable (fun ω => T ω ^ 2 - (2 * t) * T ω) μ :=
hT.integrable_sq.sub hCross
have he : (fun ω => (T ω - t) ^ 2) =
(fun ω => T ω ^ 2 - (2 * t) * T ω + t ^ 2) := by
funext ω
ring
unfold sqRisk
rw [he, integral_add hDiff (integrable_const (t ^ 2)),
integral_sub hT.integrable_sq hCross, integral_const_mul,
integral_const, probReal_univ, one_smul, variance_eq_sub hT]
simp only [Pi.pow_apply]
ring
Independent cell variances add, while their biases add before squaring, as required in the passage to equation (29). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Proof (Lean source)
lemma independentCell_sum_sqRisk_eq {Ω ι : Type*} [MeasurableSpace Ω] [Fintype ι]
(μ : Measure Ω) [IsProbabilityMeasure μ] (T : ι → Ω → ℝ) (t : ι → ℝ)
(hT : ∀ i, MemLp (T i) 2 μ) (hind : iIndepFun T μ) :
sqRisk μ (fun ω => ∑ i, T i ω) (∑ i, t i) =
(∑ i, variance (T i) μ) + (∑ i, ((∫ ω, T i ω ∂μ) - t i)) ^ 2 := by
have hsum : MemLp (fun ω => ∑ i, T i ω) 2 μ := by
exact memLp_finsetSum univ (fun i _ => hT i)
have hvar : variance (fun ω => ∑ i, T i ω) μ = ∑ i, variance (T i) μ := by
have hfun : (∑ i, T i) = (fun ω => ∑ i, T i ω) := by funext ω; simp
rw [← hfun]
exact IndepFun.variance_sum (X := T) (s := univ)
(fun i _ => hT i) (fun i _ j _ hij => hind.indepFun hij)
rw [cellStatistic_sqRisk_eq_variance_add_bias μ _ _ hsum, hvar,
integral_finsetSum _ (fun i _ => (hT i).integrable (by norm_num)),
Finset.sum_sub_distrib]
Cellwise variance envelopes and absolute bias envelopes give a sum-risk bound. The squared bias is kept intact, avoiding a spurious factor of the alphabet size. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma independentCell_sum_sqRisk_le {Ω ι : Type*} [MeasurableSpace Ω] [Fintype ι]
(μ : Measure Ω) [IsProbabilityMeasure μ] (T : ι → Ω → ℝ) (t v b : ι → ℝ)
(hT : ∀ i, MemLp (T i) 2 μ) (hind : iIndepFun T μ)
(hv : ∀ i, variance (T i) μ ≤ v i)
(hb : ∀ i, |(∫ ω, T i ω ∂μ) - t i| ≤ b i) :
sqRisk μ (fun ω => ∑ i, T i ω) (∑ i, t i) ≤
(∑ i, v i) + (∑ i, b i) ^ 2 := by
rw [independentCell_sum_sqRisk_eq μ T t hT hind]
apply add_le_add (Finset.sum_le_sum fun i _ => hv i)
have hab : |∑ i, ((∫ ω, T i ω ∂μ) - t i)| ≤ ∑ i, b i :=
(Finset.abs_sum_le_sum_abs _ _).trans (Finset.sum_le_sum fun i _ => hb i)
simpa only [sq_abs] using pow_le_pow_left₀ (abs_nonneg _) hab 2
theorem activeBranch_combined_bias_rate_le
Equations (32)–(33) absorb the entire summed bias, including the clipping displacement. The only random-scale input is the cellwise pilot moment estimate (19). In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ A : ℝ,
0 < A
conclusion 1
Ω :
Type
μ :
Measure Ω
d :
ℕ
1 ≤ d
P :
S :
Fin d → Ω → ℝ
Cs Cb a :
ℝ
0 ≤ Cs
0 < a
(d : ℝ) ≤ 8 * a * logAlphabet d
(∀ x, Integrable (S x) μ)
(∀ x, Integrable (fun ω => (S x ω) ^ 2) μ)
(∀ x, (∫ ω, (S x ω) ^ 2 ∂μ) ≤ Cs * (cellMass P x * logAlphabet d / a + logAlphabet d ^ 2 / a ^ 2))
Proof (Lean source)
lemma activeBranch_combined_bias_rate_le :
∃ A : ℝ, 0 < A ∧ ∀ {Ω : Type} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] {d : ℕ}, 1 ≤ d →
∀ (P : DiscreteLaw d) (S : Fin d → Ω → ℝ) (Cs Cb a : ℝ),
0 ≤ Cs → 0 < a → (d : ℝ) ≤ 8 * a * logAlphabet d →
(∀ x, Integrable (S x) μ) →
(∀ x, Integrable (fun ω => (S x ω) ^ 2) μ) →
(∀ x, (∫ ω, (S x ω) ^ 2 ∂μ) ≤
Cs * (cellMass P x * logAlphabet d / a + logAlphabet d ^ 2 / a ^ 2)) →
(∑ x : Fin d, Cb * (sqrt (cellMass P x / (a * logAlphabet d)) +
1 / (a * logAlphabet d) + (d : ℝ) ^ (-(3 / 16 : ℝ)) * (∫ ω, S x ω ∂μ))) ^ 2 ≤
2 * Cb ^ 2 * (18 + Cs * A) * ((d : ℝ) / (a * logAlphabet d)) := by
obtain ⟨A, hA, hclip⟩ := activeBranch_clipping_bias_rate_le
refine ⟨A, hA, ?_⟩
intro Ω _ μ _ d hd P S Cs Cb a hCs ha hactive hS hS₂ hmom
have hd0 : (0 : ℝ) < d := by exact_mod_cast (by omega : 0 < d)
have hL := logAlphabet_pos hd
have hR : (d : ℝ) / (a * logAlphabet d) ≤ 8 :=
(div_le_iff₀ (mul_pos ha hL)).2 (by nlinarith)
have hbase := activeBranch_total_bias_sq_le P (a * logAlphabet d) (mul_pos ha hL) hR
have htail := hclip μ hd P S Cs a hCs ha hactive hS hS₂ hmom
have hpow : ((d : ℝ) ^ (-(3 / 16 : ℝ))) ^ 2 = (d : ℝ) ^ (-(3 / 8 : ℝ)) := by
rw [← Real.rpow_mul_natCast hd0.le]
norm_num
have hid : (∑ x : Fin d, Cb * (sqrt (cellMass P x / (a * logAlphabet d)) +
1 / (a * logAlphabet d) + (d : ℝ) ^ (-(3 / 16 : ℝ)) * (∫ ω, S x ω ∂μ))) =
Cb * (((∑ x : Fin d, sqrt (cellMass P x / (a * logAlphabet d))) +
(d : ℝ) / (a * logAlphabet d)) +
(d : ℝ) ^ (-(3 / 16 : ℝ)) * (∑ x : Fin d, ∫ ω, S x ω ∂μ)) := by
rw [← Finset.mul_sum, Finset.sum_add_distrib, Finset.sum_add_distrib,
← Finset.mul_sum]
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul]
ring
rw [hid, mul_pow]
have htwo (u v : ℝ) : (u + v) ^ 2 ≤ 2 * u ^ 2 + 2 * v ^ 2 := by
nlinarith [sq_nonneg (u - v)]
calc
_ ≤ Cb ^ 2 * (2 * (((∑ x : Fin d, sqrt (cellMass P x / (a * logAlphabet d))) +
(d : ℝ) / (a * logAlphabet d)) ^ 2) +
2 * ((d : ℝ) ^ (-(3 / 16 : ℝ)) * (∑ x : Fin d, ∫ ω, S x ω ∂μ)) ^ 2) :=
mul_le_mul_of_nonneg_left (htwo _ _) (sq_nonneg _)
_ ≤ Cb ^ 2 * (2 * (18 * ((d : ℝ) / (a * logAlphabet d))) +
2 * (Cs * A * ((d : ℝ) / (a * logAlphabet d)))) := by
gcongr
simpa only [mul_pow, hpow] using htail
_ = _ := by ring
theorem activeBranch_independentCell_sqRisk_rate_le
Equations (29)–(34): independent cell statistics obey the active-branch rate once their integrated variance and bias estimates and the pilot second moments have been proved. The constant is uniform in the alphabet and the intensity. These are intermediate cell estimates, not a premise asserting estimator risk. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Cs Cv Cb :
ℝ
hCs :
0 ≤ Cs
hCv :
0 ≤ Cv
conclusion
∃ C : ℝ,
0 < C
conclusion 1
Ω :
Type
μ :
Measure Ω
d :
ℕ
1 ≤ d
P :
T S :
Fin d → Ω → ℝ
t :
Fin d → ℝ
a :
ℝ
0 < a
(d : ℝ) ≤ 8 * a * logAlphabet d
(∀ x, MemLp (T x) 2 μ)
iIndepFun T μ
(∀ x, Integrable (S x) μ)
(∀ x, Integrable (fun ω => (S x ω) ^ 2) μ)
(∀ x, (∫ ω, (S x ω) ^ 2 ∂μ) ≤ Cs * (cellMass P x * logAlphabet d / a + logAlphabet d ^ 2 / a ^ 2))
(∀ x, variance (T x) μ ≤ Cv * (d : ℝ) ^ (1 / 16 : ℝ) * (∫ ω, (S x ω) ^ 2 ∂μ))
(∀ x, |(∫ ω, T x ω ∂μ) - t x| ≤ Cb * (sqrt (cellMass P x / (a * logAlphabet d)) + 1 / (a * logAlphabet d) + (d : ℝ) ^ (-(3 / 16 : ℝ)) * (∫ ω, S x ω ∂μ)))
sqRisk μ (fun ω => ∑ x, T x ω) (∑ x, t x)
≤ C * ((d : ℝ) / (a * logAlphabet d))
Proof (Lean source)
lemma activeBranch_independentCell_sqRisk_rate_le (Cs Cv Cb : ℝ)
(hCs : 0 ≤ Cs) (hCv : 0 ≤ Cv) :
∃ C : ℝ, 0 < C ∧ ∀ {Ω : Type} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] {d : ℕ}, 1 ≤ d →
∀ (P : DiscreteLaw d) (T S : Fin d → Ω → ℝ) (t : Fin d → ℝ) (a : ℝ),
0 < a → (d : ℝ) ≤ 8 * a * logAlphabet d →
(∀ x, MemLp (T x) 2 μ) → iIndepFun T μ →
(∀ x, Integrable (S x) μ) →
(∀ x, Integrable (fun ω => (S x ω) ^ 2) μ) →
(∀ x, (∫ ω, (S x ω) ^ 2 ∂μ) ≤
Cs * (cellMass P x * logAlphabet d / a + logAlphabet d ^ 2 / a ^ 2)) →
(∀ x, variance (T x) μ ≤
Cv * (d : ℝ) ^ (1 / 16 : ℝ) * (∫ ω, (S x ω) ^ 2 ∂μ)) →
(∀ x, |(∫ ω, T x ω ∂μ) - t x| ≤
Cb * (sqrt (cellMass P x / (a * logAlphabet d)) +
1 / (a * logAlphabet d) + (d : ℝ) ^ (-(3 / 16 : ℝ)) * (∫ ω, S x ω ∂μ))) →
sqRisk μ (fun ω => ∑ x, T x ω) (∑ x, t x) ≤
C * ((d : ℝ) / (a * logAlphabet d)) := by
obtain ⟨Av, hAv, hvariance⟩ := activeBranch_variance_scale_rate_le
obtain ⟨Ab, hAb, hbias⟩ := activeBranch_combined_bias_rate_le
let C := Cv * Cs * Av + 2 * Cb ^ 2 * (18 + Cs * Ab) + 1
have hC : 0 < C := by dsimp [C]; positivity
refine ⟨C, hC, ?_⟩
intro Ω _ μ _ d hd P T S t a ha hactive hT hind hS hS₂ hmom hv hb
have hR : 0 ≤ (d : ℝ) / (a * logAlphabet d) :=
div_nonneg (Nat.cast_nonneg _) (mul_pos ha (logAlphabet_pos hd)).le
have hvs := hvariance hd P (fun x => ∫ ω, (S x ω) ^ 2 ∂μ)
Cs a hCs ha hactive hmom
have hbs := hbias μ hd P S Cs Cb a hCs ha hactive hS hS₂ hmom
have hsum := independentCell_sum_sqRisk_le μ T t
(fun x => Cv * (d : ℝ) ^ (1 / 16 : ℝ) * (∫ ω, (S x ω) ^ 2 ∂μ))
(fun x => Cb * (sqrt (cellMass P x / (a * logAlphabet d)) +
1 / (a * logAlphabet d) + (d : ℝ) ^ (-(3 / 16 : ℝ)) * (∫ ω, S x ω ∂μ)))
hT hind hv hb
rw [← Finset.mul_sum] at hsum
have hvs' := mul_le_mul_of_nonneg_left hvs hCv
dsimp [C]
nlinarith
Projection to the unit interval is continuous, including at both endpoints. The stated conclusion holds.
Formal statement
conclusion
Proof (Lean source)
-- @node: projectUnit_continuous
@[fun_prop] lemma projectUnit_continuous : Continuous projectUnit := by
unfold projectUnit
fun_prop
The final projection in equation (34) preserves any risk bound assembled for the unprojected sum. Its squared-loss integrability follows from the original L² statistic and pointwise contraction, so no extra regularity premise is needed. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma projectUnit_sqRisk_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (T : Ω → ℝ) (t : ℝ)
(hT : MemLp T 2 μ) (ht : t ∈ Icc (0 : ℝ) 1) :
sqRisk μ (fun ω => projectUnit (T ω)) t ≤
sqRisk μ T t := by
have hc : Continuous (fun z : ℝ => (projectUnit z - t) ^ 2) := by
fun_prop
have hs : AEStronglyMeasurable (fun ω => (projectUnit (T ω) - t) ^ 2) μ :=
hc.comp_aestronglyMeasurable hT.aestronglyMeasurable
have hraw : Integrable (fun ω => (T ω - t) ^ 2) μ :=
(hT.sub (memLp_const t)).integrable_sq
have hproj : Integrable (fun ω => (projectUnit (T ω) - t) ^ 2) μ :=
hraw.mono' hs (ae_of_all _ fun ω => by
rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)]
exact projectUnit_sq_error_le ht)
exact integral_mono hproj hraw (fun ω => projectUnit_sq_error_le ht)
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.UpperJacksonCoefficients 10 declarations The centered Jackson polynomial's cube oscillation yields the explicit Chebyshev coefficient envelope (14).
Jackson coefficient and localized second-moment bounds
The centered Jackson polynomial's cube oscillation yields the explicit Chebyshev coefficient envelope (14). Combining it with the localized Poisson factorial bound proves (26) and the clipping second-moment bound (28), with no assumed coefficient envelope.
A four-variable polynomial bounded on the cube has the sharp tensor Chebyshev coefficient envelope. Specializing the library's path result to a constant path gives a scalar bound with no variation term. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
K :
ℕ
hK :
0 < K
p :
MvPolynomial (Fin 4) ℝ
B :
ℝ
hB :
0 ≤ B
hdeg :
∀ i, p.degreeOf i ≤ 2 * (K - 1)
hb :
∀ z ∈ normalizedCube 4, |eval z p| ≤ B
conclusion
≤ 16 * (↑(2 * (K - 1) + 1) : ℝ) ^ 4 * (1 + sqrt 2) ^ (4 * (2 * (K - 1))) * B
Proof (Lean source)
lemma cubePolynomial_coeff_sum_le {K : ℕ} (hK : 0 < K)
(p : MvPolynomial (Fin 4) ℝ) (B : ℝ) (hB : 0 ≤ B)
(hdeg : ∀ i, p.degreeOf i ≤ 2 * (K - 1))
(hb : ∀ z ∈ normalizedCube 4, |eval z p| ≤ B) :
(∑ α : Fin 4 → Fin (2 * (K - 1) + 1), |tensorCoeffs p α|) ≤
16 * (↑(2 * (K - 1) + 1) : ℝ) ^ 4 *
(1 + sqrt 2) ^ (4 * (2 * (K - 1))) * B := by
have hvar (v : ℝ) : eVariationOn (fun _ : Time => v) univ = 0 :=
(eVariationOn.eq_zero_iff _).mpr (by intros; simp)
have hsize (v : ℝ) : pathSize (ContinuousMap.const Time v) = |v| := by
simp only [pathSize, pathTV]
rw [show (ContinuousMap.const Time v : Time → ℝ) = (fun _ => v) from rfl,
hvar, ENNReal.toReal_zero, add_zero]
simp [ContinuousMap.norm_eq_iSup_norm]
have he := tensorCoefficientPath_four_chebyshev_size_envelope
(fun _ => p) B hB (fun _ => hdeg) (fun _ => continuous_const)
(fun _ _ => continuous_const) (by
intro z hz
change eVariationOn (fun _ : Time => eval z p) univ < ⊤
rw [hvar]
exact ENNReal.zero_lt_top) (by
intro z hz
change pathSize (ContinuousMap.const Time (eval z p)) ≤ B
rw [hsize]
exact hb z hz)
change (∑ α, pathSize (ContinuousMap.const Time (tensorCoeffs p α))) ≤ _ at he
simpa only [hsize] using he
Equation (14) for the exact normalized polynomial chosen by the estimator. The coefficient norm is derived from (12), polynomial degree control, and Chebyshev extraction, including the centering constant. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
ε :
ℝ
hε :
0 < ε
K :
ℕ
hK :
0 < K
c r :
Fin 4 → ℝ
hr :
∀ j, 0 < r j
hQ :
∀ v ∈ centeredRectangle c r, ∀ j, 0 ≤ v j
conclusion
≤ 16 * (↑(2 * (K - 1) + 1) : ℝ) ^ 4 * (1 + sqrt 2) ^ (4 * (2 * (K - 1))) * clippingScaleFormula ε c r
Proof (Lean source)
lemma jacksonCoeff_sum_abs_le (ε : ℝ) (hε : 0 < ε) (K : ℕ) (hK : 0 < K)
(c r : Fin 4 → ℝ) (hr : ∀ j, 0 < r j)
(hQ : ∀ v ∈ centeredRectangle c r, ∀ j, 0 ≤ v j) :
(∑ α : Fin 4 → Fin (2 * (K - 1) + 1), |jacksonCoeff ε K c r α|) ≤
16 * (↑(2 * (K - 1) + 1) : ℝ) ^ 4 *
(1 + sqrt 2) ^ (4 * (2 * (K - 1))) * clippingScaleFormula ε c r := by
let p := (jacksonPolynomialPair ε K c r).2 - C (armwiseExtensionFormula ε c)
have hc (j : Fin 4) : 0 ≤ c j :=
hQ c (by intro i; simpa using (hr i).le) j
have hs := clippingScale_nonneg ε c r hc (fun j => (hr j).le)
apply cubePolynomial_coeff_sum_le hK p _ hs
· intro j
exact (MvPolynomial.degreeOf_sub_le j _ _).trans (max_le
(jacksonPolynomialPair_normalized_degree_le ε hε K hK c r hr hQ j) (by simp))
· intro z hz
have hq := affinePoint_mem_centeredRectangle c r z hr hz
have hb := jacksonCoeff_mean_oscillation_le ε hε K hK c r hr hQ
(affinePoint c r z) hq
rw [jacksonCoeff_sum_eq_eval ε hε K hK c r hr hQ] at hb
have he : normalizedPoint c r (affinePoint c r z) = z := by
funext j
dsimp [normalizedPoint, affinePoint]
field_simp [ne_of_gt (hr j)]
ring
change |eval (normalizedPoint c r (affinePoint c r z))
(jacksonPolynomialPair ε K c r).2 - armwiseExtensionFormula ε c| ≤ _ at hb
rw [he] at hb
simpa only [p, map_sub, MvPolynomial.eval_C] using hb
The coefficient envelope applies to every actual pilot rectangle, without localization of the true parameter or extra count regularity assumptions. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
b :
pilotMidpoint H hH m L Np
r :
pilotRadiusFormula H hH m L Np
≤ 16 * (↑(2 * (K - 1) + 1) : ℝ) ^ 4 * (1 + sqrt 2) ^ (4 * (2 * (K - 1))) * clippingScaleFormula ε b r
Proof (Lean source)
lemma pilotJacksonCoeff_sum_abs_le (H : ℝ) (hH : 0 < H) (m L : ℝ)
(hm : 0 < m) (hL : 0 < L) (Np : Fin 4 → ℕ)
(ε : ℝ) (hε : 0 < ε) (K : ℕ) (hK : 0 < K) :
let b := pilotMidpoint H hH m L Np
let r := pilotRadiusFormula H hH m L Np
(∑ α : Fin 4 → Fin (2 * (K - 1) + 1), |jacksonCoeff ε K b r α|) ≤
16 * (↑(2 * (K - 1) + 1) : ℝ) ^ 4 *
(1 + sqrt 2) ^ (4 * (2 * (K - 1))) * clippingScaleFormula ε b r := by
exact jacksonCoeff_sum_abs_le ε hε K hK _ _
(pilotRadius_pos H hH m L hm hL Np) (pilotCenteredRectangle_nonneg H hH m L Np)
Squaring the explicit coefficient factor costs only an exponential linear in the Jackson order. The coarse numerical constant is universal and is sufficient for the logarithmic degree calibration in (27). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma jacksonCoeff_factor_square_le_exp (K : ℕ) (hK : 0 < K) :
(16 * (↑(2 * (K - 1) + 1) : ℝ) ^ 4 *
(1 + sqrt 2) ^ (4 * (2 * (K - 1)))) ^ 2 ≤
exp (304 * (K : ℝ)) := by
let D := 2 * (K - 1)
have hd : 0 ≤ (D : ℝ) := Nat.cast_nonneg _
have hD : (D : ℝ) ≤ 2 * (K : ℝ) := by
exact_mod_cast (show D ≤ 2 * K by dsimp [D]; omega)
have hk : (1 : ℝ) ≤ K := by exact_mod_cast hK
have hr : 1 + sqrt 2 ≤ exp 2 := by
have hs : sqrt 2 ≤ 2 := by
nlinarith [Real.sq_sqrt (by norm_num : (0 : ℝ) ≤ 2), Real.sqrt_nonneg 2]
linarith [Real.add_one_le_exp (2 : ℝ)]
have hdim : (↑(D + 1) : ℝ) ≤ exp D := by
push_cast
linarith [Real.add_one_le_exp (D : ℝ)]
have hconstant : (256 : ℝ) ≤ exp 256 := by
linarith [Real.add_one_le_exp (256 : ℝ)]
have h₁ := pow_le_pow_left₀ (by positivity : 0 ≤ (↑(D + 1) : ℝ)) hdim 8
have h₂ := pow_le_pow_left₀ (by positivity : 0 ≤ 1 + sqrt 2) hr (8 * D)
calc
_ = 256 * (↑(D + 1) : ℝ) ^ 8 * (1 + sqrt 2) ^ (8 * D) := by
dsimp only [D]
rw [mul_pow, mul_pow, ← pow_mul, ← pow_mul]
rw [show 4 * (2 * (K - 1)) * 2 = 8 * (2 * (K - 1)) by omega]
norm_num
_ ≤ exp 256 * (exp D) ^ 8 * (exp 2) ^ (8 * D) := by
gcongr
_ = exp (256 + 24 * (D : ℝ)) := by
rw [← Real.exp_nat_mul, ← Real.exp_nat_mul, ← Real.exp_add, ← Real.exp_add]
congr 1
push_cast
ring
_ ≤ _ := Real.exp_le_exp.mpr (by nlinarith)
Equations (14) and (25) give the full coefficient-times-factorial envelope in (26), with fixed universal linear and quadratic growth constants. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
ε :
ℝ
hε :
0 < ε
K :
ℕ
hK :
0 < K
c r :
Fin 4 → ℝ
hr :
∀ j, 0 < r j
hQ :
∀ v ∈ centeredRectangle c r, ∀ j, 0 ≤ v j
L :
ℝ
hL :
0 < L
conclusion
(∑ α : Fin 4 → Fin (2 * (K - 1) + 1), |jacksonCoeff ε K c r α|) ^ 2 * exp (32 * (2 * (K - 1) : ℕ) ^ 2 / L)
≤ clippingScaleFormula ε c r ^ 2 * exp (304 * (K : ℝ) + 128 * (K : ℝ) ^ 2 / L)
Proof (Lean source)
lemma jacksonCoeff_square_exp_le (ε : ℝ) (hε : 0 < ε) (K : ℕ) (hK : 0 < K)
(c r : Fin 4 → ℝ) (hr : ∀ j, 0 < r j)
(hQ : ∀ v ∈ centeredRectangle c r, ∀ j, 0 ≤ v j)
(L : ℝ) (hL : 0 < L) :
(∑ α : Fin 4 → Fin (2 * (K - 1) + 1), |jacksonCoeff ε K c r α|) ^ 2 *
exp (32 * (2 * (K - 1) : ℕ) ^ 2 / L) ≤
clippingScaleFormula ε c r ^ 2 * exp (304 * (K : ℝ) + 128 * (K : ℝ) ^ 2 / L) := by
let A := 16 * (↑(2 * (K - 1) + 1) : ℝ) ^ 4 *
(1 + sqrt 2) ^ (4 * (2 * (K - 1)))
have hc := jacksonCoeff_sum_abs_le ε hε K hK c r hr hQ
have hcsq := pow_le_pow_left₀ (Finset.sum_nonneg fun _ _ => abs_nonneg _) hc 2
have hfactor := jacksonCoeff_factor_square_le_exp K hK
have hD : (↑(2 * (K - 1)) : ℝ) ≤ 2 * (K : ℝ) := by
exact_mod_cast (show 2 * (K - 1) ≤ 2 * K by omega)
have hDs := pow_le_pow_left₀ (Nat.cast_nonneg (2 * (K - 1))) hD 2
have hquad : 32 * (↑(2 * (K - 1)) : ℝ) ^ 2 / L ≤ 128 * (K : ℝ) ^ 2 / L := by
apply div_le_div_of_nonneg_right _ hL.le
nlinarith
calc
_ ≤ (A * clippingScaleFormula ε c r) ^ 2 *
exp (32 * (↑(2 * (K - 1)) : ℝ) ^ 2 / L) :=
mul_le_mul_of_nonneg_right hcsq (Real.exp_pos _).le
_ = clippingScaleFormula ε c r ^ 2 *
(A ^ 2 * exp (32 * (↑(2 * (K - 1)) : ℝ) ^ 2 / L)) := by ring
_ ≤ clippingScaleFormula ε c r ^ 2 *
(exp (304 * (K : ℝ)) * exp (128 * (K : ℝ) ^ 2 / L)) := by
apply mul_le_mul_of_nonneg_left _ (sq_nonneg _)
exact mul_le_mul hfactor (Real.exp_le_exp.mpr hquad)
(Real.exp_pos _).le (Real.exp_pos _).le
_ = _ := by rw [← Real.exp_add]
theorem pilotFactorialCellValue_square_exp_le Lemma pilotFactorialCellValue_square_exp_le in the paper ↗
The actual localized factorial statistic satisfies equation (26) with its random armwise scale and universal exponential growth constants. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
W :
Fin 4 → Ω → ℕ
q :
Fin 4 → ℝ
hqnonneg :
∀ j, 0 ≤ q j
H :
ℝ
hH :
0 < H
hHlarge :
1 ≤ H
m L :
ℝ
hm :
0 < m
hL :
0 < L
Np :
Fin 4 → ℕ
hWlaw :
hind :
iIndepFun W μ
hq :
ε :
ℝ
hε :
0 < ε
K :
ℕ
hK :
0 < K
conclusion
b :
pilotMidpoint H hH m L Np
r :
pilotRadiusFormula H hH m L Np
(∫ ω, (factorialCellValue ε K m b r (fun j => W j ω) - armwiseExtensionFormula ε b) ^ 2 ∂μ)
≤ clippingScaleFormula ε b r ^ 2 * exp (304 * (K : ℝ) + 128 * (K : ℝ) ^ 2 / L)
Proof (Lean source)
lemma pilotFactorialCellValue_square_exp_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (W : Fin 4 → Ω → ℕ)
(q : Fin 4 → ℝ) (hqnonneg : ∀ j, 0 ≤ q j)
(H : ℝ) (hH : 0 < H) (hHlarge : 1 ≤ H) (m L : ℝ)
(hm : 0 < m) (hL : 0 < L) (Np : Fin 4 → ℕ)
(hWlaw : ∀ j, HasLaw (W j)
(poissonMeasure ⟨m * q j, mul_nonneg hm.le (hqnonneg j)⟩) μ)
(hind : iIndepFun W μ)
(hq : q ∈ pilotRectangle (pilotLower H hH m L Np) (pilotUpperFormula H hH m L Np))
(ε : ℝ) (hε : 0 < ε) (K : ℕ) (hK : 0 < K) :
let b := pilotMidpoint H hH m L Np
let r := pilotRadiusFormula H hH m L Np
(∫ ω, (factorialCellValue ε K m b r (fun j => W j ω) - armwiseExtensionFormula ε b) ^ 2 ∂μ) ≤
clippingScaleFormula ε b r ^ 2 * exp (304 * (K : ℝ) + 128 * (K : ℝ) ^ 2 / L) := by
exact (pilotFactorialCellValue_square_envelope μ W q hqnonneg H hH hHlarge m L
hm hL Np hWlaw hind hq ε K).trans
(jacksonCoeff_square_exp_le ε hε K hK _ _ (pilotRadius_pos H hH m L hm hL Np)
(pilotCenteredRectangle_nonneg H hH m L Np) L hL)
theorem pilotClippedCellValue_square_exp_le Lemma pilotClippedCellValue_square_exp_le in the paper ↗
The clipping contraction in (28) has the same derived exponential scale as (26). Localization supplies mean containment through the Jackson construction. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
W :
Fin 4 → Ω → ℕ
q :
Fin 4 → ℝ
hqnonneg :
∀ j, 0 ≤ q j
H :
ℝ
hH :
0 < H
hHlarge :
1 ≤ H
m L :
ℝ
hm :
0 < m
hL :
0 < L
Np :
Fin 4 → ℕ
hWlaw :
hind :
iIndepFun W μ
hq :
ε :
ℝ
hε :
0 < ε
K d :
ℕ
hK :
0 < K
hd :
1 ≤ d
conclusion
b :
pilotMidpoint H hH m L Np
r :
pilotRadiusFormula H hH m L Np
p :
+ ∑ α : Fin 4 → Fin (2 * (K - 1) + 1), jacksonCoeff ε K b r α * ∏ j, ((q j - b j) / r j) ^ (α j).val
(∫ ω, (clippedCellValueFormula ε K d m b r (fun j => W j ω) - p) ^ 2 ∂μ)
≤ clippingScaleFormula ε b r ^ 2 * exp (304 * (K : ℝ) + 128 * (K : ℝ) ^ 2 / L)
Proof (Lean source)
lemma pilotClippedCellValue_square_exp_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (W : Fin 4 → Ω → ℕ)
(q : Fin 4 → ℝ) (hqnonneg : ∀ j, 0 ≤ q j)
(H : ℝ) (hH : 0 < H) (hHlarge : 1 ≤ H) (m L : ℝ)
(hm : 0 < m) (hL : 0 < L) (Np : Fin 4 → ℕ)
(hWlaw : ∀ j, HasLaw (W j)
(poissonMeasure ⟨m * q j, mul_nonneg hm.le (hqnonneg j)⟩) μ)
(hind : iIndepFun W μ)
(hq : q ∈ pilotRectangle (pilotLower H hH m L Np) (pilotUpperFormula H hH m L Np))
(ε : ℝ) (hε : 0 < ε) (K d : ℕ) (hK : 0 < K) (hd : 1 ≤ d) :
let b := pilotMidpoint H hH m L Np
let r := pilotRadiusFormula H hH m L Np
let p := armwiseExtensionFormula ε b + ∑ α : Fin 4 → Fin (2 * (K - 1) + 1),
jacksonCoeff ε K b r α * ∏ j, ((q j - b j) / r j) ^ (α j).val
(∫ ω, (clippedCellValueFormula ε K d m b r (fun j => W j ω) - p) ^ 2 ∂μ) ≤
clippingScaleFormula ε b r ^ 2 * exp (304 * (K : ℝ) + 128 * (K : ℝ) ^ 2 / L) := by
exact (pilotClippedCellValue_square_envelope_of_localized μ W q hqnonneg H hH hHlarge
m L hm hL Np hWlaw hind hq ε hε K d hK hd).trans
(jacksonCoeff_square_exp_le ε hε K hK _ _ (pilotRadius_pos H hH m L hm hL Np)
(pilotCenteredRectangle_nonneg H hH m L Np) L hL)
Positive radii and nonnegative centers make the clipping scale strictly positive. Each anchored arm weight is at least one. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
0 < clippingScaleFormula ε c r
Proof (Lean source)
lemma clippingScale_pos (ε : ℝ) (c r : Fin 4 → ℝ)
(hc : ∀ j, 0 ≤ c j) (hr : ∀ j, 0 < r j) :
0 < clippingScaleFormula ε c r := by
have ht : 0 ≤ totalMassVecFormula c := by
rw [totalMassVec_eq_sum_coordinates]
exact Finset.sum_nonneg fun j _ => hc j
have hw (a : Bool) : 0 < armWeightFormula ε c a := by
have hd : 0 ≤ anchoredDenomFormula ε c a :=
(add_nonneg (hc _) (hc _)).trans (le_max_left _ _)
exact add_pos_of_pos_of_nonneg (by norm_num) (div_nonneg ht hd)
have hR (a : Bool) : 0 < armRadiusFormula r a := add_pos (hr _) (hr _)
simp only [clippingScaleFormula, Fintype.sum_bool]
exact mul_pos (by norm_num) (add_pos (mul_pos (hw true) (hR true))
(mul_pos (hw false) (hR false)))
The first assertion of (28) follows from the factorial mean identity, clipping displacement, and the derived coefficient envelope. The scale positivity needed for division is provided by the pilot construction. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
W :
Fin 4 → Ω → ℕ
q :
Fin 4 → ℝ
hqnonneg :
∀ j, 0 ≤ q j
H :
ℝ
hH :
0 < H
hHlarge :
1 ≤ H
m L :
ℝ
hm :
0 < m
hL :
0 < L
Np :
Fin 4 → ℕ
hWlaw :
hind :
iIndepFun W μ
hq :
ε :
ℝ
hε :
0 < ε
K d :
ℕ
hK :
0 < K
hd :
1 ≤ d
conclusion
b :
pilotMidpoint H hH m L Np
r :
pilotRadiusFormula H hH m L Np
p :
+ ∑ α : Fin 4 → Fin (2 * (K - 1) + 1), jacksonCoeff ε K b r α * ∏ j, ((q j - b j) / r j) ^ (α j).val
|(∫ ω, clippedCellValueFormula ε K d m b r (fun j => W j ω) ∂μ) - p|
≤ clippingScaleFormula ε b r * exp (304 * (K : ℝ) + 128 * (K : ℝ) ^ 2 / L) / (d : ℝ) ^ (1 / 4 : ℝ)
Proof (Lean source)
lemma pilotClippedCellValue_bias_exp_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (W : Fin 4 → Ω → ℕ)
(q : Fin 4 → ℝ) (hqnonneg : ∀ j, 0 ≤ q j)
(H : ℝ) (hH : 0 < H) (hHlarge : 1 ≤ H) (m L : ℝ)
(hm : 0 < m) (hL : 0 < L) (Np : Fin 4 → ℕ)
(hWlaw : ∀ j, HasLaw (W j)
(poissonMeasure ⟨m * q j, mul_nonneg hm.le (hqnonneg j)⟩) μ)
(hind : iIndepFun W μ)
(hq : q ∈ pilotRectangle (pilotLower H hH m L Np) (pilotUpperFormula H hH m L Np))
(ε : ℝ) (hε : 0 < ε) (K d : ℕ) (hK : 0 < K) (hd : 1 ≤ d) :
let b := pilotMidpoint H hH m L Np
let r := pilotRadiusFormula H hH m L Np
let p := armwiseExtensionFormula ε b + ∑ α : Fin 4 → Fin (2 * (K - 1) + 1),
jacksonCoeff ε K b r α * ∏ j, ((q j - b j) / r j) ^ (α j).val
|(∫ ω, clippedCellValueFormula ε K d m b r (fun j => W j ω) ∂μ) - p| ≤
clippingScaleFormula ε b r * exp (304 * (K : ℝ) + 128 * (K : ℝ) ^ 2 / L) /
(d : ℝ) ^ (1 / 4 : ℝ) := by
dsimp only
have hs := clippingScale_pos ε _ _ (pilotMidpoint_nonneg H hH m L hm Np)
(pilotRadius_pos H hH m L hm hL Np)
have hd0 : (0 : ℝ) < d := by exact_mod_cast (by omega : 0 < d)
have hw := mul_pos (Real.rpow_pos_of_pos hd0 (1 / 4 : ℝ)) hs
apply (pilotClippedCellValue_bias_envelope μ W q hqnonneg H hH hHlarge m L
hm hL Np hWlaw hind hq ε K d hw).trans
calc
_ ≤ (clippingScaleFormula ε (pilotMidpoint H hH m L Np) (pilotRadiusFormula H hH m L Np) ^ 2 *
exp (304 * (K : ℝ) + 128 * (K : ℝ) ^ 2 / L)) /
((d : ℝ) ^ (1 / 4 : ℝ) *
clippingScaleFormula ε (pilotMidpoint H hH m L Np) (pilotRadiusFormula H hH m L Np)) :=
div_le_div_of_nonneg_right
(jacksonCoeff_square_exp_le ε hε K hK _ _ (pilotRadius_pos H hH m L hm hL Np)
(pilotCenteredRectangle_nonneg H hH m L Np) L hL) hw.le
_ = _ := by field_simp
theorem pilotJacksonCoeff_square_tuning_exists Lemma pilotJacksonCoeff_square_tuning_exists in the paper ↗
A universal logarithmic degree tuning absorbs the derived coefficient and factorial inflation into the d^(1/16) scale in (27), for every pilot rectangle. In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ κ A : ℝ,
0 < κ
0 < A
conclusion 1
d :
ℕ
1 ≤ d
H :
ℝ
hH :
0 < H
m :
ℝ
0 < m
Np :
Fin 4 → ℕ
ε :
ℝ
0 < ε
b :
pilotMidpoint H hH m (logAlphabet d) Np
r :
pilotRadiusFormula H hH m (logAlphabet d) Np
K :
jacksonDegree κ d
(∑ α : Fin 4 → Fin (2 * (K - 1) + 1), |jacksonCoeff ε K b r α|) ^ 2 * exp (32 * (2 * (K - 1) : ℕ) ^ 2 / logAlphabet d)
≤ A * (d : ℝ) ^ (1 / 16 : ℝ) * clippingScaleFormula ε b r ^ 2
Proof (Lean source)
lemma pilotJacksonCoeff_square_tuning_exists :
∃ κ A : ℝ, 0 < κ ∧ 0 < A ∧ ∀ {d : ℕ}, 1 ≤ d →
∀ (H : ℝ) (hH : 0 < H) (m : ℝ), 0 < m → ∀ (Np : Fin 4 → ℕ)
(ε : ℝ), 0 < ε →
let b := pilotMidpoint H hH m (logAlphabet d) Np
let r := pilotRadiusFormula H hH m (logAlphabet d) Np
let K := jacksonDegree κ d
(∑ α : Fin 4 → Fin (2 * (K - 1) + 1), |jacksonCoeff ε K b r α|) ^ 2 *
exp (32 * (2 * (K - 1) : ℕ) ^ 2 / logAlphabet d) ≤
A * (d : ℝ) ^ (1 / 16 : ℝ) * clippingScaleFormula ε b r ^ 2 := by
obtain ⟨κ, A, hκ, hA, he⟩ := jacksonDegree_exponential_growth_le
304 128 (by norm_num) (by norm_num)
refine ⟨κ, A, hκ, hA, ?_⟩
intro d hd H hH m hm Np ε hε
dsimp only
have hL := logAlphabet_pos hd
have hK : 0 < jacksonDegree κ d := by
unfold jacksonDegree
omega
apply (jacksonCoeff_square_exp_le ε hε _ hK _ _
(pilotRadius_pos H hH m _ hm hL Np)
(pilotCenteredRectangle_nonneg H hH m _ Np) _ hL).trans
calc
_ ≤ clippingScaleFormula ε (pilotMidpoint H hH m (logAlphabet d) Np)
(pilotRadiusFormula H hH m (logAlphabet d) Np) ^ 2 *
(A * (d : ℝ) ^ (1 / 16 : ℝ)) :=
mul_le_mul_of_nonneg_left (he hd) (sq_nonneg _)
_ = _ := by ring
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.UpperJacksonOscillation 10 declarations The estimator's chosen polynomial inherits the tensor convolution oscillation bound (12).
Jackson mean containment in the clipping interval
The estimator's chosen polynomial inherits the tensor convolution oscillation bound (12). Polynomial uniqueness on the cube recovers its coordinate degrees, so the finite coefficient expression used by the factorial mean represents that same convolution, rather than an unrelated approximation polynomial.
Nonnegative centers and radii give a nonnegative armwise clipping scale. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
0 ≤ clippingScaleFormula ε c r
Proof (Lean source)
lemma clippingScale_nonneg (ε : ℝ) (c r : Fin 4 → ℝ)
(hc : ∀ j, 0 ≤ c j) (hr : ∀ j, 0 ≤ r j) :
0 ≤ clippingScaleFormula ε c r := by
have ht : 0 ≤ totalMassVecFormula c := by
simp only [totalMassVecFormula, armMassVecFormula]
exact add_nonneg (add_nonneg (hc _) (hc _)) (add_nonneg (hc _) (hc _))
have hd (a : Bool) : 0 ≤ anchoredDenomFormula ε c a :=
(add_nonneg (hc _) (hc _)).trans (le_max_left _ _)
unfold clippingScaleFormula armWeightFormula armRadiusFormula
exact mul_nonneg (by norm_num) (Finset.sum_nonneg fun a _ =>
mul_nonneg (add_nonneg (by norm_num) (div_nonneg ht (hd a)))
(add_nonneg (hr _) (hr _)))
theorem armwiseExtension_rectangle_oscillation_le Lemma armwiseExtension_rectangle_oscillation_le in the paper ↗
The armwise modulus bounds every point in a nonnegative rectangle by its center oscillation scale, retaining the arm weights and radii together. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
ε :
ℝ
hε :
0 < ε
c r :
Fin 4 → ℝ
hc :
∀ j, 0 ≤ c j
hc0 :
c ≠ 0
hQ :
∀ v ∈ centeredRectangle c r, ∀ j, 0 ≤ v j
v :
Fin 4 → ℝ
hv :
v ∈ centeredRectangle c r
conclusion
Proof (Lean source)
lemma armwiseExtension_rectangle_oscillation_le (ε : ℝ) (hε : 0 < ε)
(c r : Fin 4 → ℝ) (hc : ∀ j, 0 ≤ c j) (hc0 : c ≠ 0)
(hQ : ∀ v ∈ centeredRectangle c r, ∀ j, 0 ≤ v j)
(v : Fin 4 → ℝ) (hv : v ∈ centeredRectangle c r) :
|armwiseExtensionFormula ε v - armwiseExtensionFormula ε c| ≤ clippingScaleFormula ε c r := by
have ht : 0 ≤ totalMassVecFormula c := by
simp only [totalMassVecFormula, armMassVecFormula]
exact add_nonneg (add_nonneg (hc _) (hc _)) (add_nonneg (hc _) (hc _))
have hw (a : Bool) : 0 ≤ armWeightFormula ε c a := by
unfold armWeightFormula
exact add_nonneg (by norm_num) (div_nonneg ht
((add_nonneg (hc _) (hc _)).trans (le_max_left _ _)))
have hdelta (a : Bool) : armDelta v c a ≤ armRadiusFormula r a := by
simpa only [armDelta, armRadiusFormula, Fintype.sum_bool, add_comm] using
add_le_add (hv (cellIdx a false)) (hv (cellIdx a true))
calc
_ ≤ 2 * (∑ a : Bool, armDelta v c a) +
2 * (∑ a : Bool, totalMassVecFormula c / anchoredDenomFormula ε c a * armDelta v c a) :=
armwise_modulus_extension.1 ε hε v c (hQ v hv) hc hc0
_ = 2 * ∑ a : Bool, armWeightFormula ε c a * armDelta v c a := by
simp only [armWeightFormula, Fintype.sum_bool]
ring
_ ≤ clippingScaleFormula ε c r := by
apply mul_le_mul_of_nonneg_left _ (by norm_num : (0 : ℝ) ≤ 2)
exact Finset.sum_le_sum fun a _ => mul_le_mul_of_nonneg_left (hdelta a) (hw a)
The actual pair selected in the estimator represents the canonical tensor convolution whenever its construction conditions hold. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
ε :
ℝ
hε :
0 < ε
K :
ℕ
hK :
0 < K
c r :
Fin 4 → ℝ
hr :
∀ j, 0 < r j
hQ :
∀ v ∈ centeredRectangle c r, ∀ j, 0 ≤ v j
x :
Fin 4 → ℝ
conclusion
eval (cosPoint x) (jacksonPolynomialPair ε K c r).2
Proof (Lean source)
lemma jacksonPolynomialPair_cos_eval (ε : ℝ) (hε : 0 < ε) (K : ℕ) (hK : 0 < K)
(c r : Fin 4 → ℝ) (hr : ∀ j, 0 < r j)
(hQ : ∀ v ∈ centeredRectangle c r, ∀ j, 0 ≤ v j) (x : Fin 4 → ℝ) :
eval (cosPoint x) (jacksonPolynomialPair ε K c r).2 =
tensorConvolution K (fun z => armwiseExtensionFormula ε (affinePoint c r z)) x := by
unfold jacksonPolynomialPair
simp only [dif_pos hK, dif_pos hε, dif_pos hr, dif_pos hQ]
exact (Classical.choose_spec (Classical.choose_spec
(affineJackson_exists_mvPolynomial_four hK c r hr (armwiseExtensionFormula ε)
(armwise_continuousOn_rectangle ε hε c r hr hQ)))).2.1 x
theorem jacksonPolynomialPair_normalized_degree_le Lemma jacksonPolynomialPair_normalized_degree_le in the paper ↗
Polynomial uniqueness on the cube recovers the normalized representative's coordinate degree bound even though the affine existence interface exposes only the physical representative's degrees. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
ε :
ℝ
hε :
0 < ε
K :
ℕ
hK :
0 < K
c r :
Fin 4 → ℝ
hr :
∀ j, 0 < r j
hQ :
∀ v ∈ centeredRectangle c r, ∀ j, 0 ≤ v j
j :
Fin 4
conclusion
(jacksonPolynomialPair ε K c r).2.degreeOf j ≤ 2 * (K - 1)
Proof (Lean source)
lemma jacksonPolynomialPair_normalized_degree_le (ε : ℝ) (hε : 0 < ε)
(K : ℕ) (hK : 0 < K) (c r : Fin 4 → ℝ) (hr : ∀ j, 0 < r j)
(hQ : ∀ v ∈ centeredRectangle c r, ∀ j, 0 ≤ v j) (j : Fin 4) :
(jacksonPolynomialPair ε K c r).2.degreeOf j ≤ 2 * (K - 1) := by
have haff : Continuous (affinePoint c r) := by
unfold affinePoint
fun_prop
have hf : ContinuousOn (fun z => armwiseExtensionFormula ε (affinePoint c r z))
(normalizedCube 4) :=
(armwise_continuousOn_rectangle ε hε c r hr hQ).comp haff.continuousOn
(fun z hz => affinePoint_mem_centeredRectangle c r z hr hz)
obtain ⟨p, hp, hdeg, _⟩ := tensorConvolution_exists_mvPolynomial hK _ hf
have heq : (jacksonPolynomialPair ε K c r).2 = p := by
apply MvPolynomial.funext_set (fun _ => Icc (-1 : ℝ) 1)
(fun _ => Set.Icc_infinite (by norm_num : (-1 : ℝ) < 1))
intro z hz
let x : Fin 4 → ℝ := fun i => arccos (z i)
have hcos : cosPoint x = z := by
funext i
exact Real.cos_arccos (hz i (Set.mem_univ i)).1 (hz i (Set.mem_univ i)).2
rw [← hcos, jacksonPolynomialPair_cos_eval ε hε K hK c r hr hQ, hp]
rw [heq]
exact MvPolynomial.degreeOf_le_iff.mpr (fun m hm => hdeg m hm j)
The finite centered coefficient sum used in the Poisson mean is exactly the normalized Jackson polynomial evaluation, without a truncation remainder. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
ε :
ℝ
hε :
0 < ε
K :
ℕ
hK :
0 < K
c r :
Fin 4 → ℝ
hr :
∀ j, 0 < r j
hQ :
∀ v ∈ centeredRectangle c r, ∀ j, 0 ≤ v j
z :
Fin 4 → ℝ
conclusion
= eval z (jacksonPolynomialPair ε K c r).2
Proof (Lean source)
lemma jacksonCoeff_sum_eq_eval (ε : ℝ) (hε : 0 < ε) (K : ℕ) (hK : 0 < K)
(c r : Fin 4 → ℝ) (hr : ∀ j, 0 < r j)
(hQ : ∀ v ∈ centeredRectangle c r, ∀ j, 0 ≤ v j) (z : Fin 4 → ℝ) :
armwiseExtensionFormula ε c + ∑ α : Fin 4 → Fin (2 * (K - 1) + 1),
jacksonCoeff ε K c r α * ∏ j, z j ^ (α j).val =
eval z (jacksonPolynomialPair ε K c r).2 := by
let p := (jacksonPolynomialPair ε K c r).2 - C (armwiseExtensionFormula ε c)
have hdeg (j : Fin 4) : p.degreeOf j ≤ 2 * (K - 1) := by
exact (MvPolynomial.degreeOf_sub_le j _ _).trans (max_le
(jacksonPolynomialPair_normalized_degree_le ε hε K hK c r hr hQ j) (by simp))
have he := congrArg (eval z) (tensorPolynomial_tensorCoeffs p hdeg)
simp only [tensorPolynomial, map_sum, MvPolynomial.eval_monomial,
tensorExponent] at he
simp_rw [Finsupp.prod_fintype _ (fun i e => z i ^ e) (fun i => pow_zero (z i))] at he
simp only [Finsupp.equivFunOnFinite_symm_apply_apply] at he
change (∑ α : Fin 4 → Fin (2 * (K - 1) + 1),
jacksonCoeff ε K c r α * ∏ j, z j ^ (α j).val) = eval z p at he
rw [he]
simp only [p, map_sub, MvPolynomial.eval_C]
ring
A normalized positive Jackson kernel preserves a uniform oscillation bound around any fixed center. Continuity supplies the integral's regularity. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d K :
ℕ
hK :
0 < K
f :
(Fin d → ℝ) → ℝ
hf :
ContinuousOn f (normalizedCube d)
c B :
ℝ
hbound :
∀ z ∈ normalizedCube d, |f z - c| ≤ B
x :
Fin d → ℝ
conclusion
|tensorConvolution K f x - c| ≤ B
Proof (Lean source)
lemma tensorConvolution_abs_sub_le {d K : ℕ} (hK : 0 < K)
(f : (Fin d → ℝ) → ℝ) (hf : ContinuousOn f (normalizedCube d))
(c B : ℝ) (hbound : ∀ z ∈ normalizedCube d, |f z - c| ≤ B)
(x : Fin d → ℝ) : |tensorConvolution K f x - c| ≤ B := by
have hc : Continuous (fun u : Fin d → ℝ => f (cosPoint (x - u))) := by
apply hf.comp_continuous
· unfold cosPoint
fun_prop
· exact fun u => cosPoint_mem_normalizedCube _
have hk : Continuous (tensorJackson K d) := by
unfold tensorJackson
fun_prop
have hcompact : IsCompact (periodBox d) :=
isCompact_pi_infinite fun _ => isCompact_Icc
have hfi : IntegrableOn (fun u => f (cosPoint (x - u)) * tensorJackson K d u)
(periodBox d) := (hc.mul hk).continuousOn.integrableOn_compact hcompact
have hci : IntegrableOn (fun u => c * tensorJackson K d u) (periodBox d) :=
(integrableOn_tensorJackson K d).const_mul c
have hdi : IntegrableOn (fun u => (f (cosPoint (x - u)) - c) * tensorJackson K d u)
(periodBox d) := ((hc.sub continuous_const).mul hk).continuousOn.integrableOn_compact hcompact
have hid : tensorConvolution K f x - c =
∫ u in periodBox d, (f (cosPoint (x - u)) - c) * tensorJackson K d u := by
calc
_ = (∫ u in periodBox d, f (cosPoint (x - u)) * tensorJackson K d u) -
∫ u in periodBox d, c * tensorJackson K d u := by
rw [integral_const_mul, tensorJackson_integral_eq_one hK, mul_one]
rfl
_ = _ := by
rw [← integral_sub hfi hci]
apply integral_congr_ae
exact ae_of_all _ fun u => by ring
rw [hid]
calc
_ ≤ ∫ u in periodBox d, |(f (cosPoint (x - u)) - c) * tensorJackson K d u| :=
abs_integral_le_integral_abs
_ ≤ ∫ u in periodBox d, B * tensorJackson K d u := by
apply setIntegral_mono hdi.abs ((integrableOn_tensorJackson K d).const_mul B)
intro u
change |(f (cosPoint (x - u)) - c) * tensorJackson K d u| ≤ B * tensorJackson K d u
rw [abs_mul, abs_of_nonneg (tensorJackson_nonneg hK u)]
exact mul_le_mul_of_nonneg_right
(hbound _ (cosPoint_mem_normalizedCube _)) (tensorJackson_nonneg hK u)
_ = B := by rw [integral_const_mul, tensorJackson_integral_eq_one hK, mul_one]
Equation (12) holds for the exact finite coefficient expression appearing as the Poisson factorial mean. The oscillation scale is derived from the modulus and the positive normalized kernel rather than assumed as a containment premise. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
ε :
ℝ
hε :
0 < ε
K :
ℕ
hK :
0 < K
c r :
Fin 4 → ℝ
hr :
∀ j, 0 < r j
hQ :
∀ v ∈ centeredRectangle c r, ∀ j, 0 ≤ v j
q :
Fin 4 → ℝ
hq :
q ∈ centeredRectangle c r
conclusion
|(armwiseExtensionFormula ε c + ∑ α : Fin 4 → Fin (2 * (K - 1) + 1), jacksonCoeff ε K c r α * ∏ j, ((q j - c j) / r j) ^ (α j).val)
- armwiseExtensionFormula ε c|
≤ clippingScaleFormula ε c r
Proof (Lean source)
lemma jacksonCoeff_mean_oscillation_le (ε : ℝ) (hε : 0 < ε)
(K : ℕ) (hK : 0 < K) (c r : Fin 4 → ℝ) (hr : ∀ j, 0 < r j)
(hQ : ∀ v ∈ centeredRectangle c r, ∀ j, 0 ≤ v j)
(q : Fin 4 → ℝ) (hq : q ∈ centeredRectangle c r) :
|(armwiseExtensionFormula ε c + ∑ α : Fin 4 → Fin (2 * (K - 1) + 1),
jacksonCoeff ε K c r α * ∏ j, ((q j - c j) / r j) ^ (α j).val) -
armwiseExtensionFormula ε c| ≤ clippingScaleFormula ε c r := by
have hlo (j : Fin 4) : 0 < c j := by
have h := hQ (affinePoint c r (fun _ => -1))
(affinePoint_mem_centeredRectangle c r _ hr (by intro i; constructor <;> norm_num)) j
change 0 ≤ c j + r j * (-1) at h
linarith [hr j]
have hc : ∀ j, 0 ≤ c j := fun j => (hlo j).le
have hc0 : c ≠ 0 := by
intro h
have hpos := hlo 0
simp only [h, Pi.zero_apply] at hpos
exact (lt_irrefl 0) hpos
have haff : Continuous (affinePoint c r) := by unfold affinePoint; fun_prop
have hf : ContinuousOn (fun z => armwiseExtensionFormula ε (affinePoint c r z))
(normalizedCube 4) :=
(armwise_continuousOn_rectangle ε hε c r hr hQ).comp haff.continuousOn
(fun z hz => affinePoint_mem_centeredRectangle c r z hr hz)
have hn := normalizedPoint_mem_normalizedCube c r q hr hq
let x : Fin 4 → ℝ := fun j => arccos (normalizedPoint c r q j)
have hcos : cosPoint x = normalizedPoint c r q := by
funext j
exact Real.cos_arccos (hn j).1 (hn j).2
rw [jacksonCoeff_sum_eq_eval ε hε K hK c r hr hQ]
change |eval (normalizedPoint c r q) (jacksonPolynomialPair ε K c r).2 - _| ≤ _
rw [← hcos, jacksonPolynomialPair_cos_eval ε hε K hK c r hr hQ]
apply tensorConvolution_abs_sub_le hK _ hf
intro z hz
exact armwiseExtension_rectangle_oscillation_le ε hε c r hc hc0 hQ _
(affinePoint_mem_centeredRectangle c r z hr hz)
The pilot's centered rectangle is nonnegative by the truncated lower endpoint, with no extra regularity or positivity premise on the counts. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma pilotCenteredRectangle_nonneg (H : ℝ) (hH : 0 < H) (m L : ℝ)
(Np : Fin 4 → ℕ) (v : Fin 4 → ℝ)
(hv : v ∈ centeredRectangle (pilotMidpoint H hH m L Np) (pilotRadiusFormula H hH m L Np))
(j : Fin 4) : 0 ≤ v j := by
have hl := (abs_le.mp (hv j)).1
have hn := le_max_left 0 (pilotCenterFormula m Np j - pilotHalfWidthFormula H hH m L Np j)
dsimp [pilotMidpoint, pilotRadiusFormula, pilotLower, pilotUpperFormula] at hl
linarith
theorem pilotJackson_mean_mem_clipping_interval Lemma pilotJackson_mean_mem_clipping_interval in the paper ↗
On the actual pilot rectangle, the exact factorial mean belongs to the clipping interval for every positive alphabet size, including null cells. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
H :
ℝ
hH :
0 < H
m L :
ℝ
hm :
0 < m
hL :
0 < L
Np :
Fin 4 → ℕ
ε :
ℝ
hε :
0 < ε
K d :
ℕ
hK :
0 < K
hd :
1 ≤ d
q :
Fin 4 → ℝ
hq :
conclusion
b :
pilotMidpoint H hH m L Np
r :
pilotRadiusFormula H hH m L Np
0 ≤ (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε b r
|(armwiseExtensionFormula ε b + ∑ α : Fin 4 → Fin (2 * (K - 1) + 1), jacksonCoeff ε K b r α * ∏ j, ((q j - b j) / r j) ^ (α j).val)
- armwiseExtensionFormula ε b|
≤ (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε b r
Proof (Lean source)
lemma pilotJackson_mean_mem_clipping_interval (H : ℝ) (hH : 0 < H) (m L : ℝ)
(hm : 0 < m) (hL : 0 < L) (Np : Fin 4 → ℕ)
(ε : ℝ) (hε : 0 < ε) (K d : ℕ) (hK : 0 < K) (hd : 1 ≤ d)
(q : Fin 4 → ℝ)
(hq : q ∈ pilotRectangle (pilotLower H hH m L Np) (pilotUpperFormula H hH m L Np)) :
let b := pilotMidpoint H hH m L Np
let r := pilotRadiusFormula H hH m L Np
0 ≤ (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε b r ∧
|(armwiseExtensionFormula ε b + ∑ α : Fin 4 → Fin (2 * (K - 1) + 1),
jacksonCoeff ε K b r α * ∏ j, ((q j - b j) / r j) ^ (α j).val) -
armwiseExtensionFormula ε b| ≤ (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε b r := by
dsimp only
have hr := pilotRadius_pos H hH m L hm hL Np
have hQ := pilotCenteredRectangle_nonneg H hH m L Np
have hs := clippingScale_nonneg ε _ _ (pilotMidpoint_nonneg H hH m L hm Np)
(fun j => (hr j).le)
have hdreal : (1 : ℝ) ≤ d := by exact_mod_cast hd
have hpow : 1 ≤ (d : ℝ) ^ (1 / 4 : ℝ) :=
Real.one_le_rpow hdreal (by norm_num)
constructor
· exact mul_nonneg (Real.rpow_nonneg (Nat.cast_nonneg _) _) hs
· exact (jacksonCoeff_mean_oscillation_le ε hε K hK _ _ hr hQ q
(pilotRectangle_center_bound H hH m L Np q hq)).trans
(le_mul_of_one_le_left hs hpow)
theorem pilotClippedCellValue_square_envelope_of_localized Lemma pilotClippedCellValue_square_envelope_of_localized in the paper ↗
Localization alone suffices for the clipping second-moment transfer in (28): the mean containment and nonnegative width are supplied by the Jackson construction. The coefficient norm remains explicit for the later bound (14). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
W :
Fin 4 → Ω → ℕ
q :
Fin 4 → ℝ
hqnonneg :
∀ j, 0 ≤ q j
H :
ℝ
hH :
0 < H
hHlarge :
1 ≤ H
m L :
ℝ
hm :
0 < m
hL :
0 < L
Np :
Fin 4 → ℕ
hWlaw :
∀ j,
hind :
iIndepFun W μ
hq :
ε :
ℝ
hε :
0 < ε
K d :
ℕ
hK :
0 < K
hd :
1 ≤ d
conclusion
b :
pilotMidpoint H hH m L Np
r :
pilotRadiusFormula H hH m L Np
p :
+ ∑ α : Fin 4 → Fin (2 * (K - 1) + 1), jacksonCoeff ε K b r α * ∏ j, ((q j - b j) / r j) ^ (α j).val
(∫ ω, (clippedCellValueFormula ε K d m b r (fun j => W j ω) - p) ^ 2 ∂μ)
≤ (∑ α : Fin 4 → Fin (2 * (K - 1) + 1), |jacksonCoeff ε K b r α|) ^ 2 * exp (32 * (2 * (K - 1) : ℕ) ^ 2 / L)
Proof (Lean source)
lemma pilotClippedCellValue_square_envelope_of_localized {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (W : Fin 4 → Ω → ℕ)
(q : Fin 4 → ℝ) (hqnonneg : ∀ j, 0 ≤ q j)
(H : ℝ) (hH : 0 < H) (hHlarge : 1 ≤ H) (m L : ℝ)
(hm : 0 < m) (hL : 0 < L) (Np : Fin 4 → ℕ)
(hWlaw : ∀ j, HasLaw (W j)
(poissonMeasure ⟨m * q j, mul_nonneg hm.le (hqnonneg j)⟩) μ)
(hind : iIndepFun W μ)
(hq : q ∈ pilotRectangle (pilotLower H hH m L Np) (pilotUpperFormula H hH m L Np))
(ε : ℝ) (hε : 0 < ε) (K d : ℕ) (hK : 0 < K) (hd : 1 ≤ d) :
let b := pilotMidpoint H hH m L Np
let r := pilotRadiusFormula H hH m L Np
let p := armwiseExtensionFormula ε b + ∑ α : Fin 4 → Fin (2 * (K - 1) + 1),
jacksonCoeff ε K b r α * ∏ j, ((q j - b j) / r j) ^ (α j).val
(∫ ω, (clippedCellValueFormula ε K d m b r (fun j => W j ω) - p) ^ 2 ∂μ) ≤
(∑ α : Fin 4 → Fin (2 * (K - 1) + 1), |jacksonCoeff ε K b r α|) ^ 2 *
exp (32 * (2 * (K - 1) : ℕ) ^ 2 / L) := by
obtain ⟨hwidth, hmean⟩ := pilotJackson_mean_mem_clipping_interval
H hH m L hm hL Np ε hε K d hK hd q hq
exact pilotClippedCellValue_square_envelope μ W q hqnonneg H hH hHlarge m L
hm hL Np hWlaw hind hq ε K d hwidth hmean
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.UpperLocalizedBias 6 declarations The true-mass radius and boundary estimates are combined with occupied-cell overlap.
Inverse-arm weighted localized bias envelope
The true-mass radius and boundary estimates are combined with occupied-cell overlap. This reduces the right side of roadmap (11) to the square-root and linear scales preceding degree calibration in (20), without detaching an inverse arm mass from its radius.
theorem observed_inverse_arm_bounds
An occupied cell's inverse arm fraction lies between one and the inverse public overlap floor. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Proof (Lean source)
lemma observed_inverse_arm_bounds {d : ℕ} (ε : ℝ) (P : DiscreteLaw d)
(hε : 0 < ε) (hP : ObservedClass ε P) (x : Fin d)
(hp : 0 < cellMass P x) (a : Bool) :
1 ≤ cellMass P x / armMass P a x ∧
cellMass P x / armMass P a x ≤ 1 / ε := by
have hmass : totalMassVecFormula (cellVector P x) = cellMass P x := by
simp [totalMassVecFormula, armMassVecFormula, cellVector, cellMass, cellIdx]
ring
have hfloor : ε * cellMass P x ≤ armMass P a x := by
rw [← anchoredDenom_cellVector_eq_armMass ε P hP x hp a, ← hmass]
exact le_max_right _ _
have ha : 0 < armMass P a x := lt_of_lt_of_le (mul_pos hε hp) hfloor
have hother : 0 ≤ armMass P (!a) x :=
Finset.sum_nonneg fun _ _ => toReal_nonneg
constructor
· apply (le_div_iff₀ ha).2
cases a <;> simp [cellMass, armMass] at * <;> linarith
· apply (div_le_div_iff₀ ha hε).2
simpa only [one_mul, mul_comm] using hfloor
Weighting the localized radii by their own inverse arm fractions retains one square-root inverse-overlap factor and one linear inverse-overlap factor. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
ε :
ℝ
P :
hε :
0 < ε
hP :
ObservedClass ε P
x :
Fin d
hp :
0 < cellMass P x
H :
ℝ
hH :
0 < H
hHlarge :
1 ≤ H
m L :
ℝ
hm :
0 < m
hL :
0 ≤ L
Np :
Fin 4 → ℕ
hloc :
conclusion
≤ 4 * (H * (H + 2)) * (sqrt (cellMass P x * L / (m * ε)) + L / (m * ε))
Proof (Lean source)
lemma observed_pilot_weightedRadius_le {d : ℕ} (ε : ℝ) (P : DiscreteLaw d)
(hε : 0 < ε) (hP : ObservedClass ε P) (x : Fin d) (hp : 0 < cellMass P x)
(H : ℝ) (hH : 0 < H) (hHlarge : 1 ≤ H) (m L : ℝ)
(hm : 0 < m) (hL : 0 ≤ L) (Np : Fin 4 → ℕ)
(hloc : cellVector P x ∈
pilotRectangle (pilotLower H hH m L Np) (pilotUpperFormula H hH m L Np)) :
(∑ a : Bool, cellMass P x / armMass P a x *
armRadiusFormula (pilotRadiusFormula H hH m L Np) a) ≤
4 * (H * (H + 2)) *
(sqrt (cellMass P x * L / (m * ε)) + L / (m * ε)) := by
have hq (j : Fin 4) : 0 ≤ cellVector P x j := ENNReal.toReal_nonneg
have hmass : totalMassVecFormula (cellVector P x) = cellMass P x := by
simp [totalMassVecFormula, armMassVecFormula, cellVector, cellMass, cellIdx]
ring
have ht (a : Bool) : 0 ≤ armMassVecFormula (cellVector P x) a :=
add_nonneg (hq _) (hq _)
have hterm (a : Bool) := anchored_weighted_radius_le
(B := totalMassVecFormula (cellVector P x)) (t := armMassVecFormula (cellVector P x) a)
(by rw [hmass]; exact hp.le) (ht a) (div_nonneg hL hm.le) hε
(show 0 ≤ 2 * (H * (H + 2)) by positivity)
(by simpa only [mul_div_assoc] using
(pilotArmRadius_le_true_scale H hH hHlarge m L hm hL Np
(cellVector P x) hq (fun j => (hloc j).1) a))
have hbound (a : Bool) : cellMass P x / armMass P a x *
armRadiusFormula (pilotRadiusFormula H hH m L Np) a ≤
2 * (H * (H + 2)) *
(sqrt (cellMass P x * L / (m * ε)) + L / (m * ε)) := by
have h := hterm a
change totalMassVecFormula (cellVector P x) /
anchoredDenomFormula ε (cellVector P x) a * _ ≤ _ at h
rw [hmass, anchoredDenom_cellVector_eq_armMass ε P hP x hp a] at h
simpa only [mul_div_assoc, div_div] using h
simp only [Fintype.sum_bool]
linarith [hbound false, hbound true]
The entire occupied-cell envelope in (11), including its unweighted term, reduces to the localized scales needed in (20). This is a deterministic bound on the approximation envelope; it does not assume a polynomial error bound. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
ε :
ℝ
P :
hε :
0 < ε
hP :
ObservedClass ε P
x :
Fin d
hp :
0 < cellMass P x
H :
ℝ
hH :
0 < H
hHlarge :
1 ≤ H
m L K :
ℝ
hm :
0 < m
hL :
0 ≤ L
hK :
0 < K
Np :
Fin 4 → ℕ
hloc :
conclusion
(∑ a : Bool, (1 + cellMass P x / armMass P a x) * ((∑ y : Bool, sqrt (((cellVector P x) (cellIdx a y) - pilotLower H hH m L Np (cellIdx a y)) * (pilotUpperFormula H hH m L Np (cellIdx a y) - (cellVector P x) (cellIdx a y)))) / K + armRadiusFormula (pilotRadiusFormula H hH m L Np) a / K ^ 2))
+ 8 * (H * (H + 2)) * (sqrt (cellMass P x * L / (m * ε)) + L / (m * ε)) / K ^ 2
Proof (Lean source)
lemma observed_pilot_biasEnvelope_le {d : ℕ} (ε : ℝ) (P : DiscreteLaw d)
(hε : 0 < ε) (hP : ObservedClass ε P) (x : Fin d) (hp : 0 < cellMass P x)
(H : ℝ) (hH : 0 < H) (hHlarge : 1 ≤ H) (m L K : ℝ)
(hm : 0 < m) (hL : 0 ≤ L) (hK : 0 < K) (Np : Fin 4 → ℕ)
(hloc : cellVector P x ∈
pilotRectangle (pilotLower H hH m L Np) (pilotUpperFormula H hH m L Np)) :
(∑ a : Bool, (1 + cellMass P x / armMass P a x) *
((∑ y : Bool, sqrt
(((cellVector P x) (cellIdx a y) - pilotLower H hH m L Np (cellIdx a y)) *
(pilotUpperFormula H hH m L Np (cellIdx a y) - (cellVector P x) (cellIdx a y)))) / K +
armRadiusFormula (pilotRadiusFormula H hH m L Np) a / K ^ 2)) ≤
8 * (H * (H + 2)) * sqrt 2 *
sqrt (cellMass P x * L / (m * ε)) / K +
8 * (H * (H + 2)) *
(sqrt (cellMass P x * L / (m * ε)) + L / (m * ε)) / K ^ 2 := by
let b (a : Bool) : ℝ := ∑ y : Bool, sqrt
(((cellVector P x) (cellIdx a y) - pilotLower H hH m L Np (cellIdx a y)) *
(pilotUpperFormula H hH m L Np (cellIdx a y) - (cellVector P x) (cellIdx a y)))
let r (a : Bool) : ℝ := armRadiusFormula (pilotRadiusFormula H hH m L Np) a
let w (a : Bool) : ℝ := cellMass P x / armMass P a x
have hr (a : Bool) : 0 ≤ r a := by
have hc (j : Fin 4) : 0 ≤ pilotCenterFormula m Np j := by dsimp [pilotCenterFormula]; positivity
have hh (j : Fin 4) : 0 ≤ pilotHalfWidthFormula H hH m L Np j := by
dsimp [pilotHalfWidthFormula]; positivity
have hj (j : Fin 4) : 0 ≤ pilotRadiusFormula H hH m L Np j := by
dsimp [pilotRadiusFormula, pilotLower, pilotUpperFormula]
rw [max_def]
split_ifs <;> linarith [hc j, hh j]
exact add_nonneg (hj _) (hj _)
have hb (a : Bool) : 0 ≤ b a := Finset.sum_nonneg fun _ _ => Real.sqrt_nonneg _
have hreduce : (∑ a : Bool, (1 + w a) * (b a / K + r a / K ^ 2)) ≤
2 * ((∑ a : Bool, w a * b a) / K + (∑ a : Bool, w a * r a) / K ^ 2) := by
calc
_ ≤ ∑ a : Bool, (2 * w a) * (b a / K + r a / K ^ 2) := by
apply Finset.sum_le_sum
intro a _
apply mul_le_mul_of_nonneg_right _
(add_nonneg (div_nonneg (hb a) hK.le) (div_nonneg (hr a) (sq_nonneg K)))
have hw := (observed_inverse_arm_bounds ε P hε hP x hp a).1
change 1 ≤ w a at hw
linarith
_ = _ := by simp only [Fintype.sum_bool]; ring
have hboundary := observed_pilot_weightedBoundary_le ε P hε hP x hp
H hH hHlarge m L hm hL Np hloc
have hradius := observed_pilot_weightedRadius_le ε P hε hP x hp
H hH hHlarge m L hm hL Np hloc
apply hreduce.trans
have hbdiv := div_le_div_of_nonneg_right hboundary hK.le
have hrdiv := div_le_div_of_nonneg_right hradius (sq_nonneg K)
change (∑ a : Bool, w a * b a) / K ≤ _ at hbdiv
change (∑ a : Bool, w a * r a) / K ^ 2 ≤ _ at hrdiv
convert mul_le_mul_of_nonneg_left (add_le_add hbdiv hrdiv)
(show (0 : ℝ) ≤ 2 by norm_num) using 1
ring
A degree at least a positive multiple of the log scale converts the boundary and radius terms into the normalized bias scales in (20). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
x y L K κ :
ℝ
hx :
0 ≤ x
hy :
0 ≤ y
hL :
1 ≤ L
hκ :
0 < κ
hdegree :
κ * L ≤ K
conclusion
L * x / K + (L * x + L ^ 2 * y) / K ^ 2 ≤ (1 / κ + 1 / κ ^ 2) * (x + y)
Proof (Lean source)
lemma logarithmic_degree_bias_scale_le {x y L K κ : ℝ}
(hx : 0 ≤ x) (hy : 0 ≤ y) (hL : 1 ≤ L) (hκ : 0 < κ)
(hdegree : κ * L ≤ K) :
L * x / K + (L * x + L ^ 2 * y) / K ^ 2 ≤
(1 / κ + 1 / κ ^ 2) * (x + y) := by
have hLp : 0 < L := by linarith
have hK : 0 < K := lt_of_lt_of_le (mul_pos hκ hLp) hdegree
have hfirst : L * x / K ≤ x / κ := by
apply (div_le_div_iff₀ hK hκ).2
convert mul_le_mul_of_nonneg_right hdegree hx using 1 <;> ring
have hsq := pow_le_pow_left₀ (mul_nonneg hκ.le hLp.le) hdegree 2
have hsecond : (L * x + L ^ 2 * y) / K ^ 2 ≤ (x + y) / κ ^ 2 := by
apply (div_le_div_iff₀ (sq_pos_of_pos hK) (sq_pos_of_pos hκ)).2
have hLsq : L ≤ L ^ 2 := by nlinarith
have hnum := mul_le_mul_of_nonneg_right hLsq hx
have hden := mul_le_mul_of_nonneg_right hsq (add_nonneg hx hy)
have hnumκ := mul_le_mul_of_nonneg_right hnum (sq_nonneg κ)
nlinarith
have hxy : x / κ ≤ (x + y) / κ :=
div_le_div_of_nonneg_right (le_add_of_nonneg_right hy) hκ.le
calc
_ ≤ (x + y) / κ + (x + y) / κ ^ 2 := add_le_add (hfirst.trans hxy) hsecond
_ = _ := by ring
The pre-calibration square-root scale is the log scale times the normalized scale; the identity also covers null cell masses. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma localized_bias_sqrt_scale_eq {p m ε L : ℝ}
(hm : 0 < m) (hε : 0 < ε) (hL : 0 < L) :
sqrt (p * L / (m * ε)) = L * sqrt (p / (m * ε * L)) := by
have he : p * L / (m * ε) = L ^ 2 * (p / (m * ε * L)) := by
field_simp
rw [he, Real.sqrt_mul (sq_nonneg L), Real.sqrt_sq hL.le]
theorem observed_pilot_biasEnvelope_normalized_le Lemma observed_pilot_biasEnvelope_normalized_le in the paper ↗
For occupied cells, the actual pilot envelope has the two bias scales in roadmap (20) once the degree is logarithmic. This bounds the envelope itself and leaves the tensor Jackson approximation step explicit upstream. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
ε :
ℝ
P :
hε :
0 < ε
hP :
ObservedClass ε P
x :
Fin d
hp :
0 < cellMass P x
H :
ℝ
hH :
0 < H
hHlarge :
1 ≤ H
m L K κ :
ℝ
hm :
0 < m
hL :
1 ≤ L
hκ :
0 < κ
hdegree :
κ * L ≤ K
Np :
Fin 4 → ℕ
hloc :
conclusion
(∑ a : Bool, (1 + cellMass P x / armMass P a x) * ((∑ y : Bool, sqrt (((cellVector P x) (cellIdx a y) - pilotLower H hH m L Np (cellIdx a y)) * (pilotUpperFormula H hH m L Np (cellIdx a y) - (cellVector P x) (cellIdx a y)))) / K + armRadiusFormula (pilotRadiusFormula H hH m L Np) a / K ^ 2))
Proof (Lean source)
lemma observed_pilot_biasEnvelope_normalized_le {d : ℕ} (ε : ℝ) (P : DiscreteLaw d)
(hε : 0 < ε) (hP : ObservedClass ε P) (x : Fin d) (hp : 0 < cellMass P x)
(H : ℝ) (hH : 0 < H) (hHlarge : 1 ≤ H) (m L K κ : ℝ)
(hm : 0 < m) (hL : 1 ≤ L) (hκ : 0 < κ) (hdegree : κ * L ≤ K)
(Np : Fin 4 → ℕ)
(hloc : cellVector P x ∈
pilotRectangle (pilotLower H hH m L Np) (pilotUpperFormula H hH m L Np)) :
(∑ a : Bool, (1 + cellMass P x / armMass P a x) *
((∑ y : Bool, sqrt
(((cellVector P x) (cellIdx a y) - pilotLower H hH m L Np (cellIdx a y)) *
(pilotUpperFormula H hH m L Np (cellIdx a y) - (cellVector P x) (cellIdx a y)))) / K +
armRadiusFormula (pilotRadiusFormula H hH m L Np) a / K ^ 2)) ≤
8 * (H * (H + 2)) * sqrt 2 * (1 / κ + 1 / κ ^ 2) *
(sqrt (cellMass P x / (m * ε * L)) + 1 / (m * ε * L)) := by
have hLp : 0 < L := by linarith
have hK : 0 < K := lt_of_lt_of_le (mul_pos hκ hLp) hdegree
have he := observed_pilot_biasEnvelope_le ε P hε hP x hp
H hH hHlarge m L K hm hLp.le hK Np hloc
have hs := localized_bias_sqrt_scale_eq (p := cellMass P x) hm hε hLp
have ht : L / (m * ε) = L ^ 2 * (1 / (m * ε * L)) := by
field_simp
have hn := logarithmic_degree_bias_scale_le
(Real.sqrt_nonneg (cellMass P x / (m * ε * L)))
(by positivity : 0 ≤ 1 / (m * ε * L)) hL hκ hdegree
have htwo : 1 ≤ sqrt 2 := by
nlinarith [Real.sq_sqrt (show (0 : ℝ) ≤ 2 by norm_num), Real.sqrt_nonneg 2]
have hr : 0 ≤ (sqrt (cellMass P x * L / (m * ε)) + L / (m * ε)) / K ^ 2 := by
positivity
have hcoef : 0 ≤ 8 * (H * (H + 2)) := by positivity
have hraise := mul_le_mul_of_nonneg_left
(mul_le_mul_of_nonneg_right htwo hr) hcoef
apply he.trans
calc
_ ≤ 8 * (H * (H + 2)) * sqrt 2 *
(sqrt (cellMass P x * L / (m * ε)) / K +
(sqrt (cellMass P x * L / (m * ε)) + L / (m * ε)) / K ^ 2) := by
convert add_le_add (le_refl
(8 * (H * (H + 2)) * sqrt 2 * sqrt (cellMass P x * L / (m * ε)) / K))
hraise using 1 <;> first | rfl | ring
_ ≤ _ := by
rw [hs, ht]
convert mul_le_mul_of_nonneg_left hn
(show 0 ≤ 8 * (H * (H + 2)) * sqrt 2 by positivity) using 1 <;> first | rfl | ring
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.UpperLocalizedClipping 4 declarations One universal degree tuning yields both assertions of roadmap (28).
Calibrated localized clipping
One universal degree tuning yields both assertions of roadmap (28). Combining its mean displacement with the occupied-cell Jackson bias (20) controls the actual clipped statistic around the cell target.
theorem pilotClippedCellValue_tuning_exists Lemma pilotClippedCellValue_tuning_exists in the paper ↗
The same universal degree choice controls clipping bias and second moment. The Poisson laws provide the factorial mean and integrability; localization provides containment. No coefficient or moment bound is assumed. In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ κ A : ℝ,
0 < κ
0 < A
conclusion 1
Ω :
Type*
μ :
Measure Ω
d :
ℕ
1 ≤ d
W :
Fin 4 → Ω → ℕ
q :
Fin 4 → ℝ
hqnonneg :
∀ j, 0 ≤ q j
H :
ℝ
hH :
0 < H
1 ≤ H
m :
ℝ
hm :
0 < m
Np :
Fin 4 → ℕ
iIndepFun W μ
q ∈ pilotRectangle (pilotLower H hH m (logAlphabet d) Np) (pilotUpperFormula H hH m (logAlphabet d) Np)
ε :
ℝ
0 < ε
b :
pilotMidpoint H hH m (logAlphabet d) Np
r :
pilotRadiusFormula H hH m (logAlphabet d) Np
K :
jacksonDegree κ d
p :
+ ∑ α : Fin 4 → Fin (2 * (K - 1) + 1), jacksonCoeff ε K b r α * ∏ j, ((q j - b j) / r j) ^ (α j).val
|(∫ ω, clippedCellValueFormula ε K d m b r (fun j => W j ω) ∂μ) - p|
≤ A * (d : ℝ) ^ (-(3 / 16 : ℝ)) * clippingScaleFormula ε b r
(∫ ω, (clippedCellValueFormula ε K d m b r (fun j => W j ω) - p) ^ 2 ∂μ)
≤ A * (d : ℝ) ^ (1 / 16 : ℝ) * clippingScaleFormula ε b r ^ 2
Proof (Lean source)
lemma pilotClippedCellValue_tuning_exists :
∃ κ A : ℝ, 0 < κ ∧ 0 < A ∧ ∀ {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] {d : ℕ}, 1 ≤ d →
∀ (W : Fin 4 → Ω → ℕ) (q : Fin 4 → ℝ) (hqnonneg : ∀ j, 0 ≤ q j)
(H : ℝ) (hH : 0 < H), 1 ≤ H → ∀ (m : ℝ) (hm : 0 < m) (Np : Fin 4 → ℕ),
(∀ j, HasLaw (W j)
(poissonMeasure ⟨m * q j, mul_nonneg hm.le (hqnonneg j)⟩) μ) →
iIndepFun W μ →
q ∈ pilotRectangle (pilotLower H hH m (logAlphabet d) Np)
(pilotUpperFormula H hH m (logAlphabet d) Np) →
∀ (ε : ℝ), 0 < ε →
let b := pilotMidpoint H hH m (logAlphabet d) Np
let r := pilotRadiusFormula H hH m (logAlphabet d) Np
let K := jacksonDegree κ d
let p := armwiseExtensionFormula ε b + ∑ α : Fin 4 → Fin (2 * (K - 1) + 1),
jacksonCoeff ε K b r α * ∏ j, ((q j - b j) / r j) ^ (α j).val
(|(∫ ω, clippedCellValueFormula ε K d m b r (fun j => W j ω) ∂μ) - p| ≤
A * (d : ℝ) ^ (-(3 / 16 : ℝ)) * clippingScaleFormula ε b r) ∧
((∫ ω, (clippedCellValueFormula ε K d m b r (fun j => W j ω) - p) ^ 2 ∂μ) ≤
A * (d : ℝ) ^ (1 / 16 : ℝ) * clippingScaleFormula ε b r ^ 2) := by
obtain ⟨κ, A, hκ, hA, htune⟩ := pilotJacksonCoeff_square_tuning_exists
refine ⟨κ, A, hκ, hA, ?_⟩
intro Ω _ μ _ d hd W q hqnonneg H hH hHlarge m hm Np hWlaw hind hq ε hε
dsimp only
have hL := logAlphabet_pos hd
have hK : 0 < jacksonDegree κ d := by unfold jacksonDegree; omega
have hs := clippingScale_pos ε (pilotMidpoint H hH m (logAlphabet d) Np)
(pilotRadiusFormula H hH m (logAlphabet d) Np) (pilotMidpoint_nonneg H hH m (logAlphabet d) hm Np)
(pilotRadius_pos H hH m _ hm hL Np)
have hd0 : (0 : ℝ) < d := by exact_mod_cast (by omega : 0 < d)
have hw := mul_pos (Real.rpow_pos_of_pos hd0 (1 / 4 : ℝ)) hs
have hc := htune hd H hH m hm Np ε hε
constructor
· apply (pilotClippedCellValue_bias_envelope μ W q hqnonneg H hH hHlarge
m _ hm hL Np hWlaw hind hq ε _ d hw).trans
apply (div_le_div_of_nonneg_right hc hw.le).trans_eq
have hp : (d : ℝ) ^ (1 / 16 : ℝ) / (d : ℝ) ^ (1 / 4 : ℝ) =
(d : ℝ) ^ (-(3 / 16 : ℝ)) := by
rw [← Real.rpow_sub hd0]
norm_num
rw [show A * (d : ℝ) ^ (1 / 16 : ℝ) *
clippingScaleFormula ε (pilotMidpoint H hH m (logAlphabet d) Np)
(pilotRadiusFormula H hH m (logAlphabet d) Np) ^ 2 /
((d : ℝ) ^ (1 / 4 : ℝ) *
clippingScaleFormula ε (pilotMidpoint H hH m (logAlphabet d) Np)
(pilotRadiusFormula H hH m (logAlphabet d) Np)) =
A * ((d : ℝ) ^ (1 / 16 : ℝ) / (d : ℝ) ^ (1 / 4 : ℝ)) *
clippingScaleFormula ε (pilotMidpoint H hH m (logAlphabet d) Np)
(pilotRadiusFormula H hH m (logAlphabet d) Np) by
field_simp]
rw [hp]
· exact (pilotClippedCellValue_square_envelope_of_localized μ W q hqnonneg
H hH hHlarge m _ hm hL Np hWlaw hind hq ε hε _ d hK hd).trans hc
theorem observed_pilotClippedCellValue_target_tuning_exists Lemma observed_pilotClippedCellValue_target_tuning_exists in the paper ↗
Combining (20) and both parts of (28) controls the actual clipped occupied-cell mean and loss around the statistical target. The polynomial approximation error is retained explicitly in the squared-loss bound. In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ κ A : ℝ,
0 < κ
0 < A
conclusion 1
Ω :
Type*
μ :
Measure Ω
d :
ℕ
1 ≤ d
ε :
ℝ
P :
0 < ε
ObservedClass ε P
x :
Fin d
0 < cellMass P x
W :
Fin 4 → Ω → ℕ
H :
ℝ
hH :
0 < H
1 ≤ H
m :
ℝ
hm :
0 < m
Np :
Fin 4 → ℕ
(∀ j, HasLaw (W j) (poissonMeasure ⟨m * cellVector P x j, mul_nonneg hm.le ENNReal.toReal_nonneg⟩) μ)
iIndepFun W μ
cellVector P x ∈ pilotRectangle (pilotLower H hH m (logAlphabet d) Np) (pilotUpperFormula H hH m (logAlphabet d) Np)
b :
pilotMidpoint H hH m (logAlphabet d) Np
r :
pilotRadiusFormula H hH m (logAlphabet d) Np
K :
jacksonDegree κ d
|(∫ ω, clippedCellValueFormula ε K d m b r (fun j => W j ω) ∂μ)
- armwiseExtensionFormula ε (cellVector P x)|
≤ B + A * (d : ℝ) ^ (-(3 / 16 : ℝ)) * clippingScaleFormula ε b r
(∫ ω, (clippedCellValueFormula ε K d m b r (fun j => W j ω) - armwiseExtensionFormula ε (cellVector P x)) ^ 2 ∂μ)
≤ 2 * A * (d : ℝ) ^ (1 / 16 : ℝ) * clippingScaleFormula ε b r ^ 2 + 2 * B ^ 2
Proof (Lean source)
lemma observed_pilotClippedCellValue_target_tuning_exists :
∃ κ A : ℝ, 0 < κ ∧ 0 < A ∧ ∀ {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] {d : ℕ}, 1 ≤ d →
∀ (ε : ℝ) (P : DiscreteLaw d), 0 < ε → ObservedClass ε P →
∀ (x : Fin d), 0 < cellMass P x →
∀ (W : Fin 4 → Ω → ℕ) (H : ℝ) (hH : 0 < H), 1 ≤ H →
∀ (m : ℝ) (hm : 0 < m) (Np : Fin 4 → ℕ),
(∀ j, HasLaw (W j)
(poissonMeasure ⟨m * cellVector P x j,
mul_nonneg hm.le ENNReal.toReal_nonneg⟩) μ) →
iIndepFun W μ →
cellVector P x ∈ pilotRectangle (pilotLower H hH m (logAlphabet d) Np)
(pilotUpperFormula H hH m (logAlphabet d) Np) →
let b := pilotMidpoint H hH m (logAlphabet d) Np
let r := pilotRadiusFormula H hH m (logAlphabet d) Np
let K := jacksonDegree κ d
let B := 512 * (H * (H + 2)) * sqrt 2 * (1 / (κ / 2) + 1 / (κ / 2) ^ 2) *
(sqrt (cellMass P x / (m * ε * logAlphabet d)) + 1 / (m * ε * logAlphabet d))
(|(∫ ω, clippedCellValueFormula ε K d m b r (fun j => W j ω) ∂μ) -
armwiseExtensionFormula ε (cellVector P x)| ≤
B + A * (d : ℝ) ^ (-(3 / 16 : ℝ)) * clippingScaleFormula ε b r) ∧
((∫ ω, (clippedCellValueFormula ε K d m b r (fun j => W j ω) -
armwiseExtensionFormula ε (cellVector P x)) ^ 2 ∂μ) ≤
2 * A * (d : ℝ) ^ (1 / 16 : ℝ) * clippingScaleFormula ε b r ^ 2 + 2 * B ^ 2) := by
obtain ⟨κ, A, hκ, hA, hclip⟩ := pilotClippedCellValue_tuning_exists
refine ⟨κ, A, hκ, hA, ?_⟩
intro Ω _ μ _ d hd ε P hε hP x hp W H hH hHlarge m hm Np hWlaw hind hloc
dsimp only
let b := pilotMidpoint H hH m (logAlphabet d) Np
let r := pilotRadiusFormula H hH m (logAlphabet d) Np
let K := jacksonDegree κ d
let p := armwiseExtensionFormula ε b + ∑ α : Fin 4 → Fin (2 * (K - 1) + 1),
jacksonCoeff ε K b r α * ∏ j, ((cellVector P x j - b j) / r j) ^ (α j).val
have hK : 0 < K := by dsimp [K]; unfold jacksonDegree; omega
have hL1 : 1 ≤ logAlphabet d := by
have hd1 : (1 : ℝ) ≤ d := by exact_mod_cast hd
rw [logAlphabet, Real.log_mul (ne_of_gt (Real.exp_pos _)) (by positivity), Real.log_exp]
linarith [Real.log_nonneg hd1]
have hb := observed_pilotJackson_mean_bias_normalized_le ε P hε hP x hp
H hH hHlarge m (logAlphabet d) (κ / 2) hm hL1 (by positivity)
K hK (jacksonDegree_ge_half_log κ d) Np hloc
obtain ⟨hbclip, hsclip⟩ := hclip μ hd W (cellVector P x)
(fun _ => toReal_nonneg) H hH hHlarge m hm Np hWlaw hind hloc ε hε
constructor
· exact (abs_sub_le _ p _).trans ((add_le_add hbclip hb).trans_eq (add_comm _ _))
· have hsq := pow_le_pow_left₀ (abs_nonneg _) hb 2
rw [sq_abs] at hsq
have hmeanmem := pilotJackson_mean_mem_clipping_interval H hH m _ hm
(logAlphabet_pos hd) Np ε hε K d hK hd (cellVector P x) hloc
have hsint : Integrable (fun ω =>
(clippedCellValueFormula ε K d m b r (fun j => W j ω) - p) ^ 2) μ := by
have hraw := factorialCellValue_memLp_two μ W _ hWlaw hind ε K m b r
have hT : Integrable (fun ω => clippedCellValueFormula ε K d m b r (fun j => W j ω)) μ := by
exact (integrable_const _).sup ((integrable_const _).inf
(hraw.integrable (by norm_num)))
have hmeas := (hT.sub (integrable_const p)).aestronglyMeasurable.pow 2
exact (factorialCellValue_centered_square_integrable μ W _ hWlaw hind
ε K m b r p).mono' hmeas (ae_of_all _ fun ω => by
rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)]
exact clippedCellValue_sq_error_le ε K d m b r _ p hmeanmem.1 hmeanmem.2)
calc
_ ≤ ∫ ω, 2 * (clippedCellValueFormula ε K d m b r (fun j => W j ω) - p) ^ 2 +
2 * (p - armwiseExtensionFormula ε (cellVector P x)) ^ 2 ∂μ := by
apply integral_mono_of_nonneg (ae_of_all _ fun _ => sq_nonneg _)
((hsint.const_mul 2).add (integrable_const _))
apply ae_of_all
intro ω
change (clippedCellValueFormula ε K d m b r (fun j => W j ω) -
armwiseExtensionFormula ε (cellVector P x)) ^ 2 ≤
2 * (clippedCellValueFormula ε K d m b r (fun j => W j ω) - p) ^ 2 +
2 * (p - armwiseExtensionFormula ε (cellVector P x)) ^ 2
nlinarith [sq_nonneg (clippedCellValueFormula ε K d m b r (fun j => W j ω) -
p - (p - armwiseExtensionFormula ε (cellVector P x)))]
_ = 2 * (∫ ω, (clippedCellValueFormula ε K d m b r (fun j => W j ω) - p) ^ 2 ∂μ) +
2 * (p - armwiseExtensionFormula ε (cellVector P x)) ^ 2 := by
rw [integral_add (hsint.const_mul 2) (integrable_const _), integral_const_mul]
simp
_ ≤ _ := by nlinarith [hsclip]
A zero-count pilot always contains the null atom, including at its lower boundary. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
H :
ℝ
hH :
0 < H
m L :
ℝ
hm :
0 < m
hL :
0 ≤ L
conclusion
(0 : Fin 4 → ℝ) ∈ pilotRectangle (pilotLower H hH m L (fun _ => 0)) (pilotUpperFormula H hH m L (fun _ => 0))
Proof (Lean source)
lemma zeroPilot_rectangle_contains_zero (H : ℝ) (hH : 0 < H) (m L : ℝ)
(hm : 0 < m) (hL : 0 ≤ L) :
(0 : Fin 4 → ℝ) ∈ pilotRectangle
(pilotLower H hH m L (fun _ => 0))
(pilotUpperFormula H hH m L (fun _ => 0)) := by
have hh : 0 ≤ H * (L / m) := by positivity
intro j
simp [pilotLower, pilotUpperFormula, pilotCenterFormula, pilotHalfWidthFormula, hh]
theorem zeroPilot_clippedCellValue_target_tuning_exists Lemma zeroPilot_clippedCellValue_target_tuning_exists in the paper ↗
The null-cell instance of (20) and (28) controls the clipped statistic's mean and second moment around the zero target, without inverse arm masses. The degree tuning is the same universal tuning used for occupied cells. In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ κ A : ℝ,
0 < κ
0 < A
conclusion 1
Ω :
Type*
μ :
Measure Ω
d :
ℕ
1 ≤ d
ε :
ℝ
0 < ε
ε ≤ 1
W :
Fin 4 → Ω → ℕ
H :
ℝ
hH :
0 < H
1 ≤ H
m :
ℝ
_hm :
0 < m
(∀ j, HasLaw (W j) (poissonMeasure 0) μ)
iIndepFun W μ
b :
pilotMidpoint H hH m (logAlphabet d) (fun _ => 0)
r :
pilotRadiusFormula H hH m (logAlphabet d) (fun _ => 0)
K :
jacksonDegree κ d
B :
(64 * H / (κ / 2) ^ 2) * (1 / (m * ε * logAlphabet d))
|(∫ ω, clippedCellValueFormula ε K d m b r (fun j => W j ω) ∂μ)
- armwiseExtensionFormula ε (0 : Fin 4 → ℝ)|
≤ B + A * (d : ℝ) ^ (-(3 / 16 : ℝ)) * clippingScaleFormula ε b r
(∫ ω, (clippedCellValueFormula ε K d m b r (fun j => W j ω) - armwiseExtensionFormula ε (0 : Fin 4 → ℝ)) ^ 2 ∂μ)
≤ 2 * A * (d : ℝ) ^ (1 / 16 : ℝ) * clippingScaleFormula ε b r ^ 2 + 2 * B ^ 2
Proof (Lean source)
lemma zeroPilot_clippedCellValue_target_tuning_exists :
∃ κ A : ℝ, 0 < κ ∧ 0 < A ∧ ∀ {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] {d : ℕ}, 1 ≤ d →
∀ (ε : ℝ), 0 < ε → ε ≤ 1 →
∀ (W : Fin 4 → Ω → ℕ) (H : ℝ) (hH : 0 < H), 1 ≤ H →
∀ (m : ℝ) (_hm : 0 < m),
(∀ j, HasLaw (W j)
(poissonMeasure 0) μ) →
iIndepFun W μ →
let b := pilotMidpoint H hH m (logAlphabet d) (fun _ => 0)
let r := pilotRadiusFormula H hH m (logAlphabet d) (fun _ => 0)
let K := jacksonDegree κ d
let B := (64 * H / (κ / 2) ^ 2) * (1 / (m * ε * logAlphabet d))
(|(∫ ω, clippedCellValueFormula ε K d m b r (fun j => W j ω) ∂μ) -
armwiseExtensionFormula ε (0 : Fin 4 → ℝ)| ≤
B + A * (d : ℝ) ^ (-(3 / 16 : ℝ)) * clippingScaleFormula ε b r) ∧
((∫ ω, (clippedCellValueFormula ε K d m b r (fun j => W j ω) -
armwiseExtensionFormula ε (0 : Fin 4 → ℝ)) ^ 2 ∂μ) ≤
2 * A * (d : ℝ) ^ (1 / 16 : ℝ) * clippingScaleFormula ε b r ^ 2 + 2 * B ^ 2) := by
obtain ⟨κ, A, hκ, hA, hclip⟩ := pilotClippedCellValue_tuning_exists
refine ⟨κ, A, hκ, hA, ?_⟩
intro Ω _ μ _ d hd ε hε hε1 W H hH hHlarge m hm hWlaw hind
have hloc := zeroPilot_rectangle_contains_zero H hH m (logAlphabet d) hm
(logAlphabet_pos hd).le
dsimp only
let b := pilotMidpoint H hH m (logAlphabet d) (fun _ => 0)
let r := pilotRadiusFormula H hH m (logAlphabet d) (fun _ => 0)
let K := jacksonDegree κ d
let p := armwiseExtensionFormula ε b + ∑ α : Fin 4 → Fin (2 * (K - 1) + 1),
jacksonCoeff ε K b r α * ∏ j, (((0 : Fin 4 → ℝ) j - b j) / r j) ^ (α j).val
have hK : 0 < K := by dsimp [K]; unfold jacksonDegree; omega
have hb := zeroPilot_jacksonCoeff_mean_normalized_le H hH m (logAlphabet d)
hm (logAlphabet_pos hd) ε hε hε1 K hK (κ / 2) (by positivity)
(jacksonDegree_ge_half_log κ d)
have hWlaw0 : ∀ j, HasLaw (W j)
(poissonMeasure ⟨m * (0 : Fin 4 → ℝ) j,
mul_nonneg hm.le (le_refl 0)⟩) μ := by
intro j
have hz : (⟨m * (0 : Fin 4 → ℝ) j,
mul_nonneg hm.le (le_refl 0)⟩ : NNReal) = 0 := by
ext
simp
rw [hz]
exact hWlaw j
obtain ⟨hbclip, hsclip⟩ := hclip μ hd W (0 : Fin 4 → ℝ)
(fun _ => le_rfl) H hH hHlarge m hm (fun _ => 0) hWlaw0 hind hloc ε hε
constructor
· exact (abs_sub_le _ p _).trans ((add_le_add hbclip hb).trans_eq (add_comm _ _))
· have hsq := pow_le_pow_left₀ (abs_nonneg _) hb 2
rw [sq_abs] at hsq
change (p - armwiseExtensionFormula ε 0) ^ 2 ≤ _ at hsq
have hmeanmem := pilotJackson_mean_mem_clipping_interval H hH m _ hm
(logAlphabet_pos hd) (fun _ => 0) ε hε K d hK hd (0 : Fin 4 → ℝ) hloc
have hsint : Integrable (fun ω =>
(clippedCellValueFormula ε K d m b r (fun j => W j ω) - p) ^ 2) μ := by
have hraw := factorialCellValue_memLp_two μ W _ hWlaw0 hind ε K m b r
have hT : Integrable (fun ω => clippedCellValueFormula ε K d m b r (fun j => W j ω)) μ := by
exact (integrable_const _).sup ((integrable_const _).inf
(hraw.integrable (by norm_num)))
have hmeas := (hT.sub (integrable_const p)).aestronglyMeasurable.pow 2
exact (factorialCellValue_centered_square_integrable μ W _ hWlaw0 hind
ε K m b r p).mono' hmeas (ae_of_all _ fun ω => by
rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)]
exact clippedCellValue_sq_error_le ε K d m b r _ p hmeanmem.1 hmeanmem.2)
calc
_ ≤ ∫ ω, 2 * (clippedCellValueFormula ε K d m b r (fun j => W j ω) - p) ^ 2 +
2 * (p - armwiseExtensionFormula ε (0 : Fin 4 → ℝ)) ^ 2 ∂μ := by
apply integral_mono_of_nonneg (ae_of_all _ fun _ => sq_nonneg _)
((hsint.const_mul 2).add (integrable_const _))
apply ae_of_all
intro ω
change (clippedCellValueFormula ε K d m b r (fun j => W j ω) -
armwiseExtensionFormula ε (0 : Fin 4 → ℝ)) ^ 2 ≤
2 * (clippedCellValueFormula ε K d m b r (fun j => W j ω) - p) ^ 2 +
2 * (p - armwiseExtensionFormula ε (0 : Fin 4 → ℝ)) ^ 2
nlinarith [sq_nonneg (clippedCellValueFormula ε K d m b r (fun j => W j ω) -
p - (p - armwiseExtensionFormula ε (0 : Fin 4 → ℝ)))]
_ = 2 * (∫ ω, (clippedCellValueFormula ε K d m b r (fun j => W j ω) - p) ^ 2 ∂μ) +
2 * (p - armwiseExtensionFormula ε (0 : Fin 4 → ℝ)) ^ 2 := by
rw [integral_add (hsint.const_mul 2) (integrable_const _), integral_const_mul]
simp
_ ≤ _ := by nlinarith [hsclip]
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.UpperLogAbsorption 7 declarations The polynomial decay in roadmap equations (31) and (33) absorbs both the second and fourth powers of the logarithmic alphabet scale.
Absorbing the logarithmic factors in upper-risk aggregation
The polynomial decay in roadmap equations (31) and (33) absorbs both the second and fourth powers of the logarithmic alphabet scale. These estimates are uniform over all nonempty alphabets, without imposing a relation to n.
theorem logAlphabet_pow_mul_decay_le
Any positive polynomial decay absorbs a fixed positive integer power of L_d. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
1 ≤ d
k :
ℕ
hk :
0 < k
δ :
ℝ
hδ :
0 < δ
conclusion
logAlphabet d ^ k * (d : ℝ) ^ (-δ) ≤ ((exp 1) ^ (δ / k) / (δ / k)) ^ k
Proof (Lean source)
lemma logAlphabet_pow_mul_decay_le {d : ℕ} (hd : 1 ≤ d)
(k : ℕ) (hk : 0 < k) (δ : ℝ) (hδ : 0 < δ) :
logAlphabet d ^ k * (d : ℝ) ^ (-δ) ≤
((exp 1) ^ (δ / k) / (δ / k)) ^ k := by
have hd0 : (0 : ℝ) < d := by exact_mod_cast (lt_of_lt_of_le Nat.zero_lt_one hd)
have hk0 : (0 : ℝ) < k := by exact_mod_cast hk
have he : 0 < δ / k := div_pos hδ hk0
have h := Real.log_le_rpow_div (x := exp 1 * d) (by positivity) he
have hpow := pow_le_pow_left₀ (logAlphabet_pos hd).le h k
change logAlphabet d ^ k ≤ ((exp 1 * d) ^ (δ / k) / (δ / k)) ^ k at hpow
have hid : ((exp 1 * d) ^ (δ / k) / (δ / k)) ^ k =
((exp 1) ^ (δ / k) / (δ / k)) ^ k * (d : ℝ) ^ δ := by
rw [Real.mul_rpow (Real.exp_pos _).le hd0.le]
rw [mul_div_right_comm, mul_pow, ← Real.rpow_mul_natCast hd0.le]
congr 2
field_simp
rw [hid] at hpow
have hm := mul_le_mul_of_nonneg_right hpow (Real.rpow_nonneg hd0.le (-δ))
have hcancel : (d : ℝ) ^ δ * (d : ℝ) ^ (-δ) = 1 := by
rw [← Real.rpow_add hd0, add_neg_cancel, Real.rpow_zero]
simpa only [mul_assoc, hcancel, mul_one] using hm
theorem logAlphabet_second_fourth_decay_bounded
The two logarithmic ratios in (31) and (33) have one universal envelope for each fixed positive decay exponent. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
δ :
ℝ
hδ :
0 < δ
conclusion
∃ A : ℝ,
0 < A
conclusion 1
d :
ℕ
1 ≤ d
(logAlphabet d ^ 2 + 8 * logAlphabet d ^ 4) * (d : ℝ) ^ (-δ) ≤ A
Proof (Lean source)
lemma logAlphabet_second_fourth_decay_bounded (δ : ℝ) (hδ : 0 < δ) :
∃ A : ℝ, 0 < A ∧ ∀ d : ℕ, 1 ≤ d →
(logAlphabet d ^ 2 + 8 * logAlphabet d ^ 4) * (d : ℝ) ^ (-δ) ≤ A := by
let A₂ := ((exp 1) ^ (δ / 2) / (δ / 2)) ^ (2 : ℕ)
let A₄ := ((exp 1) ^ (δ / 4) / (δ / 4)) ^ (4 : ℕ)
refine ⟨A₂ + 8 * A₄, by dsimp [A₂, A₄]; positivity, ?_⟩
intro d hd
have h₂ := logAlphabet_pow_mul_decay_le hd 2 (by norm_num) δ hδ
have h₄ := logAlphabet_pow_mul_decay_le hd 4 (by norm_num) δ hδ
dsimp [A₂, A₄]
nlinarith
On the active branch, the summed pilot second-moment scale is controlled by the logarithmic envelope that appears in (31) and (33). Here a = m ε. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d a L :
ℝ
ha :
0 < a
hL :
0 < L
hactive :
d ≤ 8 * a * L
conclusion
L / a + d * L ^ 2 / a ^ 2 ≤ (L ^ 2 + 8 * L ^ 4) / (a * L)
Proof (Lean source)
lemma activeBranch_pilot_scale_le (d a L : ℝ)
(ha : 0 < a) (hL : 0 < L) (hactive : d ≤ 8 * a * L) :
L / a + d * L ^ 2 / a ^ 2 ≤ (L ^ 2 + 8 * L ^ 4) / (a * L) := by
calc
_ ≤ L / a + (8 * a * L) * L ^ 2 / a ^ 2 := by gcongr
_ = _ := by field_simp
Equation (33)'s Cauchy–Schwarz step for the integrated random clipping scales. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
d :
ℕ
μ :
S :
Fin d → Ω → ℝ
hS :
∀ x, Integrable (S x) μ
hS₂ :
∀ x, Integrable (fun z => (S x z) ^ 2) μ
Proof (Lean source)
lemma clippingScale_integral_sum_sq_le {Ω : Type*} [MeasurableSpace Ω]
{d : ℕ} (μ : Measure Ω) [IsProbabilityMeasure μ]
(S : Fin d → Ω → ℝ) (hS : ∀ x, Integrable (S x) μ)
(hS₂ : ∀ x, Integrable (fun z => (S x z) ^ 2) μ) :
(∑ x : Fin d, ∫ z, S x z ∂μ) ^ 2 ≤
(d : ℝ) * ∑ x : Fin d, ∫ z, (S x z) ^ 2 ∂μ := by
have h := Finset.sum_mul_sq_le_sq_mul_sq (Finset.univ : Finset (Fin d))
(fun _ => (1 : ℝ)) (fun x => ∫ z, S x z ∂μ)
simp only [one_mul, one_pow, Finset.sum_const, Finset.card_univ,
Fintype.card_fin, nsmul_eq_mul, mul_one] at h
apply h.trans
gcongr with x
have hmem : MemLp (S x) 2 μ :=
(memLp_two_iff_integrable_sq (hS x).aestronglyMeasurable).2 (hS₂ x)
have hv := ProbabilityTheory.variance_eq_sub hmem
have hn := ProbabilityTheory.variance_nonneg (X := S x) (μ := μ)
simp only [Pi.pow_apply] at hv
linarith
theorem activeBranch_pilot_moment_sum_rate_le
Equations (31) and (33): the pilot moment sum, multiplied by any fixed sublinear alphabet power, is at most a universal constant times R = d/(m ε L). The premise is the cellwise pilot estimate (19), rather than a risk bound. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
δ :
ℝ
hδ :
0 < δ
Proof (Lean source)
lemma activeBranch_pilot_moment_sum_rate_le (δ : ℝ) (hδ : 0 < δ) :
∃ A : ℝ, 0 < A ∧ ∀ {d : ℕ}, 1 ≤ d → ∀ (P : DiscreteLaw d)
(v : Fin d → ℝ) (C a : ℝ), 0 ≤ C → 0 < a →
(d : ℝ) ≤ 8 * a * logAlphabet d →
(∀ x, v x ≤ C * (cellMass P x * logAlphabet d / a + logAlphabet d ^ 2 / a ^ 2)) →
(d : ℝ) ^ (1 - δ) * (∑ x : Fin d, v x) ≤
C * A * ((d : ℝ) / (a * logAlphabet d)) := by
obtain ⟨A, hA, hlog⟩ := logAlphabet_second_fourth_decay_bounded δ hδ
refine ⟨A, hA, ?_⟩
intro d hd P v C a hC ha hactive hv
have hd0 : (0 : ℝ) < d := by exact_mod_cast (lt_of_lt_of_le Nat.zero_lt_one hd)
have hL := logAlphabet_pos hd
have hsum := clippingScale_secondMoment_sum_le P v C (logAlphabet d) a 1 (by
intro x
simpa using hv x)
simp only [mul_one, one_pow] at hsum
have hid : (d : ℝ) ^ (1 - δ) = (d : ℝ) * (d : ℝ) ^ (-δ) := by
rw [sub_eq_add_neg, Real.rpow_add hd0, Real.rpow_one]
calc
_ ≤ (d : ℝ) ^ (1 - δ) *
(C * (logAlphabet d / a + (d : ℝ) * logAlphabet d ^ 2 / a ^ 2)) :=
mul_le_mul_of_nonneg_left hsum (Real.rpow_nonneg hd0.le _)
_ ≤ (d : ℝ) ^ (1 - δ) *
(C * ((logAlphabet d ^ 2 + 8 * logAlphabet d ^ 4) / (a * logAlphabet d))) := by
gcongr
exact activeBranch_pilot_scale_le d a (logAlphabet d) ha hL hactive
_ = C * ((logAlphabet d ^ 2 + 8 * logAlphabet d ^ 4) * (d : ℝ) ^ (-δ)) *
((d : ℝ) / (a * logAlphabet d)) := by rw [hid]; ring
_ ≤ C * A * ((d : ℝ) / (a * logAlphabet d)) := by
gcongr
exact hlog d hd
theorem activeBranch_clipping_bias_rate_le
Equation (33), including its final absorption into R, for actual integrated clipping scales. The second-moment premise is exactly the pilot estimate (19). In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ A : ℝ,
0 < A
conclusion 1
Ω :
Type
μ :
Measure Ω
d :
ℕ
1 ≤ d
P :
S :
Fin d → Ω → ℝ
C a :
ℝ
0 ≤ C
0 < a
(d : ℝ) ≤ 8 * a * logAlphabet d
(∀ x, Integrable (S x) μ)
(∀ x, Integrable (fun z => (S x z) ^ 2) μ)
(∀ x, (∫ z, (S x z) ^ 2 ∂μ) ≤ C * (cellMass P x * logAlphabet d / a + logAlphabet d ^ 2 / a ^ 2))
(d : ℝ) ^ (-(3 / 8 : ℝ)) * (∑ x : Fin d, ∫ z, S x z ∂μ) ^ 2
≤ C * A * ((d : ℝ) / (a * logAlphabet d))
Proof (Lean source)
lemma activeBranch_clipping_bias_rate_le :
∃ A : ℝ, 0 < A ∧ ∀ {Ω : Type} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] {d : ℕ}, 1 ≤ d →
∀ (P : DiscreteLaw d) (S : Fin d → Ω → ℝ) (C a : ℝ),
0 ≤ C → 0 < a → (d : ℝ) ≤ 8 * a * logAlphabet d →
(∀ x, Integrable (S x) μ) →
(∀ x, Integrable (fun z => (S x z) ^ 2) μ) →
(∀ x, (∫ z, (S x z) ^ 2 ∂μ) ≤
C * (cellMass P x * logAlphabet d / a + logAlphabet d ^ 2 / a ^ 2)) →
(d : ℝ) ^ (-(3 / 8 : ℝ)) * (∑ x : Fin d, ∫ z, S x z ∂μ) ^ 2 ≤
C * A * ((d : ℝ) / (a * logAlphabet d)) := by
obtain ⟨A, hA, hrate⟩ := activeBranch_pilot_moment_sum_rate_le (3 / 8) (by norm_num)
refine ⟨A, hA, ?_⟩
intro Ω _ μ _ d hd P S C a hC ha hactive hS hS₂ hv
have hd0 : (0 : ℝ) < d := by exact_mod_cast (lt_of_lt_of_le Nat.zero_lt_one hd)
calc
_ ≤ (d : ℝ) ^ (-(3 / 8 : ℝ)) *
((d : ℝ) * ∑ x : Fin d, ∫ z, (S x z) ^ 2 ∂μ) :=
mul_le_mul_of_nonneg_left (clippingScale_integral_sum_sq_le μ S hS hS₂)
(Real.rpow_nonneg hd0.le _)
_ = (d : ℝ) ^ (1 - (3 / 8 : ℝ)) *
(∑ x : Fin d, ∫ z, (S x z) ^ 2 ∂μ) := by
rw [sub_eq_add_neg, Real.rpow_add hd0, Real.rpow_one]
ring
_ ≤ _ := hrate hd P (fun x => ∫ z, (S x z) ^ 2 ∂μ) C a hC ha hactive hv
theorem activeBranch_variance_scale_rate_le
Equation (31) with the variance inflation d^(1/16) used by the factorial lift. In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
Proof (Lean source)
lemma activeBranch_variance_scale_rate_le :
∃ A : ℝ, 0 < A ∧ ∀ {d : ℕ}, 1 ≤ d → ∀ (P : DiscreteLaw d)
(v : Fin d → ℝ) (C a : ℝ), 0 ≤ C → 0 < a →
(d : ℝ) ≤ 8 * a * logAlphabet d →
(∀ x, v x ≤ C * (cellMass P x * logAlphabet d / a + logAlphabet d ^ 2 / a ^ 2)) →
(d : ℝ) ^ (1 / 16 : ℝ) * (∑ x : Fin d, v x) ≤
C * A * ((d : ℝ) / (a * logAlphabet d)) := by
convert activeBranch_pilot_moment_sum_rate_le (15 / 16) (by norm_num) using 1
norm_num
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.UpperMarkedAssembly 5 declarations Roadmap (29)–(34) is assembled for the actual marked-count construction.
Good-pilot inputs for the implemented marked statistic
Roadmap (29)–(34) is assembled for the actual marked-count construction. Count laws, pairwise cell independence, L² regularity, pilot-scale moments, and bad-pilot losses are discharged. Only the two integrated good-pilot content estimates remain as explicit inputs, retaining Jackson squared error.
def markedPoissonGoodPilot
The true cell vector belongs to the rectangle constructed from the actual pilot-marked observations.
Definition (Lean source)
parameters
def
given by
{s | let Np := markedCount (le_refl s.1) (fun i => (s.2 i).1) (fun i => (s.2 i).2) true x cellVector P x ∈ pilotRectangle (pilotLower H hH (poissonIntensityFormula n) (logAlphabet d) Np) (pilotUpperFormula H hH (poissonIntensityFormula n) (logAlphabet d) Np)}
theorem markedPoissonGoodPilot_measurableSet
Localization of the implemented pilot is a measurable event. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
MeasurableSet (markedPoissonGoodPilot n P H hH x)
Proof (Lean source)
lemma markedPoissonGoodPilot_measurableSet {d : ℕ} (n : ℕ) (P : DiscreteLaw d)
(H : ℝ) (hH : 0 < H) (x : Fin d) :
MeasurableSet (markedPoissonGoodPilot n P H hH x) := by
let f : FiniteSample (Obs d × Bool) → Fin 4 → ℕ := fun s =>
markedCount (le_refl s.1) (fun i => (s.2 i).1) (fun i => (s.2 i).2) true x
have hf : Measurable f := measurable_pi_lambda _
(fun j => markedPoissonCount_measurable d true x j)
change MeasurableSet (f ⁻¹' {z | cellVector P x ∈ pilotRectangle
(pilotLower H hH (poissonIntensityFormula n) (logAlphabet d) z)
(pilotUpperFormula H hH (poissonIntensityFormula n) (logAlphabet d) z)})
exact hf MeasurableSet.of_discrete
theorem markedPoissonPilot_iIndepFun
Actual pilot counts within a cell are independent Poisson coordinates. This discharges the independence input in the bad-pilot estimate (22). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
iIndepFun (fun j : Fin 4 => fun s : FiniteSample (Obs d × Bool) => markedCount (le_refl s.1) (fun i => (s.2 i).1) (fun i => (s.2 i).2) true x j) (finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4))
Proof (Lean source)
lemma markedPoissonPilot_iIndepFun {d : ℕ} (P : DiscreteLaw d) (n : ℕ) (x : Fin d) :
iIndepFun (fun j : Fin 4 => fun s : FiniteSample (Obs d × Bool) =>
markedCount (le_refl s.1) (fun i => (s.2 i).1) (fun i => (s.2 i).2) true x j)
(finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)) := by
let g : Fin 4 → Obs d × Bool := fun j =>
((x, decide (j.val / 2 = 1), decide (j.val % 2 = 1)), true)
have hg : Injective g := by
intro j k hjk
fin_cases j <;> fin_cases k <;> simp_all [g]
simp_rw [markedCount_eq_markedPoissonHistogram]
exact (markedPoissonHistogram_iIndepFun P n).precomp hg
The implemented bad-pilot cell loss satisfies (22), including null cells. No distributional or moment inputs remain for the caller. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
n d :
ℕ
hn :
1 ≤ n
hd :
1 ≤ d
P :
ε :
ℝ
hε :
0 < ε
hε1 :
ε ≤ 1
hP :
ObservedClass ε P
H :
ℝ
hHpos :
0 < H
hH :
universalH ≤ H
κ :
ℝ
x :
Fin d
conclusion
μ :
finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4)
Cd :
2 * (512 * H ^ 2 * (8 + 2 * (H / universalH) * productMomentConstant 4 1) + 1024 * sqrt (8 * scalarMomentConstant 4) * (H / universalH) ^ 2)
(∫ s, if s ∉ markedPoissonGoodPilot n P H hHpos x then (markedPoissonCellValue n d H hHpos κ ε x s - armwiseExtensionFormula ε (cellVector P x)) ^ 2 else 0 ∂μ)
≤ Cd * (d : ℝ) ^ (-4 : ℝ) * (cellMass P x * logAlphabet d / (poissonIntensityFormula n * ε) + logAlphabet d ^ 2 / (poissonIntensityFormula n * ε) ^ 2)
Proof (Lean source)
lemma markedPoissonCellValue_bad_sq_le {n d : ℕ} (hn : 1 ≤ n) (hd : 1 ≤ d)
(P : DiscreteLaw d) (ε : ℝ) (hε : 0 < ε) (hε1 : ε ≤ 1)
(hP : ObservedClass ε P) (H : ℝ) (hHpos : 0 < H) (hH : universalH ≤ H)
(κ : ℝ) (x : Fin d) :
let μ := finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)
let Cd := 2 * (512 * H ^ 2 *
(8 + 2 * (H / universalH) * productMomentConstant 4 1) +
1024 * sqrt (8 * scalarMomentConstant 4) * (H / universalH) ^ 2)
(∫ s, if s ∉ markedPoissonGoodPilot n P H hHpos x then
(markedPoissonCellValue n d H hHpos κ ε x s -
armwiseExtensionFormula ε (cellVector P x)) ^ 2 else 0 ∂μ) ≤
Cd * (d : ℝ) ^ (-4 : ℝ) *
(cellMass P x * logAlphabet d / (poissonIntensityFormula n * ε) +
logAlphabet d ^ 2 / (poissonIntensityFormula n * ε) ^ 2) := by
classical
have hm : 0 < (n : ℝ≥0) / 8 := by positivity
let μ := finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)
let W : Fin 4 → FiniteSample (Obs d × Bool) → ℕ := fun j s =>
markedCount (le_refl s.1) (fun i => (s.2 i).1) (fun i => (s.2 i).2) true x j
let Ne : FiniteSample (Obs d × Bool) → Fin 4 → ℕ := fun s =>
markedCount (le_refl s.1) (fun i => (s.2 i).1) (fun i => (s.2 i).2) false x
have hlaw (j : Fin 4) : HasLaw (W j)
(poissonMeasure (((n : ℝ≥0) / 8) *
⟨cellVector P x j, ENNReal.toReal_nonneg⟩)) μ := by
convert markedCount_poisson_hasLaw P n true x j using 1
congr 1
have ht := pilotRectangle_bad_clippedCellValue_alphabet_sq_le μ hd ε P hε hε1
hP x W Ne ((n : ℝ≥0) / 8) hm H hHpos hH
(fun j => markedPoissonCount_measurable d true x j) hlaw
(markedPoissonPilot_iIndepFun P n x) (jacksonDegree κ d)
simpa only [markedPoissonGoodPilot, Set.mem_setOf_eq, markedPoissonCellValue,
W, Ne, poissonIntensityFormula, NNReal.coe_div, NNReal.coe_natCast, NNReal.coe_ofNat,
mul_pow] using ht
theorem markedPoissonStatistic_goodPilotEstimates_rate_le Lemma markedPoissonStatistic_goodPilotEstimates_rate_le in the paper ↗
For the implemented statistic, only the good-pilot first and second moment content estimates remain in (29)–(34). The Jackson approximation term is retained; bad-pilot loss, independence, and regularity are derived. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
∃ C : ℝ,
0 < C
conclusion 1
n d :
ℕ
1 ≤ n
1 ≤ d
P :
ε :
ℝ
0 < ε
ε ≤ 1 / 2
ObservedClass ε P
(d : ℝ) / logAlphabet d ≤ (n : ℝ) * ε
μ :
finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4)
S :
markedPoissonClippingScale n d H hHpos ε
T :
markedPoissonCellValue n d H hHpos κ ε
E :
markedPoissonGoodPilot n P H hHpos
t :
fun x => armwiseExtensionFormula ε (cellVector P x)
(∀ x, (∫ s, if s ∈ E x then (T x s - t x) ^ 2 else 0 ∂μ) ≤ Cv * (d : ℝ) ^ (1 / 16 : ℝ) * (∫ s, S x s ^ 2 ∂μ) + Ce * (sqrt (cellMass P x / (poissonIntensityFormula n * ε * logAlphabet d)) + 1 / (poissonIntensityFormula n * ε * logAlphabet d)) ^ 2)
(∀ x, |∫ s, if s ∈ E x then T x s - t x else 0 ∂μ| ≤ Cb * (sqrt (cellMass P x / (poissonIntensityFormula n * ε * logAlphabet d)) + 1 / (poissonIntensityFormula n * ε * logAlphabet d) + (d : ℝ) ^ (-(3 / 16 : ℝ)) * (∫ s, S x s ∂μ)))
sqRisk μ (markedPoissonStatistic n d H hHpos κ ε) (observedValue P)
≤ C * ((d : ℝ) / (poissonIntensityFormula n * ε * logAlphabet d))
Proof (Lean source)
lemma markedPoissonStatistic_goodPilotEstimates_rate_le
(H : ℝ) (hHpos : 0 < H) (hH : universalH ≤ H)
(κ Cv Cb Ce : ℝ) (hCv : 0 ≤ Cv) (hCe : 0 ≤ Ce) :
∃ C : ℝ, 0 < C ∧ ∀ {n d : ℕ}, 1 ≤ n → 1 ≤ d →
∀ (P : DiscreteLaw d) (ε : ℝ), 0 < ε → ε ≤ 1 / 2 →
ObservedClass ε P →
(d : ℝ) / logAlphabet d ≤ (n : ℝ) * ε →
let μ := finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)
let S := markedPoissonClippingScale n d H hHpos ε
let T := markedPoissonCellValue n d H hHpos κ ε
let E := markedPoissonGoodPilot n P H hHpos
let t := fun x => armwiseExtensionFormula ε (cellVector P x)
(∀ x, (∫ s, if s ∈ E x then (T x s - t x) ^ 2 else 0 ∂μ) ≤
Cv * (d : ℝ) ^ (1 / 16 : ℝ) * (∫ s, S x s ^ 2 ∂μ) +
Ce * (sqrt (cellMass P x / (poissonIntensityFormula n * ε * logAlphabet d)) +
1 / (poissonIntensityFormula n * ε * logAlphabet d)) ^ 2) →
(∀ x, |∫ s, if s ∈ E x then T x s - t x else 0 ∂μ| ≤
Cb * (sqrt (cellMass P x / (poissonIntensityFormula n * ε * logAlphabet d)) +
1 / (poissonIntensityFormula n * ε * logAlphabet d) +
(d : ℝ) ^ (-(3 / 16 : ℝ)) * (∫ s, S x s ∂μ))) →
sqRisk μ (markedPoissonStatistic n d H hHpos κ ε)
(observedValue P) ≤ C * ((d : ℝ) / (poissonIntensityFormula n * ε * logAlphabet d)) := by
classical
let Cs := 512 * H ^ 2 * (1 + 9 * H)
let Cd := 2 * (512 * H ^ 2 *
(8 + 2 * (H / universalH) * productMomentConstant 4 1) +
1024 * sqrt (8 * scalarMomentConstant 4) * (H / universalH) ^ 2)
have hCs : 0 ≤ Cs := by dsimp [Cs]; positivity
have hCd : 0 ≤ Cd := by
have hu := universalH_pos
have hc := productMomentConstant_pos 4 1
dsimp [Cd]
positivity
obtain ⟨C, hC, hrate⟩ := activeBranch_pilotEvent_approximation_sqRisk_rate_le
Cs Cv Cb Cd Ce hCs hCv hCd hCe
refine ⟨C, hC, ?_⟩
intro n d hn hd P ε hε hεhalf hP hactive μ S T E t hg₂ hg₁
have hε1 : ε ≤ 1 := by linarith
have hm : 0 < poissonIntensityFormula n := by unfold poissonIntensityFormula; positivity
have ha : 0 < poissonIntensityFormula n * ε := mul_pos hm hε
have hactive' : (d : ℝ) ≤ 8 * (poissonIntensityFormula n * ε) * logAlphabet d := by
have hh := (div_le_iff₀ (logAlphabet_pos hd)).1 hactive
have he : 8 * (poissonIntensityFormula n * ε) * logAlphabet d =
(n : ℝ) * ε * logAlphabet d := by unfold poissonIntensityFormula; ring
rw [he]
exact hh
have hS (x : Fin d) : MemLp (S x) 2 μ :=
markedPoissonClippingScale_memLp_two hn hd P H hHpos hH ε hε hε1 x
have hT (x : Fin d) : MemLp (T x) 2 μ :=
markedPoissonCellValue_memLp_two hn hd P H hHpos hH κ ε hε hε1 x
have hmom (x : Fin d) : (∫ s, S x s ^ 2 ∂μ) ≤
Cs * (cellMass P x * logAlphabet d / (poissonIntensityFormula n * ε) +
logAlphabet d ^ 2 / (poissonIntensityFormula n * ε) ^ 2) := by
simpa only [mul_pow] using
markedPoissonClippingScale_integral_sq_le hn hd P H hHpos ε hε hεhalf x
have hb₂ (x : Fin d) : (∫ s, if s ∉ E x then (T x s - t x) ^ 2 else 0 ∂μ) ≤
Cd * (d : ℝ) ^ (-4 : ℝ) *
(cellMass P x * logAlphabet d / (poissonIntensityFormula n * ε) +
logAlphabet d ^ 2 / (poissonIntensityFormula n * ε) ^ 2) :=
markedPoissonCellValue_bad_sq_le hn hd P ε hε hε1 hP H hHpos hH κ x
have hr := hrate μ hd P T S t E (poissonIntensityFormula n * ε) ha hactive' hT
(markedPoissonCellValue_pairwise_indepFun P n H hHpos κ ε)
(fun x => markedPoissonGoodPilot_measurableSet n P H hHpos x)
(fun x => (hS x).integrable (by norm_num))
(fun x => (hS x).integrable_sq) hmom hg₂ hg₁ hb₂
have ht : (∑ x, t x) = observedValue P := by
simp only [t, armwiseExtension_cellVector ε P hε hP]
rfl
rw [ht] at hr
have hsum := markedPoissonCellValue_sum_memLp_two hn hd P H hHpos hH κ ε hε hε1
exact (projectUnit_sqRisk_le μ (fun s => ∑ x, T x s) (observedValue P)
hsum (observedValue_mem_unitInterval P)).trans hr
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.UpperMarkedConditioning 8 declarations The actual pilot and evaluation vectors are independent.
Averaging fixed-pilot bounds in the implemented experiment
The actual pilot and evaluation vectors are independent. Their joint law and Fubini decomposition justify freezing the pilot in roadmap (26)–(28), then averaging the resulting cell loss bounds.
def markedPoissonCellCounts
The four actual counts of a cell with a specified pilot/evaluation mark. For the displayed parameters, markedPoissonCellCounts is the object specified by this definition.
Definition (Lean source)
theorem markedPoissonCellCounts_measurable
Either actual count vector is measurable. The stated conclusion holds.
Formal statement
Proof (Lean source)
-- @node: markedPoissonCellCounts_measurable
@[fun_prop]
lemma markedPoissonCellCounts_measurable {d : ℕ} (a : Bool) (x : Fin d) :
Measurable (markedPoissonCellCounts a x) := by
exact measurable_pi_lambda _ (fun j => markedPoissonCount_measurable d a x j)
theorem markedPoissonCellCounts_pilot_indep_evaluation
The pilot and evaluation vectors of the same cell depend on disjoint marked atoms, so freezing the entire pilot preserves the evaluation law. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
IndepFun (markedPoissonCellCounts true x) (markedPoissonCellCounts false x) (finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4))
Proof (Lean source)
lemma markedPoissonCellCounts_pilot_indep_evaluation {d : ℕ}
(P : DiscreteLaw d) (n : ℕ) (x : Fin d) :
IndepFun (markedPoissonCellCounts true x) (markedPoissonCellCounts false x)
(finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)) := by
classical
let atoms (a : Bool) : Finset (Obs d × Bool) :=
Finset.univ.filter (fun z => z.1.1 = x ∧ z.2 = a)
let atom (a : Bool) (j : Fin 4) : atoms a :=
⟨((x, decide (j.val / 2 = 1), decide (j.val % 2 = 1)), a), by simp [atoms]⟩
let F (a : Bool) (v : atoms a → ℕ) : Fin 4 → ℕ := fun j => v (atom a j)
have heq (a : Bool) : markedPoissonCellCounts a x =
F a ∘ (fun s z => markedPoissonHistogram s (z : Obs d × Bool)) := by
funext s j
exact markedCount_eq_markedPoissonHistogram s a x j
have hdis : Disjoint (atoms true) (atoms false) := by
apply Finset.disjoint_left.2
intro z ht hf
simp only [atoms, mem_filter, Finset.mem_univ, true_and] at ht hf
exact Bool.noConfusion (ht.2.symm.trans hf.2)
rw [heq true, heq false]
exact ((markedPoissonHistogram_iIndepFun P n).indepFun_finset
(atoms true) (atoms false) hdis (fun z =>
(measurable_pi_apply z).comp (markedPoissonHistogram_measurable d))).comp
(Measurable.of_discrete (f := F true)) (Measurable.of_discrete (f := F false))
theorem markedPoissonCellCounts_hasLaw
The law of either actual cell vector is the same fourfold Poisson product. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
HasLaw (markedPoissonCellCounts a x) (Measure.pi (fun j : Fin 4 => poissonMeasure (((n : ℝ≥0) / 8) * (P.pmf (x, decide (j.val / 2 = 1), decide (j.val % 2 = 1))).toNNReal))) (finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4))
Proof (Lean source)
lemma markedPoissonCellCounts_hasLaw {d : ℕ} (P : DiscreteLaw d)
(n : ℕ) (a : Bool) (x : Fin d) :
HasLaw (markedPoissonCellCounts a x)
(Measure.pi (fun j : Fin 4 => poissonMeasure (((n : ℝ≥0) / 8) *
(P.pmf (x, decide (j.val / 2 = 1), decide (j.val % 2 = 1))).toNNReal)))
(finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)) := by
apply iIndepFun.hasLaw_pi (fun j => markedCount_poisson_hasLaw P n a x j)
cases a
· exact markedPoissonEvaluation_iIndepFun P n x
· exact markedPoissonPilot_iIndepFun P n x
theorem markedPoissonCellCounts_pair_hasLaw
The joint law needed to average the fixed-pilot clipping bounds is a product, with no conditional-law assumption left to discharge. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
ν :
Measure.pi (fun j : Fin 4 => poissonMeasure (((n : ℝ≥0) / 8) * (P.pmf (x, decide (j.val / 2 = 1), decide (j.val % 2 = 1))).toNNReal))
HasLaw (fun s => (markedPoissonCellCounts true x s, markedPoissonCellCounts false x s)) (ν.prod ν) (finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4))
Proof (Lean source)
lemma markedPoissonCellCounts_pair_hasLaw {d : ℕ} (P : DiscreteLaw d)
(n : ℕ) (x : Fin d) :
let ν := Measure.pi (fun j : Fin 4 => poissonMeasure (((n : ℝ≥0) / 8) *
(P.pmf (x, decide (j.val / 2 = 1), decide (j.val % 2 = 1))).toNNReal))
HasLaw (fun s => (markedPoissonCellCounts true x s,
markedPoissonCellCounts false x s)) (ν.prod ν)
(finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)) := by
exact (markedPoissonCellCounts_pilot_indep_evaluation P n x).hasLaw_prod
(markedPoissonCellCounts_hasLaw P n true x)
(markedPoissonCellCounts_hasLaw P n false x)
theorem markedPoissonCellCounts_integral_freeze_pilot
Integrating any integrable function of the actual two count vectors is exactly averaging its fixed-pilot evaluation integral against the pilot law. This applies to the good-pilot first and squared losses in (28). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
P :
n :
ℕ
x :
Fin d
hf :
Integrable (fun s => f (markedPoissonCellCounts true x s, markedPoissonCellCounts false x s)) (finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4))
conclusion
μ :
finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4)
ν :
Measure.pi (fun j : Fin 4 => poissonMeasure (((n : ℝ≥0) / 8) * (P.pmf (x, decide (j.val / 2 = 1), decide (j.val % 2 = 1))).toNNReal))
= ∫ v, ∫ s, f (v, markedPoissonCellCounts false x s) ∂μ ∂ν
Proof (Lean source)
lemma markedPoissonCellCounts_integral_freeze_pilot {d : ℕ}
(P : DiscreteLaw d) (n : ℕ) (x : Fin d)
(f : (Fin 4 → ℕ) × (Fin 4 → ℕ) → ℝ)
(hf : Integrable (fun s => f (markedPoissonCellCounts true x s,
markedPoissonCellCounts false x s))
(finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4))) :
let μ := finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)
let ν := Measure.pi (fun j : Fin 4 => poissonMeasure (((n : ℝ≥0) / 8) *
(P.pmf (x, decide (j.val / 2 = 1), decide (j.val % 2 = 1))).toNNReal))
(∫ s, f (markedPoissonCellCounts true x s, markedPoissonCellCounts false x s) ∂μ) =
∫ v, ∫ s, f (v, markedPoissonCellCounts false x s) ∂μ ∂ν := by
dsimp only
let ν := Measure.pi (fun j : Fin 4 => poissonMeasure (((n : ℝ≥0) / 8) *
(P.pmf (x, decide (j.val / 2 = 1), decide (j.val % 2 = 1))).toNNReal))
have hpair := markedPoissonCellCounts_pair_hasLaw P n x
have hmeas : StronglyMeasurable f := (Measurable.of_discrete).stronglyMeasurable
have hprod : Integrable f (ν.prod ν) := by
rw [← hpair.map_eq]
exact (integrable_map_measure hmeas.aestronglyMeasurable hpair.aemeasurable).2 hf
calc
_ = ∫ v, f v ∂(ν.prod ν) := hpair.integral_comp hmeas.aestronglyMeasurable
_ = ∫ v, ∫ w, f (v, w) ∂ν ∂ν := integral_prod f hprod
_ = _ := by
apply integral_congr_ae
filter_upwards with v
exact ((markedPoissonCellCounts_hasLaw P n false x).integral_comp
(Measurable.of_discrete.stronglyMeasurable.aestronglyMeasurable)).symm
theorem markedPoissonCellCounts_integral_le_of_fixedPilot
A nonnegative fixed-pilot loss bound averages to the corresponding bound for the actual marked experiment. Integrability of the pilot envelope is transported by its proved count law. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
P :
n :
ℕ
x :
Fin d
g :
(Fin 4 → ℕ) → ℝ
hf :
Integrable (fun s => f (markedPoissonCellCounts true x s, markedPoissonCellCounts false x s)) (finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4))
hg :
Integrable (fun s => g (markedPoissonCellCounts true x s)) (finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4))
hfnonneg :
∀ v, 0 ≤ f v
hfixed :
∀ v,
(∫ s, f (v, markedPoissonCellCounts false x s) ∂finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4))
≤ g v
conclusion
(∫ s, f (markedPoissonCellCounts true x s, markedPoissonCellCounts false x s) ∂finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4))
≤ ∫ s, g (markedPoissonCellCounts true x s) ∂finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4)
Proof (Lean source)
lemma markedPoissonCellCounts_integral_le_of_fixedPilot {d : ℕ}
(P : DiscreteLaw d) (n : ℕ) (x : Fin d)
(f : (Fin 4 → ℕ) × (Fin 4 → ℕ) → ℝ) (g : (Fin 4 → ℕ) → ℝ)
(hf : Integrable (fun s => f (markedPoissonCellCounts true x s,
markedPoissonCellCounts false x s))
(finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)))
(hg : Integrable (fun s => g (markedPoissonCellCounts true x s))
(finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)))
(hfnonneg : ∀ v, 0 ≤ f v)
(hfixed : ∀ v, (∫ s, f (v, markedPoissonCellCounts false x s)
∂finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)) ≤ g v) :
(∫ s, f (markedPoissonCellCounts true x s, markedPoissonCellCounts false x s)
∂finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)) ≤
∫ s, g (markedPoissonCellCounts true x s)
∂finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4) := by
let ν := Measure.pi (fun j : Fin 4 => poissonMeasure (((n : ℝ≥0) / 8) *
(P.pmf (x, decide (j.val / 2 = 1), decide (j.val % 2 = 1))).toNNReal))
have hlaw := markedPoissonCellCounts_hasLaw P n true x
have hgm : StronglyMeasurable g := Measurable.of_discrete.stronglyMeasurable
have hgi : Integrable g ν := by
dsimp only [ν]
rw [← hlaw.map_eq]
exact (integrable_map_measure hgm.aestronglyMeasurable hlaw.aemeasurable).2 hg
rw [markedPoissonCellCounts_integral_freeze_pilot P n x f hf]
calc
_ ≤ ∫ v, g v ∂ν := integral_mono_of_nonneg
(ae_of_all ν (fun v => integral_nonneg (fun s => hfnonneg _)))
hgi (ae_of_all ν hfixed)
_ = _ := (hlaw.integral_comp hgm.aestronglyMeasurable).symm
theorem markedPoissonCellCounts_abs_integral_le_of_fixedPilot
A fixed-pilot absolute bias bound averages to the corresponding bound for the actual marked experiment. Integrability of the pilot envelope is transported by its proved count law. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
P :
n :
ℕ
x :
Fin d
g :
(Fin 4 → ℕ) → ℝ
hf :
Integrable (fun s => f (markedPoissonCellCounts true x s, markedPoissonCellCounts false x s)) (finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4))
hg :
Integrable (fun s => g (markedPoissonCellCounts true x s)) (finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4))
hfixed :
∀ v,
|∫ s, f (v, markedPoissonCellCounts false x s) ∂finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4)|
≤ g v
conclusion
|∫ s, f (markedPoissonCellCounts true x s, markedPoissonCellCounts false x s) ∂finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4)|
≤ ∫ s, g (markedPoissonCellCounts true x s) ∂finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4)
Proof (Lean source)
lemma markedPoissonCellCounts_abs_integral_le_of_fixedPilot {d : ℕ}
(P : DiscreteLaw d) (n : ℕ) (x : Fin d)
(f : (Fin 4 → ℕ) × (Fin 4 → ℕ) → ℝ) (g : (Fin 4 → ℕ) → ℝ)
(hf : Integrable (fun s => f (markedPoissonCellCounts true x s,
markedPoissonCellCounts false x s))
(finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)))
(hg : Integrable (fun s => g (markedPoissonCellCounts true x s))
(finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)))
(hfixed : ∀ v, |∫ s, f (v, markedPoissonCellCounts false x s)
∂finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)| ≤ g v) :
|∫ s, f (markedPoissonCellCounts true x s, markedPoissonCellCounts false x s)
∂finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)| ≤
∫ s, g (markedPoissonCellCounts true x s)
∂finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4) := by
let ν := Measure.pi (fun j : Fin 4 => poissonMeasure (((n : ℝ≥0) / 8) *
(P.pmf (x, decide (j.val / 2 = 1), decide (j.val % 2 = 1))).toNNReal))
have hlaw := markedPoissonCellCounts_hasLaw P n true x
have hgm : StronglyMeasurable g := Measurable.of_discrete.stronglyMeasurable
have hgi : Integrable g ν := by
dsimp only [ν]
rw [← hlaw.map_eq]
exact (integrable_map_measure hgm.aestronglyMeasurable hlaw.aemeasurable).2 hg
rw [markedPoissonCellCounts_integral_freeze_pilot P n x f hf]
apply abs_integral_le_integral_abs.trans
calc
_ ≤ ∫ v, g v ∂ν := integral_mono_of_nonneg
(ae_of_all ν (fun v => abs_nonneg _)) hgi (ae_of_all ν hfixed)
_ = _ := (hlaw.integral_comp hgm.aestronglyMeasurable).symm
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.UpperMarkedCounts 13 declarations The histogram law realizes roadmap equations (7)–(8) for the counts used by markedPoissonStatistic.
Actual marked Poisson counts and cell regularity
The histogram law realizes roadmap equations (7)–(8) for the counts used by
markedPoissonStatistic. Marginal laws then discharge the unconditional
square-integrability required by the variance assembly in equation (29).
def markedPoissonHistogram
All observation–mark singleton counts in the uncapped finite sample. For the displayed parameters, markedPoissonHistogram is the object specified by this definition.
Definition (Lean source)
theorem markedPoissonHistogram_measurable
The complete histogram is measurable on each fixed-count fibre. The stated conclusion holds.
Formal statement
Proof (Lean source)
-- @node: markedPoissonHistogram_measurable
@[fun_prop]
lemma markedPoissonHistogram_measurable (d : ℕ) :
Measurable (markedPoissonHistogram (d := d)) := by
intro A hA
rw [MeasurableSpace.measurableSet_iInf]
intro k
change MeasurableSet ((fun z : Fin k → Obs d × Bool =>
markedPoissonHistogram ⟨k, z⟩) ⁻¹' A)
exact Measurable.of_discrete hA
theorem markedPoissonHistogram_hasLaw
The actual marked histogram has the product law of independent singleton Poisson counts, including zero-mass symbols. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
HasLaw (markedPoissonHistogram (d := d)) (independentPoissonCountLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4)) (finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4))
Proof (Lean source)
lemma markedPoissonHistogram_hasLaw {d : ℕ} (P : DiscreteLaw d) (n : ℕ) :
HasLaw (markedPoissonHistogram (d := d))
(independentPoissonCountLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4))
(finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)) := by
exact ⟨(markedPoissonHistogram_measurable d).aemeasurable,
finitePoissonSampleLaw_map_histogram _ _⟩
theorem markedCount_eq_markedPoissonHistogram
A coordinate of the marked histogram is exactly the implementation's pilot or evaluation count, with the same four-coordinate encoding. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
markedCount (le_refl s.1) (fun i => (s.2 i).1) (fun i => (s.2 i).2) pilot x j
Proof (Lean source)
lemma markedCount_eq_markedPoissonHistogram {d : ℕ}
(s : FiniteSample (Obs d × Bool)) (pilot : Bool) (x : Fin d) (j : Fin 4) :
markedCount (le_refl s.1) (fun i => (s.2 i).1) (fun i => (s.2 i).2) pilot x j =
markedPoissonHistogram s ((x, decide (j.val / 2 = 1),
decide (j.val % 2 = 1)), pilot) := by
classical
unfold markedCount markedPoissonHistogram finiteSampleHistogram
simp only [Fintype.card_subtype, Finset.sum_boole, Nat.cast_id]
apply congrArg card
ext i
simp only [FiniteSample.points, FiniteSample.count, Fin.eta, mem_filter,
Finset.mem_univ, true_and, Prod.eq_iff_fst_eq_snd_eq]
constructor
· intro h
exact ⟨Finset.mem_univ i, h.2, h.1⟩
· intro h
exact ⟨h.2.2, h.2.1⟩
theorem markedCount_poisson_hasLaw
Either mark gives the intended intensity n/8 times the observed atom mass. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
HasLaw (fun s : FiniteSample (Obs d × Bool) => markedCount (le_refl s.1) (fun i => (s.2 i).1) (fun i => (s.2 i).2) pilot x j) (poissonMeasure (((n : ℝ≥0) / 8) * (P.pmf (x, decide (j.val / 2 = 1), decide (j.val % 2 = 1))).toNNReal)) (finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4))
Proof (Lean source)
lemma markedCount_poisson_hasLaw {d : ℕ} (P : DiscreteLaw d) (n : ℕ)
(pilot : Bool) (x : Fin d) (j : Fin 4) :
HasLaw (fun s : FiniteSample (Obs d × Bool) =>
markedCount (le_refl s.1) (fun i => (s.2 i).1) (fun i => (s.2 i).2) pilot x j)
(poissonMeasure (((n : ℝ≥0) / 8) *
(P.pmf (x, decide (j.val / 2 = 1), decide (j.val % 2 = 1))).toNNReal))
(finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)) := by
have h := (measurePreserving_eval
(fun z : Obs d × Bool => poissonMeasure (((n : ℝ≥0) / 4) *
((P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) {z}).toNNReal))
((x, decide (j.val / 2 = 1), decide (j.val % 2 = 1)), pilot)).hasLaw.fun_comp
(markedPoissonHistogram_hasLaw P n)
simp_rw [← markedCount_eq_markedPoissonHistogram] at h
convert h using 1
congr 1
rw [← Set.singleton_prod_singleton, Measure.prod_prod,
PMF.toMeasure_apply_singleton, ENNReal.toNNReal_mul] <;> try measurability
have hb : (bernoulliBool (1 / 2) {pilot}).toNNReal = (1 / 2 : ℝ≥0) := by
cases pilot <;> apply NNReal.coe_injective <;>
simp only [bernoulliBool, Measure.add_apply, Measure.smul_apply,
Measure.dirac_apply' _ (measurableSet_singleton _)] <;>
norm_num <;> change (ofReal (1 / 2 : ℝ)).toReal = _ <;> norm_num
rw [hb]
ring
theorem markedPoissonHistogram_iIndepFun
All pilot and evaluation atom counts are mutually independent under the actual uncapped experiment, as in roadmap (8). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
iIndepFun (fun z s => markedPoissonHistogram (d := d) s z) (finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4))
Proof (Lean source)
lemma markedPoissonHistogram_iIndepFun {d : ℕ} (P : DiscreteLaw d) (n : ℕ) :
iIndepFun (fun z s => markedPoissonHistogram (d := d) s z)
(finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)) := by
let rates := fun z : Obs d × Bool => poissonMeasure (((n : ℝ≥0) / 4) *
((P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) {z}).toNNReal)
have hlaw := markedPoissonHistogram_hasLaw P n
exact (iIndepFun_iff_hasLaw_pi_pi (fun z =>
(measurePreserving_eval rates z).hasLaw.fun_comp hlaw)).2 hlaw
def markedPoissonCellValue
One implemented clipped contribution before the final unit projection. For the displayed parameters, markedPoissonCellValue is the object specified by this definition.
Definition (Lean source)
parameters
def
markedPoissonCellValue n d H hH κ ε x s :
ℝ
given by
Np :
Ne :
clippedCellValueFormula ε (jacksonDegree κ d) d (poissonIntensityFormula n) (pilotMidpoint H hH (poissonIntensityFormula n) (logAlphabet d) Np) (pilotRadiusFormula H hH (poissonIntensityFormula n) (logAlphabet d) Np) Ne
theorem markedPoissonCount_measurable
Each actual marked count is measurable, including on unbounded count fibres. The stated conclusion holds.
Formal statement
hypotheses
conclusion
Measurable (fun s : FiniteSample (Obs d × Bool) => markedCount (le_refl s.1) (fun i => (s.2 i).1) (fun i => (s.2 i).2) pilot x j)
Proof (Lean source)
-- @node: markedPoissonCount_measurable
@[fun_prop]
lemma markedPoissonCount_measurable (d : ℕ) (pilot : Bool) (x : Fin d) (j : Fin 4) :
Measurable (fun s : FiniteSample (Obs d × Bool) =>
markedCount (le_refl s.1) (fun i => (s.2 i).1) (fun i => (s.2 i).2) pilot x j) := by
simp_rw [markedCount_eq_markedPoissonHistogram]
exact (measurable_pi_apply _).comp (markedPoissonHistogram_measurable d)
theorem markedPoissonCellValue_memLp_two
Marginal Poisson laws supply unconditional L² regularity for every actual clipped cell, without assuming any moment bound for its factorial polynomial. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
n d :
ℕ
hn :
1 ≤ n
hd :
1 ≤ d
P :
H :
ℝ
hHpos :
0 < H
hH :
universalH ≤ H
κ ε :
ℝ
hε :
0 < ε
hε1 :
ε ≤ 1
x :
Fin d
conclusion
MemLp (markedPoissonCellValue n d H hHpos κ ε x) 2 (finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4))
Proof (Lean source)
lemma markedPoissonCellValue_memLp_two {n d : ℕ} (hn : 1 ≤ n) (hd : 1 ≤ d)
(P : DiscreteLaw d) (H : ℝ) (hHpos : 0 < H)
(hH : universalH ≤ H)
(κ ε : ℝ) (hε : 0 < ε) (hε1 : ε ≤ 1) (x : Fin d) :
MemLp (markedPoissonCellValue n d H hHpos κ ε x) 2
(finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)) := by
have hm : 0 < (n : ℝ≥0) / 8 := by positivity
have hL : 1 ≤ logAlphabet d := by
have hdR : (1 : ℝ) ≤ d := by exact_mod_cast hd
rw [logAlphabet, Real.log_mul (Real.exp_pos 1).ne' (by positivity), Real.log_exp]
linarith [Real.log_nonneg hdR]
unfold markedPoissonCellValue
simpa only [poissonIntensityFormula, NNReal.coe_div,
NNReal.coe_natCast, NNReal.coe_ofNat] using
pilotClippedCellValue_memLp_two
(finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4))
(fun j s => markedCount (le_refl s.1) (fun i => (s.2 i).1)
(fun i => (s.2 i).2) true x j)
(fun j s => markedCount (le_refl s.1) (fun i => (s.2 i).1)
(fun i => (s.2 i).2) false x j)
(fun j => (P.pmf (x, decide (j.val / 2 = 1), decide (j.val % 2 = 1))).toNNReal)
((n : ℝ≥0) / 8) hm H hHpos (logAlphabet d) ε hH hL hε hε1
(jacksonDegree κ d) d
(fun j => markedPoissonCount_measurable d true x j)
(fun j => markedPoissonCount_measurable d false x j)
(fun j => markedCount_poisson_hasLaw P n true x j)
theorem markedPoissonCellValue_sum_memLp_two
The sum whose projection defines the actual marked statistic is L². In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
n d :
ℕ
hn :
1 ≤ n
hd :
1 ≤ d
P :
H :
ℝ
hHpos :
0 < H
hH :
universalH ≤ H
κ ε :
ℝ
hε :
0 < ε
hε1 :
ε ≤ 1
conclusion
MemLp (fun s => ∑ x, markedPoissonCellValue n d H hHpos κ ε x s) 2 (finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4))
Proof (Lean source)
lemma markedPoissonCellValue_sum_memLp_two {n d : ℕ} (hn : 1 ≤ n) (hd : 1 ≤ d)
(P : DiscreteLaw d) (H : ℝ) (hHpos : 0 < H)
(hH : universalH ≤ H)
(κ ε : ℝ) (hε : 0 < ε) (hε1 : ε ≤ 1) :
MemLp (fun s => ∑ x, markedPoissonCellValue n d H hHpos κ ε x s) 2
(finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)) :=
memLp_finsetSum univ (fun x _ =>
markedPoissonCellValue_memLp_two hn hd P H hHpos hH κ ε hε hε1 x)
theorem markedPoissonCellValue_pairwise_indepFun
Distinct cells use disjoint sets of observation–mark atom counts. Their actual clipped contributions are consequently independent, including bad pilots. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
Pairwise (fun x y : Fin d => IndepFun (markedPoissonCellValue n d H hH κ ε x) (markedPoissonCellValue n d H hH κ ε y) (finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4)))
Proof (Lean source)
lemma markedPoissonCellValue_pairwise_indepFun {d : ℕ} (P : DiscreteLaw d)
(n : ℕ) (H : ℝ) (hH : 0 < H) (κ ε : ℝ) :
Pairwise (fun x y : Fin d => IndepFun
(markedPoissonCellValue n d H hH κ ε x)
(markedPoissonCellValue n d H hH κ ε y)
(finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4))) := by
classical
let cells (x : Fin d) : Finset (Obs d × Bool) :=
Finset.univ.filter (fun z => z.1.1 = x)
let atom (x : Fin d) (pilot : Bool) (j : Fin 4) : cells x :=
⟨((x, decide (j.val / 2 = 1), decide (j.val % 2 = 1)), pilot), by simp [cells]⟩
let F (x : Fin d) (v : cells x → ℕ) : ℝ :=
clippedCellValueFormula ε (jacksonDegree κ d) d (poissonIntensityFormula n)
(pilotMidpoint H hH (poissonIntensityFormula n) (logAlphabet d)
(fun j => v (atom x true j)))
(pilotRadiusFormula H hH (poissonIntensityFormula n) (logAlphabet d)
(fun j => v (atom x true j))) (fun j => v (atom x false j))
have heq (x : Fin d) : markedPoissonCellValue n d H hH κ ε x =
F x ∘ (fun s z => markedPoissonHistogram s (z : Obs d × Bool)) := by
funext s
have hc (pilot : Bool) :
markedCount (le_refl s.1) (fun i => (s.2 i).1) (fun i => (s.2 i).2) pilot x =
(fun j => markedPoissonHistogram s ((x, decide (j.val / 2 = 1),
decide (j.val % 2 = 1)), pilot)) := by
funext j
exact markedCount_eq_markedPoissonHistogram s pilot x j
simp only [markedPoissonCellValue, F, atom, Function.comp_apply, hc]
intro x y hxy
have hdis : Disjoint (cells x) (cells y) := by
apply Finset.disjoint_left.2
intro z hx hy
simp only [cells, mem_filter, Finset.mem_univ, true_and] at hx hy
exact hxy (hx.symm.trans hy)
rw [heq x, heq y]
exact ((markedPoissonHistogram_iIndepFun P n).indepFun_finset
(cells x) (cells y) hdis (fun z =>
(measurable_pi_apply z).comp (markedPoissonHistogram_measurable d))).comp
(Measurable.of_discrete (f := F x)) (Measurable.of_discrete (f := F y))
theorem markedPoissonCellValue_sum_sqRisk_eq
Equation (29) holds for the actual unprojected marked statistic: distinct cell variances add, and the total bias is squared only after summation. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
n d :
ℕ
hn :
1 ≤ n
hd :
1 ≤ d
P :
H :
ℝ
hHpos :
0 < H
hH :
universalH ≤ H
κ ε t :
ℝ
hε :
0 < ε
hε1 :
ε ≤ 1
conclusion
μ :
finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4)
sqRisk μ (fun s => ∑ x, markedPoissonCellValue n d H hHpos κ ε x s) t
= (∑ x, variance (markedPoissonCellValue n d H hHpos κ ε x) μ)
+ ((∑ x, ∫ s, markedPoissonCellValue n d H hHpos κ ε x s ∂μ) - t) ^ 2
Proof (Lean source)
lemma markedPoissonCellValue_sum_sqRisk_eq {n d : ℕ} (hn : 1 ≤ n) (hd : 1 ≤ d)
(P : DiscreteLaw d) (H : ℝ) (hHpos : 0 < H)
(hH : universalH ≤ H)
(κ ε t : ℝ) (hε : 0 < ε) (hε1 : ε ≤ 1) :
let μ := finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)
sqRisk μ (fun s => ∑ x, markedPoissonCellValue n d H hHpos κ ε x s) t =
(∑ x, variance (markedPoissonCellValue n d H hHpos κ ε x) μ) +
((∑ x, ∫ s, markedPoissonCellValue n d H hHpos κ ε x s ∂μ) - t) ^ 2 := by
dsimp only
let μ := finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)
let T := markedPoissonCellValue n d H hHpos κ ε
have hT (x : Fin d) : MemLp (T x) 2 μ :=
markedPoissonCellValue_memLp_two hn hd P H hHpos hH κ ε hε hε1 x
have hsum := markedPoissonCellValue_sum_memLp_two hn hd P H hHpos hH κ ε hε hε1
have hind := markedPoissonCellValue_pairwise_indepFun P n H hHpos κ ε
rw [cellStatistic_sqRisk_eq_variance_add_bias _ _ _ hsum,
integral_finsetSum _ (fun x _ => (hT x).integrable (by norm_num))]
have hfun : (∑ x, T x) = (fun s => ∑ x, T x s) := by funext s; simp
have hv := IndepFun.variance_sum (X := T) (s := univ)
(fun x _ => hT x) (fun x _ y _ hxy => hind hxy)
rw [hfun] at hv
rw [hv]
theorem markedPoissonStatistic_sqRisk_le_variance_add_bias
The unit projection in the implemented uncapped statistic preserves the variance-plus-squared-bias upper bound for its actual cell sum. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
n d :
ℕ
hn :
1 ≤ n
hd :
1 ≤ d
P :
H :
ℝ
hHpos :
0 < H
hH :
universalH ≤ H
κ ε :
ℝ
hε :
0 < ε
hε1 :
ε ≤ 1
conclusion
μ :
finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4)
sqRisk μ (markedPoissonStatistic n d H hHpos κ ε) (observedValue P)
≤ (∑ x, variance (markedPoissonCellValue n d H hHpos κ ε x) μ)
+ ((∑ x, ∫ s, markedPoissonCellValue n d H hHpos κ ε x s ∂μ) - observedValue P) ^ 2
Proof (Lean source)
lemma markedPoissonStatistic_sqRisk_le_variance_add_bias {n d : ℕ}
(hn : 1 ≤ n) (hd : 1 ≤ d) (P : DiscreteLaw d) (H : ℝ) (hHpos : 0 < H)
(hH : universalH ≤ H)
(κ ε : ℝ) (hε : 0 < ε) (hε1 : ε ≤ 1) :
let μ := finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)
sqRisk μ (markedPoissonStatistic n d H hHpos κ ε) (observedValue P) ≤
(∑ x, variance (markedPoissonCellValue n d H hHpos κ ε x) μ) +
((∑ x, ∫ s, markedPoissonCellValue n d H hHpos κ ε x s ∂μ) - observedValue P) ^ 2 := by
dsimp only
have hsum := markedPoissonCellValue_sum_memLp_two hn hd P H hHpos hH κ ε hε hε1
have hp := projectUnit_sqRisk_le _ _ (observedValue P) hsum (observedValue_mem_unitInterval P)
change sqRisk _ (markedPoissonStatistic n d H hHpos κ ε) _ ≤ _ at hp
exact hp.trans_eq (markedPoissonCellValue_sum_sqRisk_eq hn hd P H hHpos hH κ ε
(observedValue P) hε hε1)
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.UpperMarkedGoodSquare 4 declarations Fixed-pilot Jackson approximation and clipping bounds are averaged using the actual independent marked vectors.
Integrated good-pilot moments and active risk
Fixed-pilot Jackson approximation and clipping bounds are averaged using the actual independent marked vectors. This proves the occupied-cell first- and first- and second-moment inputss of roadmap (29), retaining the squared approximation term rather than absorbing it into factorial variance.
theorem markedPoissonCellValue_occupied_good_moments_tuning_exists Lemma markedPoissonCellValue_occupied_good_moments_tuning_exists in the paper ↗
One universal degree tuning proves the integrated good-pilot occupied cell bias and squared loss for the implemented statistic. All count laws, independence, localization and integrability inputs are discharged. In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ κ A : ℝ,
0 < κ
0 < A
conclusion 1
n d :
ℕ
1 ≤ n
1 ≤ d
P :
ε :
ℝ
0 < ε
ε ≤ 1
ObservedClass ε P
H :
ℝ
hHpos :
0 < H
universalH ≤ H
1 ≤ H
x :
Fin d
0 < cellMass P x
μ :
finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4)
B :
512 * (H * (H + 2)) * sqrt 2 * (1 / (κ / 2) + 1 / (κ / 2) ^ 2) * (sqrt (cellMass P x / (poissonIntensityFormula n * ε * logAlphabet d)) + 1 / (poissonIntensityFormula n * ε * logAlphabet d))
(∫ s, if s ∈ markedPoissonGoodPilot n P H hHpos x then (markedPoissonCellValue n d H hHpos κ ε x s - armwiseExtensionFormula ε (cellVector P x)) ^ 2 else 0 ∂μ)
≤ 2 * A * (d : ℝ) ^ (1 / 16 : ℝ) * (∫ s, markedPoissonClippingScale n d H hHpos ε x s ^ 2 ∂μ)
+ 2 * B ^ 2
|∫ s, if s ∈ markedPoissonGoodPilot n P H hHpos x then markedPoissonCellValue n d H hHpos κ ε x s
- armwiseExtensionFormula ε (cellVector P x) else 0 ∂μ|
≤ B
+ A * (d : ℝ) ^ (-(3 / 16 : ℝ)) * (∫ s, markedPoissonClippingScale n d H hHpos ε x s ∂μ)
Proof (Lean source)
lemma markedPoissonCellValue_occupied_good_moments_tuning_exists :
∃ κ A : ℝ, 0 < κ ∧ 0 < A ∧ ∀ {n d : ℕ}, 1 ≤ n → 1 ≤ d →
∀ (P : DiscreteLaw d) (ε : ℝ), 0 < ε → ε ≤ 1 → ObservedClass ε P →
∀ (H : ℝ) (hHpos : 0 < H), universalH ≤ H → 1 ≤ H →
∀ (x : Fin d), 0 < cellMass P x →
let μ := finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)
let B := 512 * (H * (H + 2)) * sqrt 2 *
(1 / (κ / 2) + 1 / (κ / 2) ^ 2) *
(sqrt (cellMass P x / (poissonIntensityFormula n * ε * logAlphabet d)) +
1 / (poissonIntensityFormula n * ε * logAlphabet d))
(∫ s, if s ∈ markedPoissonGoodPilot n P H hHpos x then
(markedPoissonCellValue n d H hHpos κ ε x s -
armwiseExtensionFormula ε (cellVector P x)) ^ 2 else 0 ∂μ) ≤
2 * A * (d : ℝ) ^ (1 / 16 : ℝ) *
(∫ s, markedPoissonClippingScale n d H hHpos ε x s ^ 2 ∂μ) + 2 * B ^ 2 ∧
|∫ s, if s ∈ markedPoissonGoodPilot n P H hHpos x then
markedPoissonCellValue n d H hHpos κ ε x s -
armwiseExtensionFormula ε (cellVector P x) else 0 ∂μ| ≤
B + A * (d : ℝ) ^ (-(3 / 16 : ℝ)) *
(∫ s, markedPoissonClippingScale n d H hHpos ε x s ∂μ) := by
classical
obtain ⟨κ, A, hκ, hA, htune⟩ := observed_pilotClippedCellValue_target_tuning_exists
refine ⟨κ, A, hκ, hA, ?_⟩
intro n d hn hd P ε hε hε1 hP H hHpos hH hHlarge x hp
dsimp only
let μ := finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)
let B := 512 * (H * (H + 2)) * sqrt 2 *
(1 / (κ / 2) + 1 / (κ / 2) ^ 2) *
(sqrt (cellMass P x / (poissonIntensityFormula n * ε * logAlphabet d)) +
1 / (poissonIntensityFormula n * ε * logAlphabet d))
let E : (Fin 4 → ℕ) → Prop := fun v => cellVector P x ∈
pilotRectangle (pilotLower H hHpos (poissonIntensityFormula n) (logAlphabet d) v)
(pilotUpperFormula H hHpos (poissonIntensityFormula n) (logAlphabet d) v)
let T : (Fin 4 → ℕ) → (Fin 4 → ℕ) → ℝ := fun v w =>
clippedCellValueFormula ε (jacksonDegree κ d) d (poissonIntensityFormula n)
(pilotMidpoint H hHpos (poissonIntensityFormula n) (logAlphabet d) v)
(pilotRadiusFormula H hHpos (poissonIntensityFormula n) (logAlphabet d) v) w
let S : (Fin 4 → ℕ) → ℝ := fun v => clippingScaleFormula ε
(pilotMidpoint H hHpos (poissonIntensityFormula n) (logAlphabet d) v)
(pilotRadiusFormula H hHpos (poissonIntensityFormula n) (logAlphabet d) v)
let f : (Fin 4 → ℕ) × (Fin 4 → ℕ) → ℝ := fun z =>
if E z.1 then (T z.1 z.2 - armwiseExtensionFormula ε (cellVector P x)) ^ 2 else 0
let g : (Fin 4 → ℕ) → ℝ := fun v =>
2 * A * (d : ℝ) ^ (1 / 16 : ℝ) * S v ^ 2 + 2 * B ^ 2
have hm : 0 < poissonIntensityFormula n := by unfold poissonIntensityFormula; positivity
have hT := markedPoissonCellValue_memLp_two hn hd P H hHpos hH κ ε hε hε1 x
have hS := markedPoissonClippingScale_memLp_two hn hd P H hHpos hH ε hε hε1 x
constructor
·
have hf : Integrable (fun s => f (markedPoissonCellCounts true x s,
markedPoissonCellCounts false x s)) μ := by
have hi := ((hT.sub (memLp_const (armwiseExtensionFormula ε (cellVector P x)))).indicator
(markedPoissonGoodPilot_measurableSet n P H hHpos x)).integrable_sq
convert hi using 1
funext s
rw [Set.indicator_apply]
change (if E (markedPoissonCellCounts true x s) then _ ^ 2 else 0) =
(if E (markedPoissonCellCounts true x s) then _ else 0) ^ 2
split_ifs
· rfl
· simp
have hg : Integrable (fun s => g (markedPoissonCellCounts true x s)) μ := by
exact (hS.integrable_sq.const_mul _).add (integrable_const _)
have hfixed (v : Fin 4 → ℕ) :
(∫ s, f (v, markedPoissonCellCounts false x s) ∂μ) ≤ g v := by
by_cases hv : E v
· have hlaw (j : Fin 4) : HasLaw (fun s => markedPoissonCellCounts false x s j)
(poissonMeasure ⟨poissonIntensityFormula n * cellVector P x j,
mul_nonneg hm.le ENNReal.toReal_nonneg⟩) μ := by
have hr : (⟨poissonIntensityFormula n * cellVector P x j,
mul_nonneg hm.le ENNReal.toReal_nonneg⟩ : ℝ≥0) =
((n : ℝ≥0) / (8 : ℝ≥0)) * (P.pmf (x, decide (j.val / 2 = 1),
decide (j.val % 2 = 1))).toNNReal := by
apply NNReal.coe_injective
rfl
rw [hr]
exact markedCount_poisson_hasLaw P n false x j
have ht := (htune μ hd ε P hε hP x hp
(fun j s => markedPoissonCellCounts false x s j) H hHpos hHlarge
(poissonIntensityFormula n) hm v hlaw (markedPoissonEvaluation_iIndepFun P n x) hv).2
simpa only [f, hv, if_true, T, g, S, B] using ht
· simp only [f, hv, if_false, integral_zero]
dsimp [g]
positivity
have hbound := markedPoissonCellCounts_integral_le_of_fixedPilot P n x f g hf hg
(fun z => by dsimp [f]; split_ifs <;> positivity) hfixed
have hright : (∫ s, g (markedPoissonCellCounts true x s) ∂μ) =
2 * A * (d : ℝ) ^ (1 / 16 : ℝ) *
(∫ s, markedPoissonClippingScale n d H hHpos ε x s ^ 2 ∂μ) + 2 * B ^ 2 := by
change (∫ s, 2 * A * (d : ℝ) ^ (1 / 16 : ℝ) *
markedPoissonClippingScale n d H hHpos ε x s ^ 2 + 2 * B ^ 2 ∂μ) = _
rw [integral_add (hS.integrable_sq.const_mul _) (integrable_const _),
integral_const_mul]
simp [μ]
rw [hright] at hbound
exact hbound
·
let f₁ : (Fin 4 → ℕ) × (Fin 4 → ℕ) → ℝ := fun z =>
if E z.1 then T z.1 z.2 - armwiseExtensionFormula ε (cellVector P x) else 0
let g₁ : (Fin 4 → ℕ) → ℝ := fun v =>
B + A * (d : ℝ) ^ (-(3 / 16 : ℝ)) * S v
have hf₁ : Integrable (fun s => f₁ (markedPoissonCellCounts true x s,
markedPoissonCellCounts false x s)) μ := by
exact ((hT.sub (memLp_const (armwiseExtensionFormula ε (cellVector P x)))).indicator
(markedPoissonGoodPilot_measurableSet n P H hHpos x)).integrable (by norm_num)
have hg₁ : Integrable (fun s => g₁ (markedPoissonCellCounts true x s)) μ := by
exact (integrable_const _).add ((hS.integrable (by norm_num)).const_mul _)
have hfixed (v : Fin 4 → ℕ) :
|∫ s, f₁ (v, markedPoissonCellCounts false x s) ∂μ| ≤ g₁ v := by
by_cases hv : E v
· have hlaw (j : Fin 4) : HasLaw (fun s => markedPoissonCellCounts false x s j)
(poissonMeasure ⟨poissonIntensityFormula n * cellVector P x j,
mul_nonneg hm.le ENNReal.toReal_nonneg⟩) μ := by
have hr : (⟨poissonIntensityFormula n * cellVector P x j,
mul_nonneg hm.le ENNReal.toReal_nonneg⟩ : ℝ≥0) =
((n : ℝ≥0) / (8 : ℝ≥0)) * (P.pmf (x, decide (j.val / 2 = 1),
decide (j.val % 2 = 1))).toNNReal := by
apply NNReal.coe_injective
rfl
rw [hr]
exact markedCount_poisson_hasLaw P n false x j
have ht := (htune μ hd ε P hε hP x hp
(fun j s => markedPoissonCellCounts false x s j) H hHpos hHlarge
(poissonIntensityFormula n) hm v hlaw (markedPoissonEvaluation_iIndepFun P n x) hv).1
have hi : Integrable (fun s => T v (markedPoissonCellCounts false x s)) μ := by
exact (integrable_const _).sup ((integrable_const _).inf
((factorialCellValue_memLp_two μ
(fun j s => markedPoissonCellCounts false x s j) _ hlaw
(markedPoissonEvaluation_iIndepFun P n x) ε (jacksonDegree κ d)
(poissonIntensityFormula n)
(pilotMidpoint H hHpos (poissonIntensityFormula n) (logAlphabet d) v)
(pilotRadiusFormula H hHpos (poissonIntensityFormula n) (logAlphabet d) v)).integrable
(by norm_num)))
simp only [f₁, hv, if_true]
rw [integral_sub hi (integrable_const _)]
simpa [integral_const, μ, T, g₁, S, B] using ht
· simp only [f₁, hv, if_false, integral_zero, abs_zero]
have hL := logAlphabet_pos hd
dsimp [g₁, B]
have hSpos : 0 ≤ S v := by
exact (clippingScale_pos ε
(pilotMidpoint H hHpos (poissonIntensityFormula n) (logAlphabet d) v)
(pilotRadiusFormula H hHpos (poissonIntensityFormula n) (logAlphabet d) v)
(pilotMidpoint_nonneg H hHpos _ _ hm v)
(pilotRadius_pos H hHpos _ _ hm (logAlphabet_pos hd) v)).le
positivity
have hbound := markedPoissonCellCounts_abs_integral_le_of_fixedPilot P n x
f₁ g₁ hf₁ hg₁ hfixed
have hright : (∫ s, g₁ (markedPoissonCellCounts true x s) ∂μ) =
B + A * (d : ℝ) ^ (-(3 / 16 : ℝ)) *
(∫ s, markedPoissonClippingScale n d H hHpos ε x s ∂μ) := by
change (∫ s, B + A * (d : ℝ) ^ (-(3 / 16 : ℝ)) *
markedPoissonClippingScale n d H hHpos ε x s ∂μ) = _
rw [integral_add (integrable_const _) ((hS.integrable (by norm_num)).const_mul _)]
simp only [integral_const_mul, integral_const]
simp [μ, B]
rw [hright] at hbound
exact hbound
theorem markedPoissonCellValue_occupied_good_sq_tuning_exists Lemma markedPoissonCellValue_occupied_good_sq_tuning_exists in the paper ↗
One universal degree tuning proves the integrated good-pilot occupied cell loss for the implemented statistic. All count laws, independence, localization and integrability inputs are discharged. In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ κ A : ℝ,
0 < κ
0 < A
conclusion 1
n d :
ℕ
1 ≤ n
1 ≤ d
P :
ε :
ℝ
0 < ε
ε ≤ 1
ObservedClass ε P
H :
ℝ
hHpos :
0 < H
universalH ≤ H
1 ≤ H
x :
Fin d
0 < cellMass P x
μ :
finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4)
B :
512 * (H * (H + 2)) * sqrt 2 * (1 / (κ / 2) + 1 / (κ / 2) ^ 2) * (sqrt (cellMass P x / (poissonIntensityFormula n * ε * logAlphabet d)) + 1 / (poissonIntensityFormula n * ε * logAlphabet d))
(∫ s, if s ∈ markedPoissonGoodPilot n P H hHpos x then (markedPoissonCellValue n d H hHpos κ ε x s - armwiseExtensionFormula ε (cellVector P x)) ^ 2 else 0 ∂μ)
≤ 2 * A * (d : ℝ) ^ (1 / 16 : ℝ) * (∫ s, markedPoissonClippingScale n d H hHpos ε x s ^ 2 ∂μ)
+ 2 * B ^ 2
Proof (Lean source)
lemma markedPoissonCellValue_occupied_good_sq_tuning_exists :
∃ κ A : ℝ, 0 < κ ∧ 0 < A ∧ ∀ {n d : ℕ}, 1 ≤ n → 1 ≤ d →
∀ (P : DiscreteLaw d) (ε : ℝ), 0 < ε → ε ≤ 1 → ObservedClass ε P →
∀ (H : ℝ) (hHpos : 0 < H), universalH ≤ H → 1 ≤ H →
∀ (x : Fin d), 0 < cellMass P x →
let μ := finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)
let B := 512 * (H * (H + 2)) * sqrt 2 *
(1 / (κ / 2) + 1 / (κ / 2) ^ 2) *
(sqrt (cellMass P x / (poissonIntensityFormula n * ε * logAlphabet d)) +
1 / (poissonIntensityFormula n * ε * logAlphabet d))
(∫ s, if s ∈ markedPoissonGoodPilot n P H hHpos x then
(markedPoissonCellValue n d H hHpos κ ε x s -
armwiseExtensionFormula ε (cellVector P x)) ^ 2 else 0 ∂μ) ≤
2 * A * (d : ℝ) ^ (1 / 16 : ℝ) *
(∫ s, markedPoissonClippingScale n d H hHpos ε x s ^ 2 ∂μ) + 2 * B ^ 2 := by
obtain ⟨κ, A, hκ, hA, ht⟩ := markedPoissonCellValue_occupied_good_moments_tuning_exists
refine ⟨κ, A, hκ, hA, ?_⟩
intro n d hn hd P ε hε hε1 hP H hHpos hH hHlarge x hp
exact (ht hn hd P ε hε hε1 hP H hHpos hH hHlarge x hp).1
theorem markedPoissonClippingScale_nonneg
The actual pilot clipping scale is nonnegative at every finite sample. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
0 ≤ markedPoissonClippingScale n d H hH ε x s
Proof (Lean source)
lemma markedPoissonClippingScale_nonneg {n d : ℕ} (hn : 1 ≤ n) (hd : 1 ≤ d)
(H : ℝ) (hH : 0 < H) (ε : ℝ) (x : Fin d)
(s : FiniteSample (Obs d × Bool)) :
0 ≤ markedPoissonClippingScale n d H hH ε x s := by
have hm : 0 < poissonIntensityFormula n := by unfold poissonIntensityFormula; positivity
unfold markedPoissonClippingScale
exact (clippingScale_pos ε _ _ (pilotMidpoint_nonneg H hH _ _ hm _)
(pilotRadius_pos H hH _ _ hm (logAlphabet_pos hd) _)).le
theorem markedPoissonStatistic_active_rate_tuning_exists Lemma markedPoissonStatistic_active_rate_tuning_exists in the paper ↗
Equations (19), (20), (22), and (28) yield the active-branch risk for the implemented uncapped statistic. Both good-pilot content inputs are proved here, including null cells, with the same universal degree tuning. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
∃ κ C : ℝ,
0 < κ
0 < C
conclusion 1
n d :
ℕ
1 ≤ n
1 ≤ d
P :
ε :
ℝ
0 < ε
ε ≤ 1 / 2
ObservedClass ε P
(d : ℝ) / logAlphabet d ≤ (n : ℝ) * ε
sqRisk (finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4)) (markedPoissonStatistic n d H hHpos κ ε) (observedValue P)
≤ C * ((d : ℝ) / (poissonIntensityFormula n * ε * logAlphabet d))
Proof (Lean source)
lemma markedPoissonStatistic_active_rate_tuning_exists
(H : ℝ) (hHpos : 0 < H) (hH : universalH ≤ H) (hHlarge : 1 ≤ H) :
∃ κ C : ℝ, 0 < κ ∧ 0 < C ∧ ∀ {n d : ℕ}, 1 ≤ n → 1 ≤ d →
∀ (P : DiscreteLaw d) (ε : ℝ), 0 < ε → ε ≤ 1 / 2 →
ObservedClass ε P → (d : ℝ) / logAlphabet d ≤ (n : ℝ) * ε →
sqRisk
(finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4))
(markedPoissonStatistic n d H hHpos κ ε) (observedValue P) ≤
C * ((d : ℝ) / (poissonIntensityFormula n * ε * logAlphabet d)) := by
classical
obtain ⟨κ, A, hκ, hA, ht⟩ := markedPoissonCellValue_occupied_good_moments_tuning_exists
let D := 512 * (H * (H + 2)) * sqrt 2 *
(1 / (κ / 2) + 1 / (κ / 2) ^ 2)
let Dnull := 64 * H / (κ / 2) ^ 2
let De := D + Dnull
have hD : 0 ≤ D := by dsimp [D]; positivity
have hDnull : 0 ≤ Dnull := by dsimp [Dnull]; positivity
have hDe : 0 ≤ De := add_nonneg hD hDnull
obtain ⟨C, hC, hrate⟩ := markedPoissonStatistic_goodPilotEstimates_rate_le
H hHpos hH κ (2 * A) (De + A) (2 * De ^ 2) (by positivity) (by positivity)
refine ⟨κ, C, hκ, hC, ?_⟩
intro n d hn hd P ε hε hεhalf hP hactive
have hε1 : ε ≤ 1 := by linarith
let μ := finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)
let Q (x : Fin d) := sqrt (cellMass P x /
(poissonIntensityFormula n * ε * logAlphabet d)) +
1 / (poissonIntensityFormula n * ε * logAlphabet d)
have hm : 0 < poissonIntensityFormula n := by unfold poissonIntensityFormula; positivity
have hL := logAlphabet_pos hd
have hQ (x : Fin d) : 0 ≤ Q x := by dsimp [Q]; positivity
have hs (x : Fin d) : 0 ≤ ∫ s, markedPoissonClippingScale n d H hHpos ε x s ∂μ :=
integral_nonneg (markedPoissonClippingScale_nonneg hn hd H hHpos ε x)
have hs₂ (x : Fin d) : 0 ≤ ∫ s, markedPoissonClippingScale n d H hHpos ε x s ^ 2 ∂μ :=
integral_nonneg (fun _ => sq_nonneg _)
apply hrate hn hd P ε hε hεhalf hP hactive
· intro x
by_cases hp : 0 < cellMass P x
· have hb := (ht hn hd P ε hε hε1 hP H hHpos hH hHlarge x hp).1
change (∫ s, if s ∈ markedPoissonGoodPilot n P H hHpos x then
(markedPoissonCellValue n d H hHpos κ ε x s -
armwiseExtensionFormula ε (cellVector P x)) ^ 2 else 0 ∂μ) ≤
2 * A * (d : ℝ) ^ (1 / 16 : ℝ) *
(∫ s, markedPoissonClippingScale n d H hHpos ε x s ^ 2 ∂μ) +
2 * (D * Q x) ^ 2 at hb
apply hb.trans
have hsq : (D * Q x) ^ 2 ≤ (De * Q x) ^ 2 := by
apply pow_le_pow_left₀ (mul_nonneg hD (hQ x))
exact mul_le_mul_of_nonneg_right (by dsimp [De]; linarith) (hQ x)
dsimp only [Q]
nlinarith [hsq]
· have hp0 : cellMass P x = 0 := le_antisymm (le_of_not_gt hp)
(Finset.sum_nonneg (fun a _ => Finset.sum_nonneg (fun y _ => toReal_nonneg)))
have hb := (markedPoissonCellValue_null_good_moments_le hn hd P x hp0
H hHpos κ ε hκ hε hε1).1
have hQnull : Q x = 1 / (poissonIntensityFormula n * ε * logAlphabet d) := by
simp [Q, hp0]
change (∫ s, if s ∈ markedPoissonGoodPilot n P H hHpos x then
(markedPoissonCellValue n d H hHpos κ ε x s -
armwiseExtensionFormula ε (cellVector P x)) ^ 2 else 0 ∂μ) ≤
(Dnull * (1 / (poissonIntensityFormula n * ε * logAlphabet d))) ^ 2 at hb
rw [← hQnull] at hb
apply hb.trans
have hsq : (Dnull * Q x) ^ 2 ≤ (De * Q x) ^ 2 := by
apply pow_le_pow_left₀ (mul_nonneg hDnull (hQ x))
exact mul_le_mul_of_nonneg_right (by dsimp [De]; linarith) (hQ x)
have hv : 0 ≤ 2 * A * (d : ℝ) ^ (1 / 16 : ℝ) *
(∫ s, markedPoissonClippingScale n d H hHpos ε x s ^ 2 ∂μ) := by
positivity
dsimp only [Q]
nlinarith [hsq, sq_nonneg (De * Q x)]
· intro x
have hdPow : 0 ≤ (d : ℝ) ^ (-(3 / 16 : ℝ)) := Real.rpow_nonneg (by positivity) _
have hw : 0 ≤ (d : ℝ) ^ (-(3 / 16 : ℝ)) *
(∫ s, markedPoissonClippingScale n d H hHpos ε x s ∂μ) :=
mul_nonneg hdPow (hs x)
by_cases hp : 0 < cellMass P x
· have hb := (ht hn hd P ε hε hε1 hP H hHpos hH hHlarge x hp).2
apply hb.trans
change D * Q x + A * (d : ℝ) ^ (-(3 / 16 : ℝ)) *
(∫ s, markedPoissonClippingScale n d H hHpos ε x s ∂μ) ≤ _
change D * Q x + A * (d : ℝ) ^ (-(3 / 16 : ℝ)) *
(∫ s, markedPoissonClippingScale n d H hHpos ε x s ∂μ) ≤
(De + A) * (Q x + (d : ℝ) ^ (-(3 / 16 : ℝ)) *
(∫ s, markedPoissonClippingScale n d H hHpos ε x s ∂μ))
calc
_ ≤ (De + A) * Q x + (De + A) * ((d : ℝ) ^ (-(3 / 16 : ℝ)) *
(∫ s, markedPoissonClippingScale n d H hHpos ε x s ∂μ)) := by
apply add_le_add
· exact mul_le_mul_of_nonneg_right (by dsimp only [De]; linarith) (hQ x)
· simpa only [mul_assoc] using
mul_le_mul_of_nonneg_right (show A ≤ De + A by linarith) hw
_ = _ := by ring
· have hp0 : cellMass P x = 0 := le_antisymm (le_of_not_gt hp)
(Finset.sum_nonneg (fun a _ => Finset.sum_nonneg (fun y _ => toReal_nonneg)))
have hb := (markedPoissonCellValue_null_good_moments_le hn hd P x hp0
H hHpos κ ε hκ hε hε1).2
apply hb.trans
change Dnull * (1 / (poissonIntensityFormula n * ε * logAlphabet d)) ≤
(De + A) * (Q x + (d : ℝ) ^ (-(3 / 16 : ℝ)) *
(∫ s, markedPoissonClippingScale n d H hHpos ε x s ∂μ))
have hQnull : Q x = 1 / (poissonIntensityFormula n * ε * logAlphabet d) := by
simp [Q, hp0]
rw [← hQnull]
calc
_ ≤ (De + A) * Q x := mul_le_mul_of_nonneg_right
(by dsimp only [De]; linarith) (hQ x)
_ ≤ _ := mul_le_mul_of_nonneg_left (le_add_of_nonneg_right hw)
(add_nonneg hDe hA.le)
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.UpperMarkedMoments 5 declarations The actual marked counts discharge the pilot moment input in roadmap (19) and its mass-normalized sum in (30).
Implemented marked-pilot moments
The actual marked counts discharge the pilot moment input in roadmap (19) and its mass-normalized sum in (30). These are unconditional bounds, including null cells and bad pilots.
def markedPoissonClippingScale
The clipping scale computed from the actual pilot-marked counts. For the displayed parameters, markedPoissonClippingScale is the object specified by this definition.
Definition (Lean source)
parameters
def
markedPoissonClippingScale n d H hH ε x s :
ℝ
given by
Np :
clippingScaleFormula ε (pilotMidpoint H hH (poissonIntensityFormula n) (logAlphabet d) Np) (pilotRadiusFormula H hH (poissonIntensityFormula n) (logAlphabet d) Np)
theorem markedPoissonClippingScale_memLp_two
The actual pilot scale is L² even on failed localization events. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
n d :
ℕ
hn :
1 ≤ n
hd :
1 ≤ d
P :
H :
ℝ
hHpos :
0 < H
hH :
universalH ≤ H
ε :
ℝ
hε :
0 < ε
hε1 :
ε ≤ 1
x :
Fin d
conclusion
MemLp (markedPoissonClippingScale n d H hHpos ε x) 2 (finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4))
Proof (Lean source)
lemma markedPoissonClippingScale_memLp_two {n d : ℕ} (hn : 1 ≤ n) (hd : 1 ≤ d)
(P : DiscreteLaw d) (H : ℝ) (hHpos : 0 < H)
(hH : universalH ≤ H)
(ε : ℝ) (hε : 0 < ε) (hε1 : ε ≤ 1) (x : Fin d) :
MemLp (markedPoissonClippingScale n d H hHpos ε x) 2
(finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)) := by
have hm : 0 < (n : ℝ≥0) / 8 := by positivity
have hL : 1 ≤ logAlphabet d := by
have hdR : (1 : ℝ) ≤ d := by exact_mod_cast hd
rw [logAlphabet, Real.log_mul (Real.exp_pos 1).ne' (by positivity), Real.log_exp]
linarith [Real.log_nonneg hdR]
unfold markedPoissonClippingScale
simpa only [poissonIntensityFormula, NNReal.coe_div,
NNReal.coe_natCast, NNReal.coe_ofNat] using
pilotClippingScale_memLp_two
(finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4))
(fun j s => markedCount (le_refl s.1) (fun i => (s.2 i).1)
(fun i => (s.2 i).2) true x j)
(fun j => (P.pmf (x, decide (j.val / 2 = 1), decide (j.val % 2 = 1))).toNNReal)
((n : ℝ≥0) / 8) hm H hHpos (logAlphabet d) ε hH hL hε hε1
(fun j => markedPoissonCount_measurable d true x j)
(fun j => markedCount_poisson_hasLaw P n true x j)
theorem markedPoissonClippingScale_integral_sq_le Lemma markedPoissonClippingScale_integral_sq_le in the paper ↗
Equation (19) holds for the implemented counts, with no assumed count law or pilot regularity left for a caller to discharge. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
μ :
finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4)
(∫ s, markedPoissonClippingScale n d H hH ε x s ^ 2 ∂μ)
≤ 512 * H ^ 2 * (1 + 9 * H) * (cellMass P x * logAlphabet d / (poissonIntensityFormula n * ε) + logAlphabet d ^ 2 / (poissonIntensityFormula n ^ 2 * ε ^ 2))
Proof (Lean source)
lemma markedPoissonClippingScale_integral_sq_le {n d : ℕ}
(hn : 1 ≤ n) (hd : 1 ≤ d) (P : DiscreteLaw d)
(H : ℝ) (hH : 0 < H) (ε : ℝ) (hε : 0 < ε) (hεhalf : ε ≤ 1 / 2)
(x : Fin d) :
let μ := finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)
(∫ s, markedPoissonClippingScale n d H hH ε x s ^ 2 ∂μ) ≤
512 * H ^ 2 * (1 + 9 * H) *
(cellMass P x * logAlphabet d / (poissonIntensityFormula n * ε) +
logAlphabet d ^ 2 / (poissonIntensityFormula n ^ 2 * ε ^ 2)) := by
dsimp only
have hm : 0 < poissonIntensityFormula n := by unfold poissonIntensityFormula; positivity
have hL := (logAlphabet_pos hd).le
let μ := finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)
let W : Fin 4 → FiniteSample (Obs d × Bool) → ℕ := fun j s =>
markedCount (le_refl s.1) (fun i => (s.2 i).1) (fun i => (s.2 i).2) true x j
have hlaw (j : Fin 4) : HasLaw (W j) (poissonMeasure
⟨poissonIntensityFormula n * cellVector P x j,
mul_nonneg hm.le ENNReal.toReal_nonneg⟩) μ := by
have hr : (⟨poissonIntensityFormula n * cellVector P x j,
mul_nonneg hm.le ENNReal.toReal_nonneg⟩ : ℝ≥0) =
(((n : ℝ≥0) / (8 : ℝ≥0)) *
(P.pmf (x, decide (j.val / 2 = 1), decide (j.val % 2 = 1))).toNNReal : ℝ≥0) := by
apply NNReal.coe_injective
change poissonIntensityFormula n * cellVector P x j =
((n : ℝ) / 8) * (P.pmf (x, decide (j.val / 2 = 1),
decide (j.val % 2 = 1))).toReal
rfl
rw [hr]
exact markedCount_poisson_hasLaw P n true x j
have hb := observedPilotClippingScale_integral_sq_le μ P x W
H hH (poissonIntensityFormula n) (logAlphabet d) ε hm hL hε hεhalf hlaw
simpa only [markedPoissonClippingScale, W] using hb
theorem markedPoissonClippingScale_sum_integral_sq_le Lemma markedPoissonClippingScale_sum_integral_sq_le in the paper ↗
Equation (30) for the actual pilot: the mass term sums to one, including zero-mass cells, so the envelope contains no extra factor d on that term. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
n d :
ℕ
hn :
1 ≤ n
hd :
1 ≤ d
P :
H :
ℝ
hH :
0 < H
ε :
ℝ
hε :
0 < ε
hεhalf :
ε ≤ 1 / 2
conclusion
μ :
finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4)
(∑ x, ∫ s, markedPoissonClippingScale n d H hH ε x s ^ 2 ∂μ)
≤ 512 * H ^ 2 * (1 + 9 * H) * (logAlphabet d / (poissonIntensityFormula n * ε) + (d : ℝ) * logAlphabet d ^ 2 / (poissonIntensityFormula n ^ 2 * ε ^ 2))
Proof (Lean source)
lemma markedPoissonClippingScale_sum_integral_sq_le {n d : ℕ}
(hn : 1 ≤ n) (hd : 1 ≤ d) (P : DiscreteLaw d)
(H : ℝ) (hH : 0 < H) (ε : ℝ) (hε : 0 < ε) (hεhalf : ε ≤ 1 / 2) :
let μ := finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)
(∑ x, ∫ s, markedPoissonClippingScale n d H hH ε x s ^ 2 ∂μ) ≤
512 * H ^ 2 * (1 + 9 * H) *
(logAlphabet d / (poissonIntensityFormula n * ε) +
(d : ℝ) * logAlphabet d ^ 2 / (poissonIntensityFormula n ^ 2 * ε ^ 2)) := by
dsimp only
exact clippingScale_secondMoment_sum_le P _ _ _ _ _
(fun x => markedPoissonClippingScale_integral_sq_le hn hd P H hH ε hε hεhalf x)
theorem markedPoissonStatistic_cellEstimates_rate_le Lemma markedPoissonStatistic_cellEstimates_rate_le in the paper ↗
Equations (29)–(34) for the implemented statistic. Only the still-to-be-proved cellwise variance and bias content estimates remain as inputs: count laws, independence, square-integrability, and pilot moments are discharged here. The constant is uniform in n, d, the overlap floor, and the observed law. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
∃ C : ℝ,
0 < C
conclusion 1
n d :
ℕ
1 ≤ n
1 ≤ d
P :
ε :
ℝ
0 < ε
ε ≤ 1 / 2
(d : ℝ) / logAlphabet d ≤ (n : ℝ) * ε
μ :
finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4)
S :
markedPoissonClippingScale n d H hHpos ε
T :
markedPoissonCellValue n d H hHpos κ ε
(∀ x, variance (T x) μ ≤ Cv * (d : ℝ) ^ (1 / 16 : ℝ) * (∫ s, S x s ^ 2 ∂μ))
(∀ x, |(∫ s, T x s ∂μ) - cellMass P x * max (outcomeMean P false x) (outcomeMean P true x)| ≤ Cb * (sqrt (cellMass P x / (poissonIntensityFormula n * ε * logAlphabet d)) + 1 / (poissonIntensityFormula n * ε * logAlphabet d) + (d : ℝ) ^ (-(3 / 16 : ℝ)) * (∫ s, S x s ∂μ)))
sqRisk μ (markedPoissonStatistic n d H hHpos κ ε) (observedValue P)
≤ C * ((d : ℝ) / (poissonIntensityFormula n * ε * logAlphabet d))
Proof (Lean source)
lemma markedPoissonStatistic_cellEstimates_rate_le (H : ℝ) (hHpos : 0 < H)
(hH : universalH ≤ H)
(κ Cv Cb : ℝ) (hCv : 0 ≤ Cv) :
∃ C : ℝ, 0 < C ∧ ∀ {n d : ℕ}, 1 ≤ n → 1 ≤ d →
∀ (P : DiscreteLaw d) (ε : ℝ), 0 < ε → ε ≤ 1 / 2 →
(d : ℝ) / logAlphabet d ≤ (n : ℝ) * ε →
let μ := finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)
let S := markedPoissonClippingScale n d H hHpos ε
let T := markedPoissonCellValue n d H hHpos κ ε
(∀ x, variance (T x) μ ≤
Cv * (d : ℝ) ^ (1 / 16 : ℝ) * (∫ s, S x s ^ 2 ∂μ)) →
(∀ x, |(∫ s, T x s ∂μ) -
cellMass P x * max (outcomeMean P false x) (outcomeMean P true x)| ≤
Cb * (sqrt (cellMass P x / (poissonIntensityFormula n * ε * logAlphabet d)) +
1 / (poissonIntensityFormula n * ε * logAlphabet d) +
(d : ℝ) ^ (-(3 / 16 : ℝ)) * (∫ s, S x s ∂μ))) →
sqRisk μ (markedPoissonStatistic n d H hHpos κ ε)
(observedValue P) ≤ C * ((d : ℝ) / (poissonIntensityFormula n * ε * logAlphabet d)) := by
obtain ⟨Av, hAv, hvariance⟩ := activeBranch_variance_scale_rate_le
obtain ⟨Ab, hAb, hbias⟩ := activeBranch_combined_bias_rate_le
let Cs := 512 * H ^ 2 * (1 + 9 * H)
have hCs : 0 ≤ Cs := by dsimp [Cs]; positivity
let C := Cv * Cs * Av + 2 * Cb ^ 2 * (18 + Cs * Ab) + 1
refine ⟨C, by dsimp [C]; positivity, ?_⟩
intro n d hn hd P ε hε hεhalf hactive μ S T hv hb
have hm : 0 < poissonIntensityFormula n := by unfold poissonIntensityFormula; positivity
have ha : 0 < poissonIntensityFormula n * ε := mul_pos hm hε
have hL := logAlphabet_pos hd
have hε1 : ε ≤ 1 := by linarith
have hactive' : (d : ℝ) ≤ 8 * (poissonIntensityFormula n * ε) * logAlphabet d := by
have hh := (div_le_iff₀ hL).1 hactive
have he : 8 * (poissonIntensityFormula n * ε) * logAlphabet d =
(n : ℝ) * ε * logAlphabet d := by unfold poissonIntensityFormula; ring
rw [he]
exact hh
have hS (x : Fin d) : MemLp (S x) 2 μ :=
markedPoissonClippingScale_memLp_two hn hd P H hHpos hH ε hε hε1 x
have hS₁ (x : Fin d) : Integrable (S x) μ := (hS x).integrable (by norm_num)
have hS₂ (x : Fin d) : Integrable (fun s => S x s ^ 2) μ := (hS x).integrable_sq
have hmom (x : Fin d) : (∫ s, S x s ^ 2 ∂μ) ≤
Cs * (cellMass P x * logAlphabet d / (poissonIntensityFormula n * ε) +
logAlphabet d ^ 2 / (poissonIntensityFormula n * ε) ^ 2) := by
simpa only [mul_pow] using
markedPoissonClippingScale_integral_sq_le hn hd P H hHpos ε hε hεhalf x
have hvs := hvariance hd P (fun x => ∫ s, S x s ^ 2 ∂μ)
Cs (poissonIntensityFormula n * ε) hCs ha hactive' hmom
have hbs := hbias μ hd P S Cs Cb (poissonIntensityFormula n * ε)
hCs ha hactive' hS₁ hS₂ hmom
have hab : |∑ x, ((∫ s, T x s ∂μ) -
cellMass P x * max (outcomeMean P false x) (outcomeMean P true x))| ≤
∑ x, Cb * (sqrt (cellMass P x / (poissonIntensityFormula n * ε * logAlphabet d)) +
1 / (poissonIntensityFormula n * ε * logAlphabet d) +
(d : ℝ) ^ (-(3 / 16 : ℝ)) * (∫ s, S x s ∂μ)) :=
(Finset.abs_sum_le_sum_abs _ _).trans (Finset.sum_le_sum fun x _ => hb x)
have hsq := pow_le_pow_left₀ (abs_nonneg _) hab 2
simp only [sq_abs, Finset.sum_sub_distrib] at hsq
rw [← observedValue] at hsq
have hvsum : (∑ x, variance (T x) μ) ≤
Cv * (d : ℝ) ^ (1 / 16 : ℝ) * (∑ x, ∫ s, S x s ^ 2 ∂μ) := by
rw [Finset.mul_sum]
exact Finset.sum_le_sum fun x _ => hv x
have hvs' := mul_le_mul_of_nonneg_left hvs hCv
have hproj := markedPoissonStatistic_sqRisk_le_variance_add_bias
hn hd P H hHpos hH κ ε hε hε1
change sqRisk μ (markedPoissonStatistic n d H hHpos κ ε)
(observedValue P) ≤ (∑ x, variance (T x) μ) +
((∑ x, ∫ s, T x s ∂μ) - observedValue P) ^ 2 at hproj
have hR : 0 ≤ (d : ℝ) / (poissonIntensityFormula n * ε * logAlphabet d) := by positivity
dsimp only [C]
nlinarith
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.UpperNullCell 6 declarations The null-cell case following roadmap equation (11) uses the quadratic cosine displacement at the lower corner and the kernel's second moment.
Null-cell Jackson bias
The null-cell case following roadmap equation (11) uses the quadratic cosine displacement at the lower corner and the kernel's second moment. No inverse arm mass is introduced. The bound applies to the exact factorial mean.
At a zero lower corner, each displaced coordinate has a quadratic envelope. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
conclusion 1
0 ≤ affinePoint r r (cosPoint ((fun _ => pi) - u)) j
conclusion 2
affinePoint r r (cosPoint ((fun _ => pi) - u)) j ≤ r j / 2 * (u j) ^ 2
Proof (Lean source)
lemma nullCorner_affine_cos_bounds (r u : Fin 4 → ℝ) (hr : ∀ j, 0 ≤ r j)
(j : Fin 4) :
0 ≤ affinePoint r r (cosPoint ((fun _ => pi) - u)) j ∧
affinePoint r r (cosPoint ((fun _ => pi) - u)) j ≤ r j / 2 * (u j) ^ 2 := by
have he : affinePoint r r (cosPoint ((fun _ => pi) - u)) j =
r j * (1 - cos (u j)) := by
simp only [affinePoint, cosPoint, Pi.sub_apply, Real.cos_sub,
Real.cos_pi, Real.sin_pi, neg_one_mul, zero_mul, add_zero]
ring
rw [he]
constructor
· exact mul_nonneg (hr j) (sub_nonneg.mpr (Real.cos_le_one _))
· have h := mul_le_mul_of_nonneg_left
(show 1 - cos (u j) ≤ (u j) ^ 2 / 2 by
linarith [Real.one_sub_sq_div_two_le_cos (x := u j)]) (hr j)
nlinarith
The Jackson convolution at a zero lower corner is controlled solely by the second kernel moment, including when the target functional has a tie. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
≤ 32 * (∑ j, r j) / (K : ℝ) ^ 2
Proof (Lean source)
lemma nullCorner_tensorConvolution_le (ε : ℝ) (hε : 0 < ε)
(K : ℕ) (hK : 0 < K) (r : Fin 4 → ℝ) (hr : ∀ j, 0 < r j) :
|tensorConvolution K (fun z => armwiseExtensionFormula ε (affinePoint r r z))
(fun _ => pi)| ≤ 32 * (∑ j, r j) / (K : ℝ) ^ 2 := by
have hQ : ∀ v ∈ centeredRectangle r r, ∀ j, 0 ≤ v j := by
intro v hv j
have h := (abs_le.mp (hv j)).1
linarith
have haff : Continuous (affinePoint r r) := by unfold affinePoint; fun_prop
have hf : ContinuousOn (fun z => armwiseExtensionFormula ε (affinePoint r r z))
(normalizedCube 4) :=
(armwise_continuousOn_rectangle ε hε r r hr hQ).comp haff.continuousOn
(fun z hz => affinePoint_mem_centeredRectangle r r z hr hz)
have hc : Continuous (fun u : Fin 4 → ℝ =>
armwiseExtensionFormula ε (affinePoint r r (cosPoint ((fun _ => pi) - u)))) := by
apply hf.comp_continuous
· unfold cosPoint; fun_prop
· exact fun u => cosPoint_mem_normalizedCube _
have hk : Continuous (tensorJackson K 4) := by unfold tensorJackson; fun_prop
have hcompact : IsCompact (periodBox 4) := isCompact_pi_infinite fun _ => isCompact_Icc
have hi : IntegrableOn (fun u : Fin 4 → ℝ => armwiseExtensionFormula ε
(affinePoint r r (cosPoint ((fun _ => pi) - u))) * tensorJackson K 4 u)
(periodBox 4) := (hc.mul hk).continuousOn.integrableOn_compact hcompact
have hj (j : Fin 4) : IntegrableOn
(fun u => r j / 2 * ((u j) ^ 2 * tensorJackson K 4 u)) (periodBox 4) := by
have hcont : Continuous (fun u : Fin 4 → ℝ =>
r j / 2 * ((u j) ^ 2 * tensorJackson K 4 u)) := by fun_prop
exact hcont.continuousOn.integrableOn_compact hcompact
unfold tensorConvolution
calc
_ ≤ ∫ u in periodBox 4, |armwiseExtensionFormula ε
(affinePoint r r (cosPoint ((fun _ => pi) - u))) * tensorJackson K 4 u| :=
abs_integral_le_integral_abs
_ ≤ ∫ u in periodBox 4, ∑ j, r j / 2 * ((u j) ^ 2 * tensorJackson K 4 u) := by
apply setIntegral_mono hi.abs (integrable_finsetSum _ (fun j _ => hj j))
intro u
dsimp only
have hF := armwiseExtension_nonneg_le_totalMassVec hε
(fun j => (nullCorner_affine_cos_bounds r u (fun j => (hr j).le) j).1)
rw [abs_mul, abs_of_nonneg hF.1, abs_of_nonneg (tensorJackson_nonneg hK u),
]
simp_rw [← mul_assoc]
rw [← Finset.sum_mul]
apply mul_le_mul_of_nonneg_right _ (tensorJackson_nonneg hK u)
apply hF.2.trans
rw [totalMassVec_eq_sum_coordinates]
exact Finset.sum_le_sum fun j _ =>
(nullCorner_affine_cos_bounds r u (fun j => (hr j).le) j).2
_ = ∑ j, r j / 2 * (∫ u in periodBox 4, (u j) ^ 2 * tensorJackson K 4 u) := by
rw [integral_finsetSum _ (fun j _ => hj j)]
simp only [integral_const_mul]
_ ≤ ∑ j, r j / 2 * (64 / (K : ℝ) ^ 2) := by
apply Finset.sum_le_sum
intro j _
apply mul_le_mul_of_nonneg_left _ (div_nonneg (hr j).le (by norm_num))
exact (tensorJackson_second_moment_eq hK j).le.trans (jackson_second_moment K hK)
_ = _ := by rw [← Finset.sum_mul, ← Finset.sum_div]; ring
The exact coefficient expression used by the factorial mean has the null-cell second-order bias bound; polynomial uniqueness and convolution evaluation link the bound to the estimator's chosen polynomial. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
≤ 32 * (∑ j, r j) / (K : ℝ) ^ 2
Proof (Lean source)
lemma nullCorner_jacksonCoeff_mean_le (ε : ℝ) (hε : 0 < ε)
(K : ℕ) (hK : 0 < K) (r : Fin 4 → ℝ) (hr : ∀ j, 0 < r j) :
|armwiseExtensionFormula ε r + ∑ α : Fin 4 → Fin (2 * (K - 1) + 1),
jacksonCoeff ε K r r α * ∏ j, (-1 : ℝ) ^ (α j).val| ≤
32 * (∑ j, r j) / (K : ℝ) ^ 2 := by
have hQ : ∀ v ∈ centeredRectangle r r, ∀ j, 0 ≤ v j := by
intro v hv j
have h := (abs_le.mp (hv j)).1
linarith
rw [jacksonCoeff_sum_eq_eval ε hε K hK r r hr hQ]
have hcos : cosPoint (fun _ : Fin 4 => pi) = (fun _ => (-1 : ℝ)) := by
funext j
simp [cosPoint]
rw [← hcos, jacksonPolynomialPair_cos_eval ε hε K hK r r hr hQ]
exact nullCorner_tensorConvolution_le ε hε K hK r hr
Zero pilot counts put the midpoint exactly one radius above the origin. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
H :
ℝ
hH :
0 < H
m L :
ℝ
hm :
0 < m
hL :
0 ≤ L
conclusion
conclusion 1
pilotMidpoint H hH m L (fun _ => 0) = pilotRadiusFormula H hH m L (fun _ => 0)
Proof (Lean source)
lemma zeroPilot_midpoint_eq_radius (H : ℝ) (hH : 0 < H) (m L : ℝ)
(hm : 0 < m) (hL : 0 ≤ L) :
pilotMidpoint H hH m L (fun _ => 0) = pilotRadiusFormula H hH m L (fun _ => 0) ∧
∀ j, pilotRadiusFormula H hH m L (fun _ => 0) j = H * (L / m) / 2 := by
have hh : 0 ≤ H * (L / m) := by positivity
constructor
· funext j
simp [pilotMidpoint, pilotRadiusFormula, pilotLower, pilotUpperFormula, pilotCenterFormula,
pilotHalfWidthFormula, max_eq_left (neg_nonpos.mpr hh)]
· intro j
simp [pilotRadiusFormula, pilotLower, pilotUpperFormula, pilotCenterFormula,
pilotHalfWidthFormula, max_eq_left (neg_nonpos.mpr hh)]
The actual zero-count pilot's factorial mean obeys the null-cell bias bound preceding equation (20), with no division by any cell or arm mass. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
H :
ℝ
hH :
0 < H
m L :
ℝ
hm :
0 < m
hL :
0 < L
ε :
ℝ
hε :
0 < ε
K :
ℕ
hK :
0 < K
conclusion
b :
pilotMidpoint H hH m L (fun _ => 0)
r :
pilotRadiusFormula H hH m L (fun _ => 0)
|(armwiseExtensionFormula ε b + ∑ α : Fin 4 → Fin (2 * (K - 1) + 1), jacksonCoeff ε K b r α * ∏ j, ((0 - b j) / r j) ^ (α j).val)
- armwiseExtensionFormula ε 0|
≤ 64 * H * L / (m * (K : ℝ) ^ 2)
Proof (Lean source)
lemma zeroPilot_jacksonCoeff_mean_le (H : ℝ) (hH : 0 < H) (m L : ℝ)
(hm : 0 < m) (hL : 0 < L) (ε : ℝ) (hε : 0 < ε)
(K : ℕ) (hK : 0 < K) :
let b := pilotMidpoint H hH m L (fun _ => 0)
let r := pilotRadiusFormula H hH m L (fun _ => 0)
|(armwiseExtensionFormula ε b + ∑ α : Fin 4 → Fin (2 * (K - 1) + 1),
jacksonCoeff ε K b r α * ∏ j, ((0 - b j) / r j) ^ (α j).val) -
armwiseExtensionFormula ε 0| ≤ 64 * H * L / (m * (K : ℝ) ^ 2) := by
obtain ⟨he, hr⟩ := zeroPilot_midpoint_eq_radius H hH m L hm hL.le
dsimp only
rw [he]
have hp := pilotRadius_pos H hH m L hm hL (fun _ => 0)
have hz : armwiseExtensionFormula ε 0 = 0 := by simp [armwiseExtensionFormula]
simp only [zero_sub, neg_div, div_self (ne_of_gt (hp _)), hz, sub_zero]
have hb := nullCorner_jacksonCoeff_mean_le ε hε K hK
(pilotRadiusFormula H hH m L (fun _ => 0)) hp
apply hb.trans_eq
simp only [hr, Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul]
ring
theorem zeroPilot_jacksonCoeff_mean_normalized_le Lemma zeroPilot_jacksonCoeff_mean_normalized_le in the paper ↗
Logarithmic degree calibration gives the null-cell instance of (20). The sole inverse-overlap term is the linear normalized scale. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
H :
ℝ
hH :
0 < H
m L :
ℝ
hm :
0 < m
hL :
0 < L
ε :
ℝ
hε :
0 < ε
hε1 :
ε ≤ 1
K :
ℕ
hK :
0 < K
κ :
ℝ
hκ :
0 < κ
hdegree :
κ * L ≤ K
conclusion
b :
pilotMidpoint H hH m L (fun _ => 0)
r :
pilotRadiusFormula H hH m L (fun _ => 0)
|(armwiseExtensionFormula ε b + ∑ α : Fin 4 → Fin (2 * (K - 1) + 1), jacksonCoeff ε K b r α * ∏ j, ((0 - b j) / r j) ^ (α j).val)
- armwiseExtensionFormula ε 0|
≤ (64 * H / κ ^ 2) * (1 / (m * ε * L))
Proof (Lean source)
lemma zeroPilot_jacksonCoeff_mean_normalized_le (H : ℝ) (hH : 0 < H) (m L : ℝ)
(hm : 0 < m) (hL : 0 < L) (ε : ℝ) (hε : 0 < ε) (hε1 : ε ≤ 1)
(K : ℕ) (hK : 0 < K) (κ : ℝ) (hκ : 0 < κ) (hdegree : κ * L ≤ K) :
let b := pilotMidpoint H hH m L (fun _ => 0)
let r := pilotRadiusFormula H hH m L (fun _ => 0)
|(armwiseExtensionFormula ε b + ∑ α : Fin 4 → Fin (2 * (K - 1) + 1),
jacksonCoeff ε K b r α * ∏ j, ((0 - b j) / r j) ^ (α j).val) -
armwiseExtensionFormula ε 0| ≤ (64 * H / κ ^ 2) * (1 / (m * ε * L)) := by
have hsq := pow_le_pow_left₀ (mul_pos hκ hL).le hdegree 2
have hKreal : (0 : ℝ) < K := by exact_mod_cast hK
have he : L / (K : ℝ) ^ 2 ≤ 1 / (κ ^ 2 * L) := by
apply (div_le_div_iff₀ (sq_pos_of_pos hKreal) (mul_pos (sq_pos_of_pos hκ) hL)).2
nlinarith
have hover : 1 / (m * L) ≤ 1 / (m * ε * L) := by
apply one_div_le_one_div_of_le (mul_pos (mul_pos hm hε) hL)
nlinarith [mul_le_mul_of_nonneg_left hε1 (mul_pos hm hL).le]
apply (zeroPilot_jacksonCoeff_mean_le H hH m L hm hL ε hε K hK).trans
calc
64 * H * L / (m * (K : ℝ) ^ 2) = (64 * H / m) * (L / (K : ℝ) ^ 2) := by ring
_ ≤ (64 * H / m) * (1 / (κ ^ 2 * L)) :=
mul_le_mul_of_nonneg_left he (by positivity)
_ = (64 * H / κ ^ 2) * (1 / (m * L)) := by ring
_ ≤ _ := mul_le_mul_of_nonneg_left hover (by positivity)
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.UpperNullMarked 4 declarations At a null cell both count vectors vanish almost surely.
Null-cell loss without factorial inflation
At a null cell both count vectors vanish almost surely. The factorial lift then equals its Jackson mean, which is inside the clipping interval. Thus only the null-corner approximation error remains in roadmap (20)/(28).
theorem nullCell_clippedCellValue_ae_error_le Lemma nullCell_clippedCellValue_ae_error_le in the paper ↗
At a null cell clipping has no displacement or stochastic inflation: the random-pilot loss is bounded by the null-corner Jackson error alone. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
d :
ℕ
hd :
1 ≤ d
P :
x :
Fin d
hp :
cellMass P x = 0
H :
ℝ
hH :
0 < H
m L ε :
ℝ
hm :
0 < m
hL :
0 < L
hε :
0 < ε
hε1 :
ε ≤ 1
K :
ℕ
hK :
0 < K
κ :
ℝ
hκ :
0 < κ
hdegree :
κ * L ≤ K
Np Ne :
Fin 4 → Ω → ℕ
hNp :
∀ j,
hNe :
∀ j,
hind :
iIndepFun Ne μ
conclusion
∀ᵐ ω ∂μ, |clippedCellValueFormula ε K d m (pilotMidpoint H hH m L (fun j => Np j ω)) (pilotRadiusFormula H hH m L (fun j => Np j ω)) (fun j => Ne j ω)
- armwiseExtensionFormula ε (cellVector P x)|
≤ (64 * H / κ ^ 2) * (1 / (m * ε * L))
Proof (Lean source)
lemma nullCell_clippedCellValue_ae_error_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] {d : ℕ} (hd : 1 ≤ d)
(P : DiscreteLaw d) (x : Fin d) (hp : cellMass P x = 0)
(H : ℝ) (hH : 0 < H) (m L ε : ℝ) (hm : 0 < m) (hL : 0 < L)
(hε : 0 < ε) (hε1 : ε ≤ 1) (K : ℕ) (hK : 0 < K)
(κ : ℝ) (hκ : 0 < κ) (hdegree : κ * L ≤ K)
(Np Ne : Fin 4 → Ω → ℕ)
(hNp : ∀ j, HasLaw (Np j)
(poissonMeasure ⟨m * cellVector P x j,
mul_nonneg hm.le ENNReal.toReal_nonneg⟩) μ)
(hNe : ∀ j, HasLaw (Ne j)
(poissonMeasure ⟨m * cellVector P x j,
mul_nonneg hm.le ENNReal.toReal_nonneg⟩) μ)
(hind : iIndepFun Ne μ) :
∀ᵐ ω ∂μ, |clippedCellValueFormula ε K d m
(pilotMidpoint H hH m L (fun j => Np j ω))
(pilotRadiusFormula H hH m L (fun j => Np j ω)) (fun j => Ne j ω) -
armwiseExtensionFormula ε (cellVector P x)| ≤
(64 * H / κ ^ 2) * (1 / (m * ε * L)) := by
let b := pilotMidpoint H hH m L (fun _ => 0)
let r := pilotRadiusFormula H hH m L (fun _ => 0)
let p := armwiseExtensionFormula ε b + ∑ α : Fin 4 → Fin (2 * (K - 1) + 1),
jacksonCoeff ε K b r α * ∏ j, ((0 - b j) / r j) ^ (α j).val
have hq := cellVector_eq_zero_of_cellMass_eq_zero P x hp
have hep := nullCell_poissonCounts_ae_zero μ P x hp m hm.le Np hNp
have hee := nullCell_poissonCounts_ae_zero μ P x hp m hm.le Ne hNe
have hmean := factorialCellValue_integral_eq_scaled μ Ne (cellVector P x)
(fun _ => toReal_nonneg) m hm hNe hind ε K b r
have hraw : factorialCellValue ε K m b r 0 = p := by
have he : (fun ω => factorialCellValue ε K m b r (fun j => Ne j ω)) =ᵐ[μ]
(fun _ => factorialCellValue ε K m b r 0) := by
filter_upwards [hee] with ω hω
rw [hω]
rw [integral_congr_ae he] at hmean
simpa [hq, p] using hmean
have hmem := pilotJackson_mean_mem_clipping_interval H hH m L hm hL
(fun _ => 0) ε hε K d hK hd (0 : Fin 4 → ℝ)
(zeroPilot_rectangle_contains_zero H hH m L hm hL.le)
change 0 ≤ (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε b r ∧
|p - armwiseExtensionFormula ε b| ≤ (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε b r at hmem
have hclip : clippedCellValueFormula ε K d m b r 0 = p := by
rw [clippedCellValueFormula, hraw]
rw [min_eq_right (by linarith [(abs_le.mp hmem.2).2]),
max_eq_right (by linarith [(abs_le.mp hmem.2).1])]
have hb := zeroPilot_jacksonCoeff_mean_normalized_le H hH m L hm hL
ε hε hε1 K hK κ hκ hdegree
filter_upwards [hep, hee] with ω hωp hωe
rw [hωp, hωe, hq]
change |clippedCellValueFormula ε K d m b r 0 - armwiseExtensionFormula ε 0| ≤ _
rw [hclip]
exact hb
theorem markedPoissonEvaluation_iIndepFun
Evaluation coordinates of one implemented marked cell are independent; this specializes the independent marked histogram to the false mark. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
iIndepFun (fun j : Fin 4 => fun s : FiniteSample (Obs d × Bool) => markedCount (le_refl s.1) (fun i => (s.2 i).1) (fun i => (s.2 i).2) false x j) (finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4))
Proof (Lean source)
lemma markedPoissonEvaluation_iIndepFun {d : ℕ} (P : DiscreteLaw d) (n : ℕ)
(x : Fin d) :
iIndepFun (fun j : Fin 4 => fun s :
FiniteSample (Obs d × Bool) =>
markedCount (le_refl s.1) (fun i => (s.2 i).1) (fun i => (s.2 i).2) false x j)
(finitePoissonSampleLaw
(P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)) := by
let g : Fin 4 → Obs d × Bool := fun j =>
((x, decide (j.val / 2 = 1), decide (j.val % 2 = 1)), false)
have hg : Injective g := by
intro j k hjk
fin_cases j <;> fin_cases k <;> simp_all [g]
simp_rw [markedCount_eq_markedPoissonHistogram]
exact (markedPoissonHistogram_iIndepFun P n).precomp hg
theorem markedPoissonCellValue_null_ae_error_le Lemma markedPoissonCellValue_null_ae_error_le in the paper ↗
The actual null-cell statistic satisfies the normalized approximation bound almost surely for every positive degree coefficient. All count-law and independence inputs are supplied by the implemented marked experiment. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
n d :
ℕ
hn :
1 ≤ n
hd :
1 ≤ d
P :
x :
Fin d
hp :
cellMass P x = 0
H :
ℝ
hH :
0 < H
κ ε :
ℝ
hκ :
0 < κ
hε :
0 < ε
hε1 :
ε ≤ 1
conclusion
μ :
finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4)
∀ᵐ s ∂μ, |markedPoissonCellValue n d H hH κ ε x s
- armwiseExtensionFormula ε (cellVector P x)|
≤ (64 * H / (κ / 2) ^ 2) * (1 / (poissonIntensityFormula n * ε * logAlphabet d))
Proof (Lean source)
lemma markedPoissonCellValue_null_ae_error_le {n d : ℕ} (hn : 1 ≤ n) (hd : 1 ≤ d)
(P : DiscreteLaw d) (x : Fin d) (hp : cellMass P x = 0)
(H : ℝ) (hH : 0 < H) (κ ε : ℝ) (hκ : 0 < κ)
(hε : 0 < ε) (hε1 : ε ≤ 1) :
let μ := finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)
∀ᵐ s ∂μ, |markedPoissonCellValue n d H hH κ ε x s -
armwiseExtensionFormula ε (cellVector P x)| ≤
(64 * H / (κ / 2) ^ 2) * (1 / (poissonIntensityFormula n * ε * logAlphabet d)) := by
let μ := finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)
let W : Bool → Fin 4 → FiniteSample (Obs d × Bool) → ℕ := fun a j s =>
markedCount (le_refl s.1) (fun i => (s.2 i).1) (fun i => (s.2 i).2) a x j
have hm : 0 < poissonIntensityFormula n := by unfold poissonIntensityFormula; positivity
have hlaw (a : Bool) (j : Fin 4) : HasLaw (W a j)
(poissonMeasure ⟨poissonIntensityFormula n * cellVector P x j,
mul_nonneg hm.le ENNReal.toReal_nonneg⟩) μ := by
convert markedCount_poisson_hasLaw P n a x j using 1
congr 1
have ht := nullCell_clippedCellValue_ae_error_le μ hd P x hp H hH
(poissonIntensityFormula n) (logAlphabet d) ε hm (logAlphabet_pos hd) hε hε1
(jacksonDegree κ d) (by unfold jacksonDegree; omega) (κ / 2) (by positivity)
(jacksonDegree_ge_half_log κ d) (W true) (W false) (hlaw true) (hlaw false)
(markedPoissonEvaluation_iIndepFun P n x)
exact ht
theorem markedPoissonCellValue_null_good_moments_le Lemma markedPoissonCellValue_null_good_moments_le in the paper ↗
Both good-pilot content estimates for an implemented null cell hold without a scale-moment or factorial-inflation term. Only the squared Jackson approximation scale remains in the second moment. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
n d :
ℕ
hn :
1 ≤ n
hd :
1 ≤ d
P :
x :
Fin d
hp :
cellMass P x = 0
H :
ℝ
hH :
0 < H
κ ε :
ℝ
hκ :
0 < κ
hε :
0 < ε
hε1 :
ε ≤ 1
conclusion
μ :
finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4)
B :
(64 * H / (κ / 2) ^ 2) * (1 / (poissonIntensityFormula n * ε * logAlphabet d))
(∫ s, if s ∈ markedPoissonGoodPilot n P H hH x then (markedPoissonCellValue n d H hH κ ε x s - armwiseExtensionFormula ε (cellVector P x)) ^ 2 else 0 ∂μ)
≤ B ^ 2
|∫ s, if s ∈ markedPoissonGoodPilot n P H hH x then markedPoissonCellValue n d H hH κ ε x s
- armwiseExtensionFormula ε (cellVector P x) else 0 ∂μ|
≤ B
Proof (Lean source)
lemma markedPoissonCellValue_null_good_moments_le {n d : ℕ}
(hn : 1 ≤ n) (hd : 1 ≤ d) (P : DiscreteLaw d) (x : Fin d)
(hp : cellMass P x = 0) (H : ℝ) (hH : 0 < H)
(κ ε : ℝ) (hκ : 0 < κ) (hε : 0 < ε) (hε1 : ε ≤ 1) :
let μ := finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)
let B := (64 * H / (κ / 2) ^ 2) *
(1 / (poissonIntensityFormula n * ε * logAlphabet d))
(∫ s, if s ∈ markedPoissonGoodPilot n P H hH x then
(markedPoissonCellValue n d H hH κ ε x s -
armwiseExtensionFormula ε (cellVector P x)) ^ 2 else 0 ∂μ) ≤ B ^ 2 ∧
|∫ s, if s ∈ markedPoissonGoodPilot n P H hH x then
markedPoissonCellValue n d H hH κ ε x s -
armwiseExtensionFormula ε (cellVector P x) else 0 ∂μ| ≤ B := by
classical
dsimp only
let μ := finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)
let B := (64 * H / (κ / 2) ^ 2) *
(1 / (poissonIntensityFormula n * ε * logAlphabet d))
have hm : 0 < poissonIntensityFormula n := by unfold poissonIntensityFormula; positivity
have hL := logAlphabet_pos hd
have hB : 0 ≤ B := by dsimp [B]; positivity
have he := markedPoissonCellValue_null_ae_error_le hn hd P x hp H hH κ ε hκ hε hε1
constructor
· apply (integral_mono_of_nonneg (ae_of_all μ (fun s => by
split_ifs <;> positivity)) (integrable_const (B ^ 2)) ?_).trans_eq (by simp [B])
filter_upwards [he] with s hs
split_ifs
· have hsq := pow_le_pow_left₀ (abs_nonneg _) hs 2
simpa only [sq_abs] using hsq
· exact sq_nonneg B
· apply abs_integral_le_integral_abs.trans
apply (integral_mono_of_nonneg (ae_of_all μ (fun _ => abs_nonneg _))
(integrable_const B) ?_).trans_eq (by simp [B])
filter_upwards [he] with s hs
split_ifs
· exact hs
· simpa only [abs_zero] using hB
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.UpperNullPilot 6 declarations The null-cell argument after roadmap (11) applies to the actual random pilot: zero cell mass forces all four Poisson counts to vanish almost surely.
Random pilots at null cells
The null-cell argument after roadmap (11) applies to the actual random pilot: zero cell mass forces all four Poisson counts to vanish almost surely. This transfers the calibrated zero-pilot bounds (20)/(28) to the random-pilot statistic, without conditioning on a positive-probability pilot event.
theorem cellVector_eq_zero_of_cellMass_eq_zero
Zero cell mass forces every observed atom in that cell to have zero mass. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma cellVector_eq_zero_of_cellMass_eq_zero {d : ℕ} (P : DiscreteLaw d)
(x : Fin d) (hp : cellMass P x = 0) : cellVector P x = 0 := by
have ha (a : Bool) : (∑ y : Bool, jointMass P x a y) = 0 := by
have h := (Finset.sum_eq_zero_iff_of_nonneg
(fun a (_ : a ∈ univ) =>
Finset.sum_nonneg (fun y _ => toReal_nonneg))).mp hp
exact h a (Finset.mem_univ a)
have hj (a y : Bool) : jointMass P x a y = 0 := by
exact (Finset.sum_eq_zero_iff_of_nonneg
(fun y (_ : y ∈ univ) => toReal_nonneg)).mp (ha a)
y (Finset.mem_univ y)
funext j
exact hj _ _
theorem nullCell_poissonCounts_ae_zero
The marginal Poisson laws alone force the entire null-cell count vector to vanish almost surely; independence is unnecessary for this conclusion. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma nullCell_poissonCounts_ae_zero {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) {d : ℕ} (P : DiscreteLaw d) (x : Fin d)
(hp : cellMass P x = 0) (m : ℝ) (hm : 0 ≤ m) (W : Fin 4 → Ω → ℕ)
(hW : ∀ j, HasLaw (W j)
(poissonMeasure ⟨m * cellVector P x j,
mul_nonneg hm ENNReal.toReal_nonneg⟩) μ) :
(fun ω j => W j ω) =ᵐ[μ] (fun _ => 0) := by
have hq := cellVector_eq_zero_of_cellMass_eq_zero P x hp
have hz (j : Fin 4) : W j =ᵐ[μ] (fun _ => 0) := by
have hrate : (⟨m * cellVector P x j,
mul_nonneg hm ENNReal.toReal_nonneg⟩ : NNReal) = 0 := by
ext
simp [hq]
have h := hW j
rw [hrate] at h
exact (h.ae_iff (by fun_prop)).mpr
poissonMeasure_zero_ae
filter_upwards [Filter.eventually_all.mpr hz] with ω hω
funext j
exact hω j
theorem nullCell_pilotRectangle_ae_contains Lemma nullCell_pilotRectangle_ae_contains in the paper ↗
A null cell is localized almost surely on the actual random pilot rectangle, including its lower boundary, as required in the paragraph after (11). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
Measure Ω
d :
ℕ
P :
x :
Fin d
hp :
cellMass P x = 0
H :
ℝ
hH :
0 < H
m L :
ℝ
hm :
0 < m
hL :
0 ≤ L
Np :
Fin 4 → Ω → ℕ
hNp :
∀ j,
conclusion
∀ᵐ ω ∂μ, cellVector P x ∈ pilotRectangle (pilotLower H hH m L (fun j => Np j ω)) (pilotUpperFormula H hH m L (fun j => Np j ω))
Proof (Lean source)
lemma nullCell_pilotRectangle_ae_contains {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) {d : ℕ} (P : DiscreteLaw d) (x : Fin d)
(hp : cellMass P x = 0) (H : ℝ) (hH : 0 < H) (m L : ℝ)
(hm : 0 < m) (hL : 0 ≤ L) (Np : Fin 4 → Ω → ℕ)
(hNp : ∀ j, HasLaw (Np j)
(poissonMeasure ⟨m * cellVector P x j,
mul_nonneg hm.le ENNReal.toReal_nonneg⟩) μ) :
∀ᵐ ω ∂μ, cellVector P x ∈ pilotRectangle
(pilotLower H hH m L (fun j => Np j ω))
(pilotUpperFormula H hH m L (fun j => Np j ω)) := by
filter_upwards [nullCell_poissonCounts_ae_zero μ P x hp m hm.le Np hNp] with ω hω
rw [hω, cellVector_eq_zero_of_cellMass_eq_zero P x hp]
exact zeroPilot_rectangle_contains_zero H hH m L hm hL
theorem nullCell_clippedCellValue_ae_constant Lemma nullCell_clippedCellValue_ae_constant in the paper ↗
With both pilot and evaluation Poisson counts at a null cell, the actual clipped cell statistic is almost surely a deterministic constant. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
Measure Ω
d :
ℕ
P :
x :
Fin d
hp :
cellMass P x = 0
H :
ℝ
hH :
0 < H
m L ε :
ℝ
hm :
0 < m
K :
ℕ
Np Ne :
Fin 4 → Ω → ℕ
hNp :
∀ j,
hNe :
∀ j,
conclusion
(fun ω => clippedCellValueFormula ε K d m (pilotMidpoint H hH m L (fun j => Np j ω)) (pilotRadiusFormula H hH m L (fun j => Np j ω)) (fun j => Ne j ω))
=ᵐ[μ] (fun _ => clippedCellValueFormula ε K d m (pilotMidpoint H hH m L 0) (pilotRadiusFormula H hH m L 0) 0)
Proof (Lean source)
lemma nullCell_clippedCellValue_ae_constant {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) {d : ℕ} (P : DiscreteLaw d) (x : Fin d)
(hp : cellMass P x = 0) (H : ℝ) (hH : 0 < H) (m L ε : ℝ)
(hm : 0 < m) (K : ℕ) (Np Ne : Fin 4 → Ω → ℕ)
(hNp : ∀ j, HasLaw (Np j)
(poissonMeasure ⟨m * cellVector P x j,
mul_nonneg hm.le ENNReal.toReal_nonneg⟩) μ)
(hNe : ∀ j, HasLaw (Ne j)
(poissonMeasure ⟨m * cellVector P x j,
mul_nonneg hm.le ENNReal.toReal_nonneg⟩) μ) :
(fun ω => clippedCellValueFormula ε K d m
(pilotMidpoint H hH m L (fun j => Np j ω))
(pilotRadiusFormula H hH m L (fun j => Np j ω)) (fun j => Ne j ω)) =ᵐ[μ]
(fun _ => clippedCellValueFormula ε K d m
(pilotMidpoint H hH m L 0) (pilotRadiusFormula H hH m L 0) 0) := by
filter_upwards [nullCell_poissonCounts_ae_zero μ P x hp m hm.le Np hNp,
nullCell_poissonCounts_ae_zero μ P x hp m hm.le Ne hNe] with ω hωp hωe
rw [hωp, hωe]
theorem nullCell_clippedCellValue_variance_zero Lemma nullCell_clippedCellValue_variance_zero in the paper ↗
Null cells contribute no variance to the independent-cell assembly (29). Their approximation bias need not vanish and is still controlled by (20). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
d :
ℕ
P :
x :
Fin d
hp :
cellMass P x = 0
H :
ℝ
hH :
0 < H
m L ε :
ℝ
hm :
0 < m
K :
ℕ
Np Ne :
Fin 4 → Ω → ℕ
hNp :
∀ j,
hNe :
∀ j,
conclusion
variance (fun ω => clippedCellValueFormula ε K d m (pilotMidpoint H hH m L (fun j => Np j ω)) (pilotRadiusFormula H hH m L (fun j => Np j ω)) (fun j => Ne j ω)) μ
= 0
Proof (Lean source)
lemma nullCell_clippedCellValue_variance_zero {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] {d : ℕ} (P : DiscreteLaw d) (x : Fin d)
(hp : cellMass P x = 0) (H : ℝ) (hH : 0 < H) (m L ε : ℝ)
(hm : 0 < m) (K : ℕ) (Np Ne : Fin 4 → Ω → ℕ)
(hNp : ∀ j, HasLaw (Np j)
(poissonMeasure ⟨m * cellVector P x j,
mul_nonneg hm.le ENNReal.toReal_nonneg⟩) μ)
(hNe : ∀ j, HasLaw (Ne j)
(poissonMeasure ⟨m * cellVector P x j,
mul_nonneg hm.le ENNReal.toReal_nonneg⟩) μ) :
variance (fun ω => clippedCellValueFormula ε K d m
(pilotMidpoint H hH m L (fun j => Np j ω))
(pilotRadiusFormula H hH m L (fun j => Np j ω)) (fun j => Ne j ω)) μ = 0 := by
rw [variance_congr
(nullCell_clippedCellValue_ae_constant μ P x hp H hH m L ε hm K Np Ne hNp hNe)]
simp [variance, evariance]
theorem nullCell_randomPilot_target_tuning_exists Lemma nullCell_randomPilot_target_tuning_exists in the paper ↗
The same universal degree tuning as (28) bounds the actual random-pilot null-cell mean and second moment. The marginal count laws derive the zero pilot, rather than imposing it as an extra localization assumption. In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ κ A : ℝ,
0 < κ
0 < A
conclusion 1
Ω :
Type*
μ :
Measure Ω
d :
ℕ
1 ≤ d
ε :
ℝ
0 < ε
ε ≤ 1
P :
x :
Fin d
cellMass P x = 0
Np Ne :
Fin 4 → Ω → ℕ
H :
ℝ
hH :
0 < H
1 ≤ H
m :
ℝ
hm :
0 < m
(∀ j, HasLaw (Np j) (poissonMeasure ⟨m * cellVector P x j, mul_nonneg hm.le ENNReal.toReal_nonneg⟩) μ)
(∀ j, HasLaw (Ne j) (poissonMeasure ⟨m * cellVector P x j, mul_nonneg hm.le ENNReal.toReal_nonneg⟩) μ)
iIndepFun Ne μ
b :
fun ω => pilotMidpoint H hH m (logAlphabet d) (fun j => Np j ω)
r :
fun ω => pilotRadiusFormula H hH m (logAlphabet d) (fun j => Np j ω)
K :
jacksonDegree κ d
S :
clippingScaleFormula ε (pilotMidpoint H hH m (logAlphabet d) 0) (pilotRadiusFormula H hH m (logAlphabet d) 0)
B :
(64 * H / (κ / 2) ^ 2) * (1 / (m * ε * logAlphabet d))
|(∫ ω, clippedCellValueFormula ε K d m (b ω) (r ω) (fun j => Ne j ω) ∂μ)
- armwiseExtensionFormula ε (cellVector P x)|
≤ B + A * (d : ℝ) ^ (-(3 / 16 : ℝ)) * S
(∫ ω, (clippedCellValueFormula ε K d m (b ω) (r ω) (fun j => Ne j ω) - armwiseExtensionFormula ε (cellVector P x)) ^ 2 ∂μ)
≤ 2 * A * (d : ℝ) ^ (1 / 16 : ℝ) * S ^ 2 + 2 * B ^ 2
Proof (Lean source)
lemma nullCell_randomPilot_target_tuning_exists :
∃ κ A : ℝ, 0 < κ ∧ 0 < A ∧ ∀ {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] {d : ℕ}, 1 ≤ d →
∀ (ε : ℝ), 0 < ε → ε ≤ 1 → ∀ (P : DiscreteLaw d) (x : Fin d),
cellMass P x = 0 → ∀ (Np Ne : Fin 4 → Ω → ℕ)
(H : ℝ) (hH : 0 < H), 1 ≤ H → ∀ (m : ℝ) (hm : 0 < m),
(∀ j, HasLaw (Np j)
(poissonMeasure ⟨m * cellVector P x j,
mul_nonneg hm.le ENNReal.toReal_nonneg⟩) μ) →
(∀ j, HasLaw (Ne j)
(poissonMeasure ⟨m * cellVector P x j,
mul_nonneg hm.le ENNReal.toReal_nonneg⟩) μ) →
iIndepFun Ne μ →
let b := fun ω => pilotMidpoint H hH m (logAlphabet d) (fun j => Np j ω)
let r := fun ω => pilotRadiusFormula H hH m (logAlphabet d) (fun j => Np j ω)
let K := jacksonDegree κ d
let S := clippingScaleFormula ε (pilotMidpoint H hH m (logAlphabet d) 0)
(pilotRadiusFormula H hH m (logAlphabet d) 0)
let B := (64 * H / (κ / 2) ^ 2) * (1 / (m * ε * logAlphabet d))
(|(∫ ω, clippedCellValueFormula ε K d m (b ω) (r ω) (fun j => Ne j ω) ∂μ) -
armwiseExtensionFormula ε (cellVector P x)| ≤
B + A * (d : ℝ) ^ (-(3 / 16 : ℝ)) * S) ∧
((∫ ω, (clippedCellValueFormula ε K d m (b ω) (r ω) (fun j => Ne j ω) -
armwiseExtensionFormula ε (cellVector P x)) ^ 2 ∂μ) ≤
2 * A * (d : ℝ) ^ (1 / 16 : ℝ) * S ^ 2 + 2 * B ^ 2) := by
obtain ⟨κ, A, hκ, hA, hzero⟩ := zeroPilot_clippedCellValue_target_tuning_exists
refine ⟨κ, A, hκ, hA, ?_⟩
intro Ω _ μ _ d hd ε hε hε1 P x hp Np Ne H hH hHlarge m hm hNp hNe hind
have hq := cellVector_eq_zero_of_cellMass_eq_zero P x hp
have hNe0 : ∀ j, HasLaw (Ne j) (poissonMeasure 0) μ := by
intro j
have hrate : (⟨m * cellVector P x j,
mul_nonneg hm.le ENNReal.toReal_nonneg⟩ : NNReal) = 0 := by
ext
simp [hq]
have h := hNe j
rw [hrate] at h
exact h
have hpilot := nullCell_poissonCounts_ae_zero μ P x hp m hm.le Np hNp
dsimp only
have heq :
(fun ω => clippedCellValueFormula ε (jacksonDegree κ d) d m
(pilotMidpoint H hH m (logAlphabet d) (fun j => Np j ω))
(pilotRadiusFormula H hH m (logAlphabet d) (fun j => Np j ω)) (fun j => Ne j ω)) =ᵐ[μ]
(fun ω => clippedCellValueFormula ε (jacksonDegree κ d) d m
(pilotMidpoint H hH m (logAlphabet d) 0)
(pilotRadiusFormula H hH m (logAlphabet d) 0) (fun j => Ne j ω)) := by
filter_upwards [hpilot] with ω hω
rw [hω]
have hsq := heq.fun_comp (fun z => (z - armwiseExtensionFormula ε (cellVector P x)) ^ 2)
dsimp only [Function.comp_def] at hsq
rw [integral_congr_ae heq, integral_congr_ae hsq, hq]
exact hzero μ hd ε hε hε1 Ne H hH hHlarge m hm hNe0 hind
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.UpperOccupiedCell 4 declarations A reference-point modulus is integrated against the positive tensor kernel.
Occupied-cell Jackson bias
A reference-point modulus is integrated against the positive tensor kernel. The weights remain attached to the true arm masses, as in roadmap (11), and the localized pilot envelope then yields (20).
theorem tensorConvolution_reference_boundary_le Lemma tensorConvolution_reference_boundary_le in the paper ↗
Only a modulus anchored at the evaluation point is needed for the boundary Jackson estimate; no uniform Lipschitz bound over pairs of points is imposed. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d K :
ℕ
hK :
0 < K
g :
(Fin d → ℝ) → ℝ
hg :
ContinuousOn g (normalizedCube d)
r :
Fin d → ℝ
hr :
∀ i, 0 ≤ r i
L :
ℝ
hL :
0 ≤ L
theta :
Fin d → ℝ
hlip :
∀ x ∈ normalizedCube d,
|g x - g (cosPoint theta)| ≤ L * ∑ i, r i * |x i - cosPoint theta i|
conclusion
|tensorConvolution K g theta - g (cosPoint theta)|
Proof (Lean source)
lemma tensorConvolution_reference_boundary_le {d K : ℕ} (hK : 0 < K)
(g : (Fin d → ℝ) → ℝ) (hg : ContinuousOn g (normalizedCube d))
(r : Fin d → ℝ) (hr : ∀ i, 0 ≤ r i)
(L : ℝ) (hL : 0 ≤ L)
(theta : Fin d → ℝ)
(hlip : ∀ x ∈ normalizedCube d,
|g x - g (cosPoint theta)| ≤ L * ∑ i, r i * |x i - cosPoint theta i|) :
|tensorConvolution K g theta - g (cosPoint theta)| ≤
L * ∑ i : Fin d,
(32 * r i * |sin (theta i)| / (K : ℝ) +
32 * r i / (K : ℝ) ^ 2) := by
classical
have hcompact : IsCompact (periodBox d) := by
rw [show periodBox d = {u | ∀ i, u i ∈ Icc (-pi) pi} by rfl]
exact isCompact_pi_infinite fun _ ↦ isCompact_Icc
have hcos : Continuous (fun u : Fin d → ℝ ↦ cosPoint (theta - u)) := by
apply continuous_pi
intro i
change Continuous (fun u : Fin d → ℝ ↦ cos (theta i - u i))
fun_prop
have hgcos : Continuous (fun u : Fin d → ℝ ↦ g (cosPoint (theta - u))) := by
simpa [Function.comp_def] using
hg.comp_continuous hcos (fun u ↦ cosPoint_mem_normalizedCube (theta - u))
have hkernel : Continuous (tensorJackson K d) := by
unfold tensorJackson
fun_prop
have hconvInt : IntegrableOn
(fun u : Fin d → ℝ ↦ g (cosPoint (theta - u)) * tensorJackson K d u)
(periodBox d) :=
(hgcos.mul hkernel).continuousOn.integrableOn_compact hcompact
have hconstInt : IntegrableOn
(fun u : Fin d → ℝ ↦ g (cosPoint theta) * tensorJackson K d u)
(periodBox d) :=
(continuous_const.mul hkernel).continuousOn.integrableOn_compact hcompact
have hdiffInt : IntegrableOn
(fun u : Fin d → ℝ ↦
(g (cosPoint (theta - u)) - g (cosPoint theta)) * tensorJackson K d u)
(periodBox d) :=
((hgcos.sub continuous_const).mul hkernel).continuousOn.integrableOn_compact hcompact
have hfirstInt (i : Fin d) : IntegrableOn
(fun u : Fin d → ℝ ↦ |u i| * tensorJackson K d u) (periodBox d) := by
exact ((continuous_abs.comp (continuous_apply i)).mul hkernel).continuousOn
|>.integrableOn_compact hcompact
have hsecondInt (i : Fin d) : IntegrableOn
(fun u : Fin d → ℝ ↦ (u i) ^ 2 * tensorJackson K d u) (periodBox d) := by
exact (((continuous_apply i).pow 2).mul hkernel).continuousOn
|>.integrableOn_compact hcompact
have htermInt (i : Fin d) : IntegrableOn
(fun u : Fin d → ℝ ↦
(r i * |sin (theta i)| * |u i| + r i / 2 * (u i) ^ 2) *
tensorJackson K d u) (periodBox d) := by
have hc : Continuous (fun u : Fin d → ℝ ↦
(r i * |sin (theta i)| * |u i| + r i / 2 * (u i) ^ 2) *
tensorJackson K d u) := by fun_prop
exact hc.continuousOn.integrableOn_compact hcompact
have hboundInt : IntegrableOn
(fun u : Fin d → ℝ ↦
L * ∑ i, (r i * |sin (theta i)| * |u i| +
r i / 2 * (u i) ^ 2) * tensorJackson K d u) (periodBox d) := by
have hc : Continuous (fun u : Fin d → ℝ ↦
L * ∑ i, (r i * |sin (theta i)| * |u i| +
r i / 2 * (u i) ^ 2) * tensorJackson K d u) := by
fun_prop
exact hc.continuousOn.integrableOn_compact hcompact
have hdiff : tensorConvolution K g theta - g (cosPoint theta) =
∫ u in periodBox d,
(g (cosPoint (theta - u)) - g (cosPoint theta)) * tensorJackson K d u := by
unfold tensorConvolution
calc
(∫ u in periodBox d, g (cosPoint (theta - u)) * tensorJackson K d u) -
g (cosPoint theta) =
(∫ u in periodBox d, g (cosPoint (theta - u)) * tensorJackson K d u) -
g (cosPoint theta) * (∫ u in periodBox d, tensorJackson K d u) := by
rw [tensorJackson_integral_eq_one hK, mul_one]
_ = (∫ u in periodBox d, g (cosPoint (theta - u)) * tensorJackson K d u) -
∫ u in periodBox d, g (cosPoint theta) * tensorJackson K d u := by
rw [integral_const_mul]
_ = ∫ u in periodBox d,
(g (cosPoint (theta - u)) * tensorJackson K d u -
g (cosPoint theta) * tensorJackson K d u) :=
(integral_sub hconvInt hconstInt).symm
_ = ∫ u in periodBox d,
(g (cosPoint (theta - u)) - g (cosPoint theta)) * tensorJackson K d u := by
apply integral_congr_ae
filter_upwards
intro u
ring
rw [hdiff]
calc
|∫ u in periodBox d,
(g (cosPoint (theta - u)) - g (cosPoint theta)) * tensorJackson K d u| ≤
∫ u in periodBox d,
|(g (cosPoint (theta - u)) - g (cosPoint theta)) * tensorJackson K d u| :=
abs_integral_le_integral_abs
_ ≤ ∫ u in periodBox d,
L * ∑ i, (r i * |sin (theta i)| * |u i| +
r i / 2 * (u i) ^ 2) * tensorJackson K d u := by
apply setIntegral_mono hdiffInt.abs hboundInt
intro u
change |(g (cosPoint (theta - u)) - g (cosPoint theta)) * tensorJackson K d u| ≤
L * ∑ i, (r i * |sin (theta i)| * |u i| +
r i / 2 * (u i) ^ 2) * tensorJackson K d u
rw [abs_mul, abs_of_nonneg (tensorJackson_nonneg hK u)]
calc
|g (cosPoint (theta - u)) - g (cosPoint theta)| * tensorJackson K d u ≤
(L * ∑ i, r i * |(cosPoint (theta - u)) i - (cosPoint theta) i|) *
tensorJackson K d u := by
exact mul_le_mul_of_nonneg_right
(hlip _ (cosPoint_mem_normalizedCube _)) (tensorJackson_nonneg hK u)
_ ≤ (L * ∑ i, (r i * |sin (theta i)| * |u i| +
r i / 2 * (u i) ^ 2)) * tensorJackson K d u := by
apply mul_le_mul_of_nonneg_right _ (tensorJackson_nonneg hK u)
apply mul_le_mul_of_nonneg_left _ hL
apply Finset.sum_le_sum
intro i _
have hi := abs_cos_add_sub_cos_le (theta i) (-u i)
simp only [cosPoint, Pi.sub_apply, abs_neg, neg_sq] at hi ⊢
convert mul_le_mul_of_nonneg_left hi (hr i) using 1 <;> ring_nf
_ = L * ∑ i, (r i * |sin (theta i)| * |u i| +
r i / 2 * (u i) ^ 2) * tensorJackson K d u := by
rw [mul_assoc, Finset.sum_mul]
_ = L * ∑ i : Fin d,
(r i * |sin (theta i)| *
(∫ u in periodBox d, |u i| * tensorJackson K d u) +
r i / 2 * (∫ u in periodBox d, (u i) ^ 2 * tensorJackson K d u)) := by
rw [integral_const_mul]
rw [integral_finsetSum univ (fun i _ ↦ htermInt i)]
congr 1
apply Finset.sum_congr rfl
intro i _
have heq (u : Fin d → ℝ) :
(r i * |sin (theta i)| * |u i| + r i / 2 * (u i) ^ 2) *
tensorJackson K d u =
r i * |sin (theta i)| * (|u i| * tensorJackson K d u) +
r i / 2 * ((u i) ^ 2 * tensorJackson K d u) := by ring
simp_rw [heq]
rw [integral_add, integral_const_mul, integral_const_mul]
· exact (hfirstInt i).const_mul _
· exact (hsecondInt i).const_mul _
_ ≤ L * ∑ i : Fin d,
(32 * r i * |sin (theta i)| / (K : ℝ) +
32 * r i / (K : ℝ) ^ 2) := by
apply mul_le_mul_of_nonneg_left _ hL
apply Finset.sum_le_sum
intro i _
have hfirst := (tensorJackson_first_moment_eq hK i).le.trans
(jackson_first_moment K hK)
have hsecond := (tensorJackson_second_moment_eq hK i).le.trans
(jackson_second_moment K hK)
have hcoef1 : 0 ≤ r i * |sin (theta i)| :=
mul_nonneg (hr i) (abs_nonneg _)
have hcoef2 : 0 ≤ r i / 2 := div_nonneg (hr i) (by norm_num)
have h1 := mul_le_mul_of_nonneg_left hfirst hcoef1
have h2 := mul_le_mul_of_nonneg_left hsecond hcoef2
calc
_ ≤ r i * |sin (theta i)| * (32 / (K : ℝ)) +
r i / 2 * (64 / (K : ℝ) ^ 2) := add_le_add h1 h2
_ = _ := by ring
The estimator's finite coefficient mean inherits the boundary estimate from a physical-coordinate modulus at its true reference vector. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
ε :
ℝ
hε :
0 < ε
K :
ℕ
hK :
0 < K
c r q w :
Fin 4 → ℝ
hr :
∀ j, 0 < r j
hw :
∀ j, 0 ≤ w j
hQ :
∀ v ∈ centeredRectangle c r, ∀ j, 0 ≤ v j
hq :
q ∈ centeredRectangle c r
hmod :
∀ v ∈ centeredRectangle c r,
|armwiseExtensionFormula ε v - armwiseExtensionFormula ε q| ≤ ∑ j, w j * |v j - q j|
conclusion
|(armwiseExtensionFormula ε c + ∑ α : Fin 4 → Fin (2 * (K - 1) + 1), jacksonCoeff ε K c r α * ∏ j, ((q j - c j) / r j) ^ (α j).val)
- armwiseExtensionFormula ε q|
Proof (Lean source)
lemma jacksonCoeff_reference_boundary_le (ε : ℝ) (hε : 0 < ε)
(K : ℕ) (hK : 0 < K) (c r q w : Fin 4 → ℝ)
(hr : ∀ j, 0 < r j) (hw : ∀ j, 0 ≤ w j)
(hQ : ∀ v ∈ centeredRectangle c r, ∀ j, 0 ≤ v j)
(hq : q ∈ centeredRectangle c r)
(hmod : ∀ v ∈ centeredRectangle c r,
|armwiseExtensionFormula ε v - armwiseExtensionFormula ε q| ≤ ∑ j, w j * |v j - q j|) :
|(armwiseExtensionFormula ε c + ∑ α : Fin 4 → Fin (2 * (K - 1) + 1),
jacksonCoeff ε K c r α * ∏ j, ((q j - c j) / r j) ^ (α j).val) -
armwiseExtensionFormula ε q| ≤
32 * ∑ j : Fin 4, w j *
(sqrt ((q j - (c j - r j)) * ((c j + r j) - q j)) / (K : ℝ) +
r j / (K : ℝ) ^ 2) := by
have haff : Continuous (affinePoint c r) := by unfold affinePoint; fun_prop
have hf : ContinuousOn (fun z => armwiseExtensionFormula ε (affinePoint c r z))
(normalizedCube 4) :=
(armwise_continuousOn_rectangle ε hε c r hr hQ).comp haff.continuousOn
(fun z hz => affinePoint_mem_centeredRectangle c r z hr hz)
have hn := normalizedPoint_mem_normalizedCube c r q hr hq
let θ : Fin 4 → ℝ := fun j => arccos (normalizedPoint c r q j)
have hcos : cosPoint θ = normalizedPoint c r q := by
funext j
exact Real.cos_arccos (hn j).1 (hn j).2
have hpoint : affinePoint c r (cosPoint θ) = q := by
rw [hcos, affinePoint_normalizedPoint c r q hr]
have hbound := tensorConvolution_reference_boundary_le hK _ hf
(fun j => w j * r j) (fun j => mul_nonneg (hw j) (hr j).le)
1 (by norm_num) θ (by
intro z hz
have h := hmod _ (affinePoint_mem_centeredRectangle c r z hr hz)
rw [← hpoint] at h
simpa only [affinePoint, abs_mul, abs_of_pos (hr _), one_mul,
show ∀ j, c j + r j * z j - (c j + r j * cosPoint θ j) =
r j * (z j - cosPoint θ j) by intro j; ring, mul_assoc] using h)
rw [jacksonCoeff_sum_eq_eval ε hε K hK c r hr hQ]
change |eval (normalizedPoint c r q) (jacksonPolynomialPair ε K c r).2 - _| ≤ _
rw [← hcos, jacksonPolynomialPair_cos_eval ε hε K hK c r hr hQ]
rw [hpoint, one_mul] at hbound
apply hbound.trans_eq
have hterm (j : Fin 4) :
32 * (w j * r j) * |sin (θ j)| / (K : ℝ) +
32 * (w j * r j) / (K : ℝ) ^ 2 =
32 * (w j * (sqrt ((q j - (c j - r j)) * ((c j + r j) - q j)) /
(K : ℝ) + r j / (K : ℝ) ^ 2)) := by
have hs := radius_mul_abs_sin_arccos_eq_boundary c r q hr j
change r j * |sin (θ j)| = _ at hs
calc
_ = 32 * w j * (r j * |sin (θ j)|) / (K : ℝ) +
32 * (w j * r j) / (K : ℝ) ^ 2 := by ring
_ = _ := by rw [hs]; ring
simp_rw [hterm]
rw [← Finset.mul_sum]
theorem observed_jacksonCoeff_biasEnvelope_le Lemma observed_jacksonCoeff_biasEnvelope_le in the paper ↗
Integrating the armwise modulus at an occupied cell proves equation (11) for the exact coefficient expression, including boundary means and ties. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
ε :
ℝ
P :
hε :
0 < ε
hP :
ObservedClass ε P
x :
Fin d
hp :
0 < cellMass P x
K :
ℕ
hK :
0 < K
c r :
Fin 4 → ℝ
hr :
∀ j, 0 < r j
hQ :
∀ v ∈ centeredRectangle c r, ∀ j, 0 ≤ v j
hq :
cellVector P x ∈ centeredRectangle c r
conclusion
|(armwiseExtensionFormula ε c + ∑ α : Fin 4 → Fin (2 * (K - 1) + 1), jacksonCoeff ε K c r α * ∏ j, ((cellVector P x j - c j) / r j) ^ (α j).val)
- armwiseExtensionFormula ε (cellVector P x)|
≤ 64 * ∑ a : Bool, (1 + cellMass P x / armMass P a x) * ((∑ y : Bool, sqrt ((cellVector P x (cellIdx a y) - (c (cellIdx a y) - r (cellIdx a y))) * ((c (cellIdx a y) + r (cellIdx a y)) - cellVector P x (cellIdx a y)))) / (K : ℝ) + armRadiusFormula r a / (K : ℝ) ^ 2)
Proof (Lean source)
lemma observed_jacksonCoeff_biasEnvelope_le {d : ℕ} (ε : ℝ) (P : DiscreteLaw d)
(hε : 0 < ε) (hP : ObservedClass ε P) (x : Fin d) (hp : 0 < cellMass P x)
(K : ℕ) (hK : 0 < K) (c r : Fin 4 → ℝ) (hr : ∀ j, 0 < r j)
(hQ : ∀ v ∈ centeredRectangle c r, ∀ j, 0 ≤ v j)
(hq : cellVector P x ∈ centeredRectangle c r) :
|(armwiseExtensionFormula ε c + ∑ α : Fin 4 → Fin (2 * (K - 1) + 1),
jacksonCoeff ε K c r α * ∏ j, ((cellVector P x j - c j) / r j) ^ (α j).val) -
armwiseExtensionFormula ε (cellVector P x)| ≤
64 * ∑ a : Bool, (1 + cellMass P x / armMass P a x) *
((∑ y : Bool, sqrt
((cellVector P x (cellIdx a y) - (c (cellIdx a y) - r (cellIdx a y))) *
((c (cellIdx a y) + r (cellIdx a y)) - cellVector P x (cellIdx a y)))) /
(K : ℝ) + armRadiusFormula r a / (K : ℝ) ^ 2) := by
let w : Fin 4 → ℝ := fun j =>
2 * (1 + cellMass P x / armMass P (decide (j.val / 2 = 1)) x)
have hw : ∀ j, 0 ≤ w j := by
intro j
have h := (observed_inverse_arm_bounds ε P hε hP x hp
(decide (j.val / 2 = 1))).1
dsimp [w]
linarith
have hmass : totalMassVecFormula (cellVector P x) = cellMass P x := by
simp [totalMassVecFormula, armMassVecFormula, cellVector, cellMass, cellIdx]
ring
have hq0 : cellVector P x ≠ 0 := by
intro hz
rw [hz] at hmass
simp [totalMassVecFormula, armMassVecFormula] at hmass
linarith
have hmod : ∀ v ∈ centeredRectangle c r,
|armwiseExtensionFormula ε v - armwiseExtensionFormula ε (cellVector P x)| ≤
∑ j, w j * |v j - cellVector P x j| := by
intro v hv
have h := armwise_modulus_extension.1 ε hε v (cellVector P x) (hQ v hv)
(fun _ => toReal_nonneg) hq0
rw [hmass] at h
simp_rw [anchoredDenom_cellVector_eq_armMass ε P hP x hp] at h
convert h using 1
simp [w, armDelta, Fin.sum_univ_succ, cellIdx]
ring
have h := jacksonCoeff_reference_boundary_le ε hε K hK c r (cellVector P x) w
hr hw hQ hq hmod
apply h.trans_eq
simp [w, Fin.sum_univ_succ, cellIdx, armRadiusFormula]
ring
theorem observed_pilotJackson_mean_bias_normalized_le Lemma observed_pilotJackson_mean_bias_normalized_le in the paper ↗
On the actual localized pilot rectangle, logarithmic degree calibration proves the occupied-cell bias bound (20) for the factorial mean itself. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
ε :
ℝ
P :
hε :
0 < ε
hP :
ObservedClass ε P
x :
Fin d
hp :
0 < cellMass P x
H :
ℝ
hH :
0 < H
hHlarge :
1 ≤ H
m L κ :
ℝ
hm :
0 < m
hL :
1 ≤ L
hκ :
0 < κ
K :
ℕ
hK :
0 < K
hdegree :
κ * L ≤ K
Np :
Fin 4 → ℕ
hloc :
conclusion
b :
pilotMidpoint H hH m L Np
r :
pilotRadiusFormula H hH m L Np
|(armwiseExtensionFormula ε b + ∑ α : Fin 4 → Fin (2 * (K - 1) + 1), jacksonCoeff ε K b r α * ∏ j, ((cellVector P x j - b j) / r j) ^ (α j).val)
- armwiseExtensionFormula ε (cellVector P x)|
Proof (Lean source)
lemma observed_pilotJackson_mean_bias_normalized_le {d : ℕ} (ε : ℝ)
(P : DiscreteLaw d) (hε : 0 < ε) (hP : ObservedClass ε P)
(x : Fin d) (hp : 0 < cellMass P x)
(H : ℝ) (hH : 0 < H) (hHlarge : 1 ≤ H) (m L κ : ℝ)
(hm : 0 < m) (hL : 1 ≤ L) (hκ : 0 < κ)
(K : ℕ) (hK : 0 < K) (hdegree : κ * L ≤ K) (Np : Fin 4 → ℕ)
(hloc : cellVector P x ∈
pilotRectangle (pilotLower H hH m L Np) (pilotUpperFormula H hH m L Np)) :
let b := pilotMidpoint H hH m L Np
let r := pilotRadiusFormula H hH m L Np
|(armwiseExtensionFormula ε b + ∑ α : Fin 4 → Fin (2 * (K - 1) + 1),
jacksonCoeff ε K b r α * ∏ j, ((cellVector P x j - b j) / r j) ^ (α j).val) -
armwiseExtensionFormula ε (cellVector P x)| ≤
512 * (H * (H + 2)) * sqrt 2 * (1 / κ + 1 / κ ^ 2) *
(sqrt (cellMass P x / (m * ε * L)) + 1 / (m * ε * L)) := by
have hLp : 0 < L := by linarith
have hr := pilotRadius_pos H hH m L hm hLp Np
have hq : cellVector P x ∈ centeredRectangle
(pilotMidpoint H hH m L Np) (pilotRadiusFormula H hH m L Np) := by
intro j
apply abs_le.mpr
have hj := hloc j
dsimp [pilotMidpoint, pilotRadiusFormula]
constructor <;> linarith [hj.1, hj.2]
have h := observed_jacksonCoeff_biasEnvelope_le ε P hε hP x hp K hK
(pilotMidpoint H hH m L Np) (pilotRadiusFormula H hH m L Np) hr
(pilotCenteredRectangle_nonneg H hH m L Np) hq
have hlo (j : Fin 4) : pilotMidpoint H hH m L Np j - pilotRadiusFormula H hH m L Np j =
pilotLower H hH m L Np j := by dsimp [pilotMidpoint, pilotRadiusFormula]; ring
have hhi (j : Fin 4) : pilotMidpoint H hH m L Np j + pilotRadiusFormula H hH m L Np j =
pilotUpperFormula H hH m L Np j := by dsimp [pilotMidpoint, pilotRadiusFormula]; ring
simp_rw [hlo, hhi] at h
have henvelope := observed_pilot_biasEnvelope_normalized_le ε P hε hP x hp
H hH hHlarge m L K κ hm hL hκ hdegree Np hloc
dsimp only
apply h.trans
convert mul_le_mul_of_nonneg_left henvelope (show (0 : ℝ) ≤ 64 by norm_num) using 1
ring
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.UpperPilot 19 declarations The pilot radius controls the normalized Poisson variance on every localized rectangle.
Pilot radius variance control
The pilot radius controls the normalized Poisson variance on every localized rectangle. Midpoint-based radius bounds prove roadmap (17); keeping each anchored denominator with its radius proves (18) and the deterministic clipping-scale envelope (22a). The pilot midpoint deviations and the observed-cell modulus prove (22b), retaining each inverse arm mass with its own outcome-coordinate deviations.
Truncating the lower endpoint at zero moves the midpoint above the pilot count center. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
pilotCenterFormula m Np j ≤ pilotMidpoint H hH m L Np j
Proof (Lean source)
lemma pilotCenter_le_midpoint (H : ℝ) (hH : 0 < H) (m L : ℝ)
(Np : Fin 4 → ℕ) (j : Fin 4) :
pilotCenterFormula m Np j ≤ pilotMidpoint H hH m L Np j := by
have h := le_max_right 0 (pilotCenterFormula m Np j - pilotHalfWidthFormula H hH m L Np j)
dsimp [pilotMidpoint, pilotLower, pilotUpperFormula]
linarith
The midpoint coordinates are nonnegative for positive exposure and level. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
0 ≤ pilotMidpoint H hH m L Np j
Proof (Lean source)
lemma pilotMidpoint_nonneg (H : ℝ) (hH : 0 < H) (m L : ℝ)
(hm : 0 < m) (Np : Fin 4 → ℕ) (j : Fin 4) :
0 ≤ pilotMidpoint H hH m L Np j := by
exact le_trans (by dsimp [pilotCenterFormula]; positivity)
(pilotCenter_le_midpoint H hH m L Np j)
The radius never exceeds the original half width, including intervals truncated at zero. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
pilotRadiusFormula H hH m L Np j ≤ pilotHalfWidthFormula H hH m L Np j
Proof (Lean source)
lemma pilotRadius_le_halfWidth (H : ℝ) (hH : 0 < H) (m L : ℝ)
(Np : Fin 4 → ℕ) (j : Fin 4) :
pilotRadiusFormula H hH m L Np j ≤ pilotHalfWidthFormula H hH m L Np j := by
have h := le_max_right 0 (pilotCenterFormula m Np j - pilotHalfWidthFormula H hH m L Np j)
dsimp [pilotRadiusFormula, pilotLower, pilotUpperFormula]
linarith
The coordinate radius is bounded by the square-root scale at the rectangle midpoint (17). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
pilotRadiusFormula H hH m L Np j
≤ H * (sqrt (pilotMidpoint H hH m L Np j * L / m) + L / m)
Proof (Lean source)
lemma pilotRadius_le_midpoint_scale (H : ℝ) (hH : 0 < H) (m L : ℝ)
(hm : 0 < m) (hL : 0 ≤ L) (Np : Fin 4 → ℕ) (j : Fin 4) :
pilotRadiusFormula H hH m L Np j ≤
H * (sqrt (pilotMidpoint H hH m L Np j * L / m) + L / m) := by
apply (pilotRadius_le_halfWidth H hH m L Np j).trans
dsimp only [pilotHalfWidthFormula]
apply mul_le_mul_of_nonneg_left _ hH.le
apply add_le_add _ le_rfl
apply Real.sqrt_le_sqrt
exact div_le_div_of_nonneg_right
(mul_le_mul_of_nonneg_right (pilotCenter_le_midpoint H hH m L Np j) hL) hm.le
Summing the two coordinate radii yields the armwise square-root scale in (17). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
armRadiusFormula (pilotRadiusFormula H hH m L Np) a
Proof (Lean source)
lemma pilotArmRadius_le_midpoint_scale (H : ℝ) (hH : 0 < H) (m L : ℝ)
(hm : 0 < m) (hL : 0 ≤ L) (Np : Fin 4 → ℕ) (a : Bool) :
armRadiusFormula (pilotRadiusFormula H hH m L Np) a ≤
2 * H * (sqrt (armMassVecFormula (pilotMidpoint H hH m L Np) a * L / m) + L / m) := by
let b := pilotMidpoint H hH m L Np
have hb₀ := pilotMidpoint_nonneg H hH m L hm Np (cellIdx a false)
have hb₁ := pilotMidpoint_nonneg H hH m L hm Np (cellIdx a true)
have h₀ := pilotRadius_le_midpoint_scale H hH m L hm hL Np (cellIdx a false)
have h₁ := pilotRadius_le_midpoint_scale H hH m L hm hL Np (cellIdx a true)
have hs₀ : sqrt (b (cellIdx a false) * L / m) ≤
sqrt (armMassVecFormula b a * L / m) := by
apply Real.sqrt_le_sqrt
dsimp [armMassVecFormula]
gcongr
exact le_add_of_nonneg_right hb₁
have hs₁ : sqrt (b (cellIdx a true) * L / m) ≤
sqrt (armMassVecFormula b a * L / m) := by
apply Real.sqrt_le_sqrt
dsimp [armMassVecFormula]
gcongr
exact le_add_of_nonneg_left hb₀
change _ ≤ 2 * H * (sqrt (armMassVecFormula b a * L / m) + L / m)
dsimp only [armRadiusFormula]
have ht₀ := mul_le_mul_of_nonneg_left hs₀ hH.le
have ht₁ := mul_le_mul_of_nonneg_left hs₁ hH.le
change pilotRadiusFormula H hH m L Np (cellIdx a false) ≤
H * (sqrt (b (cellIdx a false) * L / m) + L / m) at h₀
change pilotRadiusFormula H hH m L Np (cellIdx a true) ≤
H * (sqrt (b (cellIdx a true) * L / m) + L / m) at h₁
nlinarith
The anchored denominator gives the arm square-root term a single inverse-overlap scale (18). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma anchored_weighted_sqrt_le {B t δ ε : ℝ}
(hB : 0 ≤ B) (ht : 0 ≤ t) (hδ : 0 ≤ δ) (hε : 0 < ε) :
B / max t (ε * B) * sqrt (t * δ) ≤ sqrt (B * δ / ε) := by
by_cases hB0 : B = 0
· simp [hB0]
have hBp : 0 < B := lt_of_le_of_ne hB (Ne.symm hB0)
let D := max t (ε * B)
have hED : ε * B ≤ D := le_max_right _ _
have htD : t ≤ D := le_max_left _ _
have hD : 0 < D := lt_of_lt_of_le (mul_pos hε hBp) hED
have hprod : (ε * B) * t ≤ D ^ 2 := by
simpa only [pow_two] using mul_le_mul hED htD ht hD.le
have hscale := mul_le_mul_of_nonneg_left hprod
(show 0 ≤ B * δ / ε by positivity)
have hid : (B * δ / ε) * ((ε * B) * t) = B ^ 2 * (t * δ) := by
field_simp
rw [hid] at hscale
have hs := Real.sq_sqrt (show 0 ≤ t * δ by positivity)
have hu := Real.sq_sqrt (show 0 ≤ B * δ / ε by positivity)
have hsq : (B * sqrt (t * δ)) ^ 2 ≤
(sqrt (B * δ / ε) * D) ^ 2 := by
rw [mul_pow, mul_pow, hs, hu]
exact hscale
change B / D * sqrt (t * δ) ≤ _
rw [div_mul_eq_mul_div]
apply (div_le_iff₀ hD).2
nlinarith [Real.sqrt_nonneg (t * δ), Real.sqrt_nonneg (B * δ / ε),
mul_nonneg hB (Real.sqrt_nonneg (t * δ)),
mul_nonneg (Real.sqrt_nonneg (B * δ / ε)) hD.le]
Weighting an arm radius by the anchored denominator preserves its square-root scale (18). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma anchored_weighted_radius_le {B t δ ε R C : ℝ}
(hB : 0 ≤ B) (ht : 0 ≤ t) (hδ : 0 ≤ δ) (hε : 0 < ε)
(hC : 0 ≤ C) (hR : R ≤ C * (sqrt (t * δ) + δ)) :
B / max t (ε * B) * R ≤
C * (sqrt (B * δ / ε) + δ / ε) := by
have hD0 : 0 ≤ max t (ε * B) := le_trans ht (le_max_left _ _)
have hw0 : 0 ≤ B / max t (ε * B) := div_nonneg hB hD0
have hw : B / max t (ε * B) ≤ 1 / ε := by
by_cases hB0 : B = 0
· simp [hB0, hε.le]
have hBp : 0 < B := lt_of_le_of_ne hB (Ne.symm hB0)
have hD : 0 < max t (ε * B) := lt_of_lt_of_le (mul_pos hε hBp) (le_max_right _ _)
apply (div_le_div_iff₀ hD hε).2
simp [mul_comm]
have hs := anchored_weighted_sqrt_le hB ht hδ hε
have hl := mul_le_mul_of_nonneg_right hw hδ
calc
_ ≤ B / max t (ε * B) * (C * (sqrt (t * δ) + δ)) :=
mul_le_mul_of_nonneg_left hR hw0
_ = C * (B / max t (ε * B) * sqrt (t * δ) +
B / max t (ε * B) * δ) := by ring
_ ≤ C * (sqrt (B * δ / ε) + δ / ε) := by
apply mul_le_mul_of_nonneg_left _ hC
simpa only [one_div_mul_eq_div] using add_le_add hs hl
The actual pilot arm radius and random anchored weight satisfy equation (18) jointly. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
b :
pilotMidpoint H hH m L Np
totalMassVecFormula b / anchoredDenomFormula ε b a * armRadiusFormula (pilotRadiusFormula H hH m L Np) a
≤ 2 * H * (sqrt (totalMassVecFormula b * (L / m) / ε) + (L / m) / ε)
Proof (Lean source)
lemma pilotArmRadius_weighted_le (H : ℝ) (hH : 0 < H) (m L ε : ℝ)
(hm : 0 < m) (hL : 0 ≤ L) (hε : 0 < ε) (Np : Fin 4 → ℕ) (a : Bool) :
let b := pilotMidpoint H hH m L Np
totalMassVecFormula b / anchoredDenomFormula ε b a * armRadiusFormula (pilotRadiusFormula H hH m L Np) a ≤
2 * H * (sqrt (totalMassVecFormula b * (L / m) / ε) + (L / m) / ε) := by
dsimp only
let b := pilotMidpoint H hH m L Np
have hb := pilotMidpoint_nonneg H hH m L hm Np
have ht (a : Bool) : 0 ≤ armMassVecFormula b a := add_nonneg (hb _) (hb _)
have hB : 0 ≤ totalMassVecFormula b := add_nonneg (ht false) (ht true)
apply anchored_weighted_radius_le hB (ht a) (div_nonneg hL hm.le) hε
(by positivity)
simpa only [mul_div_assoc] using
pilotArmRadius_le_midpoint_scale H hH m L hm hL Np a
Each unweighted pilot arm radius is also bounded by the total-mass overlap scale. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
b :
pilotMidpoint H hH m L Np
armRadiusFormula (pilotRadiusFormula H hH m L Np) a
≤ 2 * H * (sqrt (totalMassVecFormula b * (L / m) / ε) + (L / m) / ε)
Proof (Lean source)
lemma pilotArmRadius_le_total_scale (H : ℝ) (hH : 0 < H) (m L ε : ℝ)
(hm : 0 < m) (hL : 0 ≤ L) (hε : 0 < ε) (hε1 : ε ≤ 1)
(Np : Fin 4 → ℕ) (a : Bool) :
let b := pilotMidpoint H hH m L Np
armRadiusFormula (pilotRadiusFormula H hH m L Np) a ≤
2 * H * (sqrt (totalMassVecFormula b * (L / m) / ε) + (L / m) / ε) := by
dsimp only
let b := pilotMidpoint H hH m L Np
have hb := pilotMidpoint_nonneg H hH m L hm Np
have ht (a : Bool) : 0 ≤ armMassVecFormula b a := add_nonneg (hb _) (hb _)
have hB : 0 ≤ totalMassVecFormula b := add_nonneg (ht false) (ht true)
have htB : armMassVecFormula b a ≤ totalMassVecFormula b := by
cases a
· exact le_add_of_nonneg_right (ht true)
· exact le_add_of_nonneg_left (ht false)
have hδ : 0 ≤ L / m := div_nonneg hL hm.le
have hlinear : L / m ≤ (L / m) / ε := by
apply (le_div_iff₀ hε).2
nlinarith
have harg : armMassVecFormula b a * (L / m) ≤ totalMassVecFormula b * (L / m) / ε := by
apply (le_div_iff₀ hε).2
calc
armMassVecFormula b a * (L / m) * ε ≤ armMassVecFormula b a * (L / m) :=
mul_le_of_le_one_right (mul_nonneg (ht a) hδ) hε1
_ ≤ totalMassVecFormula b * (L / m) := mul_le_mul_of_nonneg_right htB hδ
apply (pilotArmRadius_le_midpoint_scale H hH m L hm hL Np a).trans
apply mul_le_mul_of_nonneg_left _ (by positivity : 0 ≤ 2 * H)
simpa only [mul_div_assoc] using add_le_add (Real.sqrt_le_sqrt harg) hlinear
The full clipping scale has the deterministic total-mass envelope (22a), without separating random arm denominators from their radii. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
b :
pilotMidpoint H hH m L Np
clippingScaleFormula ε b (pilotRadiusFormula H hH m L Np)
≤ 16 * H * (sqrt (totalMassVecFormula b * (L / m) / ε) + (L / m) / ε)
Proof (Lean source)
lemma pilotClippingScale_le_total_scale (H : ℝ) (hH : 0 < H) (m L ε : ℝ)
(hm : 0 < m) (hL : 0 ≤ L) (hε : 0 < ε) (hε1 : ε ≤ 1)
(Np : Fin 4 → ℕ) :
let b := pilotMidpoint H hH m L Np
clippingScaleFormula ε b (pilotRadiusFormula H hH m L Np) ≤
16 * H * (sqrt (totalMassVecFormula b * (L / m) / ε) + (L / m) / ε) := by
dsimp only
have h₀ := pilotArmRadius_le_total_scale H hH m L ε hm hL hε hε1 Np false
have h₁ := pilotArmRadius_le_total_scale H hH m L ε hm hL hε hε1 Np true
have hw₀ := pilotArmRadius_weighted_le H hH m L ε hm hL hε Np false
have hw₁ := pilotArmRadius_weighted_le H hH m L ε hm hL hε Np true
dsimp only [clippingScaleFormula, armWeightFormula]
simp only [Fintype.sum_bool]
nlinarith
The localized coordinate variance is controlled by the logarithmic pilot level, including coordinates whose pilot interval meets zero (roadmap equation (24)). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
H :
ℝ
hH :
0 < H
hHlarge :
1 ≤ H
m L :
ℝ
hm :
0 < m
hL :
0 < L
Np :
Fin 4 → ℕ
j :
Fin 4
q :
ℝ
hqu :
q ≤ pilotUpperFormula H hH m L Np j
conclusion
q / (m * (pilotRadiusFormula H hH m L Np j) ^ 2) ≤ 8 / L
Proof (Lean source)
lemma pilotRadius_variance_bound (H : ℝ) (hH : 0 < H) (hHlarge : 1 ≤ H)
(m L : ℝ) (hm : 0 < m) (hL : 0 < L) (Np : Fin 4 → ℕ)
(j : Fin 4) (q : ℝ)
(hqu : q ≤ pilotUpperFormula H hH m L Np j) :
q / (m * (pilotRadiusFormula H hH m L Np j) ^ 2) ≤ 8 / L := by
let c := pilotCenterFormula m Np j
let h := pilotHalfWidthFormula H hH m L Np j
let r := pilotRadiusFormula H hH m L Np j
let level := L / m
have hc : 0 ≤ c := by dsimp [c, pilotCenterFormula]; positivity
have hlevel : 0 < level := div_pos hL hm
have harg : 0 ≤ c * level := mul_nonneg hc hlevel.le
have hs : (sqrt (c * level)) ^ 2 = c * level := Real.sq_sqrt harg
have hh : h = H * (sqrt (c * level) + level) := by
dsimp [h, pilotHalfWidthFormula, c, level]
congr 2
congr 1
ring
have hhlevel : level ≤ h := by
rw [hh]
nlinarith [Real.sqrt_nonneg (c * level)]
have hhpos : 0 < h := lt_of_lt_of_le hlevel hhlevel
have hhs : sqrt (c * level) ≤ h := by
rw [hh]
nlinarith [Real.sqrt_nonneg (c * level)]
have hclevel : c * level ≤ h ^ 2 := by
nlinarith [Real.sqrt_nonneg (c * level)]
have hr : h / 2 ≤ r := by
dsimp [r, pilotRadiusFormula, pilotLower, pilotUpperFormula]
change h / 2 ≤ (c + h - max 0 (c - h)) / 2
rw [max_def]
split_ifs <;> linarith
have hrpos : 0 < r := lt_of_lt_of_le (by linarith) hr
have hqu' : q ≤ c + h := hqu
have hscale : q * level ≤ 8 * r ^ 2 := by
have hqh : q * level ≤ (c + h) * level :=
mul_le_mul_of_nonneg_right hqu' hlevel.le
have hhlevel' : h * level ≤ h ^ 2 := by nlinarith
nlinarith
apply (div_le_div_iff₀ (mul_pos hm (sq_pos_of_pos hrpos)) hL).2
have heq : q * L = (q * level) * m := by dsimp [level]; field_simp
rw [heq]
nlinarith
theorem pilotMidpoint_sub_center_le_halfWidth Lemma pilotMidpoint_sub_center_le_halfWidth in the paper ↗
Truncating a nonnegative pilot center at zero moves its midpoint by at most its half width, including intervals that meet the boundary. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
|pilotMidpoint H hH m L Np j - pilotCenterFormula m Np j|
≤ pilotHalfWidthFormula H hH m L Np j
Proof (Lean source)
lemma pilotMidpoint_sub_center_le_halfWidth (H : ℝ) (hH : 0 < H) (m L : ℝ)
(hm : 0 < m) (hL : 0 ≤ L) (Np : Fin 4 → ℕ) (j : Fin 4) :
|pilotMidpoint H hH m L Np j - pilotCenterFormula m Np j| ≤
pilotHalfWidthFormula H hH m L Np j := by
have hc : 0 ≤ pilotCenterFormula m Np j := by dsimp [pilotCenterFormula]; positivity
have hh : 0 ≤ pilotHalfWidthFormula H hH m L Np j := by
dsimp [pilotHalfWidthFormula]
positivity
have hmid := pilotCenter_le_midpoint H hH m L Np j
rw [abs_of_nonneg (sub_nonneg.mpr hmid)]
dsimp [pilotMidpoint, pilotLower, pilotUpperFormula]
rw [max_def]
split_ifs <;> linarith
Each midpoint error is bounded by the pilot-count error plus the random half width; no localization event is needed in equation (22b). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
|pilotMidpoint H hH m L Np j - q j|
≤ |pilotCenterFormula m Np j - q j| + pilotHalfWidthFormula H hH m L Np j
Proof (Lean source)
lemma pilotMidpoint_abs_error_le (H : ℝ) (hH : 0 < H) (m L : ℝ)
(hm : 0 < m) (hL : 0 ≤ L) (Np : Fin 4 → ℕ) (q : Fin 4 → ℝ) (j : Fin 4) :
|pilotMidpoint H hH m L Np j - q j| ≤
|pilotCenterFormula m Np j - q j| + pilotHalfWidthFormula H hH m L Np j := by
calc
_ = |(pilotMidpoint H hH m L Np j - pilotCenterFormula m Np j) +
(pilotCenterFormula m Np j - q j)| := by congr 1; ring
_ ≤ |pilotMidpoint H hH m L Np j - pilotCenterFormula m Np j| +
|pilotCenterFormula m Np j - q j| := abs_add_le _ _
_ ≤ _ := by
have h := pilotMidpoint_sub_center_le_halfWidth H hH m L hm hL Np j
linarith
Summing within an arm before weighting preserves the outcome-coordinate shell bound used in equation (22b). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
armDelta (pilotMidpoint H hH m L Np) q a
≤ ∑ y : Bool, (|pilotCenterFormula m Np (cellIdx a y) - q (cellIdx a y)| + pilotHalfWidthFormula H hH m L Np (cellIdx a y))
Proof (Lean source)
lemma pilotArmDelta_le_coordinate_errors (H : ℝ) (hH : 0 < H) (m L : ℝ)
(hm : 0 < m) (hL : 0 ≤ L) (Np : Fin 4 → ℕ) (q : Fin 4 → ℝ) (a : Bool) :
armDelta (pilotMidpoint H hH m L Np) q a ≤
∑ y : Bool, (|pilotCenterFormula m Np (cellIdx a y) - q (cellIdx a y)| +
pilotHalfWidthFormula H hH m L Np (cellIdx a y)) := by
exact Finset.sum_le_sum fun y _ =>
pilotMidpoint_abs_error_le H hH m L hm hL Np q (cellIdx a y)
theorem anchoredDenom_cellVector_eq_armMass Lemma anchoredDenom_cellVector_eq_armMass in the paper ↗
On an occupied observed cell, overlap makes the anchored denominator exactly the observed arm mass, as required in roadmap equations (1) and (22b). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma anchoredDenom_cellVector_eq_armMass {d : ℕ} (ε : ℝ) (P : DiscreteLaw d)
(hP : ObservedClass ε P) (x : Fin d) (hp : 0 < cellMass P x) (a : Bool) :
anchoredDenomFormula ε (cellVector P x) a = armMass P a x := by
have hmass : totalMassVecFormula (cellVector P x) = cellMass P x := by
simp [totalMassVecFormula, armMassVecFormula, cellVector, cellMass, cellIdx]
ring
have harm : armMassVecFormula (cellVector P x) a = armMass P a x := by
cases a <;> simp [armMassVecFormula, cellVector, armMass, cellIdx] <;> ring
obtain ⟨hlo, hhi⟩ := hP.overlap x hp
have ht : ε * cellMass P x ≤ armMass P true x :=
(le_div_iff₀ hp).mp hlo
have hf : ε * cellMass P x ≤ armMass P false x := by
have hs : cellMass P x = armMass P false x + armMass P true x := by
simp [cellMass, armMass]
ring
have hu : armMass P true x ≤ (1 - ε) * cellMass P x :=
(div_le_iff₀ hp).mp hhi
nlinarith
rw [anchoredDenomFormula, hmass, harm]
apply max_eq_left
cases a <;> assumption
The cell-value perturbation keeps the inverse observed arm mass attached to its armwise deviations. The constant four absorbs the unweighted modulus term. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
ε :
ℝ
P :
hε :
0 < ε
hP :
ObservedClass ε P
x :
Fin d
hp :
0 < cellMass P x
b :
Fin 4 → ℝ
hb :
∀ j, 0 ≤ b j
conclusion
Proof (Lean source)
lemma observedCell_armwise_modulus_le {d : ℕ} (ε : ℝ) (P : DiscreteLaw d)
(hε : 0 < ε) (hP : ObservedClass ε P) (x : Fin d)
(hp : 0 < cellMass P x) (b : Fin 4 → ℝ) (hb : ∀ j, 0 ≤ b j) :
|armwiseExtensionFormula ε b - armwiseExtensionFormula ε (cellVector P x)| ≤
4 * ∑ a : Bool, cellMass P x / armMass P a x * armDelta b (cellVector P x) a := by
have hmass : totalMassVecFormula (cellVector P x) = cellMass P x := by
simp [totalMassVecFormula, armMassVecFormula, cellVector, cellMass, cellIdx]
ring
have hq : ∀ j, 0 ≤ cellVector P x j := fun _ => ENNReal.toReal_nonneg
have hq0 : cellVector P x ≠ 0 := by
intro hz
rw [hz] at hmass
simp [totalMassVecFormula, armMassVecFormula] at hmass
linarith
have hmod := armwise_modulus_extension.1 ε hε b (cellVector P x) hb hq hq0
have hweight (a : Bool) : 1 ≤ cellMass P x / armMass P a x := by
have ha0 : 0 < armMass P a x := by
have hden : 0 < anchoredDenomFormula ε (cellVector P x) a :=
lt_of_lt_of_le (mul_pos hε (by simpa only [hmass] using hp)) (le_max_right _ _)
rwa [anchoredDenom_cellVector_eq_armMass ε P hP x hp a] at hden
apply (le_div_iff₀ ha0).2
have hother : 0 ≤ armMass P (!a) x :=
Finset.sum_nonneg fun _ _ => ENNReal.toReal_nonneg
cases a <;> simp [cellMass, armMass] at * <;> linarith
have hsum : (∑ a : Bool, armDelta b (cellVector P x) a) ≤
∑ a : Bool, cellMass P x / armMass P a x * armDelta b (cellVector P x) a := by
apply Finset.sum_le_sum
intro a _
have hδ : 0 ≤ armDelta b (cellVector P x) a :=
Finset.sum_nonneg fun _ _ => abs_nonneg _
simpa only [one_mul] using mul_le_mul_of_nonneg_right (hweight a) hδ
simp only [hmass, anchoredDenom_cellVector_eq_armMass ε P hP x hp] at hmod
linarith
Equation (22b) for the actual pilot: every arm's inverse propensity remains paired with the sum of that arm's count deviations and half widths. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
ε :
ℝ
P :
hε :
0 < ε
hP :
ObservedClass ε P
x :
Fin d
hp :
0 < cellMass P x
H :
ℝ
hH :
0 < H
m L :
ℝ
hm :
0 < m
hL :
0 ≤ L
Np :
Fin 4 → ℕ
conclusion
|armwiseExtensionFormula ε (pilotMidpoint H hH m L Np)
- armwiseExtensionFormula ε (cellVector P x)|
≤ 4 * ∑ a : Bool, cellMass P x / armMass P a x * ∑ y : Bool, (|pilotCenterFormula m Np (cellIdx a y) - (cellVector P x) (cellIdx a y)| + pilotHalfWidthFormula H hH m L Np (cellIdx a y))
Proof (Lean source)
lemma pilotCellValue_abs_error_le {d : ℕ} (ε : ℝ) (P : DiscreteLaw d)
(hε : 0 < ε) (hP : ObservedClass ε P) (x : Fin d) (hp : 0 < cellMass P x)
(H : ℝ) (hH : 0 < H) (m L : ℝ) (hm : 0 < m) (hL : 0 ≤ L)
(Np : Fin 4 → ℕ) :
|armwiseExtensionFormula ε (pilotMidpoint H hH m L Np) -
armwiseExtensionFormula ε (cellVector P x)| ≤
4 * ∑ a : Bool, cellMass P x / armMass P a x *
∑ y : Bool, (|pilotCenterFormula m Np (cellIdx a y) - (cellVector P x) (cellIdx a y)| +
pilotHalfWidthFormula H hH m L Np (cellIdx a y)) := by
apply (observedCell_armwise_modulus_le ε P hε hP x hp _
(pilotMidpoint_nonneg H hH m L hm Np)).trans
apply mul_le_mul_of_nonneg_left _ (by norm_num)
apply Finset.sum_le_sum
intro a _
apply mul_le_mul_of_nonneg_left
(pilotArmDelta_le_coordinate_errors H hH m L hm hL Np (cellVector P x) a)
exact div_nonneg hp.le (Finset.sum_nonneg fun _ _ => toReal_nonneg)
theorem clippedCellValue_abs_error_le_width_add Lemma clippedCellValue_abs_error_le_width_add in the paper ↗
The clipping definition bounds cell error by its width plus the center's error, even when the true cell lies outside the pilot rectangle. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
ε :
ℝ
K d :
ℕ
m :
ℝ
c r :
Fin 4 → ℝ
Ne :
Fin 4 → ℕ
p :
ℝ
hwidth :
0 ≤ (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r
conclusion
|clippedCellValueFormula ε K d m c r Ne - p|
≤ (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r + |armwiseExtensionFormula ε c - p|
Proof (Lean source)
lemma clippedCellValue_abs_error_le_width_add (ε : ℝ) (K d : ℕ) (m : ℝ)
(c r : Fin 4 → ℝ) (Ne : Fin 4 → ℕ) (p : ℝ)
(hwidth : 0 ≤ (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r) :
|clippedCellValueFormula ε K d m c r Ne - p| ≤
(d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r + |armwiseExtensionFormula ε c - p| := by
have hlo : armwiseExtensionFormula ε c -
(d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r ≤
clippedCellValueFormula ε K d m c r Ne := le_max_left _ _
have hhi : clippedCellValueFormula ε K d m c r Ne ≤ armwiseExtensionFormula ε c +
(d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r := by
apply max_le
· linarith
· exact min_le_left _ _
have hcenter : |clippedCellValueFormula ε K d m c r Ne - armwiseExtensionFormula ε c| ≤
(d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε c r :=
abs_le.mpr ⟨by linarith, by linarith⟩
calc
_ = |(clippedCellValueFormula ε K d m c r Ne - armwiseExtensionFormula ε c) +
(armwiseExtensionFormula ε c - p)| := by congr 1; ring
_ ≤ _ := (abs_add_le _ _).trans (add_le_add hcenter le_rfl)
The actual clipped cell statistic obeys the bad-pilot deterministic envelope preceding (29), with the center term reduced to the coordinate shells in (22b). The nonnegative width follows from the construction, not an extra premise. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
ε :
ℝ
P :
hε :
0 < ε
hP :
ObservedClass ε P
x :
Fin d
hp :
0 < cellMass P x
H :
ℝ
hH :
0 < H
m L :
ℝ
hm :
0 < m
hL :
0 ≤ L
K :
ℕ
Np Ne :
Fin 4 → ℕ
conclusion
b :
pilotMidpoint H hH m L Np
r :
pilotRadiusFormula H hH m L Np
≤ (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε b r
+ 4 * ∑ a : Bool, cellMass P x / armMass P a x * ∑ y : Bool, (|pilotCenterFormula m Np (cellIdx a y) - (cellVector P x) (cellIdx a y)| + pilotHalfWidthFormula H hH m L Np (cellIdx a y))
Proof (Lean source)
lemma pilotClippedCellValue_abs_error_le {d : ℕ} (ε : ℝ) (P : DiscreteLaw d)
(hε : 0 < ε) (hP : ObservedClass ε P) (x : Fin d) (hp : 0 < cellMass P x)
(H : ℝ) (hH : 0 < H) (m L : ℝ) (hm : 0 < m) (hL : 0 ≤ L)
(K : ℕ) (Np Ne : Fin 4 → ℕ) :
let b := pilotMidpoint H hH m L Np
let r := pilotRadiusFormula H hH m L Np
|clippedCellValueFormula ε K d m b r Ne - armwiseExtensionFormula ε (cellVector P x)| ≤
(d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε b r +
4 * ∑ a : Bool, cellMass P x / armMass P a x *
∑ y : Bool, (|pilotCenterFormula m Np (cellIdx a y) - (cellVector P x) (cellIdx a y)| +
pilotHalfWidthFormula H hH m L Np (cellIdx a y)) := by
dsimp only
have hb := pilotMidpoint_nonneg H hH m L hm Np
have hr (j : Fin 4) : 0 ≤ pilotRadiusFormula H hH m L Np j := by
have hc : 0 ≤ pilotCenterFormula m Np j := by dsimp [pilotCenterFormula]; positivity
have hh : 0 ≤ pilotHalfWidthFormula H hH m L Np j := by
dsimp [pilotHalfWidthFormula]
positivity
dsimp [pilotRadiusFormula, pilotLower, pilotUpperFormula]
rw [max_def]
split_ifs <;> linarith
have ht (a : Bool) : 0 ≤ armMassVecFormula (pilotMidpoint H hH m L Np) a :=
add_nonneg (hb _) (hb _)
have hB : 0 ≤ totalMassVecFormula (pilotMidpoint H hH m L Np) :=
add_nonneg (ht false) (ht true)
have hscale : 0 ≤ clippingScaleFormula ε (pilotMidpoint H hH m L Np)
(pilotRadiusFormula H hH m L Np) := by
apply mul_nonneg (by norm_num)
apply Finset.sum_nonneg
intro a _
apply mul_nonneg _ (add_nonneg (hr _) (hr _))
exact add_nonneg (by norm_num)
(div_nonneg hB (le_trans (ht a) (le_max_left _ _)))
exact (clippedCellValue_abs_error_le_width_add ε K d m _ _ Ne _
(mul_nonneg (Real.rpow_nonneg (Nat.cast_nonneg _) _) hscale)).trans
(add_le_add le_rfl (pilotCellValue_abs_error_le ε P hε hP x hp
H hH m L hm hL Np))
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.UpperPilotMoments 7 declarations Roadmap equation (19) follows from the deterministic anchored-radius bound, a linear envelope for the pilot midpoint, and the marginal Poisson means.
Pilot clipping-scale second moments
Roadmap equation (19) follows from the deterministic anchored-radius bound, a linear envelope for the pilot midpoint, and the marginal Poisson means. The elementary square-root envelope avoids needing joint independence for this step.
A pilot midpoint is controlled by a linear count envelope, including at zero. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
pilotMidpoint H hH m L Np j ≤ (1 + H) * pilotCenterFormula m Np j + 2 * H * (L / m)
Proof (Lean source)
lemma pilotMidpoint_le_linear_center (H : ℝ) (hH : 0 < H) (m L : ℝ)
(hm : 0 < m) (hL : 0 ≤ L) (Np : Fin 4 → ℕ) (j : Fin 4) :
pilotMidpoint H hH m L Np j ≤
(1 + H) * pilotCenterFormula m Np j + 2 * H * (L / m) := by
have hc : 0 ≤ pilotCenterFormula m Np j := by unfold pilotCenterFormula; positivity
have hδ : 0 ≤ L / m := div_nonneg hL hm.le
have hs := two_mul_le_add_of_sq_le_mul hc hδ
(Real.sq_sqrt (mul_nonneg hc hδ)).le
have he := pilotMidpoint_sub_center_le_halfWidth H hH m L hm hL Np j
have ha := le_abs_self (pilotMidpoint H hH m L Np j - pilotCenterFormula m Np j)
dsimp only [pilotHalfWidthFormula] at he
rw [mul_div_assoc] at he
nlinarith [Real.sqrt_nonneg (pilotCenterFormula m Np j * (L / m))]
The four-coordinate mass is the sum of the four outcome atoms. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
Proof (Lean source)
lemma totalMassVec_eq_sum_coordinates (b : Fin 4 → ℝ) :
totalMassVecFormula b = ∑ j, b j := by
simp [totalMassVecFormula, armMassVecFormula, cellIdx, Fin.sum_univ_succ]
ring
The total pilot midpoint mass has a linear envelope with four width terms. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
totalMassVecFormula (pilotMidpoint H hH m L Np)
≤ (1 + H) * (∑ j, pilotCenterFormula m Np j) + 8 * H * (L / m)
Proof (Lean source)
lemma pilotTotalMass_le_linear_centers (H : ℝ) (hH : 0 < H) (m L : ℝ)
(hm : 0 < m) (hL : 0 ≤ L) (Np : Fin 4 → ℕ) :
totalMassVecFormula (pilotMidpoint H hH m L Np) ≤
(1 + H) * (∑ j, pilotCenterFormula m Np j) + 8 * H * (L / m) := by
rw [totalMassVec_eq_sum_coordinates]
calc
_ ≤ ∑ j, ((1 + H) * pilotCenterFormula m Np j + 2 * H * (L / m)) :=
Finset.sum_le_sum fun j _ => pilotMidpoint_le_linear_center H hH m L hm hL Np j
_ = _ := by
rw [Finset.sum_add_distrib, ← Finset.mul_sum]
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul]
ring
Squaring the anchored-radius envelope keeps only one inverse-overlap power in the term proportional to the random total midpoint mass. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
≤ 512 * H ^ 2 * (totalMassVecFormula (pilotMidpoint H hH m L Np) * (L / m) / ε + ((L / m) / ε) ^ 2)
Proof (Lean source)
lemma pilotClippingScale_sq_le (H : ℝ) (hH : 0 < H) (m L ε : ℝ)
(hm : 0 < m) (hL : 0 ≤ L) (hε : 0 < ε) (hε1 : ε ≤ 1)
(Np : Fin 4 → ℕ) :
clippingScaleFormula ε (pilotMidpoint H hH m L Np) (pilotRadiusFormula H hH m L Np) ^ 2 ≤
512 * H ^ 2 * (totalMassVecFormula (pilotMidpoint H hH m L Np) * (L / m) / ε +
((L / m) / ε) ^ 2) := by
have hB : 0 ≤ totalMassVecFormula (pilotMidpoint H hH m L Np) := by
rw [totalMassVec_eq_sum_coordinates]
exact Finset.sum_nonneg fun j _ => pilotMidpoint_nonneg H hH m L hm Np j
have hδ : 0 ≤ L / m := div_nonneg hL hm.le
have hscale := pilotClippingScale_le_total_scale H hH m L ε hm hL hε hε1 Np
have hr (j : Fin 4) : 0 ≤ pilotRadiusFormula H hH m L Np j := by
have he := le_max_right 0 (pilotCenterFormula m Np j - pilotHalfWidthFormula H hH m L Np j)
have hh : 0 ≤ pilotHalfWidthFormula H hH m L Np j := by unfold pilotHalfWidthFormula; positivity
have hc : 0 ≤ pilotCenterFormula m Np j := by unfold pilotCenterFormula; positivity
dsimp [pilotRadiusFormula, pilotLower, pilotUpperFormula]
rw [max_def]
split_ifs <;> linarith
have hS : 0 ≤ clippingScaleFormula ε (pilotMidpoint H hH m L Np)
(pilotRadiusFormula H hH m L Np) := by
unfold clippingScaleFormula armWeightFormula armRadiusFormula
apply mul_nonneg (by norm_num)
apply Finset.sum_nonneg
intro a _
apply mul_nonneg _ (add_nonneg (hr _) (hr _))
apply add_nonneg (by norm_num)
exact div_nonneg hB (le_trans
(add_nonneg (pilotMidpoint_nonneg H hH m L hm Np _)
(pilotMidpoint_nonneg H hH m L hm Np _)) (le_max_left _ _))
have hsq := pow_le_pow_left₀ hS hscale 2
have hs := Real.sq_sqrt (div_nonneg (mul_nonneg hB hδ) hε.le)
have ht (u v : ℝ) : (u + v) ^ 2 ≤ 2 * (u ^ 2 + v ^ 2) := by
nlinarith [sq_nonneg (u - v)]
calc
_ ≤ (16 * H * (sqrt (totalMassVecFormula (pilotMidpoint H hH m L Np) *
(L / m) / ε) + (L / m) / ε)) ^ 2 := hsq
_ ≤ (16 * H) ^ 2 * (2 * ((sqrt (totalMassVecFormula
(pilotMidpoint H hH m L Np) * (L / m) / ε)) ^ 2 + ((L / m) / ε) ^ 2)) := by
rw [mul_pow]
exact mul_le_mul_of_nonneg_left (ht _ _) (sq_nonneg _)
_ = _ := by rw [hs]; ring
A normalized marginal Poisson pilot count is integrable and has mean q. Both properties are transported from the first raw factorial moment. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Proof (Lean source)
lemma pilotCenter_integrable_and_mean {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) (W : Ω → ℕ) (m q : ℝ) (hm : 0 < m) (hq : 0 ≤ q)
(hlaw : HasLaw W (poissonMeasure ⟨m * q, mul_nonneg hm.le hq⟩) μ) :
Integrable (fun ω => (W ω : ℝ) / m) μ ∧
(∫ ω, (W ω : ℝ) / m ∂μ) = q := by
let rate : NNReal := ⟨m * q, mul_nonneg hm.le hq⟩
have hi : Integrable (fun N : ℕ => (N : ℝ)) (poissonMeasure rate) := by
simpa using poisson_descFactorial_integrable rate 1
have ht : Integrable (fun ω => (W ω : ℝ)) μ := by
have hmap := hlaw.map_eq
exact (hmap ▸ hi).comp_aemeasurable hlaw.aemeasurable
refine ⟨ht.div_const m, ?_⟩
rw [integral_div]
have he : (∫ ω, (W ω : ℝ) ∂μ) = m * q := by
have hx := hlaw.integral_comp
(show AEStronglyMeasurable (fun N : ℕ => (N : ℝ)) (poissonMeasure rate) from
(Measurable.of_discrete).aestronglyMeasurable)
have hr := poisson_descFactorial_moment rate 1
simp only [Nat.descFactorial_one, pow_one] at hr
change (∫ N : ℕ, (N : ℝ) ∂poissonMeasure rate) = m * q at hr
simpa only [Function.comp_def] using hx.trans hr
rw [he, mul_div_cancel_left₀ _ (ne_of_gt hm)]
Equation (19) for the actual pilot clipping scale. Only marginal Poisson laws are needed; null cells and arbitrarily small positive overlap are included. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
W :
Fin 4 → Ω → ℕ
q :
Fin 4 → ℝ
hq :
∀ j, 0 ≤ q j
H :
ℝ
hH :
0 < H
m L ε :
ℝ
hm :
0 < m
hL :
0 ≤ L
hε :
0 < ε
hε1 :
ε ≤ 1
hlaw :
conclusion
(∫ ω, clippingScaleFormula ε (pilotMidpoint H hH m L (fun j => W j ω)) (pilotRadiusFormula H hH m L (fun j => W j ω)) ^ 2 ∂μ)
≤ 512 * H ^ 2 * (1 + 9 * H) * (totalMassVecFormula q * (L / m) / ε + ((L / m) / ε) ^ 2)
Proof (Lean source)
lemma pilotClippingScale_integral_sq_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (W : Fin 4 → Ω → ℕ)
(q : Fin 4 → ℝ) (hq : ∀ j, 0 ≤ q j)
(H : ℝ) (hH : 0 < H) (m L ε : ℝ)
(hm : 0 < m) (hL : 0 ≤ L) (hε : 0 < ε) (hε1 : ε ≤ 1)
(hlaw : ∀ j, HasLaw (W j)
(poissonMeasure ⟨m * q j, mul_nonneg hm.le (hq j)⟩) μ) :
(∫ ω, clippingScaleFormula ε (pilotMidpoint H hH m L (fun j => W j ω))
(pilotRadiusFormula H hH m L (fun j => W j ω)) ^ 2 ∂μ) ≤
512 * H ^ 2 * (1 + 9 * H) *
(totalMassVecFormula q * (L / m) / ε + ((L / m) / ε) ^ 2) := by
let δ := L / m
have hδ : 0 ≤ δ := div_nonneg hL hm.le
have hc (j : Fin 4) := pilotCenter_integrable_and_mean μ (W j) m (q j) hm (hq j) (hlaw j)
let C : Ω → ℝ := fun ω => ∑ j, (W j ω : ℝ) / m
have hC : Integrable C μ := integrable_finsetSum _ (fun j _ => (hc j).1)
have hmean : (∫ ω, C ω ∂μ) = totalMassVecFormula q := by
rw [totalMassVec_eq_sum_coordinates]
dsimp [C]
rw [integral_finsetSum _ (fun j _ => (hc j).1)]
simp only [(hc _).2]
have hpoint (ω : Ω) :
clippingScaleFormula ε (pilotMidpoint H hH m L (fun j => W j ω))
(pilotRadiusFormula H hH m L (fun j => W j ω)) ^ 2 ≤
512 * H ^ 2 * (((1 + H) * C ω + 8 * H * δ) * δ / ε + (δ / ε) ^ 2) := by
apply (pilotClippingScale_sq_le H hH m L ε hm hL hε hε1 _).trans
apply mul_le_mul_of_nonneg_left _ (by positivity)
apply add_le_add _ le_rfl
apply div_le_div_of_nonneg_right _ hε.le
apply mul_le_mul_of_nonneg_right _ hδ
exact pilotTotalMass_le_linear_centers H hH m L hm hL _
have hi : Integrable (fun ω => 512 * H ^ 2 *
(((1 + H) * C ω + 8 * H * δ) * δ / ε + (δ / ε) ^ 2)) μ :=
((((hC.const_mul (1 + H)).add (integrable_const _)).mul_const δ).div_const ε
|>.add (integrable_const _)).const_mul _
calc
_ ≤ ∫ ω, 512 * H ^ 2 *
(((1 + H) * C ω + 8 * H * δ) * δ / ε + (δ / ε) ^ 2) ∂μ :=
integral_mono_of_nonneg (ae_of_all _ fun _ => sq_nonneg _) hi (ae_of_all _ hpoint)
_ = 512 * H ^ 2 * (((1 + H) * totalMassVecFormula q + 8 * H * δ) * δ / ε +
(δ / ε) ^ 2) := by
rw [integral_const_mul, integral_add, integral_div, integral_mul_const,
integral_add, integral_const_mul, hmean]
· simp
· exact hC.const_mul _
· exact integrable_const _
· exact (((hC.const_mul _).add (integrable_const _)).mul_const _).div_const _
· exact integrable_const _
_ ≤ _ := by
have hp : 0 ≤ totalMassVecFormula q := by
rw [totalMassVec_eq_sum_coordinates]
exact Finset.sum_nonneg fun j _ => hq j
have hs : δ * δ / ε ≤ (δ / ε) ^ 2 := by
apply (div_le_iff₀ hε).2
have hh : (δ / ε) ^ 2 * ε ≤ (δ / ε) ^ 2 :=
mul_le_of_le_one_right (sq_nonneg _) hε1
have hid : δ * δ = (δ / ε) ^ 2 * ε * ε := by field_simp
rw [hid]
exact mul_le_mul_of_nonneg_right hh hε.le
rw [mul_assoc (512 * H ^ 2)]
apply mul_le_mul_of_nonneg_left _ (by positivity : 0 ≤ 512 * H ^ 2)
have hterm : 0 ≤ totalMassVecFormula q * δ / ε := by positivity
change _ ≤ (1 + 9 * H) * (totalMassVecFormula q * δ / ε + (δ / ε) ^ 2)
have ht := mul_le_mul_of_nonneg_left hs (by positivity : 0 ≤ 8 * H)
have hid : ((1 + H) * totalMassVecFormula q + 8 * H * δ) * δ / ε =
(1 + H) * (totalMassVecFormula q * δ / ε) + 8 * H * (δ * δ / ε) := by ring
rw [hid]
nlinarith
theorem observedPilotClippingScale_integral_sq_le Lemma observedPilotClippingScale_integral_sq_le in the paper ↗
The observed-cell form of (19), with exactly the paper's mass and overlap scales. Positivity is not required of the cell mass, so null cells are retained. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
d :
ℕ
P :
x :
Fin d
W :
Fin 4 → Ω → ℕ
H :
ℝ
hH :
0 < H
m L ε :
ℝ
hm :
0 < m
hL :
0 ≤ L
hε :
0 < ε
hεhalf :
ε ≤ 1 / 2
hlaw :
∀ j,
conclusion
(∫ ω, clippingScaleFormula ε (pilotMidpoint H hH m L (fun j => W j ω)) (pilotRadiusFormula H hH m L (fun j => W j ω)) ^ 2 ∂μ)
≤ 512 * H ^ 2 * (1 + 9 * H) * (cellMass P x * L / (m * ε) + L ^ 2 / (m ^ 2 * ε ^ 2))
Proof (Lean source)
lemma observedPilotClippingScale_integral_sq_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] {d : ℕ} (P : DiscreteLaw d)
(x : Fin d) (W : Fin 4 → Ω → ℕ)
(H : ℝ) (hH : 0 < H) (m L ε : ℝ)
(hm : 0 < m) (hL : 0 ≤ L) (hε : 0 < ε) (hεhalf : ε ≤ 1 / 2)
(hlaw : ∀ j, HasLaw (W j) (poissonMeasure
⟨m * cellVector P x j, mul_nonneg hm.le ENNReal.toReal_nonneg⟩) μ) :
(∫ ω, clippingScaleFormula ε (pilotMidpoint H hH m L (fun j => W j ω))
(pilotRadiusFormula H hH m L (fun j => W j ω)) ^ 2 ∂μ) ≤
512 * H ^ 2 * (1 + 9 * H) *
(cellMass P x * L / (m * ε) + L ^ 2 / (m ^ 2 * ε ^ 2)) := by
have hmass : totalMassVecFormula (cellVector P x) = cellMass P x := by
simp [totalMassVecFormula, armMassVecFormula, cellVector, cellMass, cellIdx]
ring
have hbound := pilotClippingScale_integral_sq_le μ W (cellVector P x)
(fun _ => toReal_nonneg) H hH m L ε hm hL hε (by linarith) hlaw
rw [hmass] at hbound
convert hbound using 1; ring
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.UpperPilotRegularity 4 declarations The variance assembly in roadmap (29) requires square-integrable cell statistics before splitting good and bad pilots.
Unconditional pilot and clipped-cell regularity
The variance assembly in roadmap (29) requires square-integrable cell statistics before splitting good and bad pilots. Marginal Poisson score moments supply this regularity, even off the localization event. Clipping removes any need for unconditional moments of the pilot-selected factorial polynomial or independence between pilot and evaluation counts here.
The aggregate pilot deviation-and-width score is square-integrable under its marginal Poisson laws, without a joint independence assumption. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
Measure Ω
W :
Fin 4 → Ω → ℕ
q :
Fin 4 → ℝ≥0
m :
ℝ≥0
hm :
0 < m
H L :
ℝ
hH :
universalH ≤ H
hL :
1 ≤ L
hW :
∀ j, Measurable (W j)
hlaw :
∀ j, HasLaw (W j) (poissonMeasure (m * q j)) μ
conclusion
MemLp (empiricalAggregateScore W H L m q) 2 μ
Proof (Lean source)
lemma pilotAggregateScore_memLp_two {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) (W : Fin 4 → Ω → ℕ) (q : Fin 4 → ℝ≥0)
(m : ℝ≥0) (hm : 0 < m) (H L : ℝ) (hH : universalH ≤ H) (hL : 1 ≤ L)
(hW : ∀ j, Measurable (W j))
(hlaw : ∀ j, HasLaw (W j) (poissonMeasure (m * q j)) μ) :
MemLp (empiricalAggregateScore W H L m q) 2 μ := by
have hs : AEStronglyMeasurable (empiricalAggregateScore W H L m q) μ := by
unfold empiricalAggregateScore normalizedDeviation empiricalRadius
fun_prop
exact (memLp_two_iff_integrable_sq hs).2
(pilotAggregateScore_pow_integrable μ W q m hm H L hH hL hW hlaw 2
(inr (inl rfl)))
Total midpoint mass is square-integrable on the entire pilot space. The deterministic displacement bound dominates it by the true mass plus the aggregate score, so bad pilots are included. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
W :
Fin 4 → Ω → ℕ
q :
Fin 4 → ℝ≥0
m :
ℝ≥0
hm :
0 < m
H :
ℝ
hHpos :
0 < H
L :
ℝ
hH :
universalH ≤ H
hL :
1 ≤ L
hW :
∀ j, Measurable (W j)
hlaw :
∀ j, HasLaw (W j) (poissonMeasure (m * q j)) μ
conclusion
Proof (Lean source)
lemma pilotTotalMass_memLp_two {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ]
(W : Fin 4 → Ω → ℕ) (q : Fin 4 → ℝ≥0) (m : ℝ≥0) (hm : 0 < m)
(H : ℝ) (hHpos : 0 < H) (L : ℝ) (hH : universalH ≤ H) (hL : 1 ≤ L)
(hW : ∀ j, Measurable (W j))
(hlaw : ∀ j, HasLaw (W j) (poissonMeasure (m * q j)) μ) :
MemLp (fun ω => totalMassVecFormula (pilotMidpoint H hHpos m L (fun j => W j ω))) 2 μ := by
have hscore := pilotAggregateScore_memLp_two μ W q m hm H L hH hL hW hlaw
have hdom := (memLp_const (∑ j, (q j : ℝ)) (p := (2 : ENNReal)) (μ := μ)).add hscore
apply hdom.of_le
· first | fun_prop | exact ((Measurable.of_discrete (f := fun N : Fin 4 → ℕ =>
totalMassVecFormula (pilotMidpoint H hHpos m L N))).comp
(measurable_pi_lambda _ hW)).aestronglyMeasurable
· apply ae_of_all
intro ω
have hb := pilotMidpoint_nonneg H hHpos m L (NNReal.coe_pos.mpr hm)
(fun j => W j ω)
have hB : 0 ≤ totalMassVecFormula (pilotMidpoint H hHpos m L (fun j => W j ω)) := by
rw [totalMassVec_eq_sum_coordinates]
exact Finset.sum_nonneg fun j _ => hb j
have he := pilotMidpoint_total_abs_error_le_score W q m hm H hHpos L (by linarith) ω
have hsum : totalMassVecFormula (pilotMidpoint H hHpos m L (fun j => W j ω)) -
∑ j, (q j : ℝ) ≤ ∑ j, |pilotMidpoint H hHpos m L (fun i => W i ω) j - q j| := by
rw [totalMassVec_eq_sum_coordinates, ← Finset.sum_sub_distrib]
exact Finset.sum_le_sum fun j _ => le_abs_self _
rw [Real.norm_eq_abs, abs_of_nonneg hB, Real.norm_eq_abs]
exact (show _ ≤ (∑ j, (q j : ℝ)) + empiricalAggregateScore W H L m q ω by
linarith).trans (le_abs_self _)
The random clipping scale has an integrable square on good and bad pilots. Its anchored-radius square envelope is affine in the integrable midpoint mass. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
W :
Fin 4 → Ω → ℕ
q :
Fin 4 → ℝ≥0
m :
ℝ≥0
hm :
0 < m
H :
ℝ
hHpos :
0 < H
L ε :
ℝ
hH :
universalH ≤ H
hL :
1 ≤ L
hε :
0 < ε
hε1 :
ε ≤ 1
hW :
∀ j, Measurable (W j)
hlaw :
∀ j, HasLaw (W j) (poissonMeasure (m * q j)) μ
conclusion
MemLp (fun ω => clippingScaleFormula ε (pilotMidpoint H hHpos m L (fun j => W j ω)) (pilotRadiusFormula H hHpos m L (fun j => W j ω))) 2 μ
Proof (Lean source)
lemma pilotClippingScale_memLp_two {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ]
(W : Fin 4 → Ω → ℕ) (q : Fin 4 → ℝ≥0) (m : ℝ≥0) (hm : 0 < m)
(H : ℝ) (hHpos : 0 < H) (L ε : ℝ) (hH : universalH ≤ H) (hL : 1 ≤ L)
(hε : 0 < ε) (hε1 : ε ≤ 1)
(hW : ∀ j, Measurable (W j))
(hlaw : ∀ j, HasLaw (W j) (poissonMeasure (m * q j)) μ) :
MemLp (fun ω => clippingScaleFormula ε (pilotMidpoint H hHpos m L (fun j => W j ω))
(pilotRadiusFormula H hHpos m L (fun j => W j ω))) 2 μ := by
have hB := (pilotTotalMass_memLp_two μ W q m hm H hHpos L hH hL hW hlaw).integrable
(by norm_num : (1 : ENNReal) ≤ 2)
have hmeas : AEStronglyMeasurable (fun ω => clippingScaleFormula ε
(pilotMidpoint H hHpos m L (fun j => W j ω))
(pilotRadiusFormula H hHpos m L (fun j => W j ω))) μ := by
first | fun_prop | exact ((Measurable.of_discrete (f := fun N : Fin 4 → ℕ =>
clippingScaleFormula ε (pilotMidpoint H hHpos m L N) (pilotRadiusFormula H hHpos m L N))).comp
(measurable_pi_lambda _ hW)).aestronglyMeasurable
apply (memLp_two_iff_integrable_sq hmeas).2
have hi := (((hB.mul_const (L / m)).div_const ε).add
(integrable_const (((L / m) / ε) ^ 2))).const_mul (512 * H ^ 2)
apply hi.mono' (hmeas.pow 2)
apply ae_of_all
intro ω
rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)]
exact pilotClippingScale_sq_le H hHpos m L ε (NNReal.coe_pos.mpr hm)
(by linarith) hε hε1 (fun j => W j ω)
The actual clipped cell statistic is square-integrable for any measurable evaluation counts. Its random center and width dominate it pathwise, and both have already derived pilot moments; no evaluation moment gate is needed. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
W Ne :
Fin 4 → Ω → ℕ
q :
Fin 4 → ℝ≥0
m :
ℝ≥0
hm :
0 < m
H :
ℝ
hHpos :
0 < H
L ε :
ℝ
hH :
universalH ≤ H
hL :
1 ≤ L
hε :
0 < ε
hε1 :
ε ≤ 1
K d :
ℕ
hW :
∀ j, Measurable (W j)
hNe :
∀ j, Measurable (Ne j)
hlaw :
∀ j, HasLaw (W j) (poissonMeasure (m * q j)) μ
conclusion
MemLp (fun ω => clippedCellValueFormula ε K d m (pilotMidpoint H hHpos m L (fun j => W j ω)) (pilotRadiusFormula H hHpos m L (fun j => W j ω)) (fun j => Ne j ω)) 2 μ
Proof (Lean source)
lemma pilotClippedCellValue_memLp_two {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ]
(W Ne : Fin 4 → Ω → ℕ) (q : Fin 4 → ℝ≥0) (m : ℝ≥0) (hm : 0 < m)
(H : ℝ) (hHpos : 0 < H) (L ε : ℝ) (hH : universalH ≤ H) (hL : 1 ≤ L)
(hε : 0 < ε) (hε1 : ε ≤ 1) (K d : ℕ)
(hW : ∀ j, Measurable (W j)) (hNe : ∀ j, Measurable (Ne j))
(hlaw : ∀ j, HasLaw (W j) (poissonMeasure (m * q j)) μ) :
MemLp (fun ω => clippedCellValueFormula ε K d m
(pilotMidpoint H hHpos m L (fun j => W j ω))
(pilotRadiusFormula H hHpos m L (fun j => W j ω)) (fun j => Ne j ω)) 2 μ := by
have hB := pilotTotalMass_memLp_two μ W q m hm H hHpos L hH hL hW hlaw
have hS := pilotClippingScale_memLp_two μ W q m hm H hHpos L ε hH hL hε hε1 hW hlaw
have hdom := (hS.const_mul ((d : ℝ) ^ (1 / 4 : ℝ))).add hB
apply hdom.of_le
· first | fun_prop | exact ((Measurable.of_discrete (f := fun z : (Fin 4 → ℕ) × (Fin 4 → ℕ) =>
clippedCellValueFormula ε K d m (pilotMidpoint H hHpos m L z.1)
(pilotRadiusFormula H hHpos m L z.1) z.2)).comp
((measurable_pi_lambda _ hW).prodMk (measurable_pi_lambda _ hNe))).aestronglyMeasurable
· apply ae_of_all
intro ω
let b := pilotMidpoint H hHpos m L (fun j => W j ω)
let r := pilotRadiusFormula H hHpos m L (fun j => W j ω)
have hb := pilotMidpoint_nonneg H hHpos m L (NNReal.coe_pos.mpr hm) (fun j => W j ω)
have hr (j : Fin 4) : 0 ≤ r j :=
(pilotRadius_pos H hHpos m L (NNReal.coe_pos.mpr hm) (by linarith) _ j).le
have hs := clippingScale_nonneg ε b r hb hr
have hw : 0 ≤ (d : ℝ) ^ (1 / 4 : ℝ) * clippingScaleFormula ε b r :=
mul_nonneg (Real.rpow_nonneg (Nat.cast_nonneg _) _) hs
have hf := armwiseExtension_nonneg_le_totalMassVec hε hb
have ht := clippedCellValue_abs_error_le ε K d m b r (fun j => Ne j ω) 0 hw
have hf0 : 0 ≤ armwiseExtensionFormula ε b := hf.1
simp only [sub_zero, abs_of_nonneg hf0] at ht
rw [Real.norm_eq_abs, Real.norm_eq_abs]
exact (ht.trans (add_le_add le_rfl hf.2)).trans (le_abs_self _)
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.UpperPilotTail 11 declarations Roadmap (15)--(16): the actual truncated pilot interval fails only on the empirical-radius bad event.
Pilot localization and bad-coordinate moments
Roadmap (15)--(16): the actual truncated pilot interval fails only on the empirical-radius bad event. Marginal Poisson concentration bounds its probability and its second and fourth deviation-plus-width moments. A union bound controls failure of the four-coordinate rectangle, including null atoms.
The paper's pilot width is exactly the library's empirical Poisson radius. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
pilotHalfWidthFormula H hH m L Np j = empiricalRadius H m (L / m) (Np j)
Proof (Lean source)
lemma pilotHalfWidth_eq_empiricalRadius (H : ℝ) (hH : 0 < H)
(m : ℝ≥0) (L : ℝ) (Np : Fin 4 → ℕ) (j : Fin 4) :
pilotHalfWidthFormula H hH m L Np j = empiricalRadius H m (L / m) (Np j) := by
simp only [pilotHalfWidthFormula, pilotCenterFormula, empiricalRadius, mul_div_assoc]
For a nonnegative true atom, truncation at zero does not change the criterion for failure of its pilot interval. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
↔ pilotHalfWidthFormula H hH m L Np j < |pilotCenterFormula m Np j - q|
Proof (Lean source)
lemma pilotInterval_failure_iff (H : ℝ) (hH : 0 < H) (m L q : ℝ)
(hq : 0 ≤ q) (Np : Fin 4 → ℕ) (j : Fin 4) :
q ∉ Icc (pilotLower H hH m L Np j) (pilotUpperFormula H hH m L Np j) ↔
pilotHalfWidthFormula H hH m L Np j < |pilotCenterFormula m Np j - q| := by
have hmem : q ∈ Icc (pilotLower H hH m L Np j)
(pilotUpperFormula H hH m L Np j) ↔
|pilotCenterFormula m Np j - q| ≤ pilotHalfWidthFormula H hH m L Np j := by
simp only [Set.mem_Icc, pilotLower, pilotUpperFormula, max_le_iff, hq, true_and,
abs_le]
constructor <;> rintro ⟨h₁, h₂⟩ <;> constructor <;> linarith
exact hmem.not.trans not_le
theorem pilotCoordinate_score_pow_integrable Lemma pilotCoordinate_score_pow_integrable in the paper ↗
Every power through four of a coordinate's deviation-plus-width score is integrable under its Poisson law. This derives the regularity needed for (16). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
Integrable (fun w : ℕ => (|(w : ℝ) / m - q| + empiricalRadius H m (L / m) w) ^ t) (poissonMeasure (m * q))
Proof (Lean source)
lemma pilotCoordinate_score_pow_integrable (q m : ℝ≥0) (hm : 0 < m)
(H L : ℝ) (hH : universalH ≤ H) (hL : 1 ≤ L) (t : ℕ) (ht : t ≤ 4) :
Integrable (fun w : ℕ =>
(|(w : ℝ) / m - q| + empiricalRadius H m (L / m) w) ^ t)
(poissonMeasure (m * q)) := by
have hraw := integrable_score_pow (m * q) hL (t := t)
have hH0 : 0 ≤ H := universalH_pos.le.trans hH
apply (hraw.const_mul ((H / universalH / (m : ℝ)) ^ t)).mono'
(Measurable.of_discrete.aestronglyMeasurable)
apply ae_of_all
intro w
have hs := empiricalRadius_score_le_raw q m hm H L hH hL w
have hscore0 : 0 ≤ |(w : ℝ) / m - q| + empiricalRadius H m (L / m) w := by
unfold empiricalRadius
positivity
rw [Real.norm_eq_abs, abs_of_nonneg (pow_nonneg hscore0 t)]
have hp := pow_le_pow_left₀ hscore0 hs t
calc
_ ≤ (H / universalH * (score universalH L (m * q) w / (m : ℝ))) ^ t := hp
_ = _ := by rw [mul_pow, div_pow, div_pow]; ring
Roadmap (16), for moment orders two and four, on the actual coordinate failure event. The larger library bad event also controls the width score; no tail or integrability premise is added. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
q m :
ℝ≥0
hm :
0 < m
H :
ℝ
hHpos :
0 < H
L :
ℝ
hH :
universalH ≤ H
hL :
1 ≤ L
j :
Fin 4
t :
ℕ
ht :
t = 2 ∨ t = 4
conclusion
(∫ w : ℕ, if (q : ℝ) ∉ Icc (pilotLower H hHpos m L (fun _ => w) j) (pilotUpperFormula H hHpos m L (fun _ => w) j) then (|pilotCenterFormula m (fun _ => w) j - q| + pilotHalfWidthFormula H hHpos m L (fun _ => w) j) ^ t else 0 ∂poissonMeasure (m * q))
≤ (H / universalH) ^ t * scalarBadMomentConstant t * exp (-20 * L) * (sqrt ((q : ℝ) * L / m) + L / m) ^ t
Proof (Lean source)
lemma pilotInterval_bad_moment_le (q m : ℝ≥0) (hm : 0 < m)
(H : ℝ) (hHpos : 0 < H) (L : ℝ) (hH : universalH ≤ H) (hL : 1 ≤ L)
(j : Fin 4) (t : ℕ) (ht : t = 2 ∨ t = 4) :
(∫ w : ℕ, if (q : ℝ) ∉ Icc
(pilotLower H hHpos m L (fun _ => w) j)
(pilotUpperFormula H hHpos m L (fun _ => w) j) then
(|pilotCenterFormula m (fun _ => w) j - q| +
pilotHalfWidthFormula H hHpos m L (fun _ => w) j) ^ t else 0
∂poissonMeasure (m * q)) ≤
(H / universalH) ^ t * scalarBadMomentConstant t * exp (-20 * L) *
(sqrt ((q : ℝ) * L / m) + L / m) ^ t := by
classical
let B : Set ℕ := {w | normalizedDeviation m q w >
empiricalRadius H m (L / m) w / 4}
have ht4 : t ≤ 4 := by rcases ht with rfl | rfl <;> omega
have hi := (pilotCoordinate_score_pow_integrable q m hm H L hH hL t ht4).indicator
(show MeasurableSet B from Set.to_countable _ |>.measurableSet)
have hmono : (∫ w : ℕ, if (q : ℝ) ∉ Icc
(pilotLower H hHpos m L (fun _ => w) j)
(pilotUpperFormula H hHpos m L (fun _ => w) j) then
(|pilotCenterFormula m (fun _ => w) j - q| +
pilotHalfWidthFormula H hHpos m L (fun _ => w) j) ^ t else 0
∂poissonMeasure (m * q)) ≤
∫ w : ℕ, B.indicator (fun w =>
(normalizedDeviation m q w + empiricalRadius H m (L / m) w) ^ t) w
∂poissonMeasure (m * q) := by
apply integral_mono_of_nonneg (ae_of_all _ fun w => by
change (0 : ℝ) ≤ _
split_ifs
· exact le_rfl
· apply pow_nonneg
unfold pilotHalfWidthFormula
positivity) hi
apply ae_of_all
intro w
have hwidth0 : 0 ≤ empiricalRadius H m (L / m) w := by
unfold empiricalRadius
positivity
by_cases hw : (q : ℝ) ∈ Icc
(pilotLower H hHpos m L (fun _ => w) j)
(pilotUpperFormula H hHpos m L (fun _ => w) j)
· simp only [if_neg (not_not.mpr hw)]
by_cases hb : w ∈ B
· rw [Set.indicator_of_mem hb]
apply pow_nonneg
exact add_nonneg (abs_nonneg _) hwidth0
· rw [Set.indicator_of_notMem hb]
· have hfail := (pilotInterval_failure_iff H hHpos m L q q.coe_nonneg
(fun _ => w) j).mp hw
rw [pilotHalfWidth_eq_empiricalRadius] at hfail
have hb : w ∈ B := by
change empiricalRadius H m (L / m) w / 4 < |(w : ℝ) / m - q|
exact lt_of_le_of_lt (by linarith) hfail
simp only [if_pos hw, Set.indicator_of_mem hb, pilotCenterFormula,
pilotHalfWidth_eq_empiricalRadius]
exact le_rfl
apply hmono.trans
change (∫ w : ℕ, (if normalizedDeviation m q w >
empiricalRadius H m (L / m) w / 4 then
(normalizedDeviation m q w + empiricalRadius H m (L / m) w) ^ t else 0)
∂poissonMeasure (m * q)) ≤ _
rcases ht with rfl | rfl
· exact poisson_empiricalRadius_bad_second_moment q m hm H L hH hL
· exact poisson_empiricalRadius_bad_fourth_moment q m hm H L hH hL
The marginal moment estimate (16) transports to the actual random pilot counts by their Poisson laws. Independence is unnecessary for this step. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
Measure Ω
W :
Fin 4 → Ω → ℕ
q :
Fin 4 → ℝ≥0
m :
ℝ≥0
hm :
0 < m
H :
ℝ
hHpos :
0 < H
L :
ℝ
hH :
universalH ≤ H
hL :
1 ≤ L
hlaw :
∀ j, HasLaw (W j) (poissonMeasure (m * q j)) μ
j :
Fin 4
t :
ℕ
ht :
t = 2 ∨ t = 4
conclusion
(∫ ω, if (q j : ℝ) ∉ Icc (pilotLower H hHpos m L (fun i => W i ω) j) (pilotUpperFormula H hHpos m L (fun i => W i ω) j) then (|pilotCenterFormula m (fun i => W i ω) j - q j| + pilotHalfWidthFormula H hHpos m L (fun i => W i ω) j) ^ t else 0 ∂μ)
≤ (H / universalH) ^ t * scalarBadMomentConstant t * exp (-20 * L) * (sqrt ((q j : ℝ) * L / m) + L / m) ^ t
Proof (Lean source)
lemma pilotCoordinate_bad_moment_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) (W : Fin 4 → Ω → ℕ) (q : Fin 4 → ℝ≥0)
(m : ℝ≥0) (hm : 0 < m) (H : ℝ) (hHpos : 0 < H) (L : ℝ)
(hH : universalH ≤ H) (hL : 1 ≤ L)
(hlaw : ∀ j, HasLaw (W j) (poissonMeasure (m * q j)) μ)
(j : Fin 4) (t : ℕ) (ht : t = 2 ∨ t = 4) :
(∫ ω, if (q j : ℝ) ∉ Icc
(pilotLower H hHpos m L (fun i => W i ω) j)
(pilotUpperFormula H hHpos m L (fun i => W i ω) j) then
(|pilotCenterFormula m (fun i => W i ω) j - q j| +
pilotHalfWidthFormula H hHpos m L (fun i => W i ω) j) ^ t else 0 ∂μ) ≤
(H / universalH) ^ t * scalarBadMomentConstant t * exp (-20 * L) *
(sqrt ((q j : ℝ) * L / m) + L / m) ^ t := by
classical
let f : ℕ → ℝ := fun w => if (q j : ℝ) ∉ Icc
(pilotLower H hHpos m L (fun _ => w) j)
(pilotUpperFormula H hHpos m L (fun _ => w) j) then
(|pilotCenterFormula m (fun _ => w) j - q j| +
pilotHalfWidthFormula H hHpos m L (fun _ => w) j) ^ t else 0
have he := (hlaw j).integral_comp
(show AEStronglyMeasurable f (poissonMeasure (m * q j)) from
Measurable.of_discrete.aestronglyMeasurable)
simp only [Function.comp_def] at he
change (∫ ω, f (W j ω) ∂μ) ≤ _
rw [he]
exact pilotInterval_bad_moment_le (q j) m hm H hHpos L hH hL j t ht
The full four-coordinate localization rectangle fails with probability at most eight times the exponential scalar tail (15). Only marginal laws are used, so no joint independence or positive cell mass is assumed. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
Measure Ω
W :
Fin 4 → Ω → ℕ
q :
Fin 4 → ℝ≥0
m :
ℝ≥0
hm :
0 < m
H :
ℝ
hHpos :
0 < H
L :
ℝ
hH :
universalH ≤ H
hL :
1 ≤ L
hlaw :
∀ j, HasLaw (W j) (poissonMeasure (m * q j)) μ
conclusion
μ {ω | (fun j => (q j : ℝ)) ∉ pilotRectangle (pilotLower H hHpos m L (fun j => W j ω)) (pilotUpperFormula H hHpos m L (fun j => W j ω))}
Proof (Lean source)
lemma pilotRectangle_bad_probability_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) (W : Fin 4 → Ω → ℕ) (q : Fin 4 → ℝ≥0)
(m : ℝ≥0) (hm : 0 < m) (H : ℝ) (hHpos : 0 < H) (L : ℝ)
(hH : universalH ≤ H) (hL : 1 ≤ L)
(hlaw : ∀ j, HasLaw (W j) (poissonMeasure (m * q j)) μ) :
μ {ω | (fun j => (q j : ℝ)) ∉ pilotRectangle
(pilotLower H hHpos m L (fun j => W j ω))
(pilotUpperFormula H hHpos m L (fun j => W j ω))} ≤
ofReal (8 * exp (-40 * L)) := by
classical
let B : Fin 4 → Set Ω := fun j => {ω | normalizedDeviation m (q j) (W j ω) >
empiricalRadius H m (L / m) (W j ω) / 4}
have hsub : {ω | (fun j => (q j : ℝ)) ∉ pilotRectangle
(pilotLower H hHpos m L (fun j => W j ω))
(pilotUpperFormula H hHpos m L (fun j => W j ω))} ⊆ ⋃ j, B j := by
intro ω hω
change ¬ ∀ j, (q j : ℝ) ∈ Icc
(pilotLower H hHpos m L (fun j => W j ω) j)
(pilotUpperFormula H hHpos m L (fun j => W j ω) j) at hω
obtain ⟨j, hj⟩ := not_forall.mp hω
apply Set.mem_iUnion.mpr
refine ⟨j, ?_⟩
have hfail := (pilotInterval_failure_iff H hHpos m L (q j) (q j).coe_nonneg
(fun j => W j ω) j).mp hj
rw [pilotHalfWidth_eq_empiricalRadius] at hfail
have hw0 : 0 ≤ empiricalRadius H m (L / m) (W j ω) := by
unfold empiricalRadius
positivity
change empiricalRadius H m (L / m) (W j ω) / 4 <
|(W j ω : ℝ) / m - q j|
exact lt_of_le_of_lt (by linarith) hfail
have hB (j : Fin 4) : μ (B j) ≤ ofReal (2 * exp (-40 * L)) := by
rw [(hlaw j).measure_eq (show MeasurableSet {w : ℕ |
normalizedDeviation m (q j) w > empiricalRadius H m (L / m) w / 4} from
Set.to_countable _ |>.measurableSet)]
exact poisson_empiricalRadius_bad_probability (q j) m hm H L hH hL
calc
_ ≤ μ (⋃ j, B j) := measure_mono hsub
_ ≤ ∑' j, μ (B j) := measure_iUnion_le B
_ ≤ ∑' _j : Fin 4, ofReal (2 * exp (-40 * L)) :=
ENNReal.tsum_le_tsum hB
_ = ofReal (8 * exp (-40 * L)) := by
simp only [tsum_fintype, Finset.sum_const, Finset.card_univ, Fintype.card_fin,
nsmul_eq_mul]
rw [← ENNReal.ofReal_natCast]
norm_num only [Nat.cast_ofNat]
rw [← ENNReal.ofReal_mul (by norm_num : (0 : ℝ) ≤ 4)]
congr 1
ring
theorem pilotRectangle_failure_subset_empiricalBadAny Lemma pilotRectangle_failure_subset_empiricalBadAny in the paper ↗
Failure of the actual pilot rectangle is contained in the empirical-radius bad union used by the independent-product moment theorem. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
{ω | (fun j => (q j : ℝ)) ∉ pilotRectangle (pilotLower H hH m L (fun j => W j ω)) (pilotUpperFormula H hH m L (fun j => W j ω))} ⊆ empiricalBadAny W H L m q
Proof (Lean source)
lemma pilotRectangle_failure_subset_empiricalBadAny {Ω : Type*}
(W : Fin 4 → Ω → ℕ) (q : Fin 4 → ℝ≥0) (m : ℝ≥0)
(H : ℝ) (hH : 0 < H) (L : ℝ) (hL : 0 ≤ L) :
{ω | (fun j => (q j : ℝ)) ∉ pilotRectangle
(pilotLower H hH m L (fun j => W j ω))
(pilotUpperFormula H hH m L (fun j => W j ω))} ⊆
empiricalBadAny W H L m q := by
intro ω hω
change ¬ ∀ j, (q j : ℝ) ∈ Icc
(pilotLower H hH m L (fun j => W j ω) j)
(pilotUpperFormula H hH m L (fun j => W j ω) j) at hω
obtain ⟨j, hj⟩ := not_forall.mp hω
have hf := (pilotInterval_failure_iff H hH m L (q j) (q j).coe_nonneg
(fun j => W j ω) j).mp hj
rw [pilotHalfWidth_eq_empiricalRadius] at hf
have hw : 0 ≤ empiricalRadius H m (L / m) (W j ω) := by
unfold empiricalRadius
exact mul_nonneg hH.le (add_nonneg (Real.sqrt_nonneg _) (by positivity))
exact ⟨j, lt_of_le_of_lt (by linarith) hf⟩
All four-coordinate score powers needed for the bad-pilot comparison are integrable, derived from the marginal Poisson laws rather than assumed. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
Measure Ω
W :
Fin 4 → Ω → ℕ
q :
Fin 4 → ℝ≥0
m :
ℝ≥0
hm :
0 < m
H L :
ℝ
hH :
universalH ≤ H
hL :
1 ≤ L
hW :
∀ j, Measurable (W j)
hlaw :
∀ j, HasLaw (W j) (poissonMeasure (m * q j)) μ
t :
ℕ
ht :
t = 1 ∨ t = 2 ∨ t = 4
conclusion
Integrable (fun ω => (empiricalAggregateScore W H L m q ω) ^ t) μ
Proof (Lean source)
lemma pilotAggregateScore_pow_integrable {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) (W : Fin 4 → Ω → ℕ) (q : Fin 4 → ℝ≥0)
(m : ℝ≥0) (hm : 0 < m) (H L : ℝ) (hH : universalH ≤ H) (hL : 1 ≤ L)
(hW : ∀ j, Measurable (W j))
(hlaw : ∀ j, HasLaw (W j) (poissonMeasure (m * q j)) μ)
(t : ℕ) (ht : t = 1 ∨ t = 2 ∨ t = 4) :
Integrable (fun ω => (empiricalAggregateScore W H L m q ω) ^ t) μ := by
have ht4 : t ≤ 4 := by rcases ht with rfl | rfl | rfl <;> omega
have hi (j : Fin 4) : Integrable (fun ω =>
(normalizedDeviation m (q j) (W j ω) +
empiricalRadius H m (L / m) (W j ω)) ^ t) μ := by
have h := pilotCoordinate_score_pow_integrable (q j) m hm H L hH hL t ht4
rw [← (hlaw j).map_eq] at h
exact h.comp_aemeasurable (hlaw j).aemeasurable
have hdom := (integrable_finsetSum Finset.univ (fun j _ => hi j)).const_mul
((4 : ℝ) ^ (t - 1))
have hn (j : Fin 4) (ω : Ω) : 0 ≤ normalizedDeviation m (q j) (W j ω) +
empiricalRadius H m (L / m) (W j ω) := by
have hH0 := universalH_pos.le.trans hH
unfold normalizedDeviation empiricalRadius
positivity
apply hdom.mono'
· unfold empiricalAggregateScore normalizedDeviation empiricalRadius
fun_prop
· apply ae_of_all
intro ω
unfold empiricalAggregateScore
rw [Real.norm_eq_abs, abs_of_nonneg
(pow_nonneg (Finset.sum_nonneg (fun j _ => hn j ω)) t)]
rcases ht with rfl | rfl | rfl
· simp
· simpa using (pow_sum_le_card_mul_sum_pow
(s := (Finset.univ : Finset (Fin 4))) (fun j _ => hn j ω) 1)
· simpa using (pow_sum_le_card_mul_sum_pow
(s := (Finset.univ : Finset (Fin 4))) (fun j _ => hn j ω) 3)
theorem pilotRectangle_bad_aggregate_moment_le Lemma pilotRectangle_bad_aggregate_moment_le in the paper ↗
Roadmap (16) jointly controls every coordinate's score on failure of any coordinate. This retains the cross-coordinate terms needed in (22), which a sum of the scalar bad moments alone does not control. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
W :
Fin 4 → Ω → ℕ
q :
Fin 4 → ℝ≥0
m :
ℝ≥0
hm :
0 < m
H :
ℝ
hHpos :
0 < H
L :
ℝ
hH :
universalH ≤ H
hL :
1 ≤ L
hW :
∀ j, Measurable (W j)
hlaw :
∀ j, HasLaw (W j) (poissonMeasure (m * q j)) μ
hind :
iIndepFun W μ
t :
ℕ
ht :
t = 1 ∨ t = 2 ∨ t = 4
conclusion
(∫ ω, if (fun j => (q j : ℝ)) ∉ pilotRectangle (pilotLower H hHpos m L (fun j => W j ω)) (pilotUpperFormula H hHpos m L (fun j => W j ω)) then (∑ j : Fin 4, (|pilotCenterFormula m (fun i => W i ω) j - q j| + pilotHalfWidthFormula H hHpos m L (fun i => W i ω) j)) ^ t else 0 ∂μ)
≤ (H / universalH) ^ t * productMomentConstant 4 t * exp (-20 * L) * (sqrt ((∑ j, (q j : ℝ)) * L / m) + L / m) ^ t
Proof (Lean source)
lemma pilotRectangle_bad_aggregate_moment_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ]
(W : Fin 4 → Ω → ℕ) (q : Fin 4 → ℝ≥0) (m : ℝ≥0) (hm : 0 < m)
(H : ℝ) (hHpos : 0 < H) (L : ℝ) (hH : universalH ≤ H) (hL : 1 ≤ L)
(hW : ∀ j, Measurable (W j))
(hlaw : ∀ j, HasLaw (W j) (poissonMeasure (m * q j)) μ)
(hind : iIndepFun W μ) (t : ℕ) (ht : t = 1 ∨ t = 2 ∨ t = 4) :
(∫ ω, if (fun j => (q j : ℝ)) ∉ pilotRectangle
(pilotLower H hHpos m L (fun j => W j ω))
(pilotUpperFormula H hHpos m L (fun j => W j ω)) then
(∑ j : Fin 4, (|pilotCenterFormula m (fun i => W i ω) j - q j| +
pilotHalfWidthFormula H hHpos m L (fun i => W i ω) j)) ^ t else 0 ∂μ) ≤
(H / universalH) ^ t * productMomentConstant 4 t * exp (-20 * L) *
(sqrt ((∑ j, (q j : ℝ)) * L / m) + L / m) ^ t := by
classical
have hsub := pilotRectangle_failure_subset_empiricalBadAny W q m H hHpos L (by linarith)
have hi := (pilotAggregateScore_pow_integrable μ W q m hm H L hH hL hW hlaw t ht).indicator
(show MeasurableSet (empiricalBadAny W H L m q) by
unfold empiricalBadAny normalizedDeviation empiricalRadius
measurability)
have hn (ω : Ω) : 0 ≤ empiricalAggregateScore W H L m q ω := by
have hH0 := universalH_pos.le.trans hH
unfold empiricalAggregateScore normalizedDeviation empiricalRadius
positivity
have hmono : (∫ ω, if (fun j => (q j : ℝ)) ∉ pilotRectangle
(pilotLower H hHpos m L (fun j => W j ω))
(pilotUpperFormula H hHpos m L (fun j => W j ω)) then
(∑ j : Fin 4, (|pilotCenterFormula m (fun i => W i ω) j - q j| +
pilotHalfWidthFormula H hHpos m L (fun i => W i ω) j)) ^ t else 0 ∂μ) ≤
∫ ω, (empiricalBadAny W H L m q).indicator
(fun ω => (empiricalAggregateScore W H L m q ω) ^ t) ω ∂μ := by
apply integral_mono_of_nonneg
· apply ae_of_all
intro ω
simp only [pilotHalfWidth_eq_empiricalRadius, pilotCenterFormula]
change 0 ≤ if _ then (empiricalAggregateScore W H L m q ω) ^ t else 0
split_ifs
· exact le_rfl
· exact pow_nonneg (hn ω) t
· exact hi
· apply ae_of_all
intro ω
dsimp only
by_cases hf : (fun j => (q j : ℝ)) ∉ pilotRectangle
(pilotLower H hHpos m L (fun j => W j ω))
(pilotUpperFormula H hHpos m L (fun j => W j ω))
· rw [if_pos hf, Set.indicator_of_mem (hsub hf)]
simp only [empiricalAggregateScore, normalizedDeviation, pilotCenterFormula,
pilotHalfWidth_eq_empiricalRadius]
exact le_rfl
· rw [if_neg hf]
exact Set.indicator_nonneg (fun ω _ => pow_nonneg (hn ω) t) _
exact hmono.trans (independent_poisson_empiricalRadius_badAny_moment_four
μ W q m hm hW hlaw hind ht H L hH hL)
theorem pilotMidpoint_total_abs_error_le_score Lemma pilotMidpoint_total_abs_error_le_score in the paper ↗
The total pilot midpoint displacement is bounded by the joint score, including pilots whose intervals fail to contain the true vector. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
W :
Fin 4 → Ω → ℕ
q :
Fin 4 → ℝ≥0
m :
ℝ≥0
hm :
0 < m
H :
ℝ
hH :
0 < H
L :
ℝ
hL :
0 ≤ L
ω :
Ω
conclusion
(∑ j : Fin 4, |pilotMidpoint H hH m L (fun i => W i ω) j - q j|)
≤ empiricalAggregateScore W H L m q ω
Proof (Lean source)
lemma pilotMidpoint_total_abs_error_le_score {Ω : Type*}
(W : Fin 4 → Ω → ℕ) (q : Fin 4 → ℝ≥0) (m : ℝ≥0) (hm : 0 < m)
(H : ℝ) (hH : 0 < H) (L : ℝ) (hL : 0 ≤ L) (ω : Ω) :
(∑ j : Fin 4, |pilotMidpoint H hH m L (fun i => W i ω) j - q j|) ≤
empiricalAggregateScore W H L m q ω := by
unfold empiricalAggregateScore
apply Finset.sum_le_sum
intro j _
simpa only [pilotCenterFormula, normalizedDeviation, pilotHalfWidth_eq_empiricalRadius] using
pilotMidpoint_abs_error_le H hH m L (NNReal.coe_pos.mpr hm) hL
(fun i => W i ω) (fun i => (q i : ℝ)) j
theorem pilotRectangle_bad_midpoint_moment_le Lemma pilotRectangle_bad_midpoint_moment_le in the paper ↗
Joint bad-pilot moments also control movement of the actual midpoint. This supplies the unweighted center-displacement component of (22b) without separating the random midpoint from the rectangle-failure event. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
W :
Fin 4 → Ω → ℕ
q :
Fin 4 → ℝ≥0
m :
ℝ≥0
hm :
0 < m
H :
ℝ
hHpos :
0 < H
L :
ℝ
hH :
universalH ≤ H
hL :
1 ≤ L
hW :
∀ j, Measurable (W j)
hlaw :
∀ j, HasLaw (W j) (poissonMeasure (m * q j)) μ
hind :
iIndepFun W μ
t :
ℕ
ht :
t = 1 ∨ t = 2 ∨ t = 4
conclusion
(∫ ω, if (fun j => (q j : ℝ)) ∉ pilotRectangle (pilotLower H hHpos m L (fun j => W j ω)) (pilotUpperFormula H hHpos m L (fun j => W j ω)) then (∑ j : Fin 4, |pilotMidpoint H hHpos m L (fun i => W i ω) j - q j|) ^ t else 0 ∂μ)
≤ (H / universalH) ^ t * productMomentConstant 4 t * exp (-20 * L) * (sqrt ((∑ j, (q j : ℝ)) * L / m) + L / m) ^ t
Proof (Lean source)
lemma pilotRectangle_bad_midpoint_moment_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ]
(W : Fin 4 → Ω → ℕ) (q : Fin 4 → ℝ≥0) (m : ℝ≥0) (hm : 0 < m)
(H : ℝ) (hHpos : 0 < H) (L : ℝ) (hH : universalH ≤ H) (hL : 1 ≤ L)
(hW : ∀ j, Measurable (W j))
(hlaw : ∀ j, HasLaw (W j) (poissonMeasure (m * q j)) μ)
(hind : iIndepFun W μ) (t : ℕ) (ht : t = 1 ∨ t = 2 ∨ t = 4) :
(∫ ω, if (fun j => (q j : ℝ)) ∉ pilotRectangle
(pilotLower H hHpos m L (fun j => W j ω))
(pilotUpperFormula H hHpos m L (fun j => W j ω)) then
(∑ j : Fin 4, |pilotMidpoint H hHpos m L (fun i => W i ω) j - q j|) ^ t
else 0 ∂μ) ≤
(H / universalH) ^ t * productMomentConstant 4 t * exp (-20 * L) *
(sqrt ((∑ j, (q j : ℝ)) * L / m) + L / m) ^ t := by
classical
let B : Set Ω := {ω | (fun j => (q j : ℝ)) ∉ pilotRectangle
(pilotLower H hHpos m L (fun j => W j ω))
(pilotUpperFormula H hHpos m L (fun j => W j ω))}
have hB : MeasurableSet B := by
dsimp [B]
unfold pilotRectangle pilotLower pilotUpperFormula pilotHalfWidthFormula pilotCenterFormula
measurability
have hi := (pilotAggregateScore_pow_integrable μ W q m hm H L hH hL hW hlaw t ht).indicator hB
have hmono : (∫ ω, if ω ∈ B then
(∑ j : Fin 4, |pilotMidpoint H hHpos m L (fun i => W i ω) j - q j|) ^ t
else 0 ∂μ) ≤
∫ ω, B.indicator (fun ω => (empiricalAggregateScore W H L m q ω) ^ t) ω ∂μ := by
apply integral_mono_of_nonneg
· apply ae_of_all
intro ω
dsimp only
split_ifs <;> positivity
· exact hi
· apply ae_of_all
intro ω
dsimp only
by_cases hb : ω ∈ B
· rw [if_pos hb, Set.indicator_of_mem hb]
exact pow_le_pow_left₀ (Finset.sum_nonneg (fun j _ => abs_nonneg _))
(pilotMidpoint_total_abs_error_le_score W q m hm H hHpos L (by linarith) ω) t
· rw [if_neg hb, Set.indicator_of_notMem hb]
apply hmono.trans
convert pilotRectangle_bad_aggregate_moment_le μ W q m hm H hHpos L hH hL
hW hlaw hind t ht using 1
· congr 1
· apply integral_congr_ae
filter_upwards [] with ω
by_cases hb : ω ∈ B
· simp [Set.indicator_of_mem hb, show (fun j => (q j : ℝ)) ∉ pilotRectangle
(pilotLower H hHpos m L (fun j => W j ω))
(pilotUpperFormula H hHpos m L (fun j => W j ω)) from hb,
empiricalAggregateScore, normalizedDeviation, pilotCenterFormula,
pilotHalfWidth_eq_empiricalRadius]
· simp [Set.indicator_of_notMem hb, show ¬ (fun j => (q j : ℝ)) ∉ pilotRectangle
(pilotLower H hHpos m L (fun j => W j ω))
(pilotUpperFormula H hHpos m L (fun j => W j ω)) from hb]
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.UpperPoissonization 15 declarations Roadmap equations (7)–(8) use iid observation–fair-mark pairs, stopped at an independent Poisson count.
Marked Poissonization of the observable estimator
Roadmap equations (7)–(8) use iid observation–fair-mark pairs, stopped at an independent Poisson count. The statistic below is the uncapped version of the actual capped statistic, with intensity fixed by the original sample size. Its fixed-count fibres and prefix identities connect that experiment to the finite Rao–Blackwell sum in equation (37).
Fair Boolean marks form a probability law.
Definition (Lean source)
instance fairBool_isProbabilityMeasure : IsProbabilityMeasure (bernoulliBool (1 / 2)) :=
bernoulliBool_isProbabilityMeasure (by norm_num) (by norm_num)
def markedPoissonStatistic
The uncapped statistic uses all pairs in a finite marked sample, while retaining the intensity and degree chosen from the original sample size.
Definition (Lean source)
parameters
def
markedPoissonStatistic n d H₀ hH₀ κ ε s :
ℝ
given by
projectUnit <| ∑ x : Fin d, let Np := markedCount (le_refl s.1) (fun i => (s.2 i).1) (fun i => (s.2 i).2) true x let Ne := markedCount (le_refl s.1) (fun i => (s.2 i).1) (fun i => (s.2 i).2) false x clippedCellValueFormula ε (jacksonDegree κ d) d (poissonIntensityFormula n) (pilotMidpoint H₀ hH₀ (poissonIntensityFormula n) (logAlphabet d) Np) (pilotRadiusFormula H₀ hH₀ (poissonIntensityFormula n) (logAlphabet d) Np) Ne
theorem markedPoissonStatistic_mem_unitInterval
The marked statistic has unit range even when its count exceeds the sample cap. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
conclusion 1
0 ≤ markedPoissonStatistic n d H₀ hH₀ κ ε s
conclusion 2
markedPoissonStatistic n d H₀ hH₀ κ ε s ≤ 1
Proof (Lean source)
lemma markedPoissonStatistic_mem_unitInterval (n d : ℕ) (H₀ : ℝ)
(hH₀ : 0 < H₀) (κ ε : ℝ) (s : FiniteSample (Obs d × Bool)) :
0 ≤ markedPoissonStatistic n d H₀ hH₀ κ ε s ∧
markedPoissonStatistic n d H₀ hH₀ κ ε s ≤ 1 :=
projectUnit_mem_unitInterval _
theorem markedPoissonStatistic_measurable
Reading the statistic is measurable on the countable finite-sample space. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
n d :
ℕ
H₀ :
ℝ
hH₀ :
0 < H₀
κ ε :
ℝ
conclusion
Measurable (markedPoissonStatistic n d H₀ hH₀ κ ε)
Proof (Lean source)
-- @node: markedPoissonStatistic_measurable
@[fun_prop] lemma markedPoissonStatistic_measurable (n d : ℕ) (H₀ : ℝ)
(hH₀ : 0 < H₀) (κ ε : ℝ) :
Measurable (markedPoissonStatistic n d H₀ hH₀ κ ε) := by
intro A hA
rw [MeasurableSpace.measurableSet_iInf]
intro k
change MeasurableSet ((fun z : Fin k → Obs d × Bool =>
markedPoissonStatistic n d H₀ hH₀ κ ε ⟨k, z⟩) ⁻¹' A)
exact Measurable.of_discrete hA
theorem markedPoissonStatistic_sq_integrable
Boundedness supplies squared-loss integrability without an added regularity premise. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
Integrable (fun s => (markedPoissonStatistic n d H₀ hH₀ κ ε s - t) ^ 2) μ
Proof (Lean source)
lemma markedPoissonStatistic_sq_integrable (n d : ℕ) (H₀ : ℝ)
(hH₀ : 0 < H₀) (κ ε t : ℝ)
(μ : Measure (FiniteSample (Obs d × Bool))) [IsFiniteMeasure μ] :
Integrable (fun s => (markedPoissonStatistic n d H₀ hH₀ κ ε s - t) ^ 2) μ := by
apply Integrable.of_bound (by fun_prop) ((1 + |t|) ^ 2)
filter_upwards [] with s
rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)]
have hv := markedPoissonStatistic_mem_unitInterval n d H₀ hH₀ κ ε s
have ha : |markedPoissonStatistic n d H₀ hH₀ κ ε s - t| ≤ 1 + |t| := by
exact (abs_sub _ _).trans (by rw [abs_of_nonneg hv.1]; linarith [hv.2])
simpa only [sq_abs] using pow_le_pow_left₀ (abs_nonneg _) ha 2
Below the cap, the uncapped marked statistic equals the implemented capped statistic on the first k observations and their marks. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
markedPoissonStatistic n d H₀ hH₀ κ ε (prefixOfLE (fun i => (o i, marks i)) k hk)
= cappedStatisticFormula hk H₀ hH₀ κ ε o (fun i => marks ⟨i.val, lt_of_lt_of_le i.isLt hk⟩)
Proof (Lean source)
lemma markedPoissonStatistic_prefix_eq {n d k : ℕ} (hk : k ≤ n)
(H₀ : ℝ) (hH₀ : 0 < H₀) (κ ε : ℝ)
(o : Fin n → Obs d) (marks : Fin n → Bool) :
markedPoissonStatistic n d H₀ hH₀ κ ε
(prefixOfLE (fun i => (o i, marks i)) k hk) =
cappedStatisticFormula hk H₀ hH₀ κ ε o
(fun i => marks ⟨i.val, lt_of_lt_of_le i.isLt hk⟩) := by
simp only [markedPoissonStatistic, prefixOfLE, cappedStatisticFormula]
congr 1
Fair Boolean marks have constant singleton mass, including the empty array. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma fairMarkLaw_singleton_real (k : ℕ) (marks : Fin k → Bool) :
(Measure.pi (fun _ : Fin k => bernoulliBool (1 / 2))).real {marks} =
(1 / 2 : ℝ) ^ k := by
let : IsProbabilityMeasure (bernoulliBool (1 / 2)) :=
bernoulliBool_isProbabilityMeasure (by norm_num) (by norm_num)
rw [measureReal_def, Measure.pi_singleton, ENNReal.toReal_prod]
have hm (i : Fin k) : (bernoulliBool (1 / 2) {marks i}).toReal = (1 / 2 : ℝ) := by
cases marks i <;> norm_num [bernoulliBool]
simp only [hm, Finset.prod_const, Finset.card_univ, Fintype.card_fin]
Integrating independent fair marks is exactly the finite average used by the observable estimator. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
(∫ marks, f marks ∂Measure.pi (fun _ : Fin k => bernoulliBool (1 / 2)))
= ∑ marks, (1 / 2 : ℝ) ^ k * f marks
Proof (Lean source)
lemma integral_fairMarks_eq_sum (k : ℕ) (f : (Fin k → Bool) → ℝ) :
(∫ marks, f marks ∂Measure.pi (fun _ : Fin k => bernoulliBool (1 / 2))) =
∑ marks, (1 / 2 : ℝ) ^ k * f marks := by
let : IsProbabilityMeasure (bernoulliBool (1 / 2)) :=
bernoulliBool_isProbabilityMeasure (by norm_num) (by norm_num)
rw [integral_fintype Integrable.of_finite]
simp only [fairMarkLaw_singleton_real, smul_eq_mul]
theorem integral_iid_observation_fairMarks
Iid observation–mark pairs integrate as a finite fair-mark average of iid observation expectations. This identifies the auxiliary averaging law exactly. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
(∫ z : Fin k → Obs d × Bool, f (fun i => (z i).1) (fun i => (z i).2) ∂Measure.pi (fun _ => P.pmf.toMeasure.prod (bernoulliBool (1 / 2))))
= ∑ marks, (1 / 2 : ℝ) ^ k * ∫ o, f o marks ∂productLaw P k
Proof (Lean source)
lemma integral_iid_observation_fairMarks {d : ℕ} (P : DiscreteLaw d)
(k : ℕ) (f : (Fin k → Obs d) → (Fin k → Bool) → ℝ) :
(∫ z : Fin k → Obs d × Bool,
f (fun i => (z i).1) (fun i => (z i).2)
∂Measure.pi (fun _ => P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))) =
∑ marks, (1 / 2 : ℝ) ^ k * ∫ o, f o marks ∂productLaw P k := by
let : IsProbabilityMeasure (bernoulliBool (1 / 2)) :=
bernoulliBool_isProbabilityMeasure (by norm_num) (by norm_num)
have hmp := measurePreserving_arrowProdEquivProdArrow (Obs d) Bool (Fin k)
(fun _ => P.pmf.toMeasure) (fun _ => bernoulliBool (1 / 2))
have he := integral_map_of_stronglyMeasurable
(μ := Measure.pi (fun _ : Fin k => P.pmf.toMeasure.prod (bernoulliBool (1 / 2))))
hmp.measurable
(Measurable.of_discrete (f := fun z => f z.1 z.2)).stronglyMeasurable
rw [hmp.map_eq] at he
simp only [MeasurableEquiv.arrowProdEquivProdArrow, Equiv.arrowProdEquivProdArrow,
MeasurableEquiv.coe_mk, Equiv.coe_fn_mk] at he
rw [← he, integral_prod_symm _ Integrable.of_finite,
integral_fairMarks_eq_sum]
rfl
Taking the first k fair marks from n fair marks preserves their product law. This removes unused marks from each count fibre. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
n k :
ℕ
hk :
k ≤ n
conclusion
Measure.map (fun marks : Fin n → Bool => fun i : Fin k => marks ⟨i.val, lt_of_lt_of_le i.isLt hk⟩) (Measure.pi (fun _ : Fin n => bernoulliBool (1 / 2)))
= Measure.pi (fun _ : Fin k => bernoulliBool (1 / 2))
Proof (Lean source)
lemma fairMarkPrefix_map_eq {n k : ℕ} (hk : k ≤ n) :
Measure.map (fun marks : Fin n → Bool => fun i : Fin k =>
marks ⟨i.val, lt_of_lt_of_le i.isLt hk⟩)
(Measure.pi (fun _ : Fin n => bernoulliBool (1 / 2))) =
Measure.pi (fun _ : Fin k => bernoulliBool (1 / 2)) := by
rw [← iidStreamLaw_map_finPrefix (bernoulliBool (1 / 2)) n,
Measure.map_map (by fun_prop) (by fun_prop)]
exact iidStreamLaw_map_finPrefix (bernoulliBool (1 / 2)) k
The finite fair-mark average of a prefix statistic is independent of how many unused fair marks are appended. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma fairMark_average_prefix_eq {n k : ℕ} (hk : k ≤ n)
(f : (Fin k → Bool) → ℝ) :
(∑ marks : Fin n → Bool, (1 / 2 : ℝ) ^ n *
f (fun i => marks ⟨i.val, lt_of_lt_of_le i.isLt hk⟩)) =
∑ marks : Fin k → Bool, (1 / 2 : ℝ) ^ k * f marks := by
rw [← integral_fairMarks_eq_sum, ← integral_fairMarks_eq_sum,
← fairMarkPrefix_map_eq hk]
exact (integral_map_of_stronglyMeasurable (by fun_prop)
Measurable.of_discrete.stronglyMeasurable).symm
theorem markedPoissonStatistic_count_fibre_sqRisk_eq Lemma markedPoissonStatistic_count_fibre_sqRisk_eq in the paper ↗
On a count fibre that fits the sample, the marked Poisson squared-loss integral is its Poisson mass times the exact fair-mark average of capped risk. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
(∫ s : FiniteSample (Obs d × Bool) in FiniteSample.count ⁻¹' ({k} : Set ℕ), (markedPoissonStatistic n d H₀ hH₀ κ ε s - t) ^ 2 ∂finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4))
= poissonWeight (n / 4) k * ∑ marks : Fin k → Bool, (1 / 2 : ℝ) ^ k * sqRisk (productLaw P n) (fun o => cappedStatisticFormula hk H₀ hH₀ κ ε o marks) t
Proof (Lean source)
lemma markedPoissonStatistic_count_fibre_sqRisk_eq {n d k : ℕ} (hk : k ≤ n)
(H₀ : ℝ) (hH₀ : 0 < H₀) (κ ε t : ℝ) (P : DiscreteLaw d) :
(∫ s : FiniteSample (Obs d × Bool) in FiniteSample.count ⁻¹' ({k} : Set ℕ),
(markedPoissonStatistic n d H₀ hH₀ κ ε s - t) ^ 2
∂finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)) =
poissonWeight (n / 4) k * ∑ marks : Fin k → Bool, (1 / 2 : ℝ) ^ k *
sqRisk (productLaw P n)
(fun o => cappedStatisticFormula hk H₀ hH₀ κ ε o marks) t := by
let : IsProbabilityMeasure (bernoulliBool (1 / 2)) :=
bernoulliBool_isProbabilityMeasure (by norm_num) (by norm_num)
rw [poisson_prefix_count_fibre_integral _ _ n k hk _
(markedPoissonStatistic_sq_integrable n d H₀ hH₀ κ ε t _)]
simp_rw [markedPoissonStatistic_prefix_eq hk]
rw [integral_iid_observation_fairMarks P n (fun o marks =>
(cappedStatisticFormula hk H₀ hH₀ κ ε o
(fun i => marks ⟨i.val, lt_of_lt_of_le i.isLt hk⟩) - t) ^ 2)]
rw [show ((poissonMeasure ((n : ℝ≥0) / 4)) {k}).toReal =
poissonWeight (n / 4) k by
simp only [ProbabilityTheory.poissonMeasure_singleton, NNReal.coe_div,
NNReal.coe_natCast, NNReal.coe_ofNat, poissonWeight, ENNReal.toReal_ofReal_eq_iff]
positivity]
congr 1
exact fairMark_average_prefix_eq hk (fun marks =>
sqRisk (productLaw P n)
(fun o => cappedStatisticFormula hk H₀ hH₀ κ ε o marks) t)
theorem markedPoissonStatistic_nonoverflow_sqRisk_eq Lemma markedPoissonStatistic_nonoverflow_sqRisk_eq in the paper ↗
The count-capped part of the Poisson risk is the weighted auxiliary risk appearing in the finite Rao–Blackwell inequality, with no unused marks. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
(∫ s : FiniteSample (Obs d × Bool), if s.count ≤ n then (markedPoissonStatistic n d H₀ hH₀ κ ε s - t) ^ 2 else 0 ∂finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4))
= ∑ k ∈ range (n + 1), if hk : k ≤ n then poissonWeight (n / 4) k * ∑ marks : Fin k → Bool, (1 / 2 : ℝ) ^ k * sqRisk (productLaw P n) (fun o => cappedStatisticFormula hk H₀ hH₀ κ ε o marks) t else 0
Proof (Lean source)
lemma markedPoissonStatistic_nonoverflow_sqRisk_eq (n d : ℕ) (H₀ : ℝ)
(hH₀ : 0 < H₀) (κ ε t : ℝ) (P : DiscreteLaw d) :
(∫ s : FiniteSample (Obs d × Bool),
if s.count ≤ n then (markedPoissonStatistic n d H₀ hH₀ κ ε s - t) ^ 2 else 0
∂finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)) =
∑ k ∈ range (n + 1), if hk : k ≤ n then
poissonWeight (n / 4) k * ∑ marks : Fin k → Bool, (1 / 2 : ℝ) ^ k *
sqRisk (productLaw P n)
(fun o => cappedStatisticFormula hk H₀ hH₀ κ ε o marks) t else 0 := by
rw [poisson_nonoverflow_integral_eq_sum_count_fibres _ _ n _
(markedPoissonStatistic_sq_integrable n d H₀ hH₀ κ ε t _)]
have hf (k : Fin (n + 1)) := markedPoissonStatistic_count_fibre_sqRisk_eq
(le_of_lt_succ k.isLt) H₀ hH₀ κ ε t P
simp_rw [hf]
rw [← Fin.sum_univ_eq_sum_range]
apply Finset.sum_congr rfl
intro k hk
simp only [dif_pos (le_of_lt_succ k.isLt)]
theorem poissonWeight_quarter_missing_mass_le Lemma poissonWeight_quarter_missing_mass_le in the paper ↗
The missing mass in the finite count average is the quarter-mean Poisson overflow probability, so equation (35) applies to the implemented weights. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
Proof (Lean source)
lemma poissonWeight_quarter_missing_mass_le (n : ℕ) :
1 - ∑ k ∈ range (n + 1), poissonWeight (n / 4) k ≤
exp (-(n : ℝ) / 2) := by
let μ := poissonMeasure ((n : ℝ≥0) / 4)
have hw (k : ℕ) : μ.real {k} = poissonWeight (n / 4) k := by
simp only [μ, measureReal_def, ProbabilityTheory.poissonMeasure_singleton,
NNReal.coe_div, NNReal.coe_natCast, NNReal.coe_ofNat, poissonWeight,
ENNReal.toReal_ofReal_eq_iff]
positivity
have hs : μ.real (Iic n) = ∑ k ∈ range (n + 1), poissonWeight (n / 4) k := by
rw [show Iic n = (range (n + 1) : Set ℕ) by
ext k; simp only [Set.mem_Iic, Finset.mem_coe, Finset.mem_range]; omega,
← sum_measureReal_singleton]
simp_rw [hw]
have hc : μ.real (Ioi n) = 1 - μ.real (Iic n) := by
simpa only [Set.compl_Iic, probReal_univ] using
(measureReal_compl (μ := μ) measurableSet_Iic)
rw [← hs, ← hc]
exact quarterPoisson_overflow_prob_le μ n id
⟨measurable_id.aemeasurable, by simp [μ]⟩
theorem armwiseEstimator_active_le_markedPoisson_sqRisk Lemma armwiseEstimator_active_le_markedPoisson_sqRisk in the paper ↗
Equation (37) and restoration of the cap for the actual implemented estimator. Its fixed-sample risk is bounded by the uncapped marked Poisson risk plus the exponential cap penalty; no auxiliary risk bound is assumed. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
H₀ κ :
ℝ
D₀ :
ℕ
hH₀ :
0 < H₀
hκ :
0 < κ
hD₀ :
2 ≤ D₀
n d :
ℕ
ε :
ℝ
hd :
D₀ ≤ d
hactive :
(d : ℝ) / logAlphabet d ≤ (n : ℝ) * ε
P :
conclusion
sqRisk (productLaw P n) (armwiseEstimatorFormula H₀ κ D₀ hH₀ hκ hD₀ n d ε) (observedValue P)
≤ sqRisk (finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : ℝ≥0) / 4)) (markedPoissonStatistic n d H₀ hH₀ κ ε) (observedValue P)
+ exp (-(n : ℝ) / 2)
Proof (Lean source)
lemma armwiseEstimator_active_le_markedPoisson_sqRisk (H₀ κ : ℝ) (D₀ : ℕ)
(hH₀ : 0 < H₀) (hκ : 0 < κ) (hD₀ : 2 ≤ D₀)
{n d : ℕ} {ε : ℝ} (hd : D₀ ≤ d)
(hactive : (d : ℝ) / logAlphabet d ≤ (n : ℝ) * ε) (P : DiscreteLaw d) :
sqRisk (productLaw P n)
(armwiseEstimatorFormula H₀ κ D₀ hH₀ hκ hD₀ n d ε) (observedValue P) ≤
sqRisk
(finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4))
(markedPoissonStatistic n d H₀ hH₀ κ ε) (observedValue P) +
exp (-(n : ℝ) / 2) := by
let : IsProbabilityMeasure (productLaw P n) := by dsimp [productLaw]; infer_instance
have hr := armwiseEstimator_active_sqRisk_le H₀ κ D₀ hH₀ hκ hD₀
hd hactive P (productLaw P n)
rw [← markedPoissonStatistic_nonoverflow_sqRisk_eq n d H₀ hH₀ κ ε (observedValue P) P] at hr
have hnonoverflow :
(∫ s : FiniteSample (Obs d × Bool),
if s.count ≤ n then
(markedPoissonStatistic n d H₀ hH₀ κ ε s - observedValue P) ^ 2 else 0
∂finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)) ≤
sqRisk
(finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4))
(markedPoissonStatistic n d H₀ hH₀ κ ε) (observedValue P) := by
have hi := markedPoissonStatistic_sq_integrable n d H₀ hH₀ κ ε (observedValue P)
(finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4))
have hiCap : Integrable (fun s : FiniteSample (Obs d × Bool) =>
if s.count ≤ n then
(markedPoissonStatistic n d H₀ hH₀ κ ε s - observedValue P) ^ 2 else 0)
(finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : ℝ≥0) / 4)) := by
have heq : (fun s : FiniteSample (Obs d × Bool) =>
if s.count ≤ n then
(markedPoissonStatistic n d H₀ hH₀ κ ε s - observedValue P) ^ 2 else 0) =
(FiniteSample.count ⁻¹' Iic n).indicator (fun s =>
(markedPoissonStatistic n d H₀ hH₀ κ ε s - observedValue P) ^ 2) := by
funext s
simp [indicator]
rw [heq]
exact hi.indicator (measurable_finiteSample_count measurableSet_Iic)
apply integral_mono hiCap hi
intro s
dsimp only
split_ifs
· rfl
· exact sq_nonneg _
have ht := observedValue_mem_unitInterval P
have ht₂ : (observedValue P) ^ 2 ≤ 1 := by nlinarith
have hm : 0 ≤ 1 - ∑ k ∈ range (n + 1), poissonWeight (n / 4) k :=
sub_nonneg.mpr (poissonWeight_sum_le_one (by positivity) n)
have htail := (mul_le_of_le_one_right hm ht₂).trans
(poissonWeight_quarter_missing_mass_le n)
exact hr.trans (add_le_add hnonoverflow htail)
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.UpperRaoBlackwell 12 declarations The fair-mark average and the capped Poisson weights implement the auxiliary expectation in roadmap equation (37).
Finite Rao–Blackwell risk transfer
The fair-mark average and the capped Poisson weights implement the auxiliary expectation in roadmap equation (37). The omitted Poisson tail is an atom at zero, so its squared loss must be included when applying Jensen.
The projection used in every capped statistic has range in the unit interval. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
Proof (Lean source)
lemma projectUnit_mem_unitInterval (z : ℝ) :
0 ≤ projectUnit z ∧ projectUnit z ≤ 1 := by
exact ⟨le_max_left _ _, max_le (by norm_num) (min_le_left _ _)⟩
Every auxiliary capped statistic has range in the unit interval. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
conclusion 1
0 ≤ cappedStatisticFormula hk H₀ hH₀ κ ε o marks
conclusion 2
cappedStatisticFormula hk H₀ hH₀ κ ε o marks ≤ 1
Proof (Lean source)
lemma cappedStatistic_mem_unitInterval {n d k : ℕ} (hk : k ≤ n)
(H₀ : ℝ) (hH₀ : 0 < H₀) (κ ε : ℝ)
(o : Fin n → Obs d) (marks : Fin k → Bool) :
0 ≤ cappedStatisticFormula hk H₀ hH₀ κ ε o marks ∧
cappedStatisticFormula hk H₀ hH₀ κ ε o marks ≤ 1 :=
projectUnit_mem_unitInterval _
Poisson weights are nonnegative, including at zero intensity. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
rate :
ℝ
hrate :
0 ≤ rate
k :
ℕ
conclusion
0 ≤ poissonWeight rate k
Proof (Lean source)
lemma poissonWeight_nonneg {rate : ℝ} (hrate : 0 ≤ rate) (k : ℕ) :
0 ≤ poissonWeight rate k := by
unfold poissonWeight
positivity
The full Poisson weight series has total mass one. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma poissonWeight_hasSum_one {rate : ℝ} (hrate : 0 ≤ rate) :
HasSum (poissonWeight rate) 1 := by
exact ProbabilityTheory.hasSum_one_poissonMeasure ⟨rate, hrate⟩
Truncating Poisson weights at the sample cap produces a subprobability. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma poissonWeight_sum_le_one {rate : ℝ} (hrate : 0 ≤ rate) (n : ℕ) :
∑ k ∈ range (n + 1), poissonWeight rate k ≤ 1 := by
have h := poissonWeight_hasSum_one hrate
rw [← h.tsum_eq]
exact h.summable.sum_le_tsum _ (fun k _ => poissonWeight_nonneg hrate k)
Fair marks on k observations have total weight one. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Proof (Lean source)
lemma fairMarkWeight_sum_eq_one (k : ℕ) :
(∑ _marks : Fin k → Bool, (1 / 2 : ℝ) ^ k) = 1 := by
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fun,
Fintype.card_bool, Fintype.card_fin, nsmul_eq_mul]
rw [Nat.cast_pow, Nat.cast_ofNat, ← mul_pow]
norm_num
Squared loss decreases under a finite subprobability average when its missing mass is assigned the statistic zero. This explicitly retains the loss on the cap tail. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
((∑ i ∈ s, w i * v i) - t) ^ 2
≤ (∑ i ∈ s, w i * (v i - t) ^ 2) + (1 - ∑ i ∈ s, w i) * t ^ 2
Proof (Lean source)
lemma subprobability_average_sq_error_le {ι : Type*} (s : Finset ι)
(w v : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i) (hm : ∑ i ∈ s, w i ≤ 1)
(t : ℝ) :
((∑ i ∈ s, w i * v i) - t) ^ 2 ≤
(∑ i ∈ s, w i * (v i - t) ^ 2) + (1 - ∑ i ∈ s, w i) * t ^ 2 := by
have hc := weighted_inner_sq_le s w v (fun _ => 1) hw
simp only [mul_one, one_pow] at hc
have hv : 0 ≤ ∑ i ∈ s, w i * v i ^ 2 :=
Finset.sum_nonneg fun i hi => mul_nonneg (hw i hi) (sq_nonneg _)
have hb : (∑ i ∈ s, w i * v i) ^ 2 ≤ ∑ i ∈ s, w i * v i ^ 2 :=
hc.trans (mul_le_of_le_one_right hv hm)
have he : (∑ i ∈ s, w i * (v i - t) ^ 2) =
(∑ i ∈ s, w i * v i ^ 2) - 2 * t * (∑ i ∈ s, w i * v i) +
(∑ i ∈ s, w i) * t ^ 2 := by
calc
_ = ∑ i ∈ s, (w i * v i ^ 2 - (2 * t) * (w i * v i) + w i * t ^ 2) := by
apply Finset.sum_congr rfl
intro i hi
ring
_ = _ := by
rw [Finset.sum_add_distrib, Finset.sum_sub_distrib,
Finset.mul_sum, Finset.sum_mul]
rw [he]
nlinarith
Averaging over the fair marks contracts squared loss pathwise. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
Proof (Lean source)
lemma fairMark_average_sq_error_le (k : ℕ) (v : (Fin k → Bool) → ℝ) (t : ℝ) :
((∑ marks, (1 / 2 : ℝ) ^ k * v marks) - t) ^ 2 ≤
∑ marks, (1 / 2 : ℝ) ^ k * (v marks - t) ^ 2 := by
have h := subprobability_average_sq_error_le univ
(fun _ : Fin k → Bool => (1 / 2 : ℝ) ^ k) v
(fun _ _ => by positivity) (fairMarkWeight_sum_eq_one k).le t
simpa only [fairMarkWeight_sum_eq_one, sub_self, zero_mul, add_zero] using h
theorem armwiseEstimator_active_sq_error_le Lemma armwiseEstimator_active_sq_error_le in the paper ↗
In the active branch the deterministic estimator contracts the auxiliary squared loss, including the zero statistic when the Poisson count exceeds the cap. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
H₀ κ :
ℝ
D₀ :
ℕ
hH₀ :
0 < H₀
hκ :
0 < κ
hD₀ :
2 ≤ D₀
n d :
ℕ
ε :
ℝ
hd :
D₀ ≤ d
hactive :
(d : ℝ) / logAlphabet d ≤ (n : ℝ) * ε
o :
Fin n → Obs d
t :
ℝ
conclusion
(armwiseEstimatorFormula H₀ κ D₀ hH₀ hκ hD₀ n d ε o - t) ^ 2
≤ (∑ k ∈ range (n + 1), if hk : k ≤ n then poissonWeight (n / 4) k * ∑ marks : Fin k → Bool, (1 / 2 : ℝ) ^ k * (cappedStatisticFormula hk H₀ hH₀ κ ε o marks - t) ^ 2 else 0)
+ (1 - ∑ k ∈ range (n + 1), poissonWeight (n / 4) k) * t ^ 2
Proof (Lean source)
lemma armwiseEstimator_active_sq_error_le (H₀ κ : ℝ) (D₀ : ℕ)
(hH₀ : 0 < H₀) (hκ : 0 < κ) (hD₀ : 2 ≤ D₀)
{n d : ℕ} {ε : ℝ} (hd : D₀ ≤ d)
(hactive : (d : ℝ) / logAlphabet d ≤ (n : ℝ) * ε)
(o : Fin n → Obs d) (t : ℝ) :
(armwiseEstimatorFormula H₀ κ D₀ hH₀ hκ hD₀ n d ε o - t) ^ 2 ≤
(∑ k ∈ range (n + 1), if hk : k ≤ n then
poissonWeight (n / 4) k * ∑ marks : Fin k → Bool,
(1 / 2 : ℝ) ^ k * (cappedStatisticFormula hk H₀ hH₀ κ ε o marks - t) ^ 2
else 0) +
(1 - ∑ k ∈ range (n + 1), poissonWeight (n / 4) k) * t ^ 2 := by
let v : ℕ → ℝ := fun k => if hk : k ≤ n then
∑ marks : Fin k → Bool, (1 / 2 : ℝ) ^ k * cappedStatisticFormula hk H₀ hH₀ κ ε o marks
else 0
have hr : 0 ≤ (n : ℝ) / 4 := by positivity
have hc := subprobability_average_sq_error_le (range (n + 1))
(poissonWeight (n / 4)) v (fun k _ => poissonWeight_nonneg hr k)
(poissonWeight_sum_le_one hr n) t
have he : armwiseEstimatorFormula H₀ κ D₀ hH₀ hκ hD₀ n d ε o =
∑ k ∈ range (n + 1), poissonWeight (n / 4) k * v k := by
simp only [armwiseEstimatorFormula, not_lt.mpr hd, not_lt.mpr hactive, ↓reduceIte]
apply Finset.sum_congr rfl
intro k hk
have hkn : k ≤ n := le_of_lt_succ (Finset.mem_range.mp hk)
simp [v, hkn]
rw [he]
apply hc.trans
apply add_le_add _ le_rfl
apply Finset.sum_le_sum
intro k hk
have hkn : k ≤ n := le_of_lt_succ (Finset.mem_range.mp hk)
simp only [v, dif_pos hkn]
exact mul_le_mul_of_nonneg_left
(fairMark_average_sq_error_le k (cappedStatisticFormula hkn H₀ hH₀ κ ε o) t)
(poissonWeight_nonneg hr k)
The finite Rao–Blackwell estimator stays in the unit interval in every branch. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
conclusion 1
0 ≤ armwiseEstimatorFormula H₀ κ D₀ hH₀ hκ hD₀ n d ε o
conclusion 2
armwiseEstimatorFormula H₀ κ D₀ hH₀ hκ hD₀ n d ε o ≤ 1
Proof (Lean source)
lemma armwiseEstimator_mem_unitInterval (H₀ κ : ℝ) (D₀ : ℕ)
(hH₀ : 0 < H₀) (hκ : 0 < κ) (hD₀ : 2 ≤ D₀)
(n d : ℕ) (ε : ℝ) (o : Fin n → Obs d) :
0 ≤ armwiseEstimatorFormula H₀ κ D₀ hH₀ hκ hD₀ n d ε o ∧
armwiseEstimatorFormula H₀ κ D₀ hH₀ hκ hD₀ n d ε o ≤ 1 := by
unfold armwiseEstimatorFormula
split_ifs with hsmall hsat
· exact projectUnit_mem_unitInterval _
· norm_num
· have hr : 0 ≤ (n : ℝ) / 4 := by positivity
constructor
· apply Finset.sum_nonneg
intro k hk
split_ifs with hkn
· apply mul_nonneg (poissonWeight_nonneg hr k)
exact Finset.sum_nonneg fun marks _ => mul_nonneg (by positivity)
(cappedStatistic_mem_unitInterval hkn H₀ hH₀ κ ε o marks).1
· rfl
· calc
_ ≤ ∑ k ∈ range (n + 1), poissonWeight (n / 4) k := by
apply Finset.sum_le_sum
intro k hk
split_ifs with hkn
· apply mul_le_of_le_one_right (poissonWeight_nonneg hr k)
calc
_ ≤ ∑ _marks : Fin k → Bool, (1 / 2 : ℝ) ^ k := by
apply Finset.sum_le_sum
intro marks hm
exact mul_le_of_le_one_right (by positivity)
(cappedStatistic_mem_unitInterval hkn H₀ hH₀ κ ε o marks).2
_ = 1 := fairMarkWeight_sum_eq_one k
· exact poissonWeight_nonneg hr k
_ ≤ 1 := poissonWeight_sum_le_one hr n
The estimator's unit-range loss provides the uniform bounded-risk part of the minimum-form bound, independently of the unknown cell masses and propensities. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
H₀ κ :
ℝ
D₀ :
ℕ
hH₀ :
0 < H₀
hκ :
0 < κ
hD₀ :
2 ≤ D₀
n d :
ℕ
ε :
ℝ
P :
μ :
conclusion
sqRisk μ (armwiseEstimatorFormula H₀ κ D₀ hH₀ hκ hD₀ n d ε) (observedValue P)
≤ 1
Proof (Lean source)
lemma armwiseEstimator_sqRisk_le_one (H₀ κ : ℝ) (D₀ : ℕ)
(hH₀ : 0 < H₀) (hκ : 0 < κ) (hD₀ : 2 ≤ D₀)
{n d : ℕ} (ε : ℝ) (P : DiscreteLaw d)
(μ : Measure (Fin n → Obs d)) [IsProbabilityMeasure μ] :
Causalean.Stat.sqRisk μ (armwiseEstimatorFormula H₀ κ D₀ hH₀ hκ hD₀ n d ε)
(observedValue P) ≤ 1 := by
have ht := observedValue_mem_unitInterval P
have hpoint (o : Fin n → Obs d) :
(armwiseEstimatorFormula H₀ κ D₀ hH₀ hκ hD₀ n d ε o - observedValue P) ^ 2 ≤ 1 := by
have hv := armwiseEstimator_mem_unitInterval H₀ κ D₀ hH₀ hκ hD₀ n d ε o
have hab : |armwiseEstimatorFormula H₀ κ D₀ hH₀ hκ hD₀ n d ε o - observedValue P| ≤ 1 :=
abs_le.mpr ⟨by linarith, by linarith⟩
simpa only [sq_abs, one_pow] using
pow_le_pow_left₀ (abs_nonneg _) hab 2
unfold Causalean.Stat.sqRisk
calc
_ ≤ ∫ _o : Fin n → Obs d, (1 : ℝ) ∂μ :=
integral_mono Integrable.of_finite (integrable_const _) hpoint
_ = 1 := by simp
Equation (37) for the actual finite estimator: its observed risk is at most the Poisson-weighted, fair-mark auxiliary risk plus the cap-tail loss at zero. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
H₀ κ :
ℝ
D₀ :
ℕ
hH₀ :
0 < H₀
hκ :
0 < κ
hD₀ :
2 ≤ D₀
n d :
ℕ
ε :
ℝ
hd :
D₀ ≤ d
hactive :
(d : ℝ) / logAlphabet d ≤ (n : ℝ) * ε
P :
μ :
conclusion
sqRisk μ (armwiseEstimatorFormula H₀ κ D₀ hH₀ hκ hD₀ n d ε) (observedValue P)
≤ (∑ k ∈ range (n + 1), if hk : k ≤ n then poissonWeight (n / 4) k * ∑ marks : Fin k → Bool, (1 / 2 : ℝ) ^ k * sqRisk μ (fun o => cappedStatisticFormula hk H₀ hH₀ κ ε o marks) (observedValue P) else 0)
+ (1 - ∑ k ∈ range (n + 1), poissonWeight (n / 4) k) * (observedValue P) ^ 2
Proof (Lean source)
lemma armwiseEstimator_active_sqRisk_le (H₀ κ : ℝ) (D₀ : ℕ)
(hH₀ : 0 < H₀) (hκ : 0 < κ) (hD₀ : 2 ≤ D₀)
{n d : ℕ} {ε : ℝ} (hd : D₀ ≤ d)
(hactive : (d : ℝ) / logAlphabet d ≤ (n : ℝ) * ε)
(P : DiscreteLaw d) (μ : Measure (Fin n → Obs d)) [IsProbabilityMeasure μ] :
sqRisk μ (armwiseEstimatorFormula H₀ κ D₀ hH₀ hκ hD₀ n d ε)
(observedValue P) ≤
(∑ k ∈ range (n + 1), if hk : k ≤ n then
poissonWeight (n / 4) k * ∑ marks : Fin k → Bool,
(1 / 2 : ℝ) ^ k * sqRisk μ
(fun o => cappedStatisticFormula hk H₀ hH₀ κ ε o marks) (observedValue P)
else 0) +
(1 - ∑ k ∈ range (n + 1), poissonWeight (n / 4) k) * (observedValue P) ^ 2 := by
unfold sqRisk
calc
_ ≤ ∫ o, ((∑ k ∈ range (n + 1), if hk : k ≤ n then
poissonWeight (n / 4) k * ∑ marks : Fin k → Bool,
(1 / 2 : ℝ) ^ k * (cappedStatisticFormula hk H₀ hH₀ κ ε o marks - observedValue P) ^ 2
else 0) +
(1 - ∑ k ∈ range (n + 1), poissonWeight (n / 4) k) * (observedValue P) ^ 2) ∂μ :=
integral_mono Integrable.of_finite Integrable.of_finite
(fun o => armwiseEstimator_active_sq_error_le H₀ κ D₀ hH₀ hκ hD₀
hd hactive o (observedValue P))
_ = _ := by
rw [integral_add Integrable.of_finite (integrable_const _)]
simp only [integral_const, probReal_univ, one_smul]
congr 1
rw [integral_finsetSum _ (fun _ _ => Integrable.of_finite)]
apply Finset.sum_congr rfl
intro k hk
have hkn : k ≤ n := le_of_lt_succ (Finset.mem_range.mp hk)
simp only [dif_pos hkn]
rw [integral_const_mul, integral_finsetSum _ (fun _ _ => Integrable.of_finite)]
simp only [integral_const_mul]
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.UpperRiskAssembly 3 declarations Equations (34)–(37) transfer an uncapped marked-experiment estimate to the actual observed estimator and combine all three branches.
Observable upper-risk assembly
Equations (34)–(37) transfer an uncapped marked-experiment estimate to the actual observed estimator and combine all three branches. The remaining input is explicitly the uncapped experiment estimate; no headline theorem is asserted here without that analytic input.
theorem armwiseEstimator_active_marked_rate_le Lemma armwiseEstimator_active_marked_rate_le in the paper ↗
On the active branch, cap restoration and finite Rao–Blackwellization convert the uncapped intensity-scale estimate into the observed minimum rate. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
H₀ κ :
ℝ
D₀ :
ℕ
hH₀ :
0 < H₀
hκ :
0 < κ
hD₀ :
2 ≤ D₀
n d :
ℕ
ε C :
ℝ
hn :
1 ≤ n
hd :
D₀ ≤ d
hε :
0 < ε
hε1 :
ε ≤ 1 / 2
hC :
0 ≤ C
hactive :
(d : ℝ) / logAlphabet d ≤ (n : ℝ) * ε
P :
hiid :
IidSampling P μ
hmarked :
sqRisk (finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : NNReal) / 4)) (markedPoissonStatistic n d H₀ hH₀ κ ε) (observedValue P)
≤ C * ((d : ℝ) / (((n : ℝ) / 8) * ε * logAlphabet d))
conclusion
sqRisk μ (armwiseEstimatorFormula H₀ κ D₀ hH₀ hκ hD₀ n d ε) (observedValue P)
≤ (8 * C + 2) * rateScale n d ε
Proof (Lean source)
lemma armwiseEstimator_active_marked_rate_le (H₀ κ : ℝ) (D₀ : ℕ)
(hH₀ : 0 < H₀) (hκ : 0 < κ) (hD₀ : 2 ≤ D₀)
{n d : ℕ} {ε C : ℝ} (hn : 1 ≤ n) (hd : D₀ ≤ d)
(hε : 0 < ε) (hε1 : ε ≤ 1 / 2) (hC : 0 ≤ C)
(hactive : (d : ℝ) / logAlphabet d ≤ (n : ℝ) * ε)
(P : DiscreteLaw d) (μ : Measure (Fin n → Obs d)) (hiid : IidSampling P μ)
(hmarked : sqRisk
(finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : NNReal) / 4))
(markedPoissonStatistic n d H₀ hH₀ κ ε) (observedValue P) ≤
C * ((d : ℝ) / (((n : ℝ) / 8) * ε * logAlphabet d))) :
sqRisk μ (armwiseEstimatorFormula H₀ κ D₀ hH₀ hκ hD₀ n d ε)
(observedValue P) ≤ (8 * C + 2) * rateScale n d ε := by
rw [hiid]
let : IsProbabilityMeasure (productLaw P n) := by dsimp [productLaw]; infer_instance
have hunit := armwiseEstimator_sqRisk_le_one H₀ κ D₀ hH₀ hκ hD₀ ε P (productLaw P n)
have htransfer := armwiseEstimator_active_le_markedPoisson_sqRisk
H₀ κ D₀ hH₀ hκ hD₀ hd hactive P
have htail := capExponential_le_observable_rate (d := d) hn (by omega) hε hε1
have hscale : (d : ℝ) / (((n : ℝ) / 8) * ε * logAlphabet d) =
8 * ((d : ℝ) / ((n : ℝ) * ε * logAlphabet d)) := by ring
rw [hscale] at hmarked
have hraw : sqRisk (productLaw P n)
(armwiseEstimatorFormula H₀ κ D₀ hH₀ hκ hD₀ n d ε) (observedValue P) ≤
(8 * C + 2) * ((d : ℝ) / ((n : ℝ) * ε * logAlphabet d)) := by
nlinarith
unfold rateScale
rcases le_total (1 : ℝ) ((d : ℝ) / ((n : ℝ) * ε * logAlphabet d)) with hsat | hsmall
· rw [min_eq_left hsat, mul_one]
exact hunit.trans (by linarith)
· rw [min_eq_right hsmall]
exact hraw
theorem armwiseEstimator_allBranches_rate_le Lemma armwiseEstimator_allBranches_rate_le in the paper ↗
A uniform uncapped marked-experiment bound on active parameters suffices for the actual estimator on every branch; the cutoff enters only through a bounded logarithm in the empirical branch. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
H₀ κ :
ℝ
D₀ :
ℕ
hH₀ :
0 < H₀
hκ :
0 < κ
hD₀ :
2 ≤ D₀
n d :
ℕ
ε C :
ℝ
hn :
1 ≤ n
hd :
2 ≤ d
hε :
0 < ε
hε1 :
ε ≤ 1 / 2
hC :
0 ≤ C
P :
hP :
ObservedClass ε P
hiid :
IidSampling P μ
hmarked :
D₀
≤ d → (d : ℝ) / logAlphabet d ≤ (n : ℝ) * ε → sqRisk (finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : NNReal) / 4)) (markedPoissonStatistic n d H₀ hH₀ κ ε) (observedValue P) ≤ C * ((d : ℝ) / (((n : ℝ) / 8) * ε * logAlphabet d))
conclusion
sqRisk μ (armwiseEstimatorFormula H₀ κ D₀ hH₀ hκ hD₀ n d ε) (observedValue P)
≤ (3 + 8 * logAlphabet D₀ + 8 * C) * rateScale n d ε
Proof (Lean source)
lemma armwiseEstimator_allBranches_rate_le (H₀ κ : ℝ) (D₀ : ℕ)
(hH₀ : 0 < H₀) (hκ : 0 < κ) (hD₀ : 2 ≤ D₀)
{n d : ℕ} {ε C : ℝ} (hn : 1 ≤ n) (hd : 2 ≤ d)
(hε : 0 < ε) (hε1 : ε ≤ 1 / 2) (hC : 0 ≤ C)
(P : DiscreteLaw d) (hP : ObservedClass ε P)
(μ : Measure (Fin n → Obs d)) (hiid : IidSampling P μ)
(hmarked : D₀ ≤ d → (d : ℝ) / logAlphabet d ≤ (n : ℝ) * ε →
sqRisk
(finitePoissonSampleLaw (P.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : NNReal) / 4))
(markedPoissonStatistic n d H₀ hH₀ κ ε) (observedValue P) ≤
C * ((d : ℝ) / (((n : ℝ) / 8) * ε * logAlphabet d))) :
sqRisk μ (armwiseEstimatorFormula H₀ κ D₀ hH₀ hκ hD₀ n d ε)
(observedValue P) ≤
(3 + 8 * logAlphabet D₀ + 8 * C) * rateScale n d ε := by
have hL₀ := logAlphabet_pos (by omega : 1 ≤ D₀)
have hL := logAlphabet_pos (by omega : 1 ≤ d)
have hrate : 0 ≤ rateScale n d ε := by
unfold rateScale
exact le_min (by norm_num) (by positivity)
by_cases hsmall : d < D₀
· exact (armwiseEstimator_boundedAlphabet_sqRisk_le H₀ κ D₀ hH₀ hκ hD₀
hn hd hsmall hε hε1 P hP μ hiid).trans
(mul_le_mul_of_nonneg_right (by linarith) hrate)
· have hdD : D₀ ≤ d := Nat.le_of_not_gt hsmall
by_cases hsat : (n : ℝ) * ε < (d : ℝ) / logAlphabet d
· exact (armwiseEstimator_saturated_rate_bound H₀ κ D₀ hH₀ hκ hD₀
hn hdD hε hsat P μ hiid).trans
(mul_le_mul_of_nonneg_right (by linarith) hrate)
· have hactive := le_of_not_gt hsat
exact (armwiseEstimator_active_marked_rate_le H₀ κ D₀ hH₀ hκ hD₀
hn hdD hε hε1 hC hactive P μ hiid (hmarked hdD hactive)).trans
(mul_le_mul_of_nonneg_right (by linarith) hrate)
theorem armwiseEstimator_allBranches_worstRisk_le Lemma armwiseEstimator_allBranches_worstRisk_le in the paper ↗
Taking the supremum over all legal observed laws preserves the assembled three-branch rate. Measurability follows from the finite observed sample alphabet, so the remaining input is only the marked-experiment content bound. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
H₀ κ :
ℝ
D₀ :
ℕ
hH₀ :
0 < H₀
hκ :
0 < κ
hD₀ :
2 ≤ D₀
n d :
ℕ
ε C :
ℝ
hn :
1 ≤ n
hd :
2 ≤ d
hε :
0 < ε
hε1 :
ε ≤ 1 / 2
hC :
0 ≤ C
hiid :
∀ P, IidSampling P.1 (sampleLaw P)
hmarked :
D₀
≤ d → (d : ℝ) / logAlphabet d ≤ (n : ℝ) * ε → ∀ P : ModelLaw d ε, sqRisk (finitePoissonSampleLaw (P.1.pmf.toMeasure.prod (bernoulliBool (1 / 2))) ((n : NNReal) / 4)) (markedPoissonStatistic n d H₀ hH₀ κ ε) (observedValue P.1) ≤ C * ((d : ℝ) / (((n : ℝ) / 8) * ε * logAlphabet d))
conclusion
conclusion 1
Measurable (armwiseEstimatorFormula H₀ κ D₀ hH₀ hκ hD₀ n d ε)
conclusion 2
(⨆ P : ModelLaw d ε, sqRisk (sampleLaw P) (armwiseEstimatorFormula H₀ κ D₀ hH₀ hκ hD₀ n d ε) (observedValue P.1))
≤ (3 + 8 * logAlphabet D₀ + 8 * C) * rateScale n d ε
Proof (Lean source)
lemma armwiseEstimator_allBranches_worstRisk_le (H₀ κ : ℝ) (D₀ : ℕ)
(hH₀ : 0 < H₀) (hκ : 0 < κ) (hD₀ : 2 ≤ D₀)
{n d : ℕ} {ε C : ℝ} (hn : 1 ≤ n) (hd : 2 ≤ d)
(hε : 0 < ε) (hε1 : ε ≤ 1 / 2) (hC : 0 ≤ C)
(sampleLaw : ModelLaw d ε → Measure (Fin n → Obs d))
(hiid : ∀ P, IidSampling P.1 (sampleLaw P))
(hmarked : D₀ ≤ d → (d : ℝ) / logAlphabet d ≤ (n : ℝ) * ε →
∀ P : ModelLaw d ε,
sqRisk
(finitePoissonSampleLaw (P.1.pmf.toMeasure.prod (bernoulliBool (1 / 2)))
((n : NNReal) / 4))
(markedPoissonStatistic n d H₀ hH₀ κ ε) (observedValue P.1) ≤
C * ((d : ℝ) / (((n : ℝ) / 8) * ε * logAlphabet d))) :
Measurable (armwiseEstimatorFormula H₀ κ D₀ hH₀ hκ hD₀ n d ε) ∧
(⨆ P : ModelLaw d ε,
sqRisk (sampleLaw P)
(armwiseEstimatorFormula H₀ κ D₀ hH₀ hκ hD₀ n d ε) (observedValue P.1)) ≤
(3 + 8 * logAlphabet D₀ + 8 * C) * rateScale n d ε := by
constructor
· fun_prop
· classical
cases isEmpty_or_nonempty (ModelLaw d ε) with
| inl hempty =>
let := hempty
rw [iSup_of_empty']
have hL₀ := logAlphabet_pos (by omega : 1 ≤ D₀)
have hL := logAlphabet_pos (by omega : 1 ≤ d)
have hrate : 0 ≤ rateScale n d ε := by
unfold rateScale
exact le_min (by norm_num) (by positivity)
simpa using mul_nonneg (show 0 ≤ 3 + 8 * logAlphabet D₀ + 8 * C by positivity) hrate
| inr hnonempty =>
let := hnonempty
apply ciSup_le
intro P
exact armwiseEstimator_allBranches_rate_le H₀ κ D₀ hH₀ hκ hD₀
hn hd hε hε1 hC P.1 P.2 (sampleLaw P) (hiid P)
(fun hdD hactive => hmarked hdD hactive P)
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.WeightedApproximation 8 declarations Weighted scalar approximation and constrained moment priors
Weighted scalar approximation and constrained moment priors
For the displayed parameters, phiEpsFormula is the object specified by this definition.
Definition (Lean source)
For the displayed parameters, phiEps is the object specified by this definition.
Definition (Lean source)
For the displayed parameters, weightedApproxError is the object specified by this definition.
Definition (Lean source)
parameters
K :
ℕ
M ε :
ℝ
def
weightedApproxError K M ε :
ℝ
given by
sInf {e : ℝ | ∃ p : Polynomial ℝ, p.natDegree ≤ K ∧ e = sSup ((fun z : ℝ => |phiEpsFormula ε z - p.eval z| / (1 + z)) '' Icc 0 M)}
For the displayed parameters, ConstrainedPriorPair is the object specified by this definition.
For the displayed parameters, CommonAtomDomain is the object specified by this definition.
Definition (Lean source)
parameters
σp σm :
Measure ℝ
def
CommonAtomDomain σp σm :
Prop
given by
clause 1
clause 2
clause 3
Integrable (fun z : ℝ => z) σp
clause 4
Integrable (fun z : ℝ => z) σm
clause 5
σp univ < 1
def commonAtomCompletion
For the displayed parameters, commonAtomCompletion is the object specified by this definition.
def constrainedPriorSeparationFormula
For the displayed parameters, constrainedPriorSeparationFormula is the object specified by this definition.
Definition (Lean source)
parameters
K :
ℕ
M ε :
ℝ
def
constrainedPriorSeparationFormula K M ε :
ℝ
given by
sSup (Set.range fun p : ConstrainedPriorPair K M => |(∫ z, phiEpsFormula ε z ∂p.ν₁) - (∫ z, phiEpsFormula ε z ∂p.ν₀)|)
For the displayed parameters, constrainedPriorSeparation is the object specified by this definition.
Definition (Lean source)
parameters
K :
ℕ
M ε :
ℝ
_hK :
2 ≤ K
_hM :
2 ≤ M
_hMhi :
M ≤ (K : ℝ) ^ 2
_hε :
0 ≤ ε
_hεhi :
ε ≤ 1 / 2
def
constrainedPriorSeparation K M ε _hK _hM _hMhi _hε _hεhi :
ℝ
given by
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.WeightedBestApproximation 3 declarations The reverse comparison in roadmap (3)--(8) starts with a best weighted approximant.
Attainment of the weighted polynomial approximation error
The reverse comparison in roadmap (3)--(8) starts with a best weighted approximant. Multiplication by the continuous reciprocal weight embeds the degree-bounded polynomial space in the continuous functions on the compact interval. Its finite dimension gives a compact minimizing ball, hence an actual minimizer rather than an assumed dual certificate.
The weighted infimum is attained by a polynomial of the prescribed degree. This is the minimizer used to construct the active residual set in roadmap (3). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
K :
ℕ
M :
ℝ
hM :
0 ≤ M
ε :
ℝ
Proof (Lean source)
theorem weightedApproxError_attained (K : ℕ) {M : ℝ} (hM : 0 ≤ M) (ε : ℝ) :
∃ p : Polynomial ℝ, p.natDegree ≤ K ∧
sSup ((fun z : ℝ => |phiEpsFormula ε z - p.eval z| / (1 + z)) '' Icc 0 M) =
weightedApproxError K M ε := by
classical
let I := Icc (0 : ℝ) M
letI : CompactSpace I := isCompact_iff_compactSpace.mp isCompact_Icc
letI : Nonempty I := ⟨⟨0, le_rfl, hM⟩⟩
let w : C(I, ℝ) := ⟨fun z => (1 + (z : ℝ))⁻¹, by
apply Continuous.inv₀
· fun_prop
· intro z
have hz := z.property.1
dsimp [I] at hz
linarith⟩
let f : C(I, ℝ) := ⟨fun z => phiEpsFormula ε z / (1 + (z : ℝ)), by
unfold phiEpsFormula
apply Continuous.div
· apply Continuous.div
· fun_prop
· fun_prop
· intro z; have hz := z.property.1; dsimp [I] at hz; linarith
· fun_prop
· intro z; have hz := z.property.1; dsimp [I] at hz; linarith⟩
let T : Polynomial ℝ →ₗ[ℝ] C(I, ℝ) :=
{ toFun := fun p => w * p.toContinuousMapOn I
map_add' := by intro p q; ext z; simp [mul_add]
map_smul' := by intro a p; ext z; simp [mul_left_comm] }
let D := Polynomial.degreeLT ℝ (K + 1)
let S := T.domRestrict D
let V := S.range
letI : FiniteDimensional ℝ D :=
(Polynomial.degreeLT.basis ℝ (K + 1)).finiteDimensional_of_finite
letI : FiniteDimensional ℝ V :=
FiniteDimensional.of_surjective S.rangeRestrict (by
intro q
obtain ⟨p, hp⟩ := q.property
exact ⟨p, Subtype.ext hp⟩)
let e := fun p : Polynomial ℝ =>
sSup ((fun z : ℝ => |phiEpsFormula ε z - p.eval z| / (1 + z)) '' I)
have he (p : Polynomial ℝ) : e p = ‖f - T p‖ := by
have hc : ContinuousOn (fun z : ℝ =>
|phiEpsFormula ε z - p.eval z| / (1 + z)) I := by
unfold phiEpsFormula
apply ContinuousOn.div
· apply ContinuousOn.abs
apply ContinuousOn.sub
· apply ContinuousOn.div
· fun_prop
· fun_prop
· intro z hz; dsimp [I] at hz; linarith [hz.1]
· fun_prop
· fun_prop
· intro z hz; dsimp [I] at hz; linarith [hz.1]
have hpoint (z : I) : ‖(f - T p) z‖ =
|phiEpsFormula ε z - p.eval (z : ℝ)| / (1 + (z : ℝ)) := by
have hz : 0 < 1 + (z : ℝ) := by
have hz := z.property.1; dsimp [I] at hz; linarith
change |phiEpsFormula ε z / (1 + (z : ℝ)) -
(1 + (z : ℝ))⁻¹ * p.eval (z : ℝ)| = _
rw [inv_mul_eq_div, ← sub_div, abs_div, abs_of_pos hz]
apply le_antisymm
· apply csSup_le (by exact ⟨_, 0, ⟨le_rfl, hM⟩, rfl⟩)
rintro v ⟨z, hz, rfl⟩
exact (hpoint ⟨z, hz⟩).symm.le.trans (ContinuousMap.norm_coe_le_norm _ _)
· apply (ContinuousMap.norm_le_of_nonempty _).mpr
intro z
rw [hpoint]
exact le_csSup (isCompact_Icc.bddAbove_image hc)
(Set.mem_image_of_mem _ z.property)
let R : ℝ := 2 * ‖f‖ + 1
let B : Set V := closedBall 0 R
have hR : 0 ≤ R := by dsimp [R]; positivity
have hzero : (0 : V) ∈ B := by simp [B, hR]
have hcompact : IsCompact B := ProperSpace.isCompact_closedBall 0 R
obtain ⟨q, _, hqmin⟩ := hcompact.exists_isMinOn ⟨0, hzero⟩
(show Continuous (fun q : V => ‖f - (q : C(I, ℝ))‖) from by fun_prop).continuousOn
have hqzero : ‖f - (q : C(I, ℝ))‖ ≤ ‖f‖ := by simpa using hqmin hzero
have hglobal (u : V) : ‖f - (q : C(I, ℝ))‖ ≤ ‖f - (u : C(I, ℝ))‖ := by
by_cases hu : u ∈ B
· exact hqmin hu
· have hunorm : R < ‖u‖ := by
simpa only [B, Metric.mem_closedBall, dist_zero_right, not_le] using hu
have hdiff := norm_sub_norm_le (u : C(I, ℝ)) f
have hlower : ‖f‖ ≤ ‖(u : C(I, ℝ)) - f‖ := by
dsimp [R] at hunorm
change ‖(u : C(I, ℝ))‖ - ‖f‖ ≤ ‖(u : C(I, ℝ)) - f‖ at hdiff
linarith
rw [norm_sub_rev] at hlower
exact hqzero.trans hlower
obtain ⟨p, hpq⟩ := q.property
have hpdeg : (p : Polynomial ℝ).natDegree ≤ K := by
apply Polynomial.natDegree_le_iff_degree_le.mpr
apply Polynomial.mem_degreeLE.mp
rw [← Polynomial.degreeLT_succ_eq_degreeLE]
exact p.property
have hmin (u : Polynomial ℝ) (hu : u.natDegree ≤ K) : e p ≤ e u := by
have huD : u ∈ D := by
dsimp [D]
rw [Polynomial.degreeLT_succ_eq_degreeLE]
exact Polynomial.mem_degreeLE.mpr (Polynomial.natDegree_le_iff_degree_le.mp hu)
let uv : V := ⟨T u, ⟨⟨u, huD⟩, rfl⟩⟩
rw [he, he]
change ‖f - T (p : Polynomial ℝ)‖ ≤ ‖f - (uv : C(I, ℝ))‖
change T (p : Polynomial ℝ) = (q : C(I, ℝ)) at hpq
rw [hpq]
exact hglobal uv
refine ⟨p, hpdeg, le_antisymm ?_ ?_⟩
· apply le_csInf (by exact ⟨_, 0, by simp, rfl⟩)
rintro v ⟨u, hu, rfl⟩
exact hmin u hu
· apply csInf_le
· refine ⟨0, ?_⟩
rintro v ⟨u, _, rfl⟩
exact (weightedPolynomial_error_bound M ε hM u).1
· exact ⟨p, hpdeg, rfl⟩
theorem continuousResidual_strictImprovement Lemma continuousResidual_strictImprovement in the paper ↗
A continuous perturbation with the residual's strict sign at every active extremum decreases its norm. This is the compact active-set perturbation argument in roadmap (3), allowing the weighted direction q/(1+z). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
g :
ℝ → ℝ
r s :
ℝ
hrs :
r ≤ s
hg :
ContinuousOn g (Icc r s)
q :
ℝ → ℝ
hq :
ContinuousOn q (Icc r s)
hE :
0 < intervalSupNorm g r s
hsign :
∀ x ∈ Icc r s
if
|g x| = intervalSupNorm g r s
then
0 < g x * q x
conclusion
∃ t : ℝ,
conclusion 1
0 < t
conclusion 2
intervalSupNorm (fun x ↦ g x - t * q x) r s < intervalSupNorm g r s
Proof (Lean source)
theorem continuousResidual_strictImprovement
{g : ℝ → ℝ} {r s : ℝ} (hrs : r ≤ s)
(hg : ContinuousOn g (Icc r s)) (q : ℝ → ℝ)
(hq : ContinuousOn q (Icc r s))
(hE : 0 < intervalSupNorm g r s)
(hsign : ∀ x ∈ Icc r s,
|g x| = intervalSupNorm g r s → 0 < g x * q x) :
∃ t : ℝ, 0 < t ∧
intervalSupNorm (fun x ↦ g x - t * q x) r s <
intervalSupNorm g r s := by
let E := intervalSupNorm g r s
let p : ℝ → ℝ := fun x ↦ g x * q x
have hIcc_ne : (Icc r s).Nonempty := ⟨r, le_rfl, hrs⟩
have hp : ContinuousOn p (Icc r s) := hg.mul hq
have habs_le : ∀ x ∈ Icc r s, |g x| ≤ E := by
simpa [E] using
(intervalSupNorm_le_iff hg hrs).mp
(le_refl (intervalSupNorm g r s))
obtain ⟨xE, hxE, hEmax, -⟩ :=
isCompact_Icc.exists_sSup_image_eq_and_ge hIcc_ne hg.abs
have hxE_eq : |g xE| = E := by
simpa [E, intervalSupNorm] using hEmax.symm
have hp_xE : 0 < p xE := by
exact hsign xE hxE (by simpa [E] using hxE_eq)
let B : Set ℝ := {x | x ∈ Icc r s ∧ p x ≤ 0}
have hBcompact : IsCompact B := by
apply IsCompact.of_isClosed_subset isCompact_Icc
(isClosed_Icc.isClosed_le hp continuousOn_const)
exact fun _ hx ↦ hx.1
have hc : ∃ c : ℝ, c < E ∧
∀ x ∈ Icc r s, c ≤ |g x| → 0 < p x := by
by_cases hBne : B.Nonempty
· obtain ⟨xb, hxb, hxb_max⟩ :=
hBcompact.exists_isMaxOn hBne (hg.abs.mono (fun _ hx ↦ hx.1))
have hxb_lt : |g xb| < E := by
apply lt_of_le_of_ne (habs_le xb hxb.1)
intro heq
have : 0 < p xb := hsign xb hxb.1 (by simpa [E] using heq)
linarith [hxb.2]
refine ⟨(|g xb| + E) / 2, by linarith, ?_⟩
intro x hx hcx
by_contra hnot
have hxB : x ∈ B := ⟨hx, le_of_not_gt hnot⟩
have hmax_le : |g x| ≤ |g xb| := hxb_max hxB
linarith
· refine ⟨E / 2, by linarith [hE], ?_⟩
intro x hx _
by_contra hnot
exact hBne ⟨x, hx, le_of_not_gt hnot⟩
obtain ⟨c, hcE, hc⟩ := hc
let A : Set ℝ := {x | x ∈ Icc r s ∧ c ≤ |g x|}
have hAcompact : IsCompact A := by
apply IsCompact.of_isClosed_subset isCompact_Icc
(isClosed_Icc.isClosed_le continuousOn_const hg.abs)
exact fun _ hx ↦ hx.1
have hxEA : xE ∈ A := ⟨hxE, by linarith [hxE_eq]⟩
obtain ⟨xm, hxm, hxm_min⟩ :=
hAcompact.exists_isMinOn ⟨xE, hxEA⟩ (hp.mono (fun _ hx ↦ hx.1))
let m := p xm
have hm : 0 < m := by
exact hc xm hxm.1 hxm.2
obtain ⟨xM, hxM, hxM_max⟩ :=
isCompact_Icc.exists_isMaxOn hIcc_ne hq.abs
let M := |q xM|
have hq_le : ∀ x ∈ Icc r s, |q x| ≤ M := by
intro x hx
exact hxM_max hx
have hM : 0 < M := by
have hqE_ne : q xE ≠ 0 := by
intro hzero
simp [p, hzero] at hp_xE
have hqE_pos : 0 < |q xE| := abs_pos.mpr hqE_ne
exact lt_of_lt_of_le hqE_pos (hq_le xE hxE)
let t := min (m / M ^ 2) ((E - c) / (2 * M))
have ht₁ : 0 < m / M ^ 2 := div_pos hm (sq_pos_of_pos hM)
have ht₂ : 0 < (E - c) / (2 * M) :=
div_pos (sub_pos.mpr hcE) (mul_pos (by norm_num) hM)
have ht : 0 < t := lt_min ht₁ ht₂
refine ⟨t, ht, ?_⟩
unfold intervalSupNorm
apply (isCompact_Icc.sSup_lt_iff_of_continuous hIcc_ne
((hg.sub (hq.const_mul t)).abs) E).2
intro x hx
have hgx := habs_le x hx
have hqx := hq_le x hx
by_cases hhigh : c ≤ |g x|
· have hxA : x ∈ A := ⟨hx, hhigh⟩
have hmp : m ≤ p x := hxm_min hxA
have htM : t ≤ m / M ^ 2 := min_le_left _ _
have hq_sq : q x ^ 2 ≤ M ^ 2 := by
simpa only [sq_abs] using
(sq_le_sq₀ (abs_nonneg (q x)) (abs_nonneg M)).2
(by simpa [abs_of_pos hM] using hqx)
have hmove : t * q x ^ 2 < 2 * p x := by
have hM_sq : 0 < M ^ 2 := sq_pos_of_pos hM
have ht_bound : t * M ^ 2 ≤ m := by
apply (le_div_iff₀ hM_sq).mp
simpa [mul_comm] using htM
nlinarith [mul_le_mul_of_nonneg_left hq_sq (le_of_lt ht)]
have hsquares : (g x - t * q x) ^ 2 < E ^ 2 := by
calc
(g x - t * q x) ^ 2 = g x ^ 2 - 2 * t * p x + t ^ 2 * q x ^ 2 := by
simp only [p]
ring
_ < g x ^ 2 := by nlinarith
_ ≤ E ^ 2 := by
simpa only [sq_abs] using
(sq_le_sq₀ (abs_nonneg (g x)) (le_of_lt hE)).2 hgx
rw [← sq_abs] at hsquares
exact (sq_lt_sq₀ (abs_nonneg _) (le_of_lt hE)).mp hsquares
· have htM : t ≤ (E - c) / (2 * M) := min_le_right _ _
have ht_bound : t * M ≤ (E - c) / 2 := by
have htwoM : 0 < 2 * M := mul_pos (by norm_num) hM
have := (le_div_iff₀ htwoM).mp htM
nlinarith
calc
|g x - t * q x| ≤ |g x| + |t * q x| := abs_sub _ _
_ = |g x| + t * |q x| := by rw [abs_mul, abs_of_pos ht]
_ ≤ |g x| + t * M := by gcongr
_ < E := by
have : |g x| < c := lt_of_not_ge hhigh
nlinarith [sub_pos.mpr hcE]
theorem weightedApproxError_alternating_witness Lemma weightedApproxError_alternating_witness in the paper ↗
A best weighted residual has K+2 alternating active points. A missing alternation would give a sign polynomial whose weighted perturbation strictly improves the attained minimum, contradicting optimality as in roadmap (3). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
K :
ℕ
hK :
2 ≤ K
M ε :
ℝ
hM :
2 ≤ M
hMhi :
M ≤ (K : ℝ) ^ 2
hε :
0 ≤ ε
hεhi :
ε ≤ 1 / 2
conclusion
∃ (p : Polynomial ℝ) (nodes : Fin (K + 2) → ℝ) (orientation : ℝ),
p.natDegree ≤ K
StrictMono nodes
conclusion 2
orientation = 1 ∨ orientation = -1
conclusion 3
i :
(phiEpsFormula ε (nodes i) - p.eval (nodes i)) / (1 + nodes i)
= orientation * (-1 : ℝ) ^ (i : ℕ) * weightedApproxError K M ε
Proof (Lean source)
theorem weightedApproxError_alternating_witness {K : ℕ} (hK : 2 ≤ K)
{M ε : ℝ} (hM : 2 ≤ M) (hMhi : M ≤ (K : ℝ) ^ 2)
(hε : 0 ≤ ε) (hεhi : ε ≤ 1 / 2) :
∃ (p : Polynomial ℝ) (nodes : Fin (K + 2) → ℝ) (orientation : ℝ),
p.natDegree ≤ K ∧ StrictMono nodes ∧
(∀ i, nodes i ∈ Icc 0 M) ∧ (orientation = 1 ∨ orientation = -1) ∧
∀ i, (phiEpsFormula ε (nodes i) - p.eval (nodes i)) / (1 + nodes i) =
orientation * (-1 : ℝ) ^ (i : ℕ) * weightedApproxError K M ε := by
obtain ⟨p, hp, hbest⟩ := weightedApproxError_attained K (M := M) (by linarith) ε
let g := fun z : ℝ => (phiEpsFormula ε z - p.eval z) / (1 + z)
have hg : ContinuousOn g (Icc 0 M) := by
dsimp [g]
unfold phiEpsFormula
apply ContinuousOn.div
· apply ContinuousOn.sub
· apply ContinuousOn.div
· fun_prop
· fun_prop
· intro z hz; linarith [hz.1]
· fun_prop
· fun_prop
· intro z hz; linarith [hz.1]
have hnorm : intervalSupNorm g 0 M = weightedApproxError K M ε := by
unfold intervalSupNorm
rw [← hbest]
congr 1
apply Set.image_congr
intro z hz
change |g z| = |phiEpsFormula ε z - p.eval z| / (1 + z)
simp only [g, abs_div, abs_of_pos (show 0 < 1 + z from by linarith [hz.1])]
obtain ⟨c, hc, hlower⟩ := packetWeightedApproxError_lower
have hpos : 0 < intervalSupNorm g 0 M := by
rw [hnorm]
exact lt_of_lt_of_le (by
have hKR : (0 : ℝ) < K := by exact_mod_cast (by omega : 0 < K)
exact div_pos (mul_pos hc (Real.sqrt_pos.2 (by linarith))) hKR)
(hlower K M ε hK hM hMhi hε hεhi)
have halt : ∃ nodes orientation, IsAlternatingExtrema g 0 M K nodes orientation := by
by_contra hno
obtain ⟨q, hq, hsign⟩ := exists_signPolynomial_of_no_alternatingExtrema
(by linarith : (0 : ℝ) < M) hg K hpos hno
let v := fun z : ℝ => q.eval z / (1 + z)
have hv : ContinuousOn v (Icc 0 M) := by
apply ContinuousOn.div
· fun_prop
· fun_prop
· intro z hz; linarith [hz.1]
have hsignv : ∀ z ∈ Icc 0 M, |g z| = intervalSupNorm g 0 M →
0 < g z * v z := by
intro z hz he
dsimp [v]
rw [← mul_div_assoc]
exact div_pos (hsign z hz he) (by linarith [hz.1])
obtain ⟨t, _, himprove⟩ := continuousResidual_strictImprovement
(by linarith : (0 : ℝ) ≤ M) hg v hv hpos hsignv
let r := p + C t * q
have hr : r.natDegree ≤ K :=
(Polynomial.natDegree_add_le _ _).trans (max_le hp
(Polynomial.natDegree_mul_le.trans (by simpa using hq)))
have herror : sSup ((fun z : ℝ => |phiEpsFormula ε z - r.eval z| / (1 + z)) ''
Icc 0 M) = intervalSupNorm (fun z => g z - t * v z) 0 M := by
unfold intervalSupNorm
congr 1
apply Set.image_congr
intro z hz
have hzpos : 0 < 1 + z := by linarith [hz.1]
have he : g z - t * v z = (phiEpsFormula ε z - r.eval z) / (1 + z) := by
simp only [g, v, r, Polynomial.eval_add, Polynomial.eval_mul, Polynomial.eval_C]
ring
change _ = |g z - t * v z|
rw [he, abs_div, abs_of_pos hzpos]
have hb : BddBelow {e : ℝ | ∃ q : Polynomial ℝ, q.natDegree ≤ K ∧
e = sSup ((fun z : ℝ => |phiEpsFormula ε z - q.eval z| / (1 + z)) '' Icc 0 M)} := by
refine ⟨0, ?_⟩
rintro e ⟨q, _, rfl⟩
exact (weightedPolynomial_error_bound M ε (by linarith) q).1
have hle := csInf_le hb ⟨r, hr, rfl⟩
change weightedApproxError K M ε ≤ _ at hle
rw [herror, ← hnorm] at hle
exact (not_lt_of_ge hle) himprove
obtain ⟨nodes, o, hmono, hmem, ho, he⟩ := halt
exact ⟨p, nodes, o, hp, hmono, hmem, ho, fun i => by
change g (nodes i) = _
rw [he i, hnorm]⟩
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.WeightedDualCompletion 2 declarations The finite dual's positive and negative atomic parts have equal mass and first moment.
Common-atom completion of the optimal finite weighted dual
The finite dual's positive and negative atomic parts have equal mass and first moment. Their weighted normalization gives t+u=1/2. Completing them by the same atom preserves the exact optimal gap, proving E ≤ Delta.
A finite weighted-normalized annihilator completes to a feasible pair, preserving the exact functional pairing. This implements roadmap (6)--(7) for the optimal witness as well as for any finite annihilator. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
ι :
Type
Fintype ι
K :
ℕ
hK :
1 ≤ K
M :
ℝ
hM :
2 ≤ M
nodes weights :
ι → ℝ
hmem :
∀ i, nodes i ∈ Icc 0 M
hmom :
∀ j : ℕ
if
j ≤ K
then
∑ i, weights i * nodes i ^ j = 0
hnorm :
(∑ i, |weights i| * (1 + nodes i)) = 1
ε :
ℝ
conclusion
∃ p : ConstrainedPriorPair K M,
(∫ z, phiEpsFormula ε z ∂p.ν₁) - (∫ z, phiEpsFormula ε z ∂p.ν₀)
= ∑ i, weights i * phiEpsFormula ε (nodes i)
Proof (Lean source)
theorem finiteWeightedDual_completion {ι : Type} [Fintype ι]
{K : ℕ} (hK : 1 ≤ K) {M : ℝ} (hM : 2 ≤ M)
(nodes weights : ι → ℝ) (hmem : ∀ i, nodes i ∈ Icc 0 M)
(hmom : ∀ j : ℕ, j ≤ K → ∑ i, weights i * nodes i ^ j = 0)
(hnorm : (∑ i, |weights i| * (1 + nodes i)) = 1) (ε : ℝ) :
∃ p : ConstrainedPriorPair K M,
(∫ z, phiEpsFormula ε z ∂p.ν₁) - (∫ z, phiEpsFormula ε z ∂p.ν₀) =
∑ i, weights i * phiEpsFormula ε (nodes i) := by
classical
let cp := fun i => max (weights i) 0
let cm := fun i => max (-weights i) 0
have hcp (i : ι) : 0 ≤ cp i := le_max_right _ _
have hcm (i : ι) : 0 ≤ cm i := le_max_right _ _
have hdiff (i : ι) : cp i - cm i = weights i := by
dsimp [cp, cm]
rcases le_total 0 (weights i) with hi | hi
· rw [max_eq_left hi, max_eq_right (by linarith)]
ring
· rw [max_eq_right hi, max_eq_left (by linarith)]
ring
have hadd (i : ι) : cp i + cm i = |weights i| := by
dsimp [cp, cm]
rcases le_total 0 (weights i) with hi | hi
· rw [max_eq_left hi, max_eq_right (by linarith), abs_of_nonneg hi]; ring
· rw [max_eq_right hi, max_eq_left (by linarith), abs_of_nonpos hi]; ring
let σp := Measure.sum (fun i => ofReal (cp i) • Measure.dirac (nodes i))
let σm := Measure.sum (fun i => ofReal (cm i) • Measure.dirac (nodes i))
letI (i : ι) : IsFiniteMeasure (ofReal (cp i) • Measure.dirac (nodes i)) :=
Measure.smul_finite _ ENNReal.ofReal_ne_top
letI (i : ι) : IsFiniteMeasure (ofReal (cm i) • Measure.dirac (nodes i)) :=
Measure.smul_finite _ ENNReal.ofReal_ne_top
letI : IsFiniteMeasure σp := by dsimp [σp]; infer_instance
letI : IsFiniteMeasure σm := by dsimp [σm]; infer_instance
have hip (f : ℝ → ℝ) : (∫ z, f z ∂σp) = ∑ i, cp i * f (nodes i) := by
dsimp [σp]
rw [integral_sum_dirac (fun _ => ENNReal.ofReal_ne_top), tsum_fintype]
simp only [ENNReal.toReal_ofReal (hcp _), smul_eq_mul]
have him (f : ℝ → ℝ) : (∫ z, f z ∂σm) = ∑ i, cm i * f (nodes i) := by
dsimp [σm]
rw [integral_sum_dirac (fun _ => ENNReal.ofReal_ne_top), tsum_fintype]
simp only [ENNReal.toReal_ofReal (hcm _), smul_eq_mul]
have hsuppp : σp (Icc 0 M)ᶜ = 0 := by
simp [σp, Measure.sum_apply, Measure.smul_apply, hmem]
have hsuppm : σm (Icc 0 M)ᶜ = 0 := by
simp [σm, Measure.sum_apply, Measure.smul_apply, hmem]
have hmassp : (σp univ).toReal = ∑ i, cp i := by
have h := hip (fun _ => 1)
simpa only [integral_const, smul_eq_mul, mul_one, Measure.real] using h
have hmassm : (σm univ).toReal = ∑ i, cm i := by
have h := him (fun _ => 1)
simpa only [integral_const, smul_eq_mul, mul_one, Measure.real] using h
have hmoment : ∀ j : ℕ, j ≤ K → (∫ z, z ^ j ∂σp) = (∫ z, z ^ j ∂σm) := by
intro j hj
rw [hip, him]
apply sub_eq_zero.mp
rw [← Finset.sum_sub_distrib]
simp_rw [← sub_mul, hdiff]
exact hmom j hj
have hmass : σp univ = σm univ := by
have h := hmoment 0 (by omega)
simp only [pow_zero, integral_const, smul_eq_mul, mul_one, Measure.real] at h
exact (ENNReal.toReal_eq_toReal_iff' (measure_ne_top _ _) (measure_ne_top _ _)).mp h
have hhalf : (σp univ).toReal + (∫ z, z ∂σp) = 1 / 2 := by
have hzero : (∑ i, cp i) = ∑ i, cm i := by rw [← hmassp, ← hmassm, hmass]
have hone : (∑ i, cp i * nodes i) = ∑ i, cm i * nodes i := by
simpa only [pow_one, hip, him] using hmoment 1 hK
have htotal : (∑ i, cp i) + (∑ i, cm i) +
((∑ i, cp i * nodes i) + (∑ i, cm i * nodes i)) = 1 := by
rw [← hnorm]
simp_rw [← hadd]
rw [← Finset.sum_add_distrib, ← Finset.sum_add_distrib,
← Finset.sum_add_distrib]
apply Finset.sum_congr rfl
intro i _
ring
rw [hmassp, hip]
linarith
have hu : 0 ≤ ∫ z, z ∂σp := by
rw [hip]
exact Finset.sum_nonneg (fun i _ => mul_nonneg (hcp i) (hmem i).1)
have ht : (σp univ).toReal ≤ 1 / 2 := by linarith
have hidp := integrable_of_supported_Icc σp M hsuppp (fun z => z) (by fun_prop)
have hidm := integrable_of_supported_Icc σm M hsuppm (fun z => z) (by fun_prop)
have hdom : CommonAtomDomain σp σm := by
refine ⟨inferInstance, inferInstance, hidp, hidm, ?_⟩
apply (ENNReal.toReal_lt_toReal (measure_ne_top _ _) ENNReal.one_ne_top).mp
simp only [ENNReal.toReal_one]
linarith
obtain ⟨p, hp⟩ := commonAtomCompletion_constrained K M hK hM σp σm hdom
hsuppp hsuppm hmass hmoment hhalf
refine ⟨p, ?_⟩
have hg := commonAtomCompletion_phiEps_gap σp σm hdom M ε hsuppp hsuppm (by linarith)
rw [hp] at hg
rw [hg, hip, him, ← Finset.sum_sub_distrib]
simp_rw [← sub_mul, hdiff]
theorem weightedApproxError_le_constrainedPriorSeparation Lemma weightedApproxError_le_constrainedPriorSeparation in the paper ↗
Completing the optimal finite dual yields a mean-one probability-prior pair whose absolute gap is exactly E, proving the reverse comparison (8). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
K :
ℕ
hK :
2 ≤ K
M ε :
ℝ
hM :
2 ≤ M
hMhi :
M ≤ (K : ℝ) ^ 2
hε :
0 ≤ ε
hεhi :
ε ≤ 1 / 2
conclusion
weightedApproxError K M ε ≤ constrainedPriorSeparation K M ε hK hM hMhi hε hεhi
Proof (Lean source)
theorem weightedApproxError_le_constrainedPriorSeparation {K : ℕ} (hK : 2 ≤ K)
{M ε : ℝ} (hM : 2 ≤ M) (hMhi : M ≤ (K : ℝ) ^ 2)
(hε : 0 ≤ ε) (hεhi : ε ≤ 1 / 2) :
weightedApproxError K M ε ≤ constrainedPriorSeparation K M ε hK hM hMhi hε hεhi := by
obtain ⟨nodes, weights, hmem, hmom, hnorm, htarget⟩ :=
weightedApproxError_finite_dual hK hM hMhi hε hεhi
obtain ⟨p, hp⟩ := finiteWeightedDual_completion (by omega : 1 ≤ K) hM
nodes weights hmem hmom hnorm ε
have hb : BddAbove (Set.range fun p : ConstrainedPriorPair K M =>
|(∫ z, phiEpsFormula ε z ∂p.ν₁) - (∫ z, phiEpsFormula ε z ∂p.ν₀)|) := by
refine ⟨4 * sSup ((fun z : ℝ => |phiEpsFormula ε z - (0 : Polynomial ℝ).eval z| /
(1 + z)) '' Icc 0 M), ?_⟩
rintro v ⟨q, rfl⟩
exact constrainedPriorPair_gap_le_polynomial_error q ε (by linarith) 0 (by simp)
rw [← htarget, ← hp]
exact le_csSup hb (Set.mem_range_self p)
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.WeightedDuality 13 declarations Roadmap equation (2): moment matching and the mean-one normalization bound the constrained prior separation by four times the best weighted error.
Weighted dual comparison and common-atom normalization
Roadmap equation (2): moment matching and the mean-one normalization bound the constrained prior separation by four times the best weighted error. Equations (6)--(7): adding the same atom to the two Jordan parts produces probability priors of mean one without changing their moment or functional differences. The weighted dual witness for the reverse comparison remains to be built.
Weighted normalization places the common atom between one half and two. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
t u :
ℝ
ht :
0 ≤ t
hu :
0 ≤ u
hnorm :
t + u = 1 / 2
conclusion
conclusion 1
t ≤ 1 / 2
conclusion 2
(1 / 2 : ℝ) ≤ (1 - u) / (1 - t)
conclusion 3
(1 - u) / (1 - t) ≤ 2
Proof (Lean source)
lemma commonAtom_location (t u : ℝ) (ht : 0 ≤ t) (hu : 0 ≤ u)
(hnorm : t + u = 1 / 2) :
t ≤ 1 / 2 ∧ (1 / 2 : ℝ) ≤ (1 - u) / (1 - t) ∧
(1 - u) / (1 - t) ≤ 2 := by
have htupper : t ≤ 1 / 2 := by linarith
have hden : 0 < 1 - t := by linarith
refine ⟨htupper, (le_div_iff₀ hden).2 ?_, (div_le_iff₀ hden).2 ?_⟩ <;> linarith
A continuous function on the supporting interval is integrable; no extra moment-integrability assumption is needed for the compactly supported witness. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
Integrable f μ
Proof (Lean source)
lemma integrable_of_supported_Icc (μ : Measure ℝ) [IsFiniteMeasure μ]
(M : ℝ) (hsupp : μ (Icc 0 M)ᶜ = 0) (f : ℝ → ℝ)
(hf : ContinuousOn f (Icc 0 M)) : Integrable f μ := by
have hrestrict : μ.restrict (Icc 0 M) = μ :=
Measure.restrict_eq_self_of_ae_mem (by exact mem_ae_iff.mpr hsupp)
have hi := hf.integrableOn_compact (μ := μ) isCompact_Icc
simpa only [IntegrableOn, hrestrict] using hi
Filling the missing mass by a Dirac atom yields a probability measure. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma commonAtom_probability (μ : Measure ℝ) [IsFiniteMeasure μ]
(ht : (μ univ).toReal ≤ 1) (z₀ : ℝ) :
IsProbabilityMeasure
(μ + ofReal (1 - (μ univ).toReal) • Measure.dirac z₀) := by
constructor
simp only [Measure.add_apply, Measure.smul_apply, Measure.dirac_apply_of_mem
(Set.mem_univ z₀), smul_eq_mul, mul_one]
conv_lhs => lhs; rw [← ENNReal.ofReal_toReal (measure_ne_top μ univ)]
rw [← ENNReal.ofReal_add toReal_nonneg (sub_nonneg.mpr ht)]
simp
Adding the common atom preserves support when its location is in the interval. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
The integral under a completed measure is the old integral plus the atom's contribution. This applies to moments and to the weighted objective. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
(∫ z, f z ∂(μ + ofReal (1 - t) • Measure.dirac z₀))
= (∫ z, f z ∂μ) + (1 - t) * f z₀
Proof (Lean source)
lemma commonAtom_integral (μ : Measure ℝ) (t z₀ : ℝ) (ht : t ≤ 1)
(f : ℝ → ℝ) (hf : Integrable f μ) :
(∫ z, f z ∂(μ + ofReal (1 - t) • Measure.dirac z₀)) =
(∫ z, f z ∂μ) + (1 - t) * f z₀ := by
have hi : Integrable f (ofReal (1 - t) • Measure.dirac z₀) :=
(integrable_dirac (by finiteness)).smul_measure ENNReal.ofReal_ne_top
rw [integral_add_measure hf hi, integral_smul_measure, integral_dirac,
ENNReal.toReal_ofReal (sub_nonneg.mpr ht)]
rfl
theorem commonAtomCompletion_constrained
Two supported Jordan parts of a weighted-normalized moment annihilator complete to feasible priors, with the exact common-atom construction (6)--(7). All moment integrability follows from compact support. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
K :
ℕ
M :
ℝ
hK :
1 ≤ K
hM :
2 ≤ M
σp σm :
Measure ℝ
hdom :
CommonAtomDomain σp σm
hsuppp :
σp (Icc 0 M)ᶜ = 0
hsuppm :
σm (Icc 0 M)ᶜ = 0
hmom :
∀ j : ℕ
if
j ≤ K
then
(∫ z, z ^ j ∂σp) = (∫ z, z ^ j ∂σm)
conclusion
∃ p : ConstrainedPriorPair K M, commonAtomCompletion σp σm hdom = (p.ν₀, p.ν₁)
Proof (Lean source)
lemma commonAtomCompletion_constrained (K : ℕ) (M : ℝ) (hK : 1 ≤ K)
(hM : 2 ≤ M) (σp σm : Measure ℝ) (hdom : CommonAtomDomain σp σm)
(hsuppp : σp (Icc 0 M)ᶜ = 0) (hsuppm : σm (Icc 0 M)ᶜ = 0)
(hmass : σp univ = σm univ)
(hmom : ∀ j : ℕ, j ≤ K → (∫ z, z ^ j ∂σp) = (∫ z, z ^ j ∂σm))
(hnorm : (σp univ).toReal + (∫ z, z ∂σp) = 1 / 2) :
∃ p : ConstrainedPriorPair K M,
commonAtomCompletion σp σm hdom = (p.ν₀, p.ν₁) := by
let : IsFiniteMeasure σp := hdom.1
let : IsFiniteMeasure σm := hdom.2.1
let t := (σp univ).toReal
let u := ∫ z, z ∂σp
let z₀ := (1 - u) / (1 - t)
have hu : 0 ≤ u := integral_nonneg_of_ae
(by filter_upwards [show ∀ᵐ z ∂σp, z ∈ Icc 0 M from
by exact mem_ae_iff.mpr hsuppp] with z hz; exact hz.1)
have hloc := commonAtom_location t u ENNReal.toReal_nonneg hu hnorm
have ht : t ≤ 1 := by linarith [hloc.1]
have hz₀ : z₀ ∈ Icc 0 M := ⟨by linarith [hloc.2.1],
le_trans hloc.2.2 hM⟩
have hm : (σm univ).toReal = t := by dsimp [t]; rw [hmass]
have hmeanm : (∫ z, z ∂σm) = u := by
simpa only [pow_one] using (hmom 1 hK).symm
have hmean : u + (1 - t) * z₀ = 1 := by
have hden : 1 - t ≠ 0 := by linarith [hloc.1]
dsimp [z₀]
field_simp
ring
refine ⟨{
ν₀ := σm + ofReal (1 - t) • Measure.dirac z₀
ν₁ := σp + ofReal (1 - t) • Measure.dirac z₀
prob₀ := ?_
prob₁ := commonAtom_probability σp ht z₀
supp₀ := commonAtom_support σm M t z₀ hsuppm hz₀
supp₁ := commonAtom_support σp M t z₀ hsuppp hz₀
mean₀ := ?_
mean₁ := ?_
moments := ?_ }, rfl⟩
· simpa only [hm] using commonAtom_probability σm (by rw [hm]; exact ht) z₀
· rw [commonAtom_integral σm t z₀ ht _ hdom.2.2.2.1, hmeanm]
exact hmean
· rw [commonAtom_integral σp t z₀ ht _ hdom.2.2.1]
exact hmean
· intro j hj
have hp := integrable_of_supported_Icc σp M hsuppp (fun z => z ^ j)
(by fun_prop)
have hm' := integrable_of_supported_Icc σm M hsuppm (fun z => z ^ j)
(by fun_prop)
rw [commonAtom_integral σm t z₀ ht _ hm', commonAtom_integral σp t z₀ ht _ hp,
hmom j hj]
theorem commonAtomCompletion_gap
The identical added atom cancels from every integrable functional gap. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
(∫ z, f z ∂(commonAtomCompletion σp σm hdom).2)
- (∫ z, f z ∂(commonAtomCompletion σp σm hdom).1)
= (∫ z, f z ∂σp) - (∫ z, f z ∂σm)
Proof (Lean source)
lemma commonAtomCompletion_gap (σp σm : Measure ℝ) (hdom : CommonAtomDomain σp σm)
(ht : (σp univ).toReal ≤ 1) (f : ℝ → ℝ)
(hp : Integrable f σp) (hm : Integrable f σm) :
(∫ z, f z ∂(commonAtomCompletion σp σm hdom).2) -
(∫ z, f z ∂(commonAtomCompletion σp σm hdom).1) =
(∫ z, f z ∂σp) - (∫ z, f z ∂σm) := by
dsimp [commonAtomCompletion]
rw [commonAtom_integral σp _ _ ht f hp, commonAtom_integral σm _ _ ht f hm]
ring
Compact support makes the weighted objective integrable, so the common-atom completion preserves its gap without any additional regularity hypothesis. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
σp σm :
Measure ℝ
hdom :
CommonAtomDomain σp σm
M ε :
ℝ
hsuppp :
σp (Icc 0 M)ᶜ = 0
hsuppm :
σm (Icc 0 M)ᶜ = 0
conclusion
(∫ z, phiEpsFormula ε z ∂(commonAtomCompletion σp σm hdom).2)
- (∫ z, phiEpsFormula ε z ∂(commonAtomCompletion σp σm hdom).1)
= (∫ z, phiEpsFormula ε z ∂σp) - (∫ z, phiEpsFormula ε z ∂σm)
Proof (Lean source)
lemma commonAtomCompletion_phiEps_gap (σp σm : Measure ℝ)
(hdom : CommonAtomDomain σp σm) (M ε : ℝ)
(hsuppp : σp (Icc 0 M)ᶜ = 0) (hsuppm : σm (Icc 0 M)ᶜ = 0)
(ht : (σp univ).toReal ≤ 1) :
(∫ z, phiEpsFormula ε z ∂(commonAtomCompletion σp σm hdom).2) -
(∫ z, phiEpsFormula ε z ∂(commonAtomCompletion σp σm hdom).1) =
(∫ z, phiEpsFormula ε z ∂σp) - (∫ z, phiEpsFormula ε z ∂σm) := by
let : IsFiniteMeasure σp := hdom.1
let : IsFiniteMeasure σm := hdom.2.1
have hc : ContinuousOn (phiEpsFormula ε) (Icc 0 M) := by
unfold phiEpsFormula
apply ContinuousOn.div
· fun_prop
· fun_prop
· intro z hz
linarith [hz.1]
exact commonAtomCompletion_gap σp σm hdom ht (phiEpsFormula ε)
(integrable_of_supported_Icc σp M hsuppp _ hc)
(integrable_of_supported_Icc σm M hsuppm _ hc)
theorem constrainedPriorPair_polynomial_integrals
Matching monomial moments matches every polynomial of the permitted degree. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
(∫ z, p.eval z ∂P.ν₀) = (∫ z, p.eval z ∂P.ν₁)
Proof (Lean source)
lemma constrainedPriorPair_polynomial_integrals {K : ℕ} {M : ℝ}
(P : ConstrainedPriorPair K M) (p : Polynomial ℝ) (hp : p.natDegree ≤ K) :
(∫ z, p.eval z ∂P.ν₀) = (∫ z, p.eval z ∂P.ν₁) := by
let := P.prob₀
let := P.prob₁
have hpoly : (fun z => p.eval z) =
(fun z => ∑ j ∈ range (K + 1), p.coeff j * z ^ j) := by
funext z
exact Polynomial.eval_eq_sum_range' (by omega) z
have hi (μ : Measure ℝ) [IsFiniteMeasure μ]
(hs : μ (Icc 0 M)ᶜ = 0) (j : ℕ) :
Integrable (fun z => p.coeff j * z ^ j) μ :=
integrable_of_supported_Icc μ M hs _ (by fun_prop)
rw [hpoly, integral_finsetSum _ (fun j _ => hi P.ν₀ P.supp₀ j),
integral_finsetSum _ (fun j _ => hi P.ν₁ P.supp₁ j)]
apply Finset.sum_congr rfl
intro j hj
rw [integral_const_mul, integral_const_mul, P.moments j (by
have := Finset.mem_range.mp hj
omega)]
Compactness bounds each polynomial's weighted residual, and its supremum controls the residual pointwise on the supporting interval. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
Proof (Lean source)
lemma weightedPolynomial_error_bound (M ε : ℝ) (hM : 0 ≤ M)
(p : Polynomial ℝ) :
let e := sSup ((fun z : ℝ => |phiEpsFormula ε z - p.eval z| / (1 + z)) '' Icc 0 M)
0 ≤ e ∧ ∀ z ∈ Icc 0 M, |phiEpsFormula ε z - p.eval z| ≤ e * (1 + z) := by
dsimp only
have hc : ContinuousOn (fun z : ℝ =>
|phiEpsFormula ε z - p.eval z| / (1 + z)) (Icc 0 M) := by
unfold phiEpsFormula
apply ContinuousOn.div
· apply ContinuousOn.abs
apply ContinuousOn.sub
· apply ContinuousOn.div
· fun_prop
· fun_prop
· intro z hz; linarith [hz.1]
· fun_prop
· fun_prop
· intro z hz; linarith [hz.1]
have hb := isCompact_Icc.bddAbove_image hc
have hz0 : (0 : ℝ) ∈ Icc 0 M := ⟨le_rfl, hM⟩
refine ⟨le_trans (by positivity)
(le_csSup hb (Set.mem_image_of_mem _ hz0)), ?_⟩
intro z hz
exact (div_le_iff₀ (by linarith [hz.1])).mp
(le_csSup hb (Set.mem_image_of_mem _ hz))
theorem constrainedPriorPair_gap_le_polynomial_error Lemma constrainedPriorPair_gap_le_polynomial_error in the paper ↗
Mean-one probability priors integrate the residual envelope to twice its weighted error; moment matching therefore bounds their gap by four times it (roadmap equation (2), before taking the infimum). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
|(∫ z, phiEpsFormula ε z ∂P.ν₁) - (∫ z, phiEpsFormula ε z ∂P.ν₀)|
Proof (Lean source)
lemma constrainedPriorPair_gap_le_polynomial_error {K : ℕ} {M : ℝ}
(P : ConstrainedPriorPair K M) (ε : ℝ) (hM : 0 ≤ M)
(p : Polynomial ℝ) (hp : p.natDegree ≤ K) :
|(∫ z, phiEpsFormula ε z ∂P.ν₁) - (∫ z, phiEpsFormula ε z ∂P.ν₀)| ≤
4 * sSup ((fun z : ℝ => |phiEpsFormula ε z - p.eval z| / (1 + z)) '' Icc 0 M) := by
let := P.prob₀
let := P.prob₁
let e := sSup ((fun z : ℝ => |phiEpsFormula ε z - p.eval z| / (1 + z)) '' Icc 0 M)
have hb := weightedPolynomial_error_bound M ε hM p
have hc : ContinuousOn (phiEpsFormula ε) (Icc 0 M) := by
unfold phiEpsFormula
apply ContinuousOn.div
· fun_prop
· fun_prop
· intro z hz; linarith [hz.1]
have hi (μ : Measure ℝ) [IsProbabilityMeasure μ]
(hs : μ (Icc 0 M)ᶜ = 0) :
Integrable (phiEpsFormula ε) μ ∧ Integrable (fun z => p.eval z) μ :=
⟨integrable_of_supported_Icc μ M hs _ hc,
integrable_of_supported_Icc μ M hs _ (by fun_prop)⟩
have hres (μ : Measure ℝ) [IsProbabilityMeasure μ]
(hs : μ (Icc 0 M)ᶜ = 0) (hmean : (∫ z, z ∂μ) = 1) :
|∫ z, phiEpsFormula ε z - p.eval z ∂μ| ≤ 2 * e := by
have hid := integrable_of_supported_Icc μ M hs (fun z => z) (by fun_prop)
have hw : Integrable (fun z : ℝ => e * (1 + z)) μ :=
(integrable_const 1 |>.add hid).const_mul e
calc
|∫ z, phiEpsFormula ε z - p.eval z ∂μ| ≤
∫ z, |phiEpsFormula ε z - p.eval z| ∂μ := abs_integral_le_integral_abs
_ ≤ ∫ z, e * (1 + z) ∂μ := integral_mono_ae
((hi μ hs).1.sub (hi μ hs).2).abs hw (by
filter_upwards [show ∀ᵐ z ∂μ, z ∈ Icc 0 M from
mem_ae_iff.mpr hs] with z hz
exact hb.2 z hz)
_ = 2 * e := by
rw [integral_const_mul, integral_add (integrable_const 1) hid, hmean]
simp
ring
have heq : (∫ z, phiEpsFormula ε z ∂P.ν₁) - (∫ z, phiEpsFormula ε z ∂P.ν₀) =
(∫ z, phiEpsFormula ε z - p.eval z ∂P.ν₁) -
(∫ z, phiEpsFormula ε z - p.eval z ∂P.ν₀) := by
rw [integral_sub (hi P.ν₁ P.supp₁).1 (hi P.ν₁ P.supp₁).2,
integral_sub (hi P.ν₀ P.supp₀).1 (hi P.ν₀ P.supp₀).2,
constrainedPriorPair_polynomial_integrals P p hp]
ring
rw [heq]
have ha := abs_sub_le (∫ z, phiEpsFormula ε z - p.eval z ∂P.ν₁)
0 (∫ z, phiEpsFormula ε z - p.eval z ∂P.ν₀)
simp only [sub_zero, zero_sub, abs_neg] at ha
have h0 := hres P.ν₀ P.supp₀ P.mean₀
have h1 := hres P.ν₁ P.supp₁ P.mean₁
change _ ≤ 4 * e
linarith
theorem constrainedPriorPair_nonempty
The identical Dirac priors at one make the feasible set nonempty. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma constrainedPriorPair_nonempty (K : ℕ) (M : ℝ) (hM : 1 ≤ M) :
Nonempty (ConstrainedPriorPair K M) := by
refine ⟨{
ν₀ := Measure.dirac 1
ν₁ := Measure.dirac 1
prob₀ := inferInstance
prob₁ := inferInstance
supp₀ := ?_
supp₁ := ?_
mean₀ := ?_
mean₁ := ?_
moments := fun _ _ => rfl }⟩ <;> simp [hM]
theorem constrainedPriorSeparation_le_four_weightedApproxError Lemma constrainedPriorSeparation_le_four_weightedApproxError in the paper ↗
Taking the polynomial infimum and the feasible-prior supremum proves the upper half of the weighted dual comparison (roadmap equations (2) and (8)). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
K :
ℕ
M ε :
ℝ
hM :
1 ≤ M
conclusion
constrainedPriorSeparationFormula K M ε ≤ 4 * weightedApproxError K M ε
Proof (Lean source)
lemma constrainedPriorSeparation_le_four_weightedApproxError
(K : ℕ) (M ε : ℝ) (hM : 1 ≤ M) :
constrainedPriorSeparationFormula K M ε ≤ 4 * weightedApproxError K M ε := by
have hnonempty : Nonempty (ConstrainedPriorPair K M) :=
constrainedPriorPair_nonempty K M hM
unfold constrainedPriorSeparationFormula
apply csSup_le (Set.range_nonempty _)
rintro _ ⟨P, rfl⟩
have hne : {e : ℝ | ∃ p : Polynomial ℝ, p.natDegree ≤ K ∧
e = sSup ((fun z : ℝ => |phiEpsFormula ε z - p.eval z| / (1 + z)) ''
Icc 0 M)}.Nonempty := by
refine ⟨_, 0, ?_, rfl⟩
simp
have hle : |(∫ z, phiEpsFormula ε z ∂P.ν₁) - (∫ z, phiEpsFormula ε z ∂P.ν₀)| / 4 ≤
weightedApproxError K M ε := by
apply le_csInf hne
rintro e ⟨p, hp, rfl⟩
have hbound := constrainedPriorPair_gap_le_polynomial_error P ε
(by linarith) p hp
linarith
linarith
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.WeightedFiniteDual 1 declarations Alternating active residuals give finite Lagrange weights annihilating every polynomial through degree K.
An optimal finite weighted moment annihilator
Alternating active residuals give finite Lagrange weights annihilating every polynomial through degree K. Normalizing their weighted variation produces the exact optimal dual gap from roadmap (4)--(5).
There is a finite weighted-normalized moment annihilator attaining the weighted approximation error exactly, rather than only its order lower bound. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
K :
ℕ
hK :
2 ≤ K
M ε :
ℝ
hM :
2 ≤ M
hMhi :
M ≤ (K : ℝ) ^ 2
hε :
0 ≤ ε
hεhi :
ε ≤ 1 / 2
conclusion
∃ (nodes weights : Fin (K + 2) → ℝ),
conclusion 2
j :
ℕ
j ≤ K
∑ i, weights i * nodes i ^ j = 0
conclusion 3
(∑ i, |weights i| * (1 + nodes i)) = 1
conclusion 4
|∑ i, weights i * phiEpsFormula ε (nodes i)| = weightedApproxError K M ε
Proof (Lean source)
theorem weightedApproxError_finite_dual {K : ℕ} (hK : 2 ≤ K)
{M ε : ℝ} (hM : 2 ≤ M) (hMhi : M ≤ (K : ℝ) ^ 2)
(hε : 0 ≤ ε) (hεhi : ε ≤ 1 / 2) :
∃ (nodes weights : Fin (K + 2) → ℝ),
(∀ i, nodes i ∈ Icc 0 M) ∧
(∀ j : ℕ, j ≤ K → ∑ i, weights i * nodes i ^ j = 0) ∧
(∑ i, |weights i| * (1 + nodes i)) = 1 ∧
|∑ i, weights i * phiEpsFormula ε (nodes i)| = weightedApproxError K M ε := by
classical
obtain ⟨p, nodes, o, hp, hmono, hmem, ho, he⟩ :=
weightedApproxError_alternating_witness hK hM hMhi hε hεhi
let w := normalizedLagrangeWeight nodes
let D := ∑ i, |w i| * (1 + nodes i)
have hw : ∑ i, |w i| = 1 := sum_abs_normalizedLagrangeWeight_eq_one hmono.injective
have hD : 0 < D := by
have hle : 1 ≤ D := by
rw [← hw]
apply Finset.sum_le_sum
intro i _
exact le_mul_of_one_le_right (abs_nonneg _) (by linarith [(hmem i).1])
linarith
have hpoly : ∑ i, w i * p.eval (nodes i) = 0 :=
sum_normalizedLagrangeWeight_mul_eval_eq_zero hmono.injective hp
have hpow (i : Fin (K + 2)) :
(-1 : ℝ) ^ (K + 1 - (i : ℕ)) * (-1 : ℝ) ^ (i : ℕ) = (-1 : ℝ) ^ (K + 1) := by
rw [← pow_add, Nat.sub_add_cancel (by omega)]
have htarget : ∑ i, w i * phiEpsFormula ε (nodes i) =
(o * (-1 : ℝ) ^ (K + 1) * weightedApproxError K M ε) * D := by
calc
_ = (∑ i, w i * (phiEpsFormula ε (nodes i) - p.eval (nodes i))) +
∑ i, w i * p.eval (nodes i) := by
rw [← Finset.sum_add_distrib]
apply Finset.sum_congr rfl
intro i _
ring
_ = ∑ i, w i * (phiEpsFormula ε (nodes i) - p.eval (nodes i)) := by rw [hpoly, add_zero]
_ = ∑ i, (o * (-1 : ℝ) ^ (K + 1) * weightedApproxError K M ε) *
(|w i| * (1 + nodes i)) := by
apply Finset.sum_congr rfl
intro i _
have hres := (div_eq_iff (show 1 + nodes i ≠ 0 from by linarith [(hmem i).1])).mp (he i)
rw [hres]
have halt : w i = (-1 : ℝ) ^ (K + 1 - (i : ℕ)) * |w i| :=
normalizedLagrangeWeight_alternates hmono i
conv_lhs => rw [halt]
calc
_ = o * ((-1 : ℝ) ^ (K + 1 - (i : ℕ)) * (-1 : ℝ) ^ (i : ℕ)) *
weightedApproxError K M ε * (|w i| * (1 + nodes i)) := by ring
_ = _ := by rw [hpow]
_ = _ := by rw [← Finset.mul_sum]
let v := fun i => w i / D
refine ⟨nodes, v, hmem, ?_, ?_, ?_⟩
· intro j hj
simp only [v, div_mul_eq_mul_div, ← Finset.sum_div]
rw [sum_normalizedLagrangeWeight_mul_pow_eq_zero hmono.injective hj, zero_div]
· simp only [v, abs_div, abs_of_pos hD, div_mul_eq_mul_div, ← Finset.sum_div]
exact div_self hD.ne'
· have hE : 0 ≤ weightedApproxError K M ε := by
obtain ⟨q, _, hq⟩ := weightedApproxError_attained K (M := M) (by linarith) ε
rw [← hq]
exact (weightedPolynomial_error_bound M ε (by linarith) q).1
simp only [v, div_mul_eq_mul_div, ← Finset.sum_div, htarget]
rw [mul_div_cancel_right₀ _ hD.ne', abs_mul, abs_mul, abs_pow,
abs_neg, abs_one, one_pow, mul_one, abs_of_nonneg hE]
rcases ho with rfl | rfl <;> simp
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.WeightedUpperOrder 3 declarations The angular regularity of the nonlinear remainder and the promoted Jackson first-moment estimate give the universal sqrt(M)/K upper order in roadmap (14).
Weighted upper approximation by Jackson smoothing
The angular regularity of the nonlinear remainder and the promoted Jackson first-moment estimate give the universal sqrt(M)/K upper order in roadmap (14).
theorem weightedJackson_angular_approximation Lemma weightedJackson_angular_approximation in the paper ↗
An even periodic continuous function has an even Jackson-smoothed trigonometric approximant, with error controlled by its angular Lipschitz constant. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
∃ p : Polynomial ℝ,
p.natDegree ≤ 2 * (N - 1)
Proof (Lean source)
lemma weightedJackson_angular_approximation {N : ℕ} (hN : 0 < N)
(f : ℝ → ℝ) (hf : Continuous f) (he : Even f)
(hp : Periodic f (2 * pi)) (L : ℝ) (hL : 0 ≤ L)
(hl : ∀ s t, |f s - f t| ≤ L * |s - t|) :
∃ p : Polynomial ℝ, p.natDegree ≤ 2 * (N - 1) ∧
∀ t, |f t - p.eval (cos t)| ≤ 32 * L / N := by
classical
let q := fun t : ℝ => ∫ u in Icc (-pi) pi, f u * jackson N (t - u)
obtain ⟨a, b, hab⟩ := jackson_isTrigPolyLE N hN
have hjp : Periodic (jackson N) (2 * pi) := by
intro t
rw [hab, hab]
apply Finset.sum_congr rfl
intro k hk
simpa only [mul_add, Int.cast_natCast] using congrArg₂
(fun c s => a k * c + b k * s)
(Real.cos_add_int_mul_two_pi ((k : ℝ) * t) (k : ℤ))
(Real.sin_add_int_mul_two_pi ((k : ℝ) * t) (k : ℤ))
have htrig : IsTrigPolyLE (2 * (N - 1)) q := by
refine ⟨fun k => ∫ u in Icc (-pi) pi,
f u * (a k * cos ((k : ℝ) * u) - b k * sin ((k : ℝ) * u)),
fun k => ∫ u in Icc (-pi) pi,
f u * (a k * sin ((k : ℝ) * u) + b k * cos ((k : ℝ) * u)), ?_⟩
intro t
dsimp [q]
simp_rw [hab, Finset.mul_sum]
rw [integral_finsetSum _ (fun k hk =>
(show Continuous (fun u : ℝ => f u *
(a k * cos ((k : ℝ) * (t - u)) + b k * sin ((k : ℝ) * (t - u)))) from by fun_prop).continuousOn.integrableOn_compact isCompact_Icc)]
apply Finset.sum_congr rfl
intro k hk
have heq (u : ℝ) : f u *
(a k * cos ((k : ℝ) * (t - u)) + b k * sin ((k : ℝ) * (t - u))) =
(f u * (a k * cos ((k : ℝ) * u) - b k * sin ((k : ℝ) * u))) * cos ((k : ℝ) * t) +
(f u * (a k * sin ((k : ℝ) * u) + b k * cos ((k : ℝ) * u))) * sin ((k : ℝ) * t) := by
rw [mul_sub, Real.cos_sub, Real.sin_sub]
ring
simp_rw [heq]
rw [integral_add, integral_mul_const, integral_mul_const]
all_goals
apply ContinuousOn.integrableOn_compact isCompact_Icc
fun_prop
have hqe : Even q := by
intro t
let F := fun u : ℝ => f u * jackson N (t - u)
have heq (u : ℝ) : f u * jackson N (-t - u) = F (-u) := by
dsimp [F]
rw [he u]
have ha : -t - u = -(t - -u) := by ring
rw [ha, jackson_even]
dsimp [q]
simp_rw [heq]
rw [integral_Icc_eq_integral_Ioc, integral_Icc_eq_integral_Ioc,
← intervalIntegral.integral_of_le (by linarith [Real.pi_pos] : -pi ≤ pi),
← intervalIntegral.integral_of_le (by linarith [Real.pi_pos] : -pi ≤ pi),
intervalIntegral.integral_comp_neg]
simp only [neg_neg]
rfl
obtain ⟨p, hdeg, heval⟩ := even_trigPoly_exists_polynomial htrig hqe
refine ⟨p, hdeg, ?_⟩
intro t
rw [heval]
have hshift : q t = ∫ u in Icc (-pi) pi, f (u + t) * jackson N u := by
have hperiod : Periodic (fun u => f (u + t) * jackson N u) (2 * pi) := by
intro u
dsimp only
rw [show u + 2 * pi + t = (u + t) + 2 * pi by ring, hp, hjp]
convert packet_periodic_integral_translate _ hperiod t using 1
dsimp [q]
congr 1
funext u
rw [sub_add_cancel]
rw [show t - u = -(u - t) by ring, jackson_even]
have hi : IntegrableOn (fun u => (f t - f (u + t)) * jackson N u)
(Icc (-pi) pi) := by
exact ((continuous_const.sub (hf.comp (by fun_prop))).mul (continuous_jackson N)).continuousOn.integrableOn_compact isCompact_Icc
have hdiff : f t - q t = ∫ u in Icc (-pi) pi,
(f t - f (u + t)) * jackson N u := by
rw [hshift]
simp_rw [sub_mul]
rw [integral_sub, integral_const_mul, jackson_integral_eq_one N hN, mul_one]
· exact ((continuous_jackson N).const_mul _).continuousOn.integrableOn_compact isCompact_Icc
· exact ((hf.comp (by fun_prop)).mul (continuous_jackson N)).continuousOn.integrableOn_compact isCompact_Icc
rw [hdiff]
calc
_ ≤ ∫ u in Icc (-pi) pi, |(f t - f (u + t)) * jackson N u| := abs_integral_le_integral_abs
_ ≤ ∫ u in Icc (-pi) pi, L * (|u| * jackson N u) := by
apply integral_mono hi.abs
(((continuous_id.abs.mul (continuous_jackson N)).const_mul L).continuousOn.integrableOn_compact isCompact_Icc)
intro u
change |(f t - f (u + t)) * jackson N u| ≤ L * (|u| * jackson N u)
rw [abs_mul, abs_of_nonneg (jackson_nonneg N hN u)]
have h := hl t (u + t)
rw [show t - (u + t) = -u by ring, abs_neg] at h
nlinarith [jackson_nonneg N hN u]
_ ≤ L * (32 / (N : ℝ)) := by
rw [integral_const_mul]
exact mul_le_mul_of_nonneg_left (jackson_first_moment N hN) hL
_ = 32 * L / N := by ring
theorem weightedApproximation_polynomial_upper Lemma weightedApproximation_polynomial_upper in the paper ↗
Jackson smoothing of the bounded nonlinear remainder, followed by the inverse cosine-coordinate affine substitution, gives an admissible degree-K polynomial with uniform error of order sqrt(M)/K (roadmap (14)). In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
K :
ℕ
hK :
2 ≤ K
M ε :
ℝ
hM :
0 < M
hε :
0 ≤ ε
hεhi :
ε ≤ 1 / 2
conclusion
∃ p : Polynomial ℝ,
p.natDegree ≤ K
Proof (Lean source)
lemma weightedApproximation_polynomial_upper {K : ℕ} (hK : 2 ≤ K)
{M ε : ℝ} (hM : 0 < M) (hε : 0 ≤ ε) (hεhi : ε ≤ 1 / 2) :
∃ p : Polynomial ℝ, p.natDegree ≤ K ∧
∀ z ∈ Icc 0 M, |phiEpsFormula ε z - p.eval z| ≤ 64 * sqrt M / K := by
let N := K / 2 + 1
let f := fun t => packetRemainder ε (packetIntensity M t)
have hf : Continuous f := by
dsimp [f]
unfold packetRemainder
apply Continuous.add
· fun_prop
· apply Continuous.div
· fun_prop
· fun_prop
· intro t
linarith [(packetIntensity_mem_Icc hM.le t).1]
have he : Even f := by
intro t
simp only [f, packetIntensity, Real.cos_neg]
have hp : Periodic f (2 * pi) := by
intro t
simp only [f, packetIntensity, Real.cos_add_two_pi]
obtain ⟨q, hq, herror⟩ := weightedJackson_angular_approximation
(N := N) (by dsimp [N]; omega) f hf he hp (sqrt M)
(Real.sqrt_nonneg M) (packetRemainder_angular_lipschitz hM.le hε hεhi)
let a : Polynomial ℝ := C (2 / M) * X - 1
let p : Polynomial ℝ := C ε * (X - 1) + q.comp a
have ha : a.natDegree ≤ 1 := by
dsimp [a]
exact (Polynomial.natDegree_sub_le _ _).trans (max_le
(Polynomial.natDegree_mul_le.trans (by simp)) (by simp))
have hdeg : p.natDegree ≤ K := by
apply (Polynomial.natDegree_add_le _ _).trans
apply max_le
· exact Polynomial.natDegree_mul_le.trans (by
simp only [Polynomial.natDegree_C, zero_add]
exact (Polynomial.natDegree_sub_le _ _).trans (by simp; omega))
· exact Polynomial.natDegree_comp_le.trans
((Nat.mul_le_mul_right _ hq).trans
((Nat.mul_le_mul_left _ ha).trans (by dsimp [N]; simp only [mul_one]; omega)))
refine ⟨p, hdeg, ?_⟩
intro z hz
let t := arccos (2 * z / M - 1)
have ht : cos t = 2 * z / M - 1 := by
apply Real.cos_arccos
· have : 0 ≤ 2 * z / M := div_nonneg (mul_nonneg (by norm_num) hz.1) hM.le
linarith
· have : 2 * z / M ≤ 2 := (div_le_iff₀ hM).mpr (by linarith [hz.2])
linarith
have hzcoord : packetIntensity M t = z := by
unfold packetIntensity
rw [ht]
field_simp
ring
have hpe : p.eval z = ε * (z - 1) + q.eval (cos t) := by
simp [p, a, Polynomial.eval_comp, ht]
congr 2
ring
rw [hpe, phiEps_eq_packetRemainder hz.1]
have herr := herror t
dsimp only [f] at herr
rw [hzcoord] at herr
have hc : 32 * sqrt M / (N : ℝ) ≤ 64 * sqrt M / K := by
have hKR : (0 : ℝ) < K := by exact_mod_cast (by omega : 0 < K)
have hNR : (0 : ℝ) < N := by exact_mod_cast (by dsimp [N]; omega : 0 < N)
have hKN : (K : ℝ) ≤ 2 * (N : ℝ) := by exact_mod_cast (by dsimp [N]; omega : K ≤ 2 * N)
apply (div_le_div_iff₀ hNR hKR).mpr
nlinarith [Real.sqrt_nonneg M]
convert herr.trans hc using 1
congr 1
ring
theorem weightedApproxError_upper
The explicit Jackson approximant bounds the weighted polynomial infimum with a universal constant, uniformly on the closed overlap interval. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
K :
ℕ
hK :
2 ≤ K
M ε :
ℝ
hM :
0 < M
hε :
0 ≤ ε
hεhi :
ε ≤ 1 / 2
conclusion
weightedApproxError K M ε ≤ 64 * sqrt M / K
Proof (Lean source)
lemma weightedApproxError_upper {K : ℕ} (hK : 2 ≤ K) {M ε : ℝ}
(hM : 0 < M) (hε : 0 ≤ ε) (hεhi : ε ≤ 1 / 2) :
weightedApproxError K M ε ≤ 64 * sqrt M / K := by
obtain ⟨p, hp, herror⟩ := weightedApproximation_polynomial_upper hK hM hε hεhi
have hb : BddBelow {e : ℝ | ∃ q : Polynomial ℝ, q.natDegree ≤ K ∧
e = sSup ((fun z : ℝ => |phiEpsFormula ε z - q.eval z| / (1 + z)) '' Icc 0 M)} := by
refine ⟨0, ?_⟩
rintro e ⟨q, _, rfl⟩
exact (weightedPolynomial_error_bound M ε hM.le q).1
change sInf {e : ℝ | ∃ q : Polynomial ℝ, q.natDegree ≤ K ∧
e = sSup ((fun z : ℝ => |phiEpsFormula ε z - q.eval z| / (1 + z)) '' Icc 0 M)} ≤ _
apply (csInf_le hb ⟨p, hp, rfl⟩).trans
apply csSup_le (by refine ⟨_, 0, ⟨le_rfl, hM.le⟩, rfl⟩)
rintro v ⟨z, hz, rfl⟩
apply (div_le_iff₀ (by linarith [hz.1] : 0 < 1 + z)).mpr
exact (herror z hz).trans (le_mul_of_one_le_right (by positivity) (by linarith [hz.1]))
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.Helpers.WeightedUpperRegularity 4 declarations The remainder in roadmap (10) is uniformly Lipschitz in the square-root intensity coordinate.
Angular regularity for weighted Jackson approximation
The remainder in roadmap (10) is uniformly Lipschitz in the square-root intensity coordinate. The half-angle identity then gives the sqrt(M) angular Lipschitz bound needed for the weighted upper approximation (14).
Clipping the lower quadratic branch confines its slope to [-2,0]. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma weightedLowerBranch_sqrt_lipschitz {x y : ℝ} (hx : 0 ≤ x) (hxy : x ≤ y) :
|max (1 - x ^ 2) 0 - max (1 - y ^ 2) 0| ≤ 2 * (y - x) := by
have hy : 0 ≤ y := hx.trans hxy
by_cases hy1 : y ≤ 1
· have hx1 : x ≤ 1 := hxy.trans hy1
rw [max_eq_left (by nlinarith : 0 ≤ 1 - x ^ 2),
max_eq_left (by nlinarith : 0 ≤ 1 - y ^ 2),
abs_of_nonneg (by nlinarith : 0 ≤ (1 - x ^ 2) - (1 - y ^ 2))]
nlinarith [mul_nonneg (sub_nonneg.mpr hxy) (by linarith : 0 ≤ 2 - x - y)]
· rw [max_eq_right (by nlinarith : 1 - y ^ 2 ≤ 0), sub_zero,
abs_of_nonneg (le_max_right _ _)]
by_cases hx1 : x ≤ 1
· rw [max_eq_left (by nlinarith : 0 ≤ 1 - x ^ 2)]
nlinarith [sq_nonneg (1 - x)]
· rw [max_eq_right (by nlinarith : 1 - x ^ 2 ≤ 0)]
linarith
The clipped rational upper branch has slope bounded by two in the square-root intensity coordinate, including intervals crossing the kink. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma weightedUpperBranch_sqrt_lipschitz {x y : ℝ} (hx : 0 ≤ x) (hxy : x ≤ y) :
|max (x ^ 2 - 1) 0 / (1 + x ^ 2) -
max (y ^ 2 - 1) 0 / (1 + y ^ 2)| ≤ 2 * (y - x) := by
have hy : 0 ≤ y := hx.trans hxy
have hdx : 0 < 1 + x ^ 2 := by positivity
have hdy : 0 < 1 + y ^ 2 := by positivity
by_cases hx1 : 1 ≤ x
· have hy1 : 1 ≤ y := hx1.trans hxy
rw [max_eq_left (by nlinarith : 0 ≤ x ^ 2 - 1),
max_eq_left (by nlinarith : 0 ≤ y ^ 2 - 1)]
have he : (x ^ 2 - 1) / (1 + x ^ 2) - (y ^ 2 - 1) / (1 + y ^ 2) =
-(2 * (y - x) * (x + y) / ((1 + x ^ 2) * (1 + y ^ 2))) := by
field_simp
ring
rw [he, abs_neg, abs_of_nonneg (by positivity)]
apply (div_le_iff₀ (mul_pos hdx hdy)).mpr
have hb : x + y ≤ (1 + x ^ 2) * (1 + y ^ 2) := by
nlinarith [sq_nonneg (x - 1), sq_nonneg (y - 1),
mul_nonneg (sq_nonneg x) (sq_nonneg y)]
nlinarith [mul_nonneg (by linarith : 0 ≤ 2 * (y - x))
(sub_nonneg.mpr hb)]
· rw [max_eq_right (by nlinarith : x ^ 2 - 1 ≤ 0), zero_div, zero_sub, abs_neg,
abs_of_nonneg (div_nonneg (le_max_right _ _) hdy.le)]
by_cases hy1 : 1 ≤ y
· rw [max_eq_left (by nlinarith : 0 ≤ y ^ 2 - 1)]
apply (div_le_iff₀ hdy).mpr
have h1 : y + 1 ≤ 2 * (1 + y ^ 2) := by nlinarith [sq_nonneg (y - 1)]
have h2 := mul_nonneg (by linarith : 0 ≤ y - 1) (sub_nonneg.mpr h1)
have h3 := mul_nonneg (by linarith : 0 ≤ 1 - x) hdy.le
nlinarith
· rw [max_eq_right (by nlinarith : y ^ 2 - 1 ≤ 0), zero_div]
linarith
theorem packetRemainder_sqrt_lipschitz
The nonlinear remainder has a universal Lipschitz constant in sqrt(z), uniform over the entire closed overlap interval. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
ε x y :
ℝ
hε :
0 ≤ ε
hεhi :
ε ≤ 1 / 2
hx :
0 ≤ x
hy :
0 ≤ y
conclusion
|packetRemainder ε (x ^ 2) - packetRemainder ε (y ^ 2)| ≤ 2 * |x - y|
Proof (Lean source)
lemma packetRemainder_sqrt_lipschitz {ε x y : ℝ}
(hε : 0 ≤ ε) (hεhi : ε ≤ 1 / 2) (hx : 0 ≤ x) (hy : 0 ≤ y) :
|packetRemainder ε (x ^ 2) - packetRemainder ε (y ^ 2)| ≤ 2 * |x - y| := by
have ordered (x y : ℝ) (hx : 0 ≤ x) (hxy : x ≤ y) :
|packetRemainder ε (x ^ 2) - packetRemainder ε (y ^ 2)| ≤ 2 * (y - x) := by
have hl := weightedLowerBranch_sqrt_lipschitz hx hxy
have hu := weightedUpperBranch_sqrt_lipschitz hx hxy
have he : packetRemainder ε (x ^ 2) - packetRemainder ε (y ^ 2) =
ε * (max (1 - x ^ 2) 0 - max (1 - y ^ 2) 0) +
(1 - 2 * ε) * (max (x ^ 2 - 1) 0 / (1 + x ^ 2) -
max (y ^ 2 - 1) 0 / (1 + y ^ 2)) := by
unfold packetRemainder
ring
rw [he]
calc
_ ≤ |ε * (max (1 - x ^ 2) 0 - max (1 - y ^ 2) 0)| +
|(1 - 2 * ε) * (max (x ^ 2 - 1) 0 / (1 + x ^ 2) -
max (y ^ 2 - 1) 0 / (1 + y ^ 2))| := abs_add_le _ _
_ ≤ ε * (2 * (y - x)) + (1 - 2 * ε) * (2 * (y - x)) := by
rw [abs_mul, abs_mul, abs_of_nonneg hε, abs_of_nonneg (by linarith : 0 ≤ 1 - 2 * ε)]
exact add_le_add (mul_le_mul_of_nonneg_left hl hε)
(mul_le_mul_of_nonneg_left hu (by linarith))
_ ≤ 2 * (y - x) := by nlinarith
rcases le_total x y with hxy | hyx
· rw [abs_of_nonpos (sub_nonpos.mpr hxy)]
simpa only [neg_sub] using ordered x y hx hxy
· rw [abs_sub_comm, abs_of_nonneg (sub_nonneg.mpr hyx)]
exact ordered y x hy hyx
theorem packetRemainder_angular_lipschitz
The half-angle coordinate gives the uniform angular regularity asserted before roadmap (14), without differentiability at either kink. In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma packetRemainder_angular_lipschitz {M ε : ℝ} (hM : 0 ≤ M)
(hε : 0 ≤ ε) (hεhi : ε ≤ 1 / 2) (s t : ℝ) :
|packetRemainder ε (packetIntensity M s) -
packetRemainder ε (packetIntensity M t)| ≤ sqrt M * |s - t| := by
have he (u : ℝ) : (sqrt M * |cos (u / 2)|) ^ 2 = packetIntensity M u := by
rw [mul_pow, Real.sq_sqrt hM, sq_abs]
have hc := Real.cos_two_mul (u / 2)
rw [show 2 * (u / 2) = u by ring] at hc
unfold packetIntensity
nlinarith
have h := packetRemainder_sqrt_lipschitz hε hεhi
(by positivity : 0 ≤ sqrt M * |cos (s / 2)|)
(by positivity : 0 ≤ sqrt M * |cos (t / 2)|)
rw [he s, he t] at h
calc
_ ≤ 2 * abs (sqrt M * |cos (s / 2)| - sqrt M * |cos (t / 2)|) := h
_ = 2 * sqrt M * abs (|cos (s / 2)| - |cos (t / 2)|) := by
rw [← mul_sub, abs_mul, abs_of_nonneg (Real.sqrt_nonneg M)]
ring
_ ≤ 2 * sqrt M * |cos (s / 2) - cos (t / 2)| :=
mul_le_mul_of_nonneg_left (abs_abs_sub_abs_le_abs_sub _ _) (by positivity)
_ ≤ 2 * sqrt M * |s / 2 - t / 2| :=
mul_le_mul_of_nonneg_left (Real.abs_cos_sub_cos_le _ _) (by positivity)
_ = sqrt M * |s - t| := by
rw [← sub_div, abs_div, abs_of_pos (by norm_num : (0 : ℝ) < 2)]
ring
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.OpenQuestions 4 declarations Open leading-constant question
Open leading-constant question
def AdmissiblePath
For the displayed parameters, AdmissiblePath is the object specified by this definition.
Definition (Lean source)
def DistinctPathLimitsQuestion
For no explicit parameters, DistinctPathLimitsQuestion is the object specified by this definition.
Definition (Lean source)
def
DistinctPathLimitsQuestion :
Prop
given by
∃ (κ₁ κ₂ : ℝ) (N₁ d₁ N₂ d₂ : ℕ → ℕ) (ε₁ ε₂ : ℕ → ℝ),
clause 1
0 < κ₁
clause 2
0 < κ₂
clause 3
κ₁ ≠ κ₂
clause 4
AdmissiblePath N₁ d₁ ε₁
clause 5
AdmissiblePath N₂ d₂ ε₂
clause 6
Tendsto (fun k => ((N₁ k : ℝ) * ε₁ k * logAlphabet (d₁ k) / (d₁ k : ℝ)) * minimaxRisk (N₁ k) (d₁ k) (ε₁ k)) atTop (𝓝 κ₁)
clause 7
Tendsto (fun k => ((N₂ k : ℝ) * ε₂ k * logAlphabet (d₂ k) / (d₂ k : ℝ)) * minimaxRisk (N₂ k) (d₂ k) (ε₂ k)) atTop (𝓝 κ₂)
def SharpPathConstantQuestion
For no explicit parameters, SharpPathConstantQuestion is the object specified by this definition.
Definition (Lean source)
def
SharpPathConstantQuestion :
Prop
given by
∃ κ : ℝ,
0 < κ
clause 1
Tendsto (fun k => ((N k : ℝ) * ε k * logAlphabet (d k) / (d k : ℝ)) * minimaxRisk (N k) (d k) (ε k)) atTop (𝓝 κ)
def PathLimitQuestion
For no explicit parameters, PathLimitQuestion is the object specified by this definition.
Definition (Lean source)
def
PathLimitQuestion :
Prop × Prop
given by
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.TAllEstimatorLower 1 declarations All-estimator lower risk
All-estimator lower risk
theorem all_estimator_lower
A universal positive constant times the vanishing-overlap rate is a lower bound for the minimax squared error at every sample size, alphabet size, and admissible overlap level.
Formal statement
conclusion
∃ cstar : ℝ,
0 < cstar
Proof (Lean source)
theorem all_estimator_lower :
∃ cstar : ℝ, 0 < cstar ∧
∀ (n d : ℕ) (ε : ℝ),
1 ≤ n → 2 ≤ d → 0 < ε → ε ≤ 1 / 2 →
cstar * rateScale n d ε ≤ minimaxRisk n d ε := by
obtain ⟨_, ⟨cstar, Cstar, H₀, κ, D₀, hH₀, hκ, hD₀,
hcstar, _, hfrontier⟩, _⟩ := weighted_separation_and_frontier
exact ⟨cstar, hcstar, fun n d ε hn hd hε hεhalf =>
(hfrontier n d ε hn hd hε hεhalf).1⟩
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.TConsistencyThreshold 4 declarations Uniform consistency threshold and fixed-parameter reductions
Uniform consistency threshold and fixed-parameter reductions
For the displayed parameters, UniformlyConsistent is the object specified by this definition.
Definition (Lean source)
parameters
d :
ℕ → ℕ
ε :
ℕ → ℝ
def
UniformlyConsistent d ε :
Prop
given by
∃ est : ∀ n,
Estimator n (d n), Tendsto (fun n => worstCaseRiskReal (observedRisk n (d := d n) (ε := ε n)) (est n)) atTop (𝓝 0)
theorem logAlphabet_bounds
In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
conclusion
conclusion 1
0 < logAlphabet d
conclusion 2
logAlphabet d ≤ d
Proof (Lean source)
lemma logAlphabet_bounds (d : ℕ) (hd : 2 ≤ d) :
0 < logAlphabet d ∧ logAlphabet d ≤ d := by
have hdpos : (0 : ℝ) < d := by exact_mod_cast (show 0 < d by omega)
have hlog : log (d : ℝ) ≤ d - 1 :=
Real.log_le_sub_one_of_pos hdpos
have hlog0 : 0 ≤ log (d : ℝ) :=
Real.log_nonneg (by exact_mod_cast (show 1 ≤ d by omega))
have hform : logAlphabet d = 1 + log (d : ℝ) := by
rw [logAlphabet, Real.log_mul (Real.exp_ne_zero 1) (ne_of_gt hdpos), Real.log_exp]
rw [hform]
constructor <;> linarith
In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
Proof (Lean source)
lemma min_one_mul_bounds {a x : ℝ} (ha : 1 ≤ a) (hx : 0 ≤ x) :
min 1 x ≤ min 1 (a * x) ∧ min 1 (a * x) ≤ a * min 1 x := by
constructor
· exact min_le_min le_rfl (by nlinarith)
· rcases le_total x 1 with h | h
· rw [min_eq_right h]
exact min_le_right _ _
· rw [min_eq_left h]
calc
min 1 (a * x) ≤ 1 := min_le_left _ _
_ ≤ a * 1 := by simpa using ha
theorem consistency_threshold
Uniform consistency holds exactly when the alphabet-to-effective-sample-size ratio tends to zero; moreover, for a fixed alphabet the minimax risk is comparable to one capped by the inverse effective sample size, and for fixed overlap it is comparable to one capped by the alphabet-to-sample-size log ratio.
Formal statement
conclusion
conclusion 1
d :
ℕ → ℕ
ε :
ℕ → ℝ
(∀ n, 2 ≤ d n ∧ 0 < ε n ∧ ε n ≤ 1 / 2)
conclusion 2
d :
ℕ
2 ≤ d
∃ c C : ℝ,
0 < c
c ≤ C
n :
ℕ
ε :
ℝ
1 ≤ n
0 < ε
ε ≤ 1 / 2
c * min 1 (1 / ((n : ℝ) * ε)) ≤ minimaxRisk n d ε
minimaxRisk n d ε ≤ C * min 1 (1 / ((n : ℝ) * ε))
conclusion 3
ε :
ℝ
0 < ε
ε ≤ 1 / 2
∃ c C : ℝ,
0 < c
c ≤ C
n d :
ℕ
1 ≤ n
2 ≤ d
c * min 1 ((d : ℝ) / ((n : ℝ) * logAlphabet d)) ≤ minimaxRisk n d ε
minimaxRisk n d ε ≤ C * min 1 ((d : ℝ) / ((n : ℝ) * logAlphabet d))
Proof (Lean source)
theorem consistency_threshold :
(∀ (d : ℕ → ℕ) (ε : ℕ → ℝ),
(∀ n, 2 ≤ d n ∧ 0 < ε n ∧ ε n ≤ 1 / 2) →
(UniformlyConsistent d ε ↔
Tendsto (fun n => (d n : ℝ) /
((n : ℝ) * ε n * logAlphabet (d n))) atTop (𝓝 0))) ∧
(∀ d : ℕ, 2 ≤ d →
∃ c C : ℝ, 0 < c ∧ c ≤ C ∧
∀ (n : ℕ) (ε : ℝ),
1 ≤ n → 0 < ε → ε ≤ 1 / 2 →
c * min 1 (1 / ((n : ℝ) * ε)) ≤ minimaxRisk n d ε ∧
minimaxRisk n d ε ≤ C * min 1 (1 / ((n : ℝ) * ε))) ∧
(∀ ε : ℝ, 0 < ε → ε ≤ 1 / 2 →
∃ c C : ℝ, 0 < c ∧ c ≤ C ∧
∀ (n d : ℕ),
1 ≤ n → 2 ≤ d →
c * min 1 ((d : ℝ) / ((n : ℝ) * logAlphabet d)) ≤
minimaxRisk n d ε ∧
minimaxRisk n d ε ≤
C * min 1 ((d : ℝ) / ((n : ℝ) * logAlphabet d))) := by
obtain ⟨cstar, Cstar, H₀, κ, D₀, hH₀, hκ, hD₀, hcstar, hcC, hfront⟩ :=
matched_frontier
constructor
· intro d ε hd
have h := weighted_separation_and_frontier.2.2
(fun n => n) d ε tendsto_id hd
simpa only [UniformlyConsistent] using h
constructor
· intro d hd
obtain ⟨hLpos, hLle⟩ := logAlphabet_bounds d hd
have hdpos : (0 : ℝ) < d := by exact_mod_cast (show 0 < d by omega)
have ha : 1 ≤ (d : ℝ) / logAlphabet d := by
apply (le_div_iff₀ hLpos).2
simpa using hLle
refine ⟨cstar, Cstar * (d : ℝ) / logAlphabet d,
hcstar, ?_, ?_⟩
· have hCa : Cstar ≤ Cstar * ((d : ℝ) / logAlphabet d) :=
by simpa using mul_le_mul_of_nonneg_left ha (le_trans (le_of_lt hcstar) hcC)
calc cstar ≤ Cstar := hcC
_ ≤ Cstar * ((d : ℝ) / logAlphabet d) := hCa
_ = Cstar * (d : ℝ) / logAlphabet d := by ring
· intro n ε hn hε hεhalf
obtain ⟨hlower, hmid, hupper, _, _⟩ := hfront n d ε hn hd hε hεhalf
have hnpos : (0 : ℝ) < n := by exact_mod_cast (show 0 < n by omega)
have hx : 0 ≤ 1 / ((n : ℝ) * ε) := by positivity
obtain ⟨hminlo, hminhi⟩ := min_one_mul_bounds ha hx
have heq : rateScale n d ε =
min 1 (((d : ℝ) / logAlphabet d) * (1 / ((n : ℝ) * ε))) := by
rw [rateScale]
congr 1
field_simp
rw [heq] at hlower hupper
constructor
· exact (mul_le_mul_of_nonneg_left hminlo (le_of_lt hcstar)).trans hlower
· have hC : 0 ≤ Cstar := le_trans (le_of_lt hcstar) hcC
calc
minimaxRisk n d ε ≤ Cstar * min 1 (((d : ℝ) / logAlphabet d) * (1 / ((n : ℝ) * ε))) := hmid.trans hupper
_ ≤ (Cstar * (d : ℝ) / logAlphabet d) * min 1 (1 / ((n : ℝ) * ε)) := by
calc
Cstar * min 1 (((d : ℝ) / logAlphabet d) * (1 / ((n : ℝ) * ε))) ≤
Cstar * (((d : ℝ) / logAlphabet d) * min 1 (1 / ((n : ℝ) * ε))) :=
mul_le_mul_of_nonneg_left hminhi hC
_ = _ := by ring
· intro ε hε hεhalf
refine ⟨cstar, Cstar / ε, hcstar, ?_, ?_⟩
· have ha : 1 ≤ 1 / ε := by
apply (le_div_iff₀ hε).2
linarith
have hCa : Cstar ≤ Cstar * (1 / ε) :=
by simpa using mul_le_mul_of_nonneg_left ha (le_trans (le_of_lt hcstar) hcC)
calc cstar ≤ Cstar := hcC
_ ≤ Cstar * (1 / ε) := hCa
_ = Cstar / ε := by ring
· intro n d hn hd
obtain ⟨hlower, hmid, hupper, _, _⟩ := hfront n d ε hn hd hε hεhalf
obtain ⟨hLpos, _⟩ := logAlphabet_bounds d hd
have hnpos : (0 : ℝ) < n := by exact_mod_cast (show 0 < n by omega)
have hx : 0 ≤ (d : ℝ) / ((n : ℝ) * logAlphabet d) := by positivity
have ha : 1 ≤ 1 / ε := by
apply (le_div_iff₀ hε).2
linarith
obtain ⟨hminlo, hminhi⟩ := min_one_mul_bounds ha hx
have heq : rateScale n d ε =
min 1 ((1 / ε) * ((d : ℝ) / ((n : ℝ) * logAlphabet d))) := by
rw [rateScale]
congr 1
field_simp
rw [heq] at hlower hupper
constructor
· exact (mul_le_mul_of_nonneg_left hminlo (le_of_lt hcstar)).trans hlower
· have hC : 0 ≤ Cstar := le_trans (le_of_lt hcstar) hcC
calc
minimaxRisk n d ε ≤ Cstar * min 1 ((1 / ε) * ((d : ℝ) / ((n : ℝ) * logAlphabet d))) := hmid.trans hupper
_ ≤ (Cstar / ε) * min 1 ((d : ℝ) / ((n : ℝ) * logAlphabet d)) := by
calc
Cstar * min 1 ((1 / ε) * ((d : ℝ) / ((n : ℝ) * logAlphabet d))) ≤
Cstar * ((1 / ε) * min 1 ((d : ℝ) / ((n : ℝ) * logAlphabet d))) :=
mul_le_mul_of_nonneg_left hminhi hC
_ = _ := by ring
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.TIdentification 25 declarations Identification and completion of finite observed laws
Identification and completion of finite observed laws
theorem observedClass_factorization
In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
conclusion 1
jointMass P x a true
= cellMass P x * (if a then propensity P x else 1 - propensity P x) * outcomeMean P a x
conclusion 2
jointMass P x a false
= cellMass P x * (if a then propensity P x else 1 - propensity P x) * (1 - outcomeMean P a x)
Proof (Lean source)
lemma observedClass_factorization {d : ℕ} (ε : ℝ) (P : DiscreteLaw d)
(_hP : ObservedClass ε P) :
∀ x a,
jointMass P x a true =
cellMass P x * (if a then propensity P x else 1 - propensity P x) *
outcomeMean P a x ∧
jointMass P x a false =
cellMass P x * (if a then propensity P x else 1 - propensity P x) *
(1 - outcomeMean P a x) := by
intro x a
have hq (a y : Bool) : 0 ≤ jointMass P x a y := toReal_nonneg
have harm (a : Bool) :
armMass P a x = jointMass P x a false + jointMass P x a true := by
simp [armMass]
ring
have hcell : cellMass P x = armMass P false x + armMass P true x := by
simp [cellMass, armMass]
ring
have hnonneg (a : Bool) : 0 ≤ armMass P a x := by
rw [harm]
exact add_nonneg (hq a false) (hq a true)
have hmean (a : Bool) :
armMass P a x * outcomeMean P a x = jointMass P x a true := by
by_cases ha : armMass P a x = 0
· have hzero : jointMass P x a true = 0 := by
have h := harm a
rw [ha] at h
linarith [hq a false, hq a true]
simp [ha, hzero]
· rw [outcomeMean]
field_simp [ha]
have hfail (a : Bool) :
armMass P a x * (1 - outcomeMean P a x) = jointMass P x a false := by
rw [mul_sub, mul_one, hmean, harm]
ring
have hpropTrue : cellMass P x * propensity P x = armMass P true x := by
by_cases hp : cellMass P x = 0
· have ht : armMass P true x = 0 := by
rw [hcell] at hp
linarith [hnonneg false, hnonneg true]
simp [hp, ht]
· rw [propensity]
field_simp [hp]
have hpropFalse : cellMass P x * (1 - propensity P x) = armMass P false x := by
rw [mul_sub, mul_one, hpropTrue, hcell]
ring
cases a with
| false =>
constructor
· simpa [hpropFalse] using (hmean false).symm
· simpa [hpropFalse] using (hfail false).symm
| true =>
constructor
· simpa [hpropTrue] using (hmean true).symm
· simpa [hpropTrue] using (hfail true).symm
In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
armwiseExtensionFormula ε (cellVector P x)
Proof (Lean source)
lemma armwiseExtension_cellVector {d : ℕ} (ε : ℝ) (P : DiscreteLaw d)
(hε : 0 < ε) (hP : ObservedClass ε P) (x : Fin d) :
armwiseExtensionFormula ε (cellVector P x) =
cellMass P x * max (outcomeMean P false x) (outcomeMean P true x) := by
have hmass : totalMassVecFormula (cellVector P x) = cellMass P x := by
simp [totalMassVecFormula, armMassVecFormula, cellVector, cellMass, cellIdx]
ring
have harm (a : Bool) : armMassVecFormula (cellVector P x) a = armMass P a x := by
cases a <;> simp [armMassVecFormula, cellVector, armMass, cellIdx] <;> ring
have hcoord (a : Bool) : (cellVector P x) (cellIdx a true) = jointMass P x a true := by
cases a <;> simp [cellVector, cellIdx]
have hpnonneg : 0 ≤ cellMass P x := by
unfold cellMass
exact Finset.sum_nonneg (fun a _ => Finset.sum_nonneg (fun y _ => toReal_nonneg))
by_cases hp : cellMass P x = 0
· have hzero (a y : Bool) : jointMass P x a y = 0 := by
have hq (b z : Bool) : 0 ≤ jointMass P x b z := toReal_nonneg
simp [cellMass] at hp
rcases a with _ | _ <;> rcases y with _ | _ <;>
linarith [hq false false, hq false true, hq true false, hq true true]
have hv : cellVector P x = 0 := by
funext j
fin_cases j <;> simp [cellVector, hzero]
simp [hv, hp, armwiseExtensionFormula]
· have hppos : 0 < cellMass P x := lt_of_le_of_ne hpnonneg (Ne.symm hp)
obtain ⟨hlo, hhi⟩ := hP.overlap x hppos
have htrue : ε * cellMass P x ≤ armMass P true x := by
exact (le_div_iff₀ hppos).mp (by simpa [propensity] using hlo)
have hfalse : ε * cellMass P x ≤ armMass P false x := by
have hs : cellMass P x = armMass P false x + armMass P true x := by
simp [cellMass, armMass]
ring
have ht : armMass P true x ≤ (1 - ε) * cellMass P x := by
rw [propensity, div_le_iff₀ hppos] at hhi
nlinarith
nlinarith
have hterm (a : Bool) :
totalMassVecFormula (cellVector P x) * (cellVector P x) (cellIdx a true) /
anchoredDenomFormula ε (cellVector P x) a =
cellMass P x * outcomeMean P a x := by
have ha : ε * cellMass P x ≤ armMass P a x := by
cases a <;> assumption
have ha_pos : 0 < armMass P a x := lt_of_lt_of_le (mul_pos hε hppos) ha
rw [anchoredDenomFormula, hmass, harm, max_eq_left ha, hcoord, outcomeMean]
ring
have hv : cellVector P x ≠ 0 := by
intro hz
rw [hz] at hmass
simp [totalMassVecFormula, armMassVecFormula] at hmass
exact hp hmass.symm
rw [armwiseExtensionFormula, if_neg hv, hterm false, hterm true]
exact (mul_max_of_nonneg _ _ (le_of_lt hppos)).symm
For the displayed parameters, completionBernoulliMass is the object specified by this definition. For the displayed parameters, completionBernoulliMass is the defined object.
Definition (Lean source)
def completionFullMass
For the displayed parameters, completionFullMass is the object specified by this definition. For the displayed parameters, completionFullMass is the defined object.
Definition (Lean source)
parameters
def
completionFullMass P z :
ℝ
given by
if z.2.2.1
= (if z.2.1 then z.2.2.2.2 else z.2.2.2.1) then jointMass P z.1 z.2.1 z.2.2.1 * completionBernoulliMass (outcomeMean P (!z.2.1) z.1) (if z.2.1 then z.2.2.2.1 else z.2.2.2.2) else 0
theorem outcomeMean_mem_unitInterval
In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
Proof (Lean source)
lemma outcomeMean_mem_unitInterval {d : ℕ} (P : DiscreteLaw d) (a : Bool) (x : Fin d) :
outcomeMean P a x ∈ Icc (0 : ℝ) 1 := by
have hnum : 0 ≤ jointMass P x a true := ENNReal.toReal_nonneg
have hden : 0 ≤ armMass P a x := by
exact Finset.sum_nonneg fun y _ => ENNReal.toReal_nonneg
have hle : jointMass P x a true ≤ armMass P a x := by
simp [armMass]
exact ENNReal.toReal_nonneg
exact ⟨div_nonneg hnum hden, div_le_one_of_le₀ hle hden⟩
theorem completionFullMass_nonneg_sum
In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
P :
conclusion
conclusion 2
Proof (Lean source)
lemma completionFullMass_nonneg_sum {d : ℕ} (P : DiscreteLaw d) :
(∀ z, 0 ≤ completionFullMass P z) ∧
∑ z : FullObs d, ofReal (completionFullMass P z) = 1 := by
classical
have hnonneg : ∀ z, 0 ≤ completionFullMass P z := by
rintro ⟨x, a, y, y0, y1⟩
have hm0 := outcomeMean_mem_unitInterval P (!a) x
cases a <;> simp at hm0 ⊢ <;>
simp [completionFullMass, completionBernoulliMass] <;>
split_ifs
all_goals
first
| exact le_rfl
| apply mul_nonneg toReal_nonneg
linarith [hm0.1, hm0.2]
refine ⟨hnonneg, ?_⟩
rw [← ENNReal.ofReal_sum_of_nonneg (fun z _hz => hnonneg z)]
have hsum : ∑ z : FullObs d, completionFullMass P z =
∑ z : Obs d, jointMass P z.1 z.2.1 z.2.2 := by
simp [Fintype.sum_prod_type, completionFullMass, completionBernoulliMass]
ring_nf
rw [hsum]
simpa [jointMass] using
(PMF.integral_eq_sum P.pmf (fun _ : Obs d => (1 : ℝ))).symm
def bernoulliCompletion
For the displayed parameters, bernoulliCompletion is the object specified by this definition. For the displayed parameters, bernoulliCompletion is the defined object.
Definition (Lean source)
parameters
d :
ℕ
P :
def
given by
theorem bernoulliCompletion_fullMass
In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma bernoulliCompletion_fullMass {d : ℕ} (P : DiscreteLaw d) (z : FullObs d) :
fullMass (bernoulliCompletion P) z = completionFullMass P z := by
rw [fullMass, bernoulliCompletion, PMF.ofFintype_apply, ENNReal.toReal_ofReal]
exact (completionFullMass_nonneg_sum P).1 z
theorem bernoulliCompletion_pmf_toReal
In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma bernoulliCompletion_pmf_toReal {d : ℕ} (P : DiscreteLaw d) (z : FullObs d) :
((bernoulliCompletion P).pmf z).toReal = completionFullMass P z :=
bernoulliCompletion_fullMass P z
theorem bernoulliCompletion_consistency
In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
Proof (Lean source)
lemma bernoulliCompletion_consistency {d : ℕ} (P : DiscreteLaw d) :
Consistency (bernoulliCompletion P) := by
intro z hz
rw [bernoulliCompletion, PMF.ofFintype_apply, completionFullMass, if_neg hz]
simp
theorem bernoulliCompletion_poAtom
In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
poAtom (bernoulliCompletion P) x a y0 y1
= jointMass P x a (if a then y1 else y0) * completionBernoulliMass (outcomeMean P (!a) x) (if a then y0 else y1)
Proof (Lean source)
lemma bernoulliCompletion_poAtom {d : ℕ} (P : DiscreteLaw d)
(x : Fin d) (a y0 y1 : Bool) :
poAtom (bernoulliCompletion P) x a y0 y1 =
jointMass P x a (if a then y1 else y0) *
completionBernoulliMass (outcomeMean P (!a) x) (if a then y0 else y1) := by
simp [poAtom, bernoulliCompletion_fullMass, completionFullMass]
theorem bernoulliCompletion_poAtom_factorization
In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
poAtom (bernoulliCompletion P) x a y0 y1
= cellMass P x * completionBernoulliMass (propensity P x) a * completionBernoulliMass (outcomeMean P false x) y0 * completionBernoulliMass (outcomeMean P true x) y1
Proof (Lean source)
lemma bernoulliCompletion_poAtom_factorization {d : ℕ} {ε : ℝ}
(P : DiscreteLaw d) (hP : ObservedClass ε P)
(x : Fin d) (a y0 y1 : Bool) :
poAtom (bernoulliCompletion P) x a y0 y1 =
cellMass P x * completionBernoulliMass (propensity P x) a *
completionBernoulliMass (outcomeMean P false x) y0 *
completionBernoulliMass (outcomeMean P true x) y1 := by
have hf := observedClass_factorization ε P hP x false
have ht := observedClass_factorization ε P hP x true
fin_cases a <;> fin_cases y0 <;> fin_cases y1 <;>
simp only [bernoulliCompletion_poAtom, completionBernoulliMass,
Bool.not_true, Bool.not_false, if_true, if_false,
Bool.false_eq_true] at hf ht ⊢
all_goals
first | rw [ht.1] | rw [ht.2] | rw [hf.1] | rw [hf.2]
all_goals ring
theorem bernoulliCompletion_observedMarginal
In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
Proof (Lean source)
lemma bernoulliCompletion_observedMarginal {d : ℕ} (P : DiscreteLaw d) :
observedMarginal (bernoulliCompletion P) = P := by
cases P with
| mk pmf =>
change ⟨_⟩ = (⟨pmf⟩ : DiscreteLaw d)
congr 1
apply PMF.ext
rintro ⟨x, a, y⟩
rw [← ENNReal.toReal_eq_toReal_iff'
(PMF.apply_ne_top _ _) (PMF.apply_ne_top _ _)]
change (PMF.map _ (bernoulliCompletion ⟨pmf⟩).pmf (x, a, y)).toReal = _
rw [PMF.map_apply, ENNReal.tsum_toReal_eq (fun z => by
split
· exact PMF.apply_ne_top _ _
· simp), tsum_fintype]
have hsum (f : FullObs d → ℝ) :
∑ z : FullObs d, f z =
∑ x : Fin d, ∑ a : Bool, ∑ y : Bool, ∑ y0 : Bool, ∑ y1 : Bool,
f (x, a, y, y0, y1) := by
simp [Fintype.sum_prod_type]
rw [hsum]
simp only [apply_ite, ENNReal.toReal_zero]
simp only [bernoulliCompletion_pmf_toReal]
simp [completionFullMass, completionBernoulliMass]
ring_nf
fin_cases a <;> fin_cases y <;> simp [jointMass]
theorem poCellMass_eq_sum_poAtom
In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma poCellMass_eq_sum_poAtom {d : ℕ} (Q : PotentialLaw d) (x : Fin d) :
poCellMass Q x = ∑ a : Bool, ∑ y0 : Bool, ∑ y1 : Bool, poAtom Q x a y0 y1 := by
simp [poCellMass, poAtom, Fintype.sum_bool]
ring
theorem poTreatMass_eq_sum_poAtom
In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma poTreatMass_eq_sum_poAtom {d : ℕ} (Q : PotentialLaw d) (x : Fin d) (a : Bool) :
poTreatMass Q x a = ∑ y0 : Bool, ∑ y1 : Bool, poAtom Q x a y0 y1 := by
simp [poTreatMass, poAtom, Fintype.sum_bool]
ring
theorem bernoulliCompletion_exchangeability
In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma bernoulliCompletion_exchangeability {d : ℕ} {ε : ℝ}
(P : DiscreteLaw d) (hP : ObservedClass ε P) :
ConditionalExchangeability (bernoulliCompletion P) := by
intro x r a ya
rw [poCellMass_eq_sum_poAtom, poTreatMass_eq_sum_poAtom]
fin_cases r <;> fin_cases a <;> fin_cases ya <;>
simp [poArmAtom, poPotentialMass,
bernoulliCompletion_poAtom_factorization P hP,
completionBernoulliMass] <;> ring
theorem observedMarginal_jointMass
In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
jointMass (observedMarginal Q) x a y
Proof (Lean source)
lemma observedMarginal_jointMass {d : ℕ} (Q : PotentialLaw d) (x : Fin d) (a y : Bool) :
jointMass (observedMarginal Q) x a y =
∑ y0 : Bool, ∑ y1 : Bool, fullMass Q (x, a, y, y0, y1) := by
classical
have hmap : (observedMarginal Q).pmf (x, a, y) =
∑ y0 : Bool, ∑ y1 : Bool, Q.pmf (x, a, y, y0, y1) := by
change (Q.pmf.map (fun z => (z.1, z.2.1, z.2.2.1))) (x, a, y) = _
rw [PMF.map_apply, tsum_fintype]
rw [Fintype.sum_prod_type, Finset.sum_eq_single x]
· fin_cases a <;> fin_cases y <;>
simp [Fintype.sum_prod_type]
· intro b _hb hbx
simp [Ne.symm hbx]
· simp
rw [jointMass, hmap]
simp [fullMass, ENNReal.toReal_add, PMF.apply_ne_top]
theorem observedMarginal_cellMass
In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
cellMass (observedMarginal Q) x = poCellMass Q x
Proof (Lean source)
lemma observedMarginal_cellMass {d : ℕ} (Q : PotentialLaw d) (x : Fin d) :
cellMass (observedMarginal Q) x = poCellMass Q x := by
simp [cellMass, poCellMass, observedMarginal_jointMass]
theorem consistent_observed_selected_atom
In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
jointMass (observedMarginal Q) x (r = 1) y = poArmAtom Q x r (r = 1) y
Proof (Lean source)
lemma consistent_observed_selected_atom {d : ℕ} (Q : PotentialLaw d)
(h : Consistency Q) (x : Fin d) (r : Fin 2) (y : Bool) :
jointMass (observedMarginal Q) x (r = 1) y =
poArmAtom Q x r (r = 1) y := by
have hs : ∀ z, z.2.2.1 ≠ (if z.2.1 then z.2.2.2.2 else z.2.2.2.1) →
fullMass Q z = 0 := by
intro z hz
simp [fullMass, h z hz]
fin_cases r <;> fin_cases y <;>
simp [observedMarginal_jointMass, poArmAtom, poAtom, hs]
theorem poTreatMass_eq_observed_armMass
In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma poTreatMass_eq_observed_armMass {d : ℕ} (Q : PotentialLaw d)
(x : Fin d) (a : Bool) :
poTreatMass Q x a = armMass (observedMarginal Q) a x := by
simp [poTreatMass, armMass, observedMarginal_jointMass, Fintype.sum_bool]
theorem poRegression_eq_poArmAtom_ratio
In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma poRegression_eq_poArmAtom_ratio {d : ℕ} (Q : PotentialLaw d)
(x : Fin d) (r : Fin 2) :
poRegression Q r x =
(∑ arm : Bool, poArmAtom Q x r arm true) / poCellMass Q x := by
fin_cases r <;>
simp [poRegression, poCellMass, poArmAtom, poAtom] <;>
ring
theorem consistent_observed_armMass
In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma consistent_observed_armMass {d : ℕ} (Q : PotentialLaw d) (h : Consistency Q)
(x : Fin d) (r : Fin 2) :
armMass (observedMarginal Q) (r = 1) x =
∑ y : Bool, poArmAtom Q x r (r = 1) y := by
simp [armMass, consistent_observed_selected_atom Q h x r]
theorem poArmAtom_total_eq_cellMass
In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma poArmAtom_total_eq_cellMass {d : ℕ} (Q : PotentialLaw d)
(x : Fin d) (r : Fin 2) :
(∑ a : Bool, ∑ y : Bool, poArmAtom Q x r a y) = poCellMass Q x := by
fin_cases r <;>
simp [poCellMass, poArmAtom, poAtom] <;>
ring
theorem poRegression_eq_outcomeMean_of_pos
In the econometric construction, the displayed inputs and assumptions imply the stated mathematical conclusion. Under the stated hypotheses, the stated conclusion holds.
Formal statement
hypotheses
d :
ℕ
Q :
hcons :
Consistency Q
hexch :
x :
Fin d
r :
Fin 2
hcell :
0 < cellMass (observedMarginal Q) x
harm :
0 < armMass (observedMarginal Q) (r = 1) x
conclusion
Proof (Lean source)
lemma poRegression_eq_outcomeMean_of_pos {d : ℕ} (Q : PotentialLaw d)
(hcons : Consistency Q) (hexch : ConditionalExchangeability Q)
(x : Fin d) (r : Fin 2)
(hcell : 0 < cellMass (observedMarginal Q) x)
(harm : 0 < armMass (observedMarginal Q) (r = 1) x) :
poRegression Q r x = outcomeMean (observedMarginal Q) (r = 1) x := by
have heq := hexch x r (r = 1) true
rw [poTreatMass_eq_observed_armMass, ← observedMarginal_cellMass] at heq
change poArmAtom Q x r (r = 1) true * cellMass (observedMarginal Q) x =
armMass (observedMarginal Q) (r = 1) x *
(∑ arm : Bool, poArmAtom Q x r arm true) at heq
rw [consistent_observed_armMass Q hcons] at heq
rw [consistent_observed_armMass Q hcons] at harm
rw [poRegression_eq_poArmAtom_ratio, outcomeMean,
consistent_observed_selected_atom Q hcons,
consistent_observed_armMass Q hcons,
← observedMarginal_cellMass]
field_simp [ne_of_gt hcell, ne_of_gt harm]
nlinarith
theorem identification
Every admissible observed law has a compatible potential-outcome completion, the oracle value of every admissible causal law equals the value identified from its observed marginal, and each covariate cell's armwise extension equals its mass times the larger conditional outcome mean.
Formal statement
conclusion
conclusion 2
oracleValue Q = observedValue (observedMarginal Q)
conclusion 3
Proof (Lean source)
theorem identification :
(∀ (d : ℕ) (ε : ℝ) (P : DiscreteLaw d),
2 ≤ d → 0 < ε → ε ≤ 1 / 2 → ObservedClass ε P →
∃ Q : PotentialLaw d, CausalClass ε Q ∧ observedMarginal Q = P) ∧
(∀ (d : ℕ) (ε : ℝ) (Q : PotentialLaw d),
2 ≤ d → 0 < ε → ε ≤ 1 / 2 → CausalClass ε Q →
oracleValue Q = observedValue (observedMarginal Q)) ∧
(∀ (d : ℕ) (ε : ℝ) (P : DiscreteLaw d),
2 ≤ d → 0 < ε → ε ≤ 1 / 2 → ObservedClass ε P →
∀ x : Fin d, armwiseExtension ε (cellVector P x) (cellVector_nonnegative P x) =
cellMass P x * max (outcomeMean P false x) (outcomeMean P true x)) := by
refine ⟨?_, ?_, ?_⟩
· intro d ε P _hd _hε _hεhalf hP
refine ⟨bernoulliCompletion P, ?_, bernoulliCompletion_observedMarginal P⟩
exact ⟨bernoulliCompletion_consistency P,
bernoulliCompletion_exchangeability P hP,
by simpa [bernoulliCompletion_observedMarginal P] using hP.overlap⟩
· intro d ε Q _hd hε _hεhalf hQ
rw [oracleValue, observedValue]
apply Finset.sum_congr rfl
intro x _hx
rw [← observedMarginal_cellMass]
by_cases hx : cellMass (observedMarginal Q) x = 0
· simp [hx]
· have hnonneg : 0 ≤ cellMass (observedMarginal Q) x := by
unfold cellMass jointMass
positivity
have hxpos : 0 < cellMass (observedMarginal Q) x :=
lt_of_le_of_ne hnonneg (Ne.symm hx)
obtain ⟨hlo, hhi⟩ := hQ.overlap x hxpos
have ht : 0 < armMass (observedMarginal Q) true x := by
have he : 0 < ε * cellMass (observedMarginal Q) x := mul_pos hε hxpos
exact lt_of_lt_of_le he ((le_div_iff₀ hxpos).mp (by
simpa [propensity] using hlo))
have hsum : cellMass (observedMarginal Q) x =
armMass (observedMarginal Q) false x +
armMass (observedMarginal Q) true x := by
simp [cellMass, armMass]
ring
have hf : 0 < armMass (observedMarginal Q) false x := by
rw [propensity, div_le_iff₀ hxpos] at hhi
nlinarith [mul_pos hε hxpos]
rw [poRegression_eq_outcomeMean_of_pos Q hQ.consistency hQ.exchangeability
x 0 hxpos (by simpa using hf),
poRegression_eq_outcomeMean_of_pos Q hQ.consistency hQ.exchangeability
x 1 hxpos (by simpa using ht)]
rfl
· intro d ε P _hd hε _hεhalf hP x
exact armwiseExtension_cellVector ε P hε hP x
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.TMatchedFrontier 1 declarations Matched observed and causal frontiers
Matched observed and causal frontiers
theorem matched_frontier
Universal constants sandwich both the observed and causal minimax risks at the vanishing-overlap rate, while the explicit armwise estimator lies between the observed minimax risk and the same upper rate and its causal risk lies between the causal minimax risk and that upper rate.
Formal statement
conclusion
∃ (cstar Cstar H₀ κ : ℝ) (D₀ : ℕ), ∃ hH₀ : 0 < H₀, ∃ hκ : 0 < κ, ∃ hD₀ : 2 ≤ D₀,
0 < cstar
cstar ≤ Cstar
conclusion 1
n d :
ℕ
ε :
ℝ
1 ≤ n
2 ≤ d
0 < ε
ε ≤ 1 / 2
cstar * rateScale n d ε ≤ minimaxRisk n d ε
minimaxRisk n d ε ≤ armwiseWorstRisk H₀ κ D₀ hH₀ hκ hD₀ n d ε
armwiseWorstRisk H₀ κ D₀ hH₀ hκ hD₀ n d ε ≤ Cstar * rateScale n d ε
cstar * rateScale n d ε ≤ causalMinimaxRisk n d ε
causalMinimaxRisk n d ε ≤ Cstar * rateScale n d ε
causalMinimaxRisk n d ε ≤ causalArmwiseWorstRisk H₀ κ D₀ hH₀ hκ hD₀ n d ε
causalArmwiseWorstRisk H₀ κ D₀ hH₀ hκ hD₀ n d ε ≤ Cstar * rateScale n d ε
Proof (Lean source)
theorem matched_frontier :
∃ (cstar Cstar H₀ κ : ℝ) (D₀ : ℕ),
∃ hH₀ : 0 < H₀, ∃ hκ : 0 < κ, ∃ hD₀ : 2 ≤ D₀,
0 < cstar ∧ cstar ≤ Cstar ∧
∀ (n d : ℕ) (ε : ℝ),
1 ≤ n → 2 ≤ d → 0 < ε → ε ≤ 1 / 2 →
cstar * rateScale n d ε ≤ minimaxRisk n d ε ∧
minimaxRisk n d ε ≤ armwiseWorstRisk H₀ κ D₀ hH₀ hκ hD₀ n d ε ∧
armwiseWorstRisk H₀ κ D₀ hH₀ hκ hD₀ n d ε ≤ Cstar * rateScale n d ε ∧
cstar * rateScale n d ε ≤ causalMinimaxRisk n d ε ∧
causalMinimaxRisk n d ε ≤ Cstar * rateScale n d ε ∧
causalMinimaxRisk n d ε ≤
causalArmwiseWorstRisk H₀ κ D₀ hH₀ hκ hD₀ n d ε ∧
causalArmwiseWorstRisk H₀ κ D₀ hH₀ hκ hD₀ n d ε ≤
Cstar * rateScale n d ε := by
obtain ⟨cstar, Cstar, H₀, κ, D₀, hH₀, hκ, hD₀, hcstar, hcC, hfront⟩ :=
weighted_separation_and_frontier.2.1
refine ⟨cstar, Cstar, H₀, κ, D₀, hH₀, hκ, hD₀, hcstar, hcC, ?_⟩
intro n d ε hn hd hε hεhalf
obtain ⟨hlo, hmid, hhi, hclo, hchi⟩ := hfront n d ε hn hd hε hεhalf
refine ⟨hlo, hmid, hhi, hclo, hchi, ?_, ?_⟩
· rw [causalMinimaxRisk_eq_minimaxRisk hd hε hεhalf,
causalArmwiseWorstRisk_eq_armwiseWorstRisk H₀ κ D₀ hH₀ hκ hD₀ n d ε hd hε hεhalf]
exact hmid
· rw [causalArmwiseWorstRisk_eq_armwiseWorstRisk H₀ κ D₀ hH₀ hκ hD₀ n d ε hd hε hεhalf]
exact hhi
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.TObservableUpper 1 declarations Uniform observable upper risk
Uniform observable upper risk
theorem observable_upper
There are universal positive tuning constants and a measurable armwise estimator whose worst-case squared error over every admissible observed model is bounded above by a universal constant times the vanishing-overlap rate.
Formal statement
conclusion
∃ (H₀ κ Cstar : ℝ) (D₀ : ℕ), ∃ hH₀ : 0 < H₀, ∃ hκ : 0 < κ, ∃ hD₀ : 2 ≤ D₀,
0 < Cstar
conclusion 1
n d :
ℕ
ε :
ℝ
(hn : 1 ≤ n)
(hd : 2 ≤ d)
(hε : 0 < ε)
(hεhalf : ε ≤ 1 / 2)
Measurable (armwiseEstimator H₀ κ D₀ hH₀ hκ hD₀ n d ε hn hd ⟨hε, hεhalf⟩)
(∀ P, IidSampling P.1 (sampleLaw P))
(⨆ P : ModelLaw d ε, sqRisk (sampleLaw P) (armwiseEstimator H₀ κ D₀ hH₀ hκ hD₀ n d ε hn hd ⟨hε, hεhalf⟩) (observedValue P.1))
≤ Cstar * rateScale n d ε
Proof (Lean source)
theorem observable_upper :
∃ (H₀ κ Cstar : ℝ) (D₀ : ℕ),
∃ hH₀ : 0 < H₀, ∃ hκ : 0 < κ, ∃ hD₀ : 2 ≤ D₀,
0 < Cstar ∧
∀ (n d : ℕ) (ε : ℝ),
(hn : 1 ≤ n) → (hd : 2 ≤ d) → (hε : 0 < ε) → (hεhalf : ε ≤ 1 / 2) →
Measurable (armwiseEstimator H₀ κ D₀ hH₀ hκ hD₀ n d ε hn hd ⟨hε, hεhalf⟩) ∧
(∀ sampleLaw : ModelLaw d ε → Measure (Fin n → Obs d),
(∀ P, IidSampling P.1 (sampleLaw P)) →
(⨆ P : ModelLaw d ε,
sqRisk (sampleLaw P)
(armwiseEstimator H₀ κ D₀ hH₀ hκ hD₀ n d ε hn hd ⟨hε, hεhalf⟩) (observedValue P.1)) ≤
Cstar * rateScale n d ε) := by
let H₀ : ℝ := universalH + 1
have hH₀ : 0 < H₀ := by
have h := universalH_pos
dsimp [H₀]
linarith
have hH : universalH ≤ H₀ := by
dsimp [H₀]
linarith
have hHlarge : 1 ≤ H₀ := by
have h := universalH_pos
dsimp [H₀]
linarith
obtain ⟨κ, C, hκ, hC, hmarked⟩ :=
markedPoissonStatistic_active_rate_tuning_exists H₀ hH₀ hH hHlarge
have hL := logAlphabet_pos (d := 2) (by omega)
refine ⟨H₀, κ, 3 + 8 * logAlphabet 2 + 8 * C, 2, hH₀, hκ, by omega,
by positivity, ?_⟩
intro n d ε hn hd hε hεhalf
constructor
· fun_prop
· intro sampleLaw hiid
exact (armwiseEstimator_allBranches_worstRisk_le H₀ κ 2 hH₀ hκ (by omega)
hn hd hε hεhalf hC.le sampleLaw hiid
(fun _ hactive P => hmarked hn (by omega) P.1 ε hε hεhalf P.2 hactive)).2
CausalSmith.Stat.STAT_OptvalueVanishingoverlapRate_Research.TWeightedSeparationAndFrontier 2 declarations Weighted moment-prior separation and the minimax frontier
Weighted moment-prior separation and the minimax frontier
The full weighted approximation comparison and its explicit completed Jackson-packet witness, with a common universal lower constant. The optimal finite dual proves E ≤ Delta; the packet proves the lower order and supplies the required supported, mean-one, moment-matched witness. In the econometric construction, no additional assumptions imply the stated mathematical conclusion. The stated conclusion holds.
Formal statement
conclusion
∃ (c C : ℝ) (A₀ : ℕ),
0 < c
c < C
1 ≤ A₀
conclusion 1
K :
ℕ
M ε :
ℝ
(hK : 2 ≤ K)
(hM : 2 ≤ M)
(hMhi : M ≤ (K : ℝ) ^ 2)
(hε : 0 ≤ ε)
(hεhi : ε ≤ 1 / 2)
c * sqrt M / K ≤ weightedApproxError K M ε
weightedApproxError K M ε ≤ constrainedPriorSeparation K M ε hK hM hMhi hε hεhi
constrainedPriorSeparation K M ε hK hM hMhi hε hεhi ≤ 4 * weightedApproxError K M ε
4 * weightedApproxError K M ε ≤ C * sqrt M / K
∃ (p : ConstrainedPriorPair K M),
(1 / 2 : ℝ)
Proof (Lean source)
theorem weighted_approximation_and_packet :
(∃ (c C : ℝ) (A₀ : ℕ), 0 < c ∧ c < C ∧ 1 ≤ A₀ ∧
∀ (K : ℕ) (M ε : ℝ),
(hK : 2 ≤ K) → (hM : 2 ≤ M) → (hMhi : M ≤ (K : ℝ) ^ 2) →
(hε : 0 ≤ ε) → (hεhi : ε ≤ 1 / 2) →
c * sqrt M / K ≤ weightedApproxError K M ε ∧
weightedApproxError K M ε ≤ constrainedPriorSeparation K M ε hK hM hMhi hε hεhi ∧
constrainedPriorSeparation K M ε hK hM hMhi hε hεhi ≤ 4 * weightedApproxError K M ε ∧
4 * weightedApproxError K M ε ≤ C * sqrt M / K ∧
(∃ (p : ConstrainedPriorPair K M),
∃ hdom : CommonAtomDomain
(packetPositive A₀ K M) (packetNegative A₀ K M),
((packetPositive A₀ K M) univ).toReal ≤ 1 / 2 ∧
(1 / 2 : ℝ) ≤
(1 - ∫ z, z ∂packetPositive A₀ K M) /
(1 - ((packetPositive A₀ K M) univ).toReal) ∧
(1 - ∫ z, z ∂packetPositive A₀ K M) /
(1 - ((packetPositive A₀ K M) univ).toReal) ≤ 2 ∧
commonAtomCompletion
(packetPositive A₀ K M) (packetNegative A₀ K M) hdom =
(p.ν₀, p.ν₁) ∧
c * sqrt M / K ≤
|(∫ z, phiEpsFormula ε z ∂p.ν₁) - (∫ z, phiEpsFormula ε z ∂p.ν₀)|)) := by
obtain ⟨ce, hce, he⟩ := packetWeightedApproxError_lower
obtain ⟨cp, A, hcp, hA, hpacket⟩ := packetCompletion_separation_lower
let c := min (min ce cp) 1
have hc : 0 < c := lt_min (lt_min hce hcp) (by norm_num)
have hce' : c ≤ ce := (min_le_left _ _).trans (min_le_left _ _)
have hcp' : c ≤ cp := (min_le_left _ _).trans (min_le_right _ _)
refine ⟨c, 257, A, hc, lt_of_le_of_lt (min_le_right _ _) (by norm_num), hA, ?_⟩
intro K M ε hK hM hMhi hε hεhi
have hscale : 0 ≤ sqrt M / (K : ℝ) := by positivity
have hlower : c * sqrt M / K ≤ weightedApproxError K M ε := by
calc
_ ≤ ce * sqrt M / K := by
simpa only [mul_div_assoc] using mul_le_mul_of_nonneg_right hce' hscale
_ ≤ _ := he K M ε hK hM hMhi hε hεhi
refine ⟨hlower, weightedApproxError_le_constrainedPriorSeparation hK hM hMhi hε hεhi,
constrainedPriorSeparation_le_four_weightedApproxError K M ε (by linarith), ?_, ?_⟩
· have hu := weightedApproxError_upper hK (M := M) (by linarith) hε hεhi
have h := mul_le_mul_of_nonneg_left hu (by norm_num : (0 : ℝ) ≤ 4)
calc
_ ≤ 4 * (64 * sqrt M / K) := h
_ = 256 * sqrt M / K := by ring
_ ≤ 257 * sqrt M / K := by
simpa only [mul_div_assoc] using
mul_le_mul_of_nonneg_right (by norm_num : (256 : ℝ) ≤ 257) hscale
· obtain ⟨p, hdom, ht, hzlo, hzhi, hcompletion, hgap⟩ :=
hpacket K M ε hK hM hMhi hε hεhi
refine ⟨p, hdom, ht, hzlo, hzhi, hcompletion, le_trans ?_ hgap⟩
simpa only [mul_div_assoc] using mul_le_mul_of_nonneg_right hcp' hscale
theorem weighted_separation_and_frontier
The weighted approximation error, constrained-prior separation, and an explicit completed packet pair all have the square-root mass scale; universal constants then match the observed and causal minimax risks to the vanishing-overlap rate, and an estimator sequence is uniformly consistent exactly when the alphabet-to-effective-sample-size ratio tends to zero.
Formal statement
conclusion
conclusion 1
∃ (c C : ℝ) (A₀ : ℕ),
0 < c
c < C
1 ≤ A₀
K :
ℕ
M ε :
ℝ
(hK : 2 ≤ K)
(hM : 2 ≤ M)
(hMhi : M ≤ (K : ℝ) ^ 2)
(hε : 0 ≤ ε)
(hεhi : ε ≤ 1 / 2)
c * sqrt M / K ≤ weightedApproxError K M ε
weightedApproxError K M ε ≤ constrainedPriorSeparation K M ε hK hM hMhi hε hεhi
constrainedPriorSeparation K M ε hK hM hMhi hε hεhi ≤ 4 * weightedApproxError K M ε
4 * weightedApproxError K M ε ≤ C * sqrt M / K
∃ (p : ConstrainedPriorPair K M),
(1 / 2 : ℝ)
conclusion 2
∃ (cstar Cstar H₀ κ : ℝ) (D₀ : ℕ), ∃ hH₀ : 0 < H₀, ∃ hκ : 0 < κ, ∃ hD₀ : 2 ≤ D₀,
0 < cstar
cstar ≤ Cstar
n d :
ℕ
ε :
ℝ
1 ≤ n
2 ≤ d
0 < ε
ε ≤ 1 / 2
cstar * rateScale n d ε ≤ minimaxRisk n d ε
minimaxRisk n d ε ≤ armwiseWorstRisk H₀ κ D₀ hH₀ hκ hD₀ n d ε
armwiseWorstRisk H₀ κ D₀ hH₀ hκ hD₀ n d ε ≤ Cstar * rateScale n d ε
cstar * rateScale n d ε ≤ causalMinimaxRisk n d ε
causalMinimaxRisk n d ε ≤ Cstar * rateScale n d ε
conclusion 3
(∃ est : ∀ k, Estimator (N k) (d k), Tendsto (fun k => worstCaseRiskReal (observedRisk (N k) (d := d k) (ε := ε k)) (est k)) atTop (𝓝 0))
Proof (Lean source)
theorem weighted_separation_and_frontier :
(∃ (c C : ℝ) (A₀ : ℕ), 0 < c ∧ c < C ∧ 1 ≤ A₀ ∧
∀ (K : ℕ) (M ε : ℝ),
(hK : 2 ≤ K) → (hM : 2 ≤ M) → (hMhi : M ≤ (K : ℝ) ^ 2) →
(hε : 0 ≤ ε) → (hεhi : ε ≤ 1 / 2) →
c * sqrt M / K ≤ weightedApproxError K M ε ∧
weightedApproxError K M ε ≤ constrainedPriorSeparation K M ε hK hM hMhi hε hεhi ∧
constrainedPriorSeparation K M ε hK hM hMhi hε hεhi ≤ 4 * weightedApproxError K M ε ∧
4 * weightedApproxError K M ε ≤ C * sqrt M / K ∧
(∃ (p : ConstrainedPriorPair K M),
∃ hdom : CommonAtomDomain
(packetPositive A₀ K M) (packetNegative A₀ K M),
((packetPositive A₀ K M) univ).toReal ≤ 1 / 2 ∧
(1 / 2 : ℝ) ≤
(1 - ∫ z, z ∂packetPositive A₀ K M) /
(1 - ((packetPositive A₀ K M) univ).toReal) ∧
(1 - ∫ z, z ∂packetPositive A₀ K M) /
(1 - ((packetPositive A₀ K M) univ).toReal) ≤ 2 ∧
commonAtomCompletion
(packetPositive A₀ K M) (packetNegative A₀ K M) hdom =
(p.ν₀, p.ν₁) ∧
c * sqrt M / K ≤
|(∫ z, phiEpsFormula ε z ∂p.ν₁) - (∫ z, phiEpsFormula ε z ∂p.ν₀)|)) ∧
(∃ (cstar Cstar H₀ κ : ℝ) (D₀ : ℕ),
∃ hH₀ : 0 < H₀, ∃ hκ : 0 < κ, ∃ hD₀ : 2 ≤ D₀,
0 < cstar ∧ cstar ≤ Cstar ∧
∀ (n d : ℕ) (ε : ℝ),
1 ≤ n → 2 ≤ d → 0 < ε → ε ≤ 1 / 2 →
cstar * rateScale n d ε ≤ minimaxRisk n d ε ∧
minimaxRisk n d ε ≤ armwiseWorstRisk H₀ κ D₀ hH₀ hκ hD₀ n d ε ∧
armwiseWorstRisk H₀ κ D₀ hH₀ hκ hD₀ n d ε ≤ Cstar * rateScale n d ε ∧
cstar * rateScale n d ε ≤ causalMinimaxRisk n d ε ∧
causalMinimaxRisk n d ε ≤ Cstar * rateScale n d ε) ∧
(∀ (N d : ℕ → ℕ) (ε : ℕ → ℝ),
Tendsto N atTop atTop →
(∀ k, 2 ≤ d k ∧ 0 < ε k ∧ ε k ≤ 1 / 2) →
((∃ est : ∀ k, Estimator (N k) (d k),
Tendsto
(fun k => worstCaseRiskReal
(observedRisk (N k) (d := d k) (ε := ε k)) (est k))
atTop (𝓝 0)) ↔
Tendsto (fun k => (d k : ℝ) /
((N k : ℝ) * ε k * logAlphabet (d k))) atTop (𝓝 0))) := by
refine ⟨weighted_approximation_and_packet, ?_⟩
obtain ⟨cl, hcl, hlower⟩ := allAlphabet_minimaxRisk_lower
obtain ⟨H₀, κ, Cu, D₀, hH₀, hκ, hD₀, hCu, hupper⟩ := observable_upper
let cstar := min cl Cu
have hcstar : 0 < cstar := lt_min hcl hCu
have hl : ∀ n d ε, 1 ≤ n → 2 ≤ d → 0 < ε → ε ≤ 1 / 2 →
cstar * rateScale n d ε ≤ minimaxRisk n d ε := by
intro n d ε hn hd hε hεhi
exact (mul_le_mul_of_nonneg_right (min_le_left _ _)
(rateScale_nonneg (by omega) hε.le)).trans (hlower n d ε hn hd hε hεhi)
have hu : ∀ n d ε, 1 ≤ n → 2 ≤ d → 0 < ε → ε ≤ 1 / 2 →
armwiseWorstRisk H₀ κ D₀ hH₀ hκ hD₀ n d ε ≤ Cu * rateScale n d ε := by
intro n d ε hn hd hε hεhi
exact (hupper n d ε hn hd hε hεhi).2 (fun P => productLaw P.1 n) (fun _ => rfl)
constructor
· refine ⟨cstar, Cu, H₀, κ, D₀, hH₀, hκ, hD₀, hcstar, min_le_right _ _, ?_⟩
intro n d ε hn hd hε hεhi
have hlo := hl n d ε hn hd hε hεhi
have hmid := minimaxRisk_le_armwiseWorstRisk H₀ κ D₀ hH₀ hκ hD₀ n d ε
have hhi := hu n d ε hn hd hε hεhi
have hcausal := causalMinimaxRisk_bounds_of_observed hd hε hεhi hlo (hmid.trans hhi)
exact ⟨hlo, hmid, hhi, hcausal⟩
· intro N d ε hN hlegal
exact uniformConsistency_iff_of_frontier cstar Cu H₀ κ D₀ hcstar hH₀ hκ hD₀
hl hu N d ε hN hlegal