Formalization: Private Estimation and Honest Inference for Conditional Treatment Effects with Rough Nuisance Functions
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.
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Basic 51 declarations Concrete causal laws, selected observational versions, private kernels and extended-real minimax criteria.
Concrete causal laws, selected observational versions, private kernels and extended-real minimax criteria. Published inequalities are explicit conditional gates.
The covariate takes values in the closed unit interval.
Definition (Lean source)
def
given by
def O
Observed records consist of a covariate and two binary marks for treatment and outcome.
def Record
Full causal records also carry the two binary potential outcomes.
def Dataset
A dataset is a finite tuple of observed records indexed by the public sample size.
Definition (Lean source)
def bit
A binary mark is represented numerically by zero or one.
Definition (Lean source)
def X
The covariate coordinate of a full causal record.
The binary treatment coordinate of a full causal record.
def Ypot
The control and treated potential outcomes of a full causal record.
The observed binary outcome coordinate of a full causal record.
def observe
Observation forgets the potential outcomes and retains covariate, treatment and outcome.
Definition (Lean source)
The covariate coordinate is measurable. The displayed conclusion follows.
Formal statement
conclusion
Proof (Lean source)
-- @node: measurable_X
@[fun_prop] lemma measurable_X : Measurable X := by
unfold X
fun_prop
The public regularity radius is fixed at three.
Definition (Lean source)
def
L :
ℝ
given by
3
The evaluation point one half belongs to the closed unit interval. the asserted conclusion follows.
Formal statement
conclusion
(1 / 2 : ℝ) ∈ Icc 0 1
Proof (Lean source)
-- @node: half_mem_unitInterval
lemma half_mem_unitInterval : (1 / 2 : ℝ) ∈ Icc 0 1 := by
constructor <;> norm_num
def x0
The public evaluation point is one half.
Definition (Lean source)
def
given by
⟨1/2, half_mem_unitInterval⟩
The version identities pin the density and the three observational regressions to the law. The arm means are integrable for their arm-specific covariate measures, without interior bounds.
Definition (Lean source)
fields
probability :
f :
Covariate → ℝ
e :
Covariate → ℝ
mu0 :
Covariate → ℝ
mu1 :
Covariate → ℝ
f_measurable :
e_measurable :
mu0_measurable :
Measurable mu0
mu1_measurable :
Measurable mu1
e_integrable :
Integrable e (law.map X)
mu0_integrable :
mu1_integrable :
e_version :
if
then
∫ x in s, e x ∂(law.map X) = law.real {z | X z ∈ s ∧ A z = true}
mu0_version :
instance instIsProbabilityMeasureRecordLaw
The bundled causal law carries its probability-measure instance.
Definition (Lean source)
instance (P : CausalLaw) : IsProbabilityMeasure P.law := P.probability
def Pobs
The observed marginal is the pushforward of the causal law by observation.
def PX
The design marginal is the pushforward of the causal law by the covariate coordinate.
def tau
The selected observational contrast is the treated regression minus the control regression.
The point target evaluates the selected contrast at one half.
The dataset law is the finite product of the observed marginal.
The sampling hypothesis identifies the dataset distribution with that observed product law.
The observed outcome agrees almost surely with the potential outcome selected by treatment.
The potential-outcome pair and treatment are conditionally independent given the covariate.
Definition (Lean source)
parameters
def
Exchangeability P :
Prop
given by
The design marginal is absolutely continuous and its selected density is between one half and three halves almost everywhere for Lebesgue measure.
The selected propensity is between one quarter and three quarters at every covariate.
Definition (Lean source)
parameters
x :
def
Overlap P :
Prop
given by
clause 1
1/4 ≤ P.e x
clause 2
P.e x ≤ 3/4
The selected propensity obeys the one-tenth Holder modulus with radius three.
Definition (Lean source)
parameters
x x' :
def
PropensityHolder P :
Prop
given by
|P.e x - P.e x'| ≤ L * |(x : ℝ) - x'| ^ (1/10 : ℝ)
The selected control regression obeys the one-tenth Holder modulus with radius three.
Definition (Lean source)
parameters
x x' :
def
ControlHolder P :
Prop
given by
|P.mu0 x - P.mu0 x'| ≤ L * |(x : ℝ) - x'| ^ (1/10 : ℝ)
The selected contrast obeys the Lipschitz modulus with radius three.
Definition (Lean source)
parameters
x x' :
def
ContrastLipschitz P :
Prop
given by
|tau P x - tau P x'| ≤ L * |(x : ℝ) - x'|
structure CompleteModel
The complete law class bundles consistency, exchangeability, density, overlap and the three specified regularity conditions.
Definition (Lean source)
parameters
assumes
consistency :
Consistency P
exchangeability :
density :
overlap :
Overlap P
propensity_holder :
control_holder :
contrast_lipschitz :
def dHam
The Hamming distance counts the records at which two datasets differ.
Definition (Lean source)
Replacement privacy bounds every measurable output event on every adjacent pair of input datasets.
structure PrivateKernel
The private-kernel predicate bundles Markov normalization and replacement privacy.
Definition (Lean source)
parameters
assumes
markov :
pure_privacy :
PurePrivacy n epsilon M
def TV
Total variation is the substrate distance between probability measures on a common measurable space.
Definition (Lean source)
def WH
Hamming transport is the extended-real infimum of expected dataset Hamming distance over all couplings.
def IntervalCode
Connected Borel intervals are represented by extended endpoints and inclusion flags, with a separate empty code.
The empty-or-endpoints representation has the Borel sigma algebra of its sum topology. This representation property is derived locally, rather than assumed by headline consumers. the asserted conclusion follows.
Formal statement
conclusion
Proof (Lean source)
-- @node: intervalCode_borelSpace
lemma intervalCode_borelSpace : BorelSpace IntervalCode := by
constructor
apply le_antisymm
· intro s hs
have hparts := measurableSet_sum_iff.mp hs
letI : MeasurableSpace IntervalCode := borel IntervalCode
letI : BorelSpace IntervalCode := ⟨rfl⟩
have hel : MeasurableEmbedding (Sum.inl : Unit → IntervalCode) :=
Topology.IsOpenEmbedding.inl.measurableEmbedding
have her : MeasurableEmbedding (Sum.inr : EReal × EReal × Bool × Bool → IntervalCode) :=
Topology.IsOpenEmbedding.inr.measurableEmbedding
have hl := hel.measurableSet_image.mpr hparts.1
have hr := her.measurableSet_image.mpr hparts.2
simpa only [Set.image_preimage_inl_union_image_preimage_inr] using hl.union hr
· apply MeasurableSpace.generateFrom_le
intro s hs
exact measurableSet_sum_iff.mpr
⟨(hs.preimage continuous_inl).measurableSet,
(hs.preimage continuous_inr).measurableSet⟩
instance instBorelSpaceIntervalCode
The interval representation carries its derived Borel-space instance.
Definition (Lean source)
def toSet
An interval code denotes the corresponding connected subset of the real line.
def leb
The length of a coded interval is its possibly infinite Lebesgue measure.
Definition (Lean source)
def scalarRisk
Absolute risk integrates both the observed dataset distribution and randomized scalar output.
Expected interval length integrates both the observed dataset distribution and randomized interval output.
Definition (Lean source)
Coverage is the probability that the randomized coded interval contains the law-specific point target.
Maximal absolute risk is the extended-real supremum across the complete law class.
Definition (Lean source)
Maximal expected length is the extended-real supremum across the complete law class.
Definition (Lean source)
parameters
n :
ℕ
I :
def
worstLength n I :
ℝ≥0∞
given by
⨆ P, ⨆ (_ : CompleteModel P), expectedLength n I P
def nonprivateMinimaxRisk
The same absolute-risk criterion without a privacy restriction.
Definition (Lean source)
parameters
n :
ℕ
def
nonprivateMinimaxRisk n :
ℝ≥0∞
given by
Private minimax absolute risk takes the infimum of maximal risk over all total private Markov scalar releases.
Definition (Lean source)
parameters
n :
ℕ
epsilon :
ℝ
def
privateMinimaxRisk n epsilon :
ℝ≥0∞
given by
The honest private length frontier takes the infimum of maximal expected length over private Markov interval releases with class-wide coverage at least nine tenths.
Definition (Lean source)
parameters
n :
ℕ
epsilon :
ℝ
def
privateHonestLength n epsilon :
ℝ≥0∞
given by
⨅ (I : Kernel (Dataset n) IntervalCode) (_ : IsMarkovKernel I ∧ PrivateKernel n epsilon I ∧ ∀ P, CompleteModel P → 9/10 ≤ coverage n I P), worstLength n I
Boucheron, Lugosi and Massart (2013), Theorem 3.1, printed p.54, replacement-copy form. Source: https://math.hse.ru/data/2016/11/24/1113029206/Concentration%20inequalities.pdf DOI: 10.1093/acprof:oso/9780199535255.001.0001. This published claim is an assumed gate.
Definition (Lean source)
def
EfronSteinReplacement :
Sort 0
given by
if
letI : ∀ i, MeasurableSpace (E i) := mE ∀ (_ : ∀ i, IsProbabilityMeasure (μ i)) (Ψ : (∀ i, E i) → ℝ), Measurable Ψ
and
MemLp Ψ 2 (Measure.pi μ)
Boucheron, Lugosi and Massart (2013), Theorem 6.2, displayed log-mgf bound in the proof, printed p.167. Source: https://math.hse.ru/data/2016/11/24/1113029206/Concentration%20inequalities.pdf . DOI: 10.1093/acprof:oso/9780199535255.001.0001. Positive parameters are exponentiated and the zero parameter is included by equality. This is an assumed gate.
Definition (Lean source)
def
BoundedDifferencesMGF :
Sort 0
given by
letI : ∀ i, MeasurableSpace (E i) := mE ∀ (_ : ∀ i, IsProbabilityMeasure (μ i)) (Ψ : (∀ i, E i) → ℝ), Measurable Ψ → (∃ C : ℝ, ∀ z, |Ψ z| ≤ C) → ∀ c : Fin s → ℝ, (∀ i, 0 ≤ c i) → (∀ i z b, |Ψ z - Ψ (update z i b)| ≤ c i) → ∀ u : ℝ, 0
Keller and Trotter (2017), Applied Combinatorics, Section 5.6, Theorem 5.40. Source: https://appliedcombinatorics.org/book/s_graphs_counting-trees.html . Nat.card counts the finite subtype of undirected labeled trees. This is an assumed gate.
Definition (Lean source)
def
CayleyLabeledTreeCount :
Sort 0
given by
∀ s : ℕ
if
2 ≤ s
then
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.BinaryExchangeability 4 declarations Conditional exchangeability of the explicit independent Bernoulli construction.
Conditional exchangeability of the explicit independent Bernoulli construction.
Density coordinates are sent to full records by selecting the observed outcome.
Definition (Lean source)
parameters
def
given by
(z.2.2, z.1, z.2.1, if z.1 then z.2.1.2 else z.2.1.1)
theorem measurable_binaryRecordOfDensity
The consistency selection map is measurable. The displayed conclusion follows.
Formal statement
conclusion
Proof (Lean source)
-- @node: measurable_binaryRecordOfDensity
@[fun_prop] lemma measurable_binaryRecordOfDensity : Measurable binaryRecordOfDensity := by
unfold binaryRecordOfDensity
have hy : Measurable (fun z : DensityCoord Covariate =>
if z.1 then z.2.1.2 else z.2.1.1) :=
Measurable.ite (measurable_fst (MeasurableSet.singleton true)) (by fun_prop) (by fun_prop)
exact measurable_snd.snd.prodMk (measurable_fst.prodMk
(measurable_snd.fst.prodMk hy))
theorem binaryLaw_eq_map_densityLaw
The paper's finite conditional law is the pushforward of the library's product density law. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: binaryLaw_eq_map_densityLaw
lemma binaryLaw_eq_map_densityLaw (a b c : Covariate → ℝ)
(ha : Measurable a) (hb : Measurable b) (hc : Measurable c) :
binaryLaw a b c = (densityLaw volume a b c).map binaryRecordOfDensity := by
classical
have hbit (p : Covariate → ℝ) (hp : Measurable p)
(f : DensityCoord Covariate → Bool) (hf : Measurable f) :
Measurable (fun z : DensityCoord Covariate => bitMass (p z.2.2) (f z)) := by
unfold bitMass
exact Measurable.ite (hf (MeasurableSet.singleton true))
(hp.comp (by fun_prop)) (measurable_const.sub (hp.comp (by fun_prop)))
have hd : Measurable (density a b c) := by
unfold density tripleMass
exact ENNReal.measurable_ofReal.comp
(((hbit a ha _ (by fun_prop)).mul (hbit b hb _ (by fun_prop))).mul
(hbit c hc _ (by fun_prop)))
ext s hs
have hpre : MeasurableSet (binaryRecordOfDensity ⁻¹' s) := measurable_binaryRecordOfDensity hs
rw [binaryLaw, Measure.bind_apply hs
(measurable_binaryRecordLaw a b c ha hb hc).aemeasurable,
Measure.map_apply measurable_binaryRecordOfDensity hs, densityLaw,
withDensity_apply _ (measurable_binaryRecordOfDensity hs),
← lintegral_indicator (measurable_binaryRecordOfDensity hs), reference,
lintegral_prod_symm' _ (hd.indicator (measurable_binaryRecordOfDensity hs))]
simp only [lintegral_count, tsum_fintype]
rw [lintegral_prod_symm' _ (by fun_prop)]
apply lintegral_congr
intro x
rw [lintegral_prod _ (by exact (measurable_of_finite _).aemeasurable)]
simp only [lintegral_count, tsum_fintype, Measure.finsetSum_apply,
Measure.smul_apply, Measure.dirac_apply' _ hs, smul_eq_mul,
Fintype.sum_prod_type]
simp only [Fintype.sum_bool, Set.indicator_apply, density, tripleMass,
bitMass, bernoulliMass, binaryRecordOfDensity, Set.mem_preimage,
Bool.false_eq_true, if_false, if_true, Pi.one_apply, mul_ite, mul_zero, mul_one]
ac_rfl
theorem binaryCausalLaw_exchangeability
Independent treatment and potential-outcome marks remain conditionally independent when the observed consistency outcome is appended to the record. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Exchangeability (binaryCausalLaw a b c ha hb hc hr)
Proof (Lean source)
-- @node: binaryCausalLaw_exchangeability
lemma binaryCausalLaw_exchangeability (a b c : Covariate → ℝ)
(ha : Measurable a) (hb : Measurable b) (hc : Measurable c)
(hr : ∀ x, a x ∈ Icc 0 1 ∧ b x ∈ Icc 0 1 ∧ c x ∈ Icc 0 1) :
Exchangeability (binaryCausalLaw a b c ha hb hc hr) := by
letI : IsProbabilityMeasure (densityLaw volume a b c) :=
densityLaw_probability volume a b c ha hb hc
(fun x => (hr x).1) (fun x => (hr x).2.1) (fun x => (hr x).2.2)
letI : IsProbabilityMeasure ((densityLaw volume a b c).map binaryRecordOfDensity) :=
Measure.isProbabilityMeasure_map measurable_binaryRecordOfDensity.aemeasurable
letI : IsProbabilityMeasure (binaryLaw a b c) := binaryLaw_probability a b c ha hb hc hr
have hind : CondIndepFun (comap X inferInstance)
measurable_X.comap_le Ypot A
((densityLaw volume a b c).map binaryRecordOfDensity) := by
apply condIndepFun_of_map
measurable_binaryRecordOfDensity (by unfold Ypot; fun_prop)
(by unfold A; fun_prop) measurable_X
exact (densityLaw_condIndepFun volume a b c ha hb hc
(fun x => (hr x).1) (fun x => (hr x).2.1) (fun x => (hr x).2.2)).symm
simpa only [Exchangeability, binaryCausalLaw,
← binaryLaw_eq_map_densityLaw a b c ha hb hc] using hind
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.BinaryLaw 26 declarations Uniform-design binary causal laws and their conditional regression identities.
Uniform-design binary causal laws and their conditional regression identities.
def bernoulliMass
The binary probability mass is the specified success probability for one and its complement for zero.
Definition (Lean source)
A valid Bernoulli parameter gives nonnegative mass to either binary outcome. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: bernoulliMass_nonneg
lemma bernoulliMass_nonneg {p : ℝ} (hp : p ∈ Icc 0 1) (z : Bool) :
0 ≤ bernoulliMass p z := by
cases z <;> simp only [bernoulliMass, Bool.false_eq_true, if_false, if_true]
· linarith [hp.2]
· exact hp.1
The two nonnegative Bernoulli masses add to one in extended real arithmetic. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: bernoulliMass_sum
lemma bernoulliMass_sum {p : ℝ} (hp : p ∈ Icc 0 1) :
∑ z : Bool, ofReal (bernoulliMass p z) = 1 := by
rw [Fintype.sum_bool, ← ENNReal.ofReal_add
(bernoulliMass_nonneg hp true) (bernoulliMass_nonneg hp false)]
simp [bernoulliMass]
Three independent valid Bernoulli masses have unit total mass. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
a b c :
ℝ
ha :
a ∈ Icc 0 1
hb :
b ∈ Icc 0 1
hc :
c ∈ Icc 0 1
conclusion
∑ z : Bool × Bool × Bool, ofReal (bernoulliMass a z.1 * bernoulliMass b z.2.1 * bernoulliMass c z.2.2)
= 1
Proof (Lean source)
-- @node: bernoulliTripleMass_sum
lemma bernoulliTripleMass_sum {a b c : ℝ} (ha : a ∈ Icc 0 1)
(hb : b ∈ Icc 0 1) (hc : c ∈ Icc 0 1) :
∑ z : Bool × Bool × Bool,
ofReal (bernoulliMass a z.1 * bernoulliMass b z.2.1 *
bernoulliMass c z.2.2) = 1 := by
have hn (z : Bool × Bool × Bool) :
ofReal (bernoulliMass a z.1 * bernoulliMass b z.2.1 *
bernoulliMass c z.2.2) =
ofReal (bernoulliMass a z.1) *
ofReal (bernoulliMass b z.2.1) *
ofReal (bernoulliMass c z.2.2) := by
rw [ENNReal.ofReal_mul (mul_nonneg (bernoulliMass_nonneg ha _)
(bernoulliMass_nonneg hb _)),
ENNReal.ofReal_mul (bernoulliMass_nonneg ha _)]
simp_rw [hn, Fintype.sum_prod_type, ← Finset.mul_sum,
bernoulliMass_sum hc, mul_one, ← Finset.mul_sum,
bernoulliMass_sum hb, mul_one, bernoulliMass_sum ha]
def binaryLaw
A uniform covariate draw is followed by independent Bernoulli treatment and potential outcomes and the consistency observation map.
Definition (Lean source)
parameters
a b c :
Covariate → ℝ
given by
(volume : Measure Covariate).bind (fun x => ∑ z : Bool × Bool × Bool, ofReal (bernoulliMass (a x) z.1 * bernoulliMass (b x) z.2.1 * bernoulliMass (c x) z.2.2) • Measure.dirac (x, z.1, (z.2.1,z.2.2), if z.1 then z.2.2 else z.2.1))
The finite conditional Bernoulli record law varies measurably with the covariate. the asserted conclusion follows.
Formal statement
hypotheses
a :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
b :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
c :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
conclusion
Measurable (fun x : Covariate => ∑ z : Bool × Bool × Bool, ofReal (bernoulliMass (a x) z.1 * bernoulliMass (b x) z.2.1 * bernoulliMass (c x) z.2.2) • Measure.dirac (x, z.1, (z.2.1,z.2.2), if z.1 then z.2.2 else z.2.1))
Proof (Lean source)
-- @node: measurable_binaryRecordLaw
lemma measurable_binaryRecordLaw : Measurable (fun x : Covariate =>
∑ z : Bool × Bool × Bool,
ofReal (bernoulliMass (a x) z.1 * bernoulliMass (b x) z.2.1 *
bernoulliMass (c x) z.2.2) •
Measure.dirac (x, z.1, (z.2.1,z.2.2), if z.1 then z.2.2 else z.2.1)) := by
apply Measure.measurable_of_measurable_coe
intro s hs
simp only [Measure.finsetSum_apply, Measure.smul_apply, smul_eq_mul]
apply Finset.measurable_fun_sum
intro z hz
apply mul
· unfold bernoulliMass
cases z.1 <;> cases z.2.1 <;> cases z.2.2 <;> simp only [if_true, if_false,
Bool.false_eq_true] <;> fun_prop
· exact (Measure.measurable_coe hs).comp (Measure.measurable_dirac.comp (by fun_prop))
The finite Bernoulli construction is a probability law for measurable probability-valued parameters. the asserted conclusion follows.
Formal statement
hypotheses
a :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
b :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
c :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
conclusion
IsProbabilityMeasure (binaryLaw a b c)
Proof (Lean source)
-- @node: binaryLaw_probability
lemma binaryLaw_probability : IsProbabilityMeasure (binaryLaw a b c) := by
rw [isProbabilityMeasure_iff, binaryLaw, Measure.bind_apply MeasurableSet.univ]
· have hm : ∀ x : Covariate,
(∑ z : Bool × Bool × Bool,
ofReal (bernoulliMass (a x) z.1 * bernoulliMass (b x) z.2.1 *
bernoulliMass (c x) z.2.2) •
Measure.dirac (x, z.1, (z.2.1,z.2.2), if z.1 then z.2.2 else z.2.1))
univ = 1 := by
intro x
simp only [Measure.finsetSum_apply, Measure.smul_apply, Measure.dirac_apply_of_mem
(Set.mem_univ _), smul_eq_mul, mul_one]
exact bernoulliTripleMass_sum (hr x).1 (hr x).2.1 (hr x).2.2
simp_rw [hm]
simp
· exact (measurable_binaryRecordLaw a b c ha hb hc).aemeasurable
The finite Bernoulli construction has the uniform design density. the asserted conclusion follows.
Formal statement
hypotheses
a :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
b :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
c :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
Proof (Lean source)
-- @node: binaryLaw_design
lemma binaryLaw_design :
(binaryLaw a b c).map X = volume := by
classical
ext s hs
rw [Measure.map_apply measurable_X hs, binaryLaw,
Measure.bind_apply (measurable_X hs)
(measurable_binaryRecordLaw a b c ha hb hc).aemeasurable]
have hm : ∀ x : Covariate,
(∑ z : Bool × Bool × Bool,
ofReal (bernoulliMass (a x) z.1 * bernoulliMass (b x) z.2.1 *
bernoulliMass (c x) z.2.2) •
Measure.dirac (x, z.1, (z.2.1,z.2.2), if z.1 then z.2.2 else z.2.1))
(X ⁻¹' s) = s.indicator (fun _ => (1 : ℝ≥0∞)) x := by
intro x
simp only [Measure.finsetSum_apply, Measure.smul_apply,
Measure.dirac_apply' _ (measurable_X hs), Set.indicator_apply,
Set.mem_preimage, X, smul_eq_mul, Pi.one_apply]
change (∑ z : Bool × Bool × Bool,
ofReal (bernoulliMass (a x) z.1 * bernoulliMass (b x) z.2.1 *
bernoulliMass (c x) z.2.2) * (if x ∈ s then 1 else 0)) =
s.indicator (fun _ => (1 : ℝ≥0∞)) x
by_cases hx : x ∈ s
· simp only [hx, if_true, mul_one, Set.indicator_of_mem hx]
exact bernoulliTripleMass_sum (hr x).1 (hr x).2.1 (hr x).2.2
· simp [hx]
simp_rw [hm]
simp [hs]
The uniform design has the selected constant density one. The displayed conclusion follows.
Formal statement
hypotheses
a :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
b :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
c :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
hr :
∀ (x : ↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate),
The selected propensity is integrable under the constructed design marginal. the asserted conclusion follows.
Formal statement
hypotheses
a :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
b :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
c :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
conclusion
Proof (Lean source)
-- @node: binaryLaw_e_integrable
lemma binaryLaw_e_integrable : Integrable a ((binaryLaw a b c).map X) := by
let := binaryLaw_probability a b c ha hb hc hr
apply Integrable.of_mem_Icc 0 1 (by fun_prop)
exact ae_of_all _ (fun x => (hr x).1)
The selected control mean is integrable under the constructed control-arm covariate measure. the asserted conclusion follows.
Formal statement
hypotheses
a :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
b :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
c :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
conclusion
Proof (Lean source)
-- @node: binaryLaw_mu0_integrable
lemma binaryLaw_mu0_integrable :
Integrable b (((binaryLaw a b c).restrict {z | A z = false}).map X) := by
let := binaryLaw_probability a b c ha hb hc hr
apply Integrable.of_mem_Icc 0 1 (by fun_prop)
exact ae_of_all _ (fun x => (hr x).2.1)
The selected treated mean is integrable under the constructed treated-arm covariate measure. the asserted conclusion follows.
Formal statement
hypotheses
a :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
b :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
c :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
conclusion
Proof (Lean source)
-- @node: binaryLaw_mu1_integrable
lemma binaryLaw_mu1_integrable :
Integrable c (((binaryLaw a b c).restrict {z | A z = true}).map X) := by
let := binaryLaw_probability a b c ha hb hc hr
apply Integrable.of_mem_Icc 0 1 (by fun_prop)
exact ae_of_all _ (fun x => (hr x).2.2)
The constructed propensity satisfies the conditional probability identity on every measurable covariate set. the asserted conclusion follows.
Formal statement
hypotheses
a :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
b :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
c :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
hr :
∀ (x : ↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate),
Proof (Lean source)
-- @node: binaryLaw_e_version
lemma binaryLaw_e_version : ∀ s : Set Covariate, MeasurableSet s →
∫ x in s, a x ∂((binaryLaw a b c).map X) =
(binaryLaw a b c).real {z | X z ∈ s ∧ A z = true} := by
classical
intro s hs
have hset : MeasurableSet {z : Record | X z ∈ s ∧ A z = true} := by
apply MeasurableSet.inter (measurable_X hs)
exact measurableSet_eq_fun (by unfold A; fun_prop) measurable_const
rw [binaryLaw_design a b c ha hb hc hr,
integral_eq_lintegral_of_nonneg_ae (ae_of_all _ (fun x => (hr x).1.1))
ha.aestronglyMeasurable, measureReal_def, binaryLaw,
Measure.bind_apply hset (measurable_binaryRecordLaw a b c ha hb hc).aemeasurable]
congr 1
have hm : ∀ x : Covariate,
(∑ z : Bool × Bool × Bool,
ofReal (bernoulliMass (a x) z.1 * bernoulliMass (b x) z.2.1 *
bernoulliMass (c x) z.2.2) •
Measure.dirac (x, z.1, (z.2.1,z.2.2), if z.1 then z.2.2 else z.2.1))
{z | X z ∈ s ∧ A z = true} =
s.indicator (fun x => ofReal (a x)) x := by
intro x
simp only [Measure.finsetSum_apply, Measure.smul_apply,
Measure.dirac_apply' _ hset, smul_eq_mul]
by_cases hx : x ∈ s
· have ha0 := (hr x).1.1
have hb0 := (hr x).2.1.1
have hc0 := (hr x).2.2.1
have hb1 : 0 ≤ 1 - b x := sub_nonneg.mpr (hr x).2.1.2
have hc1 : 0 ≤ 1 - c x := sub_nonneg.mpr (hr x).2.2.2
simp only [Fintype.sum_prod_type, Fintype.sum_bool, bernoulliMass, X, A,
Set.indicator_apply, Pi.one_apply, Set.mem_setOf_eq,
hx, Bool.false_eq_true, and_false, and_true, if_true, if_false,
Set.indicator_of_mem hx, mul_one, mul_zero, add_zero, zero_add,
Set.mem_setOf_eq]
rw [← ENNReal.ofReal_add (by positivity) (by positivity)]
rw [← ENNReal.ofReal_add (by positivity) (by positivity)]
rw [← ENNReal.ofReal_add (by positivity) (by positivity)]
congr 1
ring
· simp [X, A, hx, Set.indicator_of_notMem hx]
simp_rw [hm]
rw [lintegral_indicator hs]
Integration under the generated record law is integration of the finite conditional masses. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
a :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
b :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
c :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
f :
Record → ℝ≥0∞
hf :
conclusion
∫⁻ z, f z ∂binaryLaw a b c
= ∫⁻ x : Covariate, ∑ z : Bool × Bool × Bool, ofReal (bernoulliMass (a x) z.1 * bernoulliMass (b x) z.2.1 * bernoulliMass (c x) z.2.2) * f (x, z.1, (z.2.1, z.2.2), if z.1 then z.2.2 else z.2.1)
Proof (Lean source)
-- @node: binaryLaw_lintegral
lemma binaryLaw_lintegral (f : Record → ℝ≥0∞) (hf : Measurable f) :
∫⁻ z, f z ∂binaryLaw a b c =
∫⁻ x : Covariate, ∑ z : Bool × Bool × Bool,
ofReal (bernoulliMass (a x) z.1 * bernoulliMass (b x) z.2.1 *
bernoulliMass (c x) z.2.2) *
f (x, z.1, (z.2.1, z.2.2), if z.1 then z.2.2 else z.2.1) := by
rw [binaryLaw, Measure.lintegral_bind
(measurable_binaryRecordLaw a b c ha hb hc).aemeasurable hf.aemeasurable]
simp only [lintegral_finsetSum_measure, lintegral_smul_measure, lintegral_dirac,
smul_eq_mul]
In each arm, the conditional mean of the observed binary response is its Bernoulli parameter. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
a :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
b :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
c :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
hr :
∀ (x : ↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate),
arm :
hs :
Proof (Lean source)
-- @node: binaryLaw_arm_version
lemma binaryLaw_arm_version (arm : Bool) (s : Set Covariate) (hs : MeasurableSet s) :
∫ x in s, (if arm then c x else b x)
∂(((binaryLaw a b c).restrict {z | A z = arm}).map X) =
∫ z in {z | X z ∈ s ∧ A z = arm}, bit (Y z) ∂binaryLaw a b c := by
classical
have hm : Measurable (fun x => if arm then c x else b x) := by
cases arm <;> simp <;> fun_prop
have ht : MeasurableSet {z : Record | A z = arm} := by
exact measurableSet_eq_fun (by unfold A; fun_prop) measurable_const
have hset : MeasurableSet {z : Record | X z ∈ s ∧ A z = arm} :=
(measurable_X hs).inter ht
rw [setIntegral_map hs hm.aestronglyMeasurable measurable_X.aemeasurable,
Measure.restrict_restrict (measurable_X hs)]
change (∫ z in {z | X z ∈ s ∧ A z = arm}, (if arm then c (X z) else b (X z))
∂binaryLaw a b c) = _
have hmy : Measurable (fun z : Record => bit (Y z)) := by
exact (by fun_prop : Measurable bit).comp (by unfold Y; fun_prop)
have hmf : Measurable (fun z : Record => if arm then c (X z) else b (X z)) :=
hm.comp measurable_X
rw [integral_eq_lintegral_of_nonneg_ae
(ae_of_all _ (fun z => by
cases arm
· exact (hr (X z)).2.1.1
· exact (hr (X z)).2.2.1))
hmf.aestronglyMeasurable,
integral_eq_lintegral_of_nonneg_ae
(ae_of_all _ (fun z => by unfold bit; split <;> norm_num))
hmy.aestronglyMeasurable,
← lintegral_indicator hset, ← lintegral_indicator hset]
congr 1
rw [binaryLaw_lintegral a b c ha hb hc
(fun z => {z | X z ∈ s ∧ A z = arm}.indicator
(fun z => ofReal (if arm then c (X z) else b (X z))) z)
(by simpa only [Function.comp_def] using
(ENNReal.measurable_ofReal.comp hmf).indicator hset),
binaryLaw_lintegral a b c ha hb hc
(fun z => {z | X z ∈ s ∧ A z = arm}.indicator
(fun z => ofReal (bit (Y z))) z)
(by simpa only [Function.comp_def] using
(ENNReal.measurable_ofReal.comp hmy).indicator hset)]
apply lintegral_congr
intro x
have ha0 := (hr x).1.1
have hb0 := (hr x).2.1.1
have hc0 := (hr x).2.2.1
have ha1 : 0 ≤ 1 - a x := sub_nonneg.mpr (hr x).1.2
have hb1 : 0 ≤ 1 - b x := sub_nonneg.mpr (hr x).2.1.2
have hc1 : 0 ≤ 1 - c x := sub_nonneg.mpr (hr x).2.2.2
by_cases hx : x ∈ s
· cases arm <;>
simp only [Fintype.sum_prod_type, Fintype.sum_bool, Set.indicator_apply,
Set.mem_setOf_eq, bernoulliMass, X, A, Y, bit, hx, and_true, and_false,
Bool.false_eq_true, if_true, if_false, ENNReal.ofReal_one, ENNReal.ofReal_zero,
mul_one, mul_zero, zero_add, add_zero]
all_goals
try simp only [Bool.true_eq_false, and_false, if_false, mul_zero, zero_add]
repeat rw [← ENNReal.ofReal_mul (by positivity)]
repeat rw [← ENNReal.ofReal_add (by positivity) (by positivity)]
congr 1
ring
· simp [Set.indicator_apply, X, A, hx]
The constructed control regression satisfies the conditional mean identity on every measurable covariate set. the asserted conclusion follows.
Formal statement
hypotheses
a :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
b :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
c :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
hr :
∀ (x : ↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate),
Proof (Lean source)
-- @node: binaryLaw_mu0_version
lemma binaryLaw_mu0_version : ∀ s : Set Covariate, MeasurableSet s →
∫ x in s, b x ∂(((binaryLaw a b c).restrict {z | A z = false}).map X) =
∫ z in {z | X z ∈ s ∧ A z = false}, bit (Y z) ∂binaryLaw a b c := by
intro s hs
exact binaryLaw_arm_version a b c ha hb hc hr false s hs
The constructed treated regression satisfies the conditional mean identity on every measurable covariate set. the asserted conclusion follows.
Formal statement
hypotheses
a :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
b :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
c :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
hr :
∀ (x : ↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate),
Proof (Lean source)
-- @node: binaryLaw_mu1_version
lemma binaryLaw_mu1_version : ∀ s : Set Covariate, MeasurableSet s →
∫ x in s, c x ∂(((binaryLaw a b c).restrict {z | A z = true}).map X) =
∫ z in {z | X z ∈ s ∧ A z = true}, bit (Y z) ∂binaryLaw a b c := by
intro s hs
exact binaryLaw_arm_version a b c ha hb hc hr true s hs
def binaryCausalLaw
The finite Bernoulli product followed by consistency gives a concrete causal-law bundle.
Definition (Lean source)
parameters
a :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
b :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
c :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
def
given by
clause 2
probability := binaryLaw_probability a b c ha hb hc hr
clause 3
f := fun _ => 1
clause 4
e := a
clause 5
mu0 := b
clause 6
mu1 := c
clause 7
f_measurable := measurable_const
clause 8
e_measurable := ha
clause 9
mu0_measurable := hb
clause 10
mu1_measurable := hc
clause 11
density_version := binaryLaw_density_version a b c ha hb hc hr
clause 12
e_integrable := binaryLaw_e_integrable a b c ha hb hc hr
clause 13
mu0_integrable := binaryLaw_mu0_integrable a b c ha hb hc hr
clause 14
mu1_integrable := binaryLaw_mu1_integrable a b c ha hb hc hr
clause 15
e_version := binaryLaw_e_version a b c ha hb hc hr
clause 16
mu0_version := binaryLaw_mu0_version a b c ha hb hc hr
clause 17
mu1_version := binaryLaw_mu1_version a b c ha hb hc hr
theorem binaryCausalLaw_consistency
Every generated record uses the potential outcome selected by its treatment. the asserted conclusion follows.
Formal statement
hypotheses
a :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
b :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
c :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
conclusion
Consistency (binaryCausalLaw a b c ha hb hc hr)
Proof (Lean source)
-- @node: binaryCausalLaw_consistency
lemma binaryCausalLaw_consistency : Consistency (binaryCausalLaw a b c ha hb hc hr) := by
classical
change ∀ᵐ z ∂binaryLaw a b c, Y z = if A z then (Ypot z).2 else (Ypot z).1
rw [ae_iff]
have hs : MeasurableSet {z : Record | ¬ (Y z = if A z then (Ypot z).2 else (Ypot z).1)} := by
have hA : Measurable A := by unfold A; fun_prop
have hY : Measurable Y := by unfold Y; fun_prop
have hp0 : Measurable (fun z => (Ypot z).1) := by unfold Ypot; fun_prop
have hp1 : Measurable (fun z => (Ypot z).2) := by unfold Ypot; fun_prop
exact (measurableSet_eq_fun hY
(hp1.ite (measurableSet_eq_fun hA measurable_const) hp0)).compl
rw [binaryLaw, Measure.bind_apply hs
(measurable_binaryRecordLaw a b c ha hb hc).aemeasurable]
have hf : ∀ x : Covariate,
(∑ z : Bool × Bool × Bool,
ofReal (bernoulliMass (a x) z.1 * bernoulliMass (b x) z.2.1 *
bernoulliMass (c x) z.2.2) •
Measure.dirac (x, z.1, (z.2.1,z.2.2), if z.1 then z.2.2 else z.2.1))
{z : Record | ¬ (Y z = if A z then (Ypot z).2 else (Ypot z).1)} = 0 := by
intro x
simp only [Measure.finsetSum_apply, Measure.smul_apply, smul_eq_mul]
apply Finset.sum_eq_zero
intro z hz
rw [Measure.dirac_apply' _ hs]
simp only [Set.mem_setOf_eq, Y, A, Ypot]
simp
exact inr rfl
simp_rw [hf]
simp
theorem binaryCausalLaw_densityBounds
The constructed law's selected density is one and its design is exactly uniform. The displayed conclusion follows.
Formal statement
hypotheses
a :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
b :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
c :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
conclusion
DensityBounds (binaryCausalLaw a b c ha hb hc hr)
Proof (Lean source)
-- @node: binaryCausalLaw_densityBounds
lemma binaryCausalLaw_densityBounds : DensityBounds (binaryCausalLaw a b c ha hb hc hr) := by
constructor
· change (binaryLaw a b c).map X ≪ volume
rw [binaryLaw_design a b c ha hb hc hr]
· filter_upwards with x
change (1 / 2 : ℝ) ≤ 1 ∧ (1 : ℝ) ≤ 3 / 2
norm_num
theorem binaryCausalLaw_completeModel
Analytic regularity and conditional exchangeability complete the binary law's model certificate; consistency and the uniform density follow from its construction. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
a :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
b :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
c :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate → ℝ
hr :
∀ (x : ↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate),
hex :
Exchangeability (binaryCausalLaw a b c ha hb hc hr)
hov :
∀ x, 1 / 4 ≤ a x ∧ a x ≤ 3 / 4
haH :
∀ x y, |a x - a y| ≤ L * |(x : ℝ) - y| ^ (1/10 : ℝ)
hbH :
∀ x y, |b x - b y| ≤ L * |(x : ℝ) - y| ^ (1/10 : ℝ)
htL :
∀ x y, |(c x - b x) - (c y - b y)| ≤ L * |(x : ℝ) - y|
conclusion
CompleteModel (binaryCausalLaw a b c ha hb hc hr)
Proof (Lean source)
-- @node: binaryCausalLaw_completeModel
lemma binaryCausalLaw_completeModel
(hex : Exchangeability (binaryCausalLaw a b c ha hb hc hr))
(hov : ∀ x, 1 / 4 ≤ a x ∧ a x ≤ 3 / 4)
(haH : ∀ x y, |a x - a y| ≤ L * |(x : ℝ) - y| ^ (1/10 : ℝ))
(hbH : ∀ x y, |b x - b y| ≤ L * |(x : ℝ) - y| ^ (1/10 : ℝ))
(htL : ∀ x y, |(c x - b x) - (c y - b y)| ≤ L * |(x : ℝ) - y|) :
CompleteModel (binaryCausalLaw a b c ha hb hc hr) := by
exact ⟨binaryCausalLaw_consistency a b c ha hb hc hr, hex,
binaryCausalLaw_densityBounds a b c ha hb hc hr, hov, haH, hbH, htL⟩
An observed event under the binary construction integrates the four conditional mark masses; the unused potential outcome sums out exactly. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
a b c :
Covariate → ℝ
ha :
hb :
hc :
hr :
∀ x, a x ∈ Icc 0 1 ∧ b x ∈ Icc 0 1 ∧ c x ∈ Icc 0 1
s :
Set O
hs :
conclusion
Pobs (binaryCausalLaw a b c ha hb hc hr) s
= ∫⁻ x : Covariate, ∑ z : Bool × Bool, ofReal (bernoulliMass (a x) z.1 * bernoulliMass (if z.1 then c x else b x) z.2) * s.indicator (fun _ => (1 : ℝ≥0∞)) (x, z.1, z.2)
Proof (Lean source)
-- @node: binaryCausalLaw_observed_apply
lemma binaryCausalLaw_observed_apply (a b c : Covariate → ℝ) (ha : Measurable a)
(hb : Measurable b) (hc : Measurable c)
(hr : ∀ x, a x ∈ Icc 0 1 ∧ b x ∈ Icc 0 1 ∧ c x ∈ Icc 0 1)
(s : Set O) (hs : MeasurableSet s) :
Pobs (binaryCausalLaw a b c ha hb hc hr) s =
∫⁻ x : Covariate, ∑ z : Bool × Bool,
ofReal (bernoulliMass (a x) z.1 *
bernoulliMass (if z.1 then c x else b x) z.2) *
s.indicator (fun _ => (1 : ℝ≥0∞)) (x, z.1, z.2) := by
classical
have hm : Measurable observe := by unfold observe X A Y; fun_prop
change (binaryLaw a b c).map observe s = _
rw [Measure.map_apply hm hs, ← lintegral_indicator_one (hm hs)]
rw [binaryLaw_lintegral a b c ha hb hc ((observe ⁻¹' s).indicator (1 : Record → ℝ≥0∞))
(measurable_const.indicator (hm hs))]
apply lintegral_congr
intro x
have ha0 := (hr x).1.1
have hb0 := (hr x).2.1.1
have hc0 := (hr x).2.2.1
have ha1 : 0 ≤ 1 - a x := sub_nonneg.mpr (hr x).1.2
have hb1 : 0 ≤ 1 - b x := sub_nonneg.mpr (hr x).2.1.2
have hc1 : 0 ≤ 1 - c x := sub_nonneg.mpr (hr x).2.2.2
have hmass (av yv : Bool) :
(∑ z : Bool,
ofReal (bernoulliMass (a x) av * bernoulliMass (b x) z *
bernoulliMass (c x) yv)) =
ofReal (bernoulliMass (a x) av * bernoulliMass (c x) yv) := by
cases av <;> cases yv <;>
simp only [Fintype.sum_bool, bernoulliMass, Bool.false_eq_true, if_true, if_false] <;>
rw [← ENNReal.ofReal_add (by positivity) (by positivity)] <;> congr 1 <;> ring
have hmass' (av yv : Bool) :
(∑ z : Bool,
ofReal (bernoulliMass (a x) av * bernoulliMass (b x) yv *
bernoulliMass (c x) z)) =
ofReal (bernoulliMass (a x) av * bernoulliMass (b x) yv) := by
cases av <;> cases yv <;>
simp only [Fintype.sum_bool, bernoulliMass, Bool.false_eq_true, if_true, if_false] <;>
rw [← ENNReal.ofReal_add (by positivity) (by positivity)] <;> congr 1 <;> ring
simp only [Fintype.sum_prod_type, Fintype.sum_bool, Set.indicator_apply,
Set.mem_preimage, observe, X, A, Y, Bool.false_eq_true, if_false, if_true, Pi.one_apply]
have ht := hmass true true
have ht' := hmass true false
have hf := hmass' false true
have hf' := hmass' false false
simp only [Fintype.sum_bool, bernoulliMass, if_true, Bool.false_eq_true, if_false] at ht ht' hf hf'
simp only [bernoulliMass, if_true, Bool.false_eq_true, if_false]
calc
_ = (ofReal (a x * b x * c x) + ofReal (a x * (1-b x) * c x)) *
(if (x, true, true) ∈ s then 1 else 0) +
(ofReal (a x * b x * (1-c x)) + ofReal (a x * (1-b x) * (1-c x))) *
(if (x, true, false) ∈ s then 1 else 0) +
(ofReal ((1-a x) * b x * c x) + ofReal ((1-a x) * b x * (1-c x))) *
(if (x, false, true) ∈ s then 1 else 0) +
(ofReal ((1-a x) * (1-b x) * c x) +
ofReal ((1-a x) * (1-b x) * (1-c x))) *
(if (x, false, false) ∈ s then 1 else 0) := by ring
_ = _ := by rw [ht, ht', hf, hf']; ring
Coordinatewise measure domination is preserved by a finite product. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Measure.pi μ ≤ Measure.pi ν
Proof (Lean source)
-- @node: finite_product_measure_mono
lemma finite_product_measure_mono {ι : Type*} [Fintype ι] {α : ι → Type*}
[∀ i, MeasurableSpace (α i)] (μ ν : ∀ i, Measure (α i))
(hle : ∀ i, μ i ≤ ν i) : Measure.pi μ ≤ Measure.pi ν := by
apply Measure.le_iff.mpr
intro s hs
rw [Measure.pi, Measure.pi, toMeasure_apply _ _ hs,
toMeasure_apply _ _ hs]
have ho : OuterMeasure.pi (fun i => (μ i).toOuterMeasure) ≤
OuterMeasure.pi (fun i => (ν i).toOuterMeasure) := by
apply OuterMeasure.le_pi.mpr
intro t _ht
exact (OuterMeasure.pi_pi_le _ _).trans
(Finset.prod_le_prod' (fun i _ => hle i (t i)))
exact ho s
A common finite domination factor becomes its nth power for an n-record experiment. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
Proof (Lean source)
-- @node: finite_product_measure_domination
lemma finite_product_measure_domination (n : ℕ) {α : Type*} [MeasurableSpace α]
(μ ν : Measure α) [SigmaFinite ν] (c : ℝ≥0∞) [SigmaFinite (c • ν)]
(hle : μ ≤ c • ν) :
Measure.pi (fun _ : Fin n => μ) ≤ c ^ n • Measure.pi (fun _ : Fin n => ν) := by
have heq : Measure.pi (fun _ : Fin n => c • ν) =
c ^ n • Measure.pi (fun _ : Fin n => ν) := by
apply Measure.pi_eq
intro s hs
rw [Measure.smul_apply, Measure.pi_pi]
simp only [Measure.smul_apply, smul_eq_mul]
rw [Finset.prod_mul_distrib]
simp
rw [← heq]
exact finite_product_measure_mono _ _ (fun _ => hle)
Every probability-valued binary construction is dominated by eight times the fair full-record law. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
Proof (Lean source)
-- @node: binaryLaw_fair_domination
lemma binaryLaw_fair_domination (a b c : Covariate → ℝ) (ha : Measurable a)
(hb : Measurable b) (hc : Measurable c)
(hr : ∀ x, a x ∈ Icc 0 1 ∧ b x ∈ Icc 0 1 ∧ c x ∈ Icc 0 1) :
binaryLaw a b c ≤ (8 : ℝ≥0∞) • binaryLaw (fun _ => 1/2) (fun _ => 1/2) (fun _ => 1/2) := by
apply Measure.le_iff.mpr
intro s hs
rw [binaryLaw, Measure.bind_apply hs
(measurable_binaryRecordLaw a b c ha hb hc).aemeasurable,
Measure.smul_apply, binaryLaw, Measure.bind_apply hs
(measurable_binaryRecordLaw _ _ _ measurable_const measurable_const measurable_const).aemeasurable,
smul_eq_mul, ← lintegral_const_mul' _ _ (by norm_num : (8 : ℝ≥0∞) ≠ ∞)]
apply lintegral_mono
intro x
simp only [Measure.finsetSum_apply, Measure.smul_apply, smul_eq_mul, Finset.mul_sum]
apply Finset.sum_le_sum
intro z hz
have hp (p : ℝ) (hp : p ∈ Icc 0 1) (v : Bool) : bernoulliMass p v ≤ 1 := by
cases v <;> simp [bernoulliMass] <;> linarith [hp.1, hp.2]
have hm : bernoulliMass (a x) z.1 * bernoulliMass (b x) z.2.1 *
bernoulliMass (c x) z.2.2 ≤ 1 := by
exact mul_le_one₀
(mul_le_one₀ (hp _ (hr x).1 _) (bernoulliMass_nonneg (hr x).2.1 _)
(hp _ (hr x).2.1 _)) (bernoulliMass_nonneg (hr x).2.2 _)
(hp _ (hr x).2.2 _)
have hmass : ofReal (bernoulliMass (a x) z.1 * bernoulliMass (b x) z.2.1 *
bernoulliMass (c x) z.2.2) ≤ 1 := by
simpa using ENNReal.ofReal_le_ofReal hm
have hf (v : Bool) : bernoulliMass (1/2) v = 1/2 := by cases v <;> norm_num [bernoulliMass]
simp only [hf]
rw [← mul_assoc]
have hfair : (8 : ℝ≥0∞) * ofReal ((1/2 : ℝ) * (1/2) * (1/2)) = 1 := by
rw [← ENNReal.ofReal_ofNat 8, ← ENNReal.ofReal_mul (by norm_num : (0 : ℝ) ≤ 8)]
norm_num
rw [hfair]
exact mul_le_mul' hmass le_rfl
theorem square_rnDeriv_integrable_of_domination Lemma square_rnDeriv_integrable_of_domination in the paper ↗
Finite domination makes the squared density deviation integrable under a finite reference law. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
α :
Type*
μ ν :
c :
ℝ≥0∞
hc0 :
c ≠ 0
hct :
c ≠ ∞
hle :
μ ≤ c • ν
conclusion
Integrable (fun z => ((μ.rnDeriv ν z).toReal - 1)^2) ν
Proof (Lean source)
-- @node: square_rnDeriv_integrable_of_domination
lemma square_rnDeriv_integrable_of_domination {α : Type*} [MeasurableSpace α]
(μ ν : Measure α) [IsFiniteMeasure ν] (c : ℝ≥0∞) (hc0 : c ≠ 0) (hct : c ≠ ∞)
(hle : μ ≤ c • ν) :
Integrable (fun z => ((μ.rnDeriv ν z).toReal - 1)^2) ν := by
letI : IsFiniteMeasure (c • ν) := Measure.smul_finite ν hct
letI : IsFiniteMeasure μ := isFiniteMeasure_of_le (c • ν) hle
have hsmall : μ.rnDeriv (c • ν) ≤ᵐ[ν] 1 :=
(Measure.absolutelyContinuous_smul hc0).ae_le (Measure.rnDeriv_le_one_of_le hle)
have hscale := Measure.rnDeriv_smul_right_of_ne_top μ ν hc0 hct
have hbound : ∀ᵐ z ∂ν, (μ.rnDeriv ν z).toReal ≤ c.toReal := by
filter_upwards [hsmall, hscale] with z hz hs
rw [hs, Pi.smul_apply, smul_eq_mul] at hz
have hz' : μ.rnDeriv ν z ≤ c := by
calc
_ = c * (c⁻¹ * μ.rnDeriv ν z) := by
rw [← mul_assoc, ENNReal.mul_inv_cancel hc0 hct, one_mul]
_ ≤ c * 1 := mul_le_mul' le_rfl hz
_ = c := mul_one _
exact ENNReal.toReal_mono hct hz'
apply Integrable.of_mem_Icc 0 ((c.toReal + 1)^2) (by fun_prop)
filter_upwards [hbound] with z hz
have hn := ENNReal.toReal_nonneg (a := μ.rnDeriv ν z)
have hc := ENNReal.toReal_nonneg (a := c)
constructor
· exact sq_nonneg _
· nlinarith
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.CellMoments 30 declarations Positive-mass conditional cell laws, independent binary pair moments and occupancy-certificate assembly.
Positive-mass conditional cell laws, independent binary pair moments and occupancy-certificate assembly.
theorem observed_cell_measure
The observed cell event has exactly the mass of the corresponding design cell. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: observed_cell_measure
lemma observed_cell_measure (k : ℕ) (h : ℝ) (P : CausalLaw) (j : Fin k) :
(Pobs P) {z | z.1 ∈ cell h k j} = (PX P) (cell h k j) := by
change (P.law.map observe) (fst ⁻¹' cell h k j) = (P.law.map X) (cell h k j)
rw [Measure.map_apply measurable_observe
((measurableSet_cell h k j).preimage measurable_fst),
Measure.map_apply measurable_X (measurableSet_cell h k j)]
rfl
theorem conditionalCellLaw_probability
Density bounded below ensures that conditioning on any public cell gives a probability law. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
IsProbabilityMeasure (conditionalCellLaw h k P j)
Proof (Lean source)
-- @node: conditionalCellLaw_probability
lemma conditionalCellLaw_probability (k : ℕ) (h : ℝ) (hk : 0 < k)
(hh : 0 < h ∧ h ≤ 1 / 4) (P : CausalLaw) (hP : CompleteModel P) (j : Fin k) :
IsProbabilityMeasure (conditionalCellLaw h k P j) := by
let : IsProbabilityMeasure (PX P) := Measure.isProbabilityMeasure_map measurable_X.aemeasurable
let : IsProbabilityMeasure (PX P) := Measure.isProbabilityMeasure_map measurable_X.aemeasurable
have hp : 0 < cellMass h k P j := lt_of_lt_of_le
(half_pos (cell_endpoint_bounds k h hk hh j).2.2.2)
(cellMass_density_bounds k h hk hh P hP j).1
have hzero : (PX P) (cell h k j) ≠ 0 := by
intro hz
simp [cellMass, measureReal_def, hz] at hp
refine ⟨?_⟩
simp only [conditionalCellLaw, Measure.smul_apply, Measure.restrict_apply
MeasurableSet.univ, Set.univ_inter, smul_eq_mul, observed_cell_measure]
exact ENNReal.inv_mul_cancel hzero (measure_ne_top _ _)
The bounded binary numerator kernel is integrable under any finite pair measure. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: integrable_causal_pair_numerator
@[fun_prop] lemma integrable_causal_pair_numerator (μ : Measure (O × O)) [IsFiniteMeasure μ] :
Integrable (fun z => KN z.1 z.2) μ := by
apply Integrable.of_bound (by unfold KN; fun_prop) 1
filter_upwards [] with z
simpa only [Real.norm_eq_abs] using (causal_pair_kernel_unit_bounds z.1 z.2).1
The bounded binary denominator kernel is integrable under any finite pair measure. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: integrable_causal_pair_denominator
@[fun_prop] lemma integrable_causal_pair_denominator (μ : Measure (O × O)) [IsFiniteMeasure μ] :
Integrable (fun z => KD z.1 z.2) μ := by
apply Integrable.of_bound (by unfold KD; fun_prop) 1
filter_upwards [] with z
simpa only [Real.norm_eq_abs] using (causal_pair_kernel_unit_bounds z.1 z.2).2
theorem conditional_cell_numerator_abs_le_denominator
The conditional pair numerator cannot exceed the denominator in absolute value. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
Proof (Lean source)
-- @node: conditional_cell_numerator_abs_le_denominator
lemma conditional_cell_numerator_abs_le_denominator (k : ℕ) (h : ℝ) (hk : 0 < k)
(hh : 0 < h ∧ h ≤ 1 / 4) (P : CausalLaw) (hP : CompleteModel P) (j : Fin k) :
|nu h k P j| ≤ dj h k P j := by
let := conditionalCellLaw_probability k h hk hh P hP j
let μ := (conditionalCellLaw h k P j).prod (conditionalCellLaw h k P j)
have hN := integrable_causal_pair_numerator μ
have hD := integrable_causal_pair_denominator μ
calc
|nu h k P j| ≤ ∫ z, |KN z.1 z.2| ∂μ := abs_integral_le_integral_abs
_ ≤ ∫ z, KD z.1 z.2 ∂μ := integral_mono hN.abs hD
(fun z => (causal_pair_kernel_bounds z.1 z.2).2.2.1)
_ = dj h k P j := rfl
theorem conditional_cell_treatment_mean
The normalized cell treatment mean is the selected propensity averaged over that cell. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: conditional_cell_treatment_mean
lemma conditional_cell_treatment_mean (k : ℕ) (h : ℝ) (P : CausalLaw) (j : Fin k) :
(∫ z, bit z.2.1 ∂conditionalCellLaw h k P j) =
(∫ x in cell h k j, P.e x ∂PX P) / cellMass h k P j := by
have ht : MeasurableSet {z : O | z.2.1 = true} :=
measurableSet_eq_fun (by fun_prop) measurable_const
have heq : (fun z : O => bit z.2.1) =
{z : O | z.2.1 = true}.indicator (fun _ => (1 : ℝ)) := by
funext z
cases hz : z.2.1 <;> simp [bit, indicator, hz]
rw [conditionalCellLaw, integral_smul_measure, heq, integral_indicator ht,
setIntegral_one_eq_measureReal]
simp only [measureReal_def, Measure.restrict_apply ht, ENNReal.toReal_inv]
rw [observed_cell_measure]
have hj : MeasurableSet ({z : O | z.2.1 = true} ∩ {z | z.1 ∈ cell h k j}) :=
ht.inter ((measurableSet_cell h k j).preimage measurable_fst)
rw [Pobs, Measure.map_apply measurable_observe hj]
have hs : observe ⁻¹' ({z : O | z.2.1 = true} ∩ {z | z.1 ∈ cell h k j}) =
{z | X z ∈ cell h k j ∧ A z = true} := by
ext z
simp [observe, and_comm]
rw [hs]
change ((PX P).real (cell h k j))⁻¹ *
P.law.real {z | X z ∈ cell h k j ∧ A z = true} = _
rw [← P.e_version _ (measurableSet_cell h k j)]
simp only [cellMass, PX, div_eq_mul_inv, mul_comm]
theorem conditional_cell_treatment_overlap
Averaging the everywhere overlap bounds under a positive-mass cell preserves overlap. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
Proof (Lean source)
-- @node: conditional_cell_treatment_overlap
lemma conditional_cell_treatment_overlap (k : ℕ) (h : ℝ) (hk : 0 < k)
(hh : 0 < h ∧ h ≤ 1 / 4) (P : CausalLaw) (hP : CompleteModel P) (j : Fin k) :
1 / 4 ≤ (∫ z, bit z.2.1 ∂conditionalCellLaw h k P j) ∧
(∫ z, bit z.2.1 ∂conditionalCellLaw h k P j) ≤ 3 / 4 := by
let : IsProbabilityMeasure (PX P) := Measure.isProbabilityMeasure_map measurable_X.aemeasurable
have hp : 0 < cellMass h k P j := lt_of_lt_of_le
(half_pos (cell_endpoint_bounds k h hk hh j).2.2.2)
(cellMass_density_bounds k h hk hh P hP j).1
have hlo : ∫ x in cell h k j, (1 / 4 : ℝ) ∂PX P ≤
∫ x in cell h k j, P.e x ∂PX P := by
exact integral_mono_ae (integrable_const _) P.e_integrable.integrableOn
(ae_of_all _ (fun x => (hP.overlap x).1))
have hhi : ∫ x in cell h k j, P.e x ∂PX P ≤
∫ x in cell h k j, (3 / 4 : ℝ) ∂PX P := by
exact integral_mono_ae P.e_integrable.integrableOn (integrable_const _)
(ae_of_all _ (fun x => (hP.overlap x).2))
simp only [setIntegral_const, smul_eq_mul] at hlo hhi
rw [conditional_cell_treatment_mean]
constructor
· apply (le_div_iff₀ hp).mpr
simpa only [cellMass, mul_comm] using hlo
· apply (div_le_iff₀ hp).mpr
simpa only [cellMass, mul_comm] using hhi
theorem independent_binary_denominator_moment Lemma independent_binary_denominator_moment in the paper ↗
Independent binary treatments have denominator moment twice p times one minus p. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
μ :
Proof (Lean source)
-- @node: independent_binary_denominator_moment
lemma independent_binary_denominator_moment (μ : Measure O) [IsProbabilityMeasure μ] :
(∫ z, KD z.1 z.2 ∂μ.prod μ) =
2 * (∫ z, bit z.2.1 ∂μ) * (1 - ∫ z, bit z.2.1 ∂μ) := by
have hb := integrable_bit_comp μ (fun z : O => z.2.1) (by fun_prop)
rw [integral_prod _ (integrable_causal_pair_denominator (μ.prod μ))]
have hk (x y : O) : KD x y =
bit x.2.1 + bit y.2.1 - (2 * bit x.2.1) * bit y.2.1 := by
cases hx : x.2.1 <;> cases hy : y.2.1 <;> norm_num [KD, bit, hx, hy]
simp_rw [hk]
have hinner (x : O) :
(∫ y, bit x.2.1 + bit y.2.1 - (2 * bit x.2.1) * bit y.2.1 ∂μ) =
bit x.2.1 + (∫ y, bit y.2.1 ∂μ) - (2 * bit x.2.1) * (∫ y, bit y.2.1 ∂μ) := by
integral_linearity
simp
simp_rw [hinner]
integral_linearity
simp only [integral_const, probReal_univ, one_smul]
ring
theorem conditional_cell_denominator_lower
The conditional pair denominator satisfies the paper's three-eighths lower bound. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
3 / 8 ≤ dj h k P j
Proof (Lean source)
-- @node: conditional_cell_denominator_lower
lemma conditional_cell_denominator_lower (k : ℕ) (h : ℝ) (hk : 0 < k)
(hh : 0 < h ∧ h ≤ 1 / 4) (P : CausalLaw) (hP : CompleteModel P) (j : Fin k) :
3 / 8 ≤ dj h k P j := by
let := conditionalCellLaw_probability k h hk hh P hP j
have he := conditional_cell_treatment_overlap k h hk hh P hP j
have hd := (cross_treatment_denominator_bounds _ _ he he).1
rw [dj, independent_binary_denominator_moment]
nlinarith
theorem conditional_cell_integral
Conditioning rescales an observed cell integral by its actual design probability. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: conditional_cell_integral
lemma conditional_cell_integral (k : ℕ) (h : ℝ) (P : CausalLaw) (j : Fin k)
(f : O → ℝ) :
(∫ z, f z ∂conditionalCellLaw h k P j) =
(∫ z in {z : O | z.1 ∈ cell h k j}, f z ∂Pobs P) / cellMass h k P j := by
rw [conditionalCellLaw, integral_smul_measure, ENNReal.toReal_inv,
observed_cell_measure]
simp only [cellMass, measureReal_def, smul_eq_mul, div_eq_mul_inv, mul_comm]
theorem observed_arm_response_design_integral
The arm-response integral equals the design integral of the arm probability times its mean. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
(∫ z in {z : O | z.1 ∈ s ∧ z.2.1 = d}, bit z.2.2 ∂Pobs P)
= ∫ x in s, (if d then P.e x else 1 - P.e x) * (if d then P.mu1 x else P.mu0 x) ∂PX P
Proof (Lean source)
-- @node: observed_arm_response_design_integral
lemma observed_arm_response_design_integral (P : CausalLaw) (hP : CompleteModel P)
(d : Bool) (s : Set Covariate) (hs : MeasurableSet s) :
(∫ z in {z : O | z.1 ∈ s ∧ z.2.1 = d}, bit z.2.2 ∂Pobs P) =
∫ x in s, (if d then P.e x else 1 - P.e x) *
(if d then P.mu1 x else P.mu0 x) ∂PX P := by
rw [← observed_arm_outcome_integral P d s hs]
have hv : (∫ z in {z | X z ∈ s ∧ A z = d}, bit (Y z) ∂P.law) =
∫ x in s, (if d then P.mu1 x else P.mu0 x)
∂((P.law.restrict {z | A z = d}).map X) := by
cases d
· exact (P.mu0_version s hs).symm
· exact (P.mu1_version s hs).symm
rw [hv, arm_design_withDensity P hP d]
have hq : Measurable (fun x => if d then P.e x else 1 - P.e x) := by
have he := P.e_measurable
cases d <;> simp only [Bool.false_eq_true, if_false, if_true] <;> fun_prop
change (∫ x in s, (if d then P.mu1 x else P.mu0 x)
∂(PX P).withDensity (ofReal ∘ (fun x => if d then P.e x else 1 - P.e x))) = _
rw [setIntegral_withDensity_eq_setIntegral_toReal_smul₀ (μ := PX P)
(ENNReal.measurable_ofReal.comp hq).aemeasurable
(ae_of_all _ (fun x => ENNReal.ofReal_lt_top)) _ hs]
apply integral_congr_ae
filter_upwards with x
have hn : 0 ≤ (if d then P.e x else 1 - P.e x) := by
cases d <;> simp only [Bool.false_eq_true, if_false, if_true] <;>
linarith [(hP.overlap x).1, (hP.overlap x).2]
simp only [Function.comp_def, ENNReal.toReal_ofReal hn, smul_eq_mul]
theorem conditional_cell_arm_response_mean
Each conditional arm-response moment averages the selected arm regression with its propensity. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
(∫ z, (if z.2.1 = d then bit z.2.2 else 0) ∂conditionalCellLaw h k P j)
= (∫ x in cell h k j, (if d then P.e x else 1 - P.e x) * (if d then P.mu1 x else P.mu0 x) ∂PX P) / cellMass h k P j
Proof (Lean source)
-- @node: conditional_cell_arm_response_mean
lemma conditional_cell_arm_response_mean (k : ℕ) (h : ℝ) (P : CausalLaw)
(hP : CompleteModel P) (j : Fin k) (d : Bool) :
(∫ z, (if z.2.1 = d then bit z.2.2 else 0) ∂conditionalCellLaw h k P j) =
(∫ x in cell h k j, (if d then P.e x else 1 - P.e x) *
(if d then P.mu1 x else P.mu0 x) ∂PX P) / cellMass h k P j := by
have ht : MeasurableSet {z : O | z.2.1 = d} :=
measurableSet_eq_fun (by fun_prop) measurable_const
have heq : (fun z : O => if z.2.1 = d then bit z.2.2 else 0) =
{z : O | z.2.1 = d}.indicator (fun z => bit z.2.2) := by
funext z
simp only [Set.indicator_apply, Set.mem_ofPred_eq]
rw [conditional_cell_integral, heq, setIntegral_indicator ht]
have hs : {z : O | z.1 ∈ cell h k j} ∩ {z : O | z.2.1 = d} =
{z : O | z.1 ∈ cell h k j ∧ z.2.1 = d} := rfl
rw [hs, observed_arm_response_design_integral P hP d _ (measurableSet_cell h k j)]
theorem conditional_cell_treated_response_mean
The conditional treated-response moment is the cell average of propensity times treated mean. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
Proof (Lean source)
-- @node: conditional_cell_treated_response_mean
lemma conditional_cell_treated_response_mean (k : ℕ) (h : ℝ) (P : CausalLaw)
(hP : CompleteModel P) (j : Fin k) :
(∫ z, bit z.2.1 * bit z.2.2 ∂conditionalCellLaw h k P j) =
(∫ x in cell h k j, P.e x * P.mu1 x ∂PX P) / cellMass h k P j := by
have heq : (fun z : O => bit z.2.1 * bit z.2.2) =
(fun z : O => if z.2.1 = true then bit z.2.2 else 0) := by
funext z
cases z.2.1 <;> simp [bit]
rw [heq, conditional_cell_arm_response_mean k h P hP j true]
simp only [if_true]
theorem conditional_cell_response_mean
The conditional response mean sums the propensity-weighted means in the two treatment arms. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
(∫ z, bit z.2.2 ∂conditionalCellLaw h k P j)
= (∫ x in cell h k j, (1 - P.e x) * P.mu0 x ∂PX P) / cellMass h k P j
+ (∫ x in cell h k j, P.e x * P.mu1 x ∂PX P) / cellMass h k P j
Proof (Lean source)
-- @node: conditional_cell_response_mean
lemma conditional_cell_response_mean (k : ℕ) (h : ℝ) (hk : 0 < k)
(hh : 0 < h ∧ h ≤ 1 / 4) (P : CausalLaw)
(hP : CompleteModel P) (j : Fin k) :
(∫ z, bit z.2.2 ∂conditionalCellLaw h k P j) =
(∫ x in cell h k j, (1 - P.e x) * P.mu0 x ∂PX P) / cellMass h k P j +
(∫ x in cell h k j, P.e x * P.mu1 x ∂PX P) / cellMass h k P j := by
let μ := conditionalCellLaw h k P j
let : IsProbabilityMeasure μ := conditionalCellLaw_probability k h hk hh P hP j
have hy := integrable_bit_comp μ (fun z : O => z.2.2) (by fun_prop)
have ht (d : Bool) : MeasurableSet {z : O | z.2.1 = d} :=
measurableSet_eq_fun (by fun_prop) measurable_const
have hi (d : Bool) : Integrable (fun z : O => if z.2.1 = d then bit z.2.2 else 0) μ := by
have heq : (fun z : O => if z.2.1 = d then bit z.2.2 else 0) =
{z : O | z.2.1 = d}.indicator (fun z => bit z.2.2) := by
funext z
simp only [Set.indicator_apply, Set.mem_ofPred_eq]
rw [heq]
exact hy.indicator (ht d)
have heq : (fun z : O => bit z.2.2) =
(fun z : O => (if z.2.1 = false then bit z.2.2 else 0) +
(if z.2.1 = true then bit z.2.2 else 0)) := by
funext z
cases z.2.1 <;> simp
rw [heq, integral_add (hi false) (hi true),
conditional_cell_arm_response_mean k h P hP j false,
conditional_cell_arm_response_mean k h P hP j true]
simp only [Bool.false_eq_true, if_false, if_true]
theorem independent_binary_numerator_moment Lemma independent_binary_numerator_moment in the paper ↗
Independent binary records have numerator moment twice the treatment-response covariance. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: independent_binary_numerator_moment
lemma independent_binary_numerator_moment (μ : Measure O) [IsProbabilityMeasure μ] :
(∫ z, KN z.1 z.2 ∂μ.prod μ) =
2 * ((∫ z, bit z.2.1 * bit z.2.2 ∂μ) -
(∫ z, bit z.2.1 ∂μ) * (∫ z, bit z.2.2 ∂μ)) := by
have ha := integrable_bit_comp μ (fun z : O => z.2.1) (by fun_prop)
have hy := integrable_bit_comp μ (fun z : O => z.2.2) (by fun_prop)
have hay : Integrable (fun z : O => bit z.2.1 * bit z.2.2) μ := by
apply Integrable.of_mem_Icc 0 1 (by fun_prop)
filter_upwards with z
cases z.2.1 <;> cases z.2.2 <;> norm_num [bit]
rw [integral_prod _ (integrable_causal_pair_numerator (μ.prod μ))]
have hk (x y : O) : KN x y =
bit x.2.1 * bit x.2.2 - bit x.2.1 * bit y.2.2 -
bit y.2.1 * bit x.2.2 + bit y.2.1 * bit y.2.2 := by
unfold KN
ring
simp_rw [hk]
have hinner (x : O) :
(∫ y, bit x.2.1 * bit x.2.2 - bit x.2.1 * bit y.2.2 -
bit y.2.1 * bit x.2.2 + bit y.2.1 * bit y.2.2 ∂μ) =
bit x.2.1 * bit x.2.2 - bit x.2.1 * (∫ y, bit y.2.2 ∂μ) -
(∫ y, bit y.2.1 ∂μ) * bit x.2.2 + ∫ y, bit y.2.1 * bit y.2.2 ∂μ := by
integral_linearity
simp
simp_rw [hinner]
integral_linearity
simp only [integral_const, probReal_univ, one_smul]
ring
theorem conditional_cell_numerator_covariance
The conditional numerator is the treatment-response covariance within the actual cell law. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: conditional_cell_numerator_covariance
lemma conditional_cell_numerator_covariance (k : ℕ) (h : ℝ) (hk : 0 < k)
(hh : 0 < h ∧ h ≤ 1 / 4) (P : CausalLaw) (hP : CompleteModel P) (j : Fin k) :
nu h k P j =
2 * ((∫ z, bit z.2.1 * bit z.2.2 ∂conditionalCellLaw h k P j) -
(∫ z, bit z.2.1 ∂conditionalCellLaw h k P j) *
(∫ z, bit z.2.2 ∂conditionalCellLaw h k P j)) := by
let := conditionalCellLaw_probability k h hk hh P hP j
exact independent_binary_numerator_moment _
theorem conditional_cell_numerator_design_formula
The conditional numerator uses the actual design-weighted cell moments, with no density approximation. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
nu h k P j
= 2 * ((∫ x in cell h k j, P.e x * P.mu1 x ∂PX P) / cellMass h k P j - ((∫ x in cell h k j, P.e x ∂PX P) / cellMass h k P j) * ((∫ x in cell h k j, (1 - P.e x) * P.mu0 x ∂PX P) / cellMass h k P j + (∫ x in cell h k j, P.e x * P.mu1 x ∂PX P) / cellMass h k P j))
Proof (Lean source)
-- @node: conditional_cell_numerator_design_formula
lemma conditional_cell_numerator_design_formula (k : ℕ) (h : ℝ) (hk : 0 < k)
(hh : 0 < h ∧ h ≤ 1 / 4) (P : CausalLaw) (hP : CompleteModel P) (j : Fin k) :
nu h k P j = 2 *
((∫ x in cell h k j, P.e x * P.mu1 x ∂PX P) / cellMass h k P j -
((∫ x in cell h k j, P.e x ∂PX P) / cellMass h k P j) *
((∫ x in cell h k j, (1 - P.e x) * P.mu0 x ∂PX P) / cellMass h k P j +
(∫ x in cell h k j, P.e x * P.mu1 x ∂PX P) / cellMass h k P j)) := by
rw [conditional_cell_numerator_covariance k h hk hh P hP j,
conditional_cell_treated_response_mean k h P hP j,
conditional_cell_treatment_mean k h P j,
conditional_cell_response_mean k h hk hh P hP j]
theorem conditional_cell_numerator_design_pair
The observed conditional numerator equals the cross-treatment moment of two actual conditional covariates. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
nu h k P j
Proof (Lean source)
-- @node: conditional_cell_numerator_design_pair
lemma conditional_cell_numerator_design_pair (k : ℕ) (h : ℝ) (hk : 0 < k)
(hh : 0 < h ∧ h ≤ 1 / 4) (P : CausalLaw) (hP : CompleteModel P) (j : Fin k) :
nu h k P j = ∫ z, designPairNumerator P z
∂(conditionalDesignLaw h k P j).prod (conditionalDesignLaw h k P j) := by
let := conditionalDesignLaw_probability k h hk hh P hP j
rw [(design_pair_numerator_moment P hP (conditionalDesignLaw h k P j)).2]
simp only [conditionalDesignLaw_integral]
exact conditional_cell_numerator_design_formula k h hk hh P hP j
theorem conditional_cell_denominator_design_pair
The observed conditional denominator equals the cross-treatment probability for two actual conditional covariates. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
dj h k P j
= ∫ z, designPairDenominator P z ∂(conditionalDesignLaw h k P j).prod (conditionalDesignLaw h k P j)
Proof (Lean source)
-- @node: conditional_cell_denominator_design_pair
lemma conditional_cell_denominator_design_pair (k : ℕ) (h : ℝ) (hk : 0 < k)
(hh : 0 < h ∧ h ≤ 1 / 4) (P : CausalLaw) (hP : CompleteModel P) (j : Fin k) :
dj h k P j = ∫ z, designPairDenominator P z
∂(conditionalDesignLaw h k P j).prod (conditionalDesignLaw h k P j) := by
let := conditionalDesignLaw_probability k h hk hh P hP j
let := conditionalCellLaw_probability k h hk hh P hP j
rw [(design_pair_denominator_moment P hP (conditionalDesignLaw h k P j)).2,
dj, independent_binary_denominator_moment, conditional_cell_treatment_mean]
simp only [conditionalDesignLaw_integral]
theorem conditional_cell_bias_bound
The actual conditional cell moments satisfy the paper's complete causal bias bound. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: conditional_cell_bias_bound
lemma conditional_cell_bias_bound (k : ℕ) (h : ℝ) (hk : 0 < k)
(hh : 0 < h ∧ h ≤ 1 / 4) (P : CausalLaw) (hP : CompleteModel P) (j : Fin k) :
|nu h k P j - theta P * dj h k P j| ≤ bias h k * dj h k P j := by
rw [conditional_cell_numerator_design_pair k h hk hh P hP j,
conditional_cell_denominator_design_pair k h hk hh P hP j]
exact conditional_design_pair_bias k h hk hh P hP j
Two distinct coordinates of an iid array have the product of the record law as their law. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: iid_pair_coordinates
lemma iid_pair_coordinates (s : ℕ) (μ : Measure O) [IsProbabilityMeasure μ]
(i l : Fin s) (hil : i ≠ l) :
MeasurePreserving (fun z : Fin s → O => (z i, z l))
(Measure.pi (fun _ : Fin s => μ)) (μ.prod μ) := by
have hind := ProbabilityTheory.iIndepFun_pi
(μ := fun _ : Fin s => μ) (X := fun _ (z : O) => z)
(fun _ => measurable_id.aemeasurable)
have hm := (hind.indepFun hil).map_prod_eq_prod_map_map
(measurable_pi_apply i).aemeasurable (measurable_pi_apply l).aemeasurable
rw [(measurePreserving_eval (fun _ : Fin s => μ) i).map_eq,
(measurePreserving_eval (fun _ : Fin s => μ) l).map_eq] at hm
exact ⟨by fun_prop, hm⟩
Every distinct iid coordinate pair has the same integrable kernel moment. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
s :
ℕ
μ :
i l :
Fin s
hil :
i ≠ l
K :
O → O → ℝ
hK :
Integrable (fun z : O × O => K z.1 z.2) (μ.prod μ)
conclusion
conclusion 1
Proof (Lean source)
-- @node: iid_pair_kernel_integral
lemma iid_pair_kernel_integral (s : ℕ) (μ : Measure O) [IsProbabilityMeasure μ]
(i l : Fin s) (hil : i ≠ l) (K : O → O → ℝ)
(hK : Integrable (fun z : O × O => K z.1 z.2) (μ.prod μ)) :
Integrable (fun z : Fin s → O => K (z i) (z l)) (Measure.pi (fun _ : Fin s => μ)) ∧
(∫ z : Fin s → O, K (z i) (z l) ∂Measure.pi (fun _ : Fin s => μ)) =
∫ z, K z.1 z.2 ∂μ.prod μ := by
have hp := iid_pair_coordinates s μ i l hil
constructor
· exact hp.integrable_comp_of_integrable hK
· rw [← hp.map_eq] at hK ⊢
exact (integral_map hp.measurable.aemeasurable hK.aestronglyMeasurable).symm
At a fixed iid cell count, the count-weighted pair sum has expectation s times the conditional pair moment; empty and singleton samples contribute zero. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
Proof (Lean source)
-- @node: iid_pairContribution_expectation
lemma iid_pairContribution_expectation (s : ℕ) (μ : Measure O) [IsProbabilityMeasure μ]
(K : O → O → ℝ)
(hK : Integrable (fun z : O × O => K z.1 z.2) (μ.prod μ)) :
(∫ z : Fin s → O, pairContribution K z ∂Measure.pi (fun _ : Fin s => μ)) =
(if s < 2 then 0 else (s : ℝ)) * (∫ z, K z.1 z.2 ∂μ.prod μ) := by
classical
by_cases hs : s < 2
· simp [pairContribution, hs]
have hc : (choose s 2 : ℝ) ≠ 0 := by
exact_mod_cast (Nat.choose_pos (show 2 ≤ s by omega)).ne'
have hi (i l : Fin s) : Integrable (fun z : Fin s → O =>
if i < l then K (z i) (z l) else 0) (Measure.pi (fun _ : Fin s => μ)) := by
by_cases hil : i < l
· simpa only [if_pos hil] using (iid_pair_kernel_integral s μ i l hil.ne K hK).1
· simp only [if_neg hil]
exact integrable_zero _ _ _
have hm (i l : Fin s) :
(∫ z : Fin s → O, (if i < l then K (z i) (z l) else 0)
∂Measure.pi (fun _ : Fin s => μ)) =
(if i < l then (1 : ℝ) else 0) * (∫ z, K z.1 z.2 ∂μ.prod μ) := by
by_cases hil : i < l
· simp only [if_pos hil, one_mul]
exact (iid_pair_kernel_integral s μ i l hil.ne K hK).2
· simp [hil]
simp only [pairContribution, if_neg hs]
rw [integral_const_mul, integral_finset_sum _ (fun i _ =>
integrable_finset_sum _ (fun l _ => hi i l))]
simp_rw [integral_finset_sum _ (fun l _ => hi _ l), hm]
simp_rw [← Finset.sum_mul]
rw [ordered_pair_count]
field_simp [hc]
Every numerator cell contribution is integrable under a finite dataset law. The displayed conclusion follows.
Formal statement
hypotheses
conclusion
Integrable (fun D => numeratorCell n h k D j) Q
Proof (Lean source)
-- @node: integrable_numeratorCell
@[fun_prop] lemma integrable_numeratorCell (n k : ℕ) (h : ℝ) (j : Fin k)
(Q : Measure (Dataset n)) [IsFiniteMeasure Q] :
Integrable (fun D => numeratorCell n h k D j) Q := by
apply Integrable.mono' (integrable_const (μ := Q) (n : ℝ)) (by fun_prop)
filter_upwards with D
have hc : cellCount n h k D j ≤ n := by
simpa only [cellCount, Fintype.card_fin] using
(Finset.card_le_univ (cellRecords n h k D j))
simpa only [Real.norm_eq_abs] using
(cell_contribution_bounds n k h D j).1.trans (by exact_mod_cast hc)
Every denominator cell contribution is integrable under a finite dataset law. The displayed conclusion follows.
Formal statement
hypotheses
conclusion
Integrable (fun D => denominatorCell n h k D j) Q
Proof (Lean source)
-- @node: integrable_denominatorCell
@[fun_prop] lemma integrable_denominatorCell (n k : ℕ) (h : ℝ) (j : Fin k)
(Q : Measure (Dataset n)) [IsFiniteMeasure Q] :
Integrable (fun D => denominatorCell n h k D j) Q := by
apply Integrable.mono' (integrable_const (μ := Q) (n : ℝ)) (by fun_prop)
filter_upwards with D
have hc : cellCount n h k D j ≤ n := by
simpa only [cellCount, Fintype.card_fin] using
(Finset.card_le_univ (cellRecords n h k D j))
simpa only [Real.norm_eq_abs] using
(cell_contribution_bounds n k h D j).2.trans (by exact_mod_cast hc)
Population statistics sum the cell expectations, without independence between cell counts. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
conclusion 1
conclusion 2
Proof (Lean source)
-- @node: population_cell_sum
lemma population_cell_sum (n k : ℕ) (h : ℝ) (Q : Measure (Dataset n))
[IsFiniteMeasure Q] :
Nbar n h k Q = ∑ j : Fin k, ∫ D, numeratorCell n h k D j ∂Q ∧
Dbar n h k Q = ∑ j : Fin k, ∫ D, denominatorCell n h k D j ∂Q := by
constructor
· exact integral_finset_sum _ (fun j _ => integrable_numeratorCell n k h j Q)
· exact integral_finset_sum _ (fun j _ => integrable_denominatorCell n k h j Q)
theorem conditional_cell_contribution_expectation
Conditional on exact membership labels, the two cell contributions equal the eligible occupancy times their respective conditional pair moments. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Q :
((dataLaw n P) {D | cellRecords n h k D j = I})⁻¹ • (dataLaw n P).restrict {D | cellRecords n h k D j = I}
(∫ D, numeratorCell n h k D j ∂Q)
= (∫ D, (if cellCount n h k D j < 2 then 0 else (cellCount n h k D j : ℝ)) ∂Q) * nu h k P j
(∫ D, denominatorCell n h k D j ∂Q)
= (∫ D, (if cellCount n h k D j < 2 then 0 else (cellCount n h k D j : ℝ)) ∂Q) * dj h k P j
Proof (Lean source)
-- @node: conditional_cell_contribution_expectation
lemma conditional_cell_contribution_expectation (n k : ℕ) (h : ℝ) (hk : 2 ≤ k)
(hh : 0 < h ∧ h ≤ 1 / 4) (P : CausalLaw) (hP : CompleteModel P)
(j : Fin k) (I : Finset (Fin n)) :
let Q := ((dataLaw n P) {D | cellRecords n h k D j = I})⁻¹ •
(dataLaw n P).restrict {D | cellRecords n h k D j = I}
(∫ D, numeratorCell n h k D j ∂Q) =
(∫ D, (if cellCount n h k D j < 2 then 0 else
(cellCount n h k D j : ℝ)) ∂Q) * nu h k P j ∧
(∫ D, denominatorCell n h k D j ∂Q) =
(∫ D, (if cellCount n h k D j < 2 then 0 else
(cellCount n h k D j : ℝ)) ∂Q) * dj h k P j := by
classical
dsimp only
let Q := ((dataLaw n P) {D | cellRecords n h k D j = I})⁻¹ •
(dataLaw n P).restrict {D | cellRecords n h k D j = I}
have hp := cell_membership_label_positive n k h hk hh P hP j I
let : IsProbabilityMeasure Q := conditional_membership_probability n k h P j I hp
let := conditionalCellLaw_probability k h (by omega) hh P hP j
let e : Fin I.card ≃ {i // i ∈ I} := (Fintype.equivFinOfCardEq (by simp)).symm
have hae : ∀ᵐ D ∂Q, cellRecords n h k D j = I :=
Measure.ae_smul_measure (ae_restrict_mem (measurableSet_cell_membership n k h j I)) _
have hcount : (∫ D, (if cellCount n h k D j < 2 then 0 else
(cellCount n h k D j : ℝ)) ∂Q) = if I.card < 2 then 0 else (I.card : ℝ) := by
calc
_ = ∫ _D : Dataset n, (if I.card < 2 then 0 else (I.card : ℝ)) ∂Q := by
apply integral_congr_ae
filter_upwards [hae] with D hD
simp only [cellCount, hD]
_ = _ := by simp
have hN : (fun D => numeratorCell n h k D j) =ᵐ[Q]
(fun D => pairContribution KN (fun i => D (e i))) := by
filter_upwards [hae] with D hD
rw [(selected_pair_formulas I D e).1]
simp only [numeratorCell, selectedNumerator, cellCount, cellSum, hD]
have hD : (fun D => denominatorCell n h k D j) =ᵐ[Q]
(fun D => pairContribution KD (fun i => D (e i))) := by
filter_upwards [hae] with D hD
rw [(selected_pair_formulas I D e).2]
simp only [denominatorCell, selectedDenominator, cellCount, cellSum, hD]
change (∫ D, numeratorCell n h k D j ∂Q) = _ ∧
(∫ D, denominatorCell n h k D j ∂Q) = _
rw [integral_congr_ae hN, integral_congr_ae hD, hcount]
constructor
· exact conditional_selected_pairContribution_expectation n k h P j I hp e KN
(integrable_causal_pair_numerator _)
· exact conditional_selected_pairContribution_expectation n k h P j I hp e KD
(integrable_causal_pair_denominator _)
Integrating a function over all datasets is summing its integrals on the disjoint exact membership-label events. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: integral_cell_membership_partition
lemma integral_cell_membership_partition (n k : ℕ) (h : ℝ) (j : Fin k)
(μ : Measure (Dataset n)) (f : Dataset n → ℝ) (hf : Integrable f μ) :
(∫ D, f D ∂μ) = ∑ I : Finset (Fin n),
∫ D in {D | cellRecords n h k D j = I}, f D ∂μ := by
have huniv : (⋃ I : Finset (Fin n), {D | cellRecords n h k D j = I}) = univ := by
ext D
simp
rw [← integral_iUnion_fintype
(fun I => measurableSet_cell_membership n k h j I)
(fun I J hIJ => Set.disjoint_left.mpr (fun D hI hJ => hIJ (hI.symm.trans hJ)))
(fun _ => hf.integrableOn), huniv, Measure.restrict_univ]
theorem cell_membership_setIntegral_normalize
Undoing normalization multiplies each conditional integral by the probability of its membership-label event. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
(∫ D in {D | cellRecords n h k D j = I}, f D ∂dataLaw n P)
= (dataLaw n P).real {D | cellRecords n h k D j = I} * ∫ D, f D ∂(((dataLaw n P) {D | cellRecords n h k D j = I})⁻¹ • (dataLaw n P).restrict {D | cellRecords n h k D j = I})
Proof (Lean source)
-- @node: cell_membership_setIntegral_normalize
lemma cell_membership_setIntegral_normalize (n k : ℕ) (h : ℝ) (hk : 2 ≤ k)
(hh : 0 < h ∧ h ≤ 1 / 4) (P : CausalLaw) (hP : CompleteModel P)
(j : Fin k) (I : Finset (Fin n)) (f : Dataset n → ℝ) :
(∫ D in {D | cellRecords n h k D j = I}, f D ∂dataLaw n P) =
(dataLaw n P).real {D | cellRecords n h k D j = I} *
∫ D, f D ∂(((dataLaw n P) {D | cellRecords n h k D j = I})⁻¹ •
(dataLaw n P).restrict {D | cellRecords n h k D j = I}) := by
classical
let : IsProbabilityMeasure (Pobs P) :=
Measure.isProbabilityMeasure_map measurable_observe.aemeasurable
let : IsProbabilityMeasure (dataLaw n P) := by unfold dataLaw; infer_instance
have hp : (dataLaw n P) {D | cellRecords n h k D j = I} ≠ 0 := by
rw [cell_membership_rectangle, dataLaw, Measure.pi_pi]
apply Finset.prod_ne_zero_iff.mpr
intro i hi
exact cell_membership_label_positive n k h hk hh P hP j I i
have hr : (dataLaw n P).real {D | cellRecords n h k D j = I} ≠ 0 :=
ENNReal.toReal_ne_zero.mpr ⟨hp, measure_ne_top _ _⟩
rw [integral_smul_measure, ENNReal.toReal_inv]
simp only [smul_eq_mul, ← mul_assoc]
rw [show ((dataLaw n P) {D | cellRecords n h k D j = I}).toReal =
(dataLaw n P).real {D | cellRecords n h k D j = I} from rfl,
mul_inv_cancel₀ hr, one_mul]
theorem occupancy_cancellation
The common occupancy weights cancel in the population ratio for every admissible density. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
conclusion 1
conclusion 2
Proof (Lean source)
-- @node: lem:occupancy-cancellation
lemma occupancy_cancellation (n k : ℕ) (h : ℝ) (hn : 2 ≤ n) (hk : 2 ≤ k)
(hh : 0 < h ∧ h ≤ 1 / 4) (P : CausalLaw) (Q : Measure (Dataset n))
(hiid : IIDSampling n P Q) (hP : CompleteModel P) :
Nbar n h k Q = ∑ j : Fin k, occupancyWeight n h k P j * nu h k P j ∧
Dbar n h k Q = ∑ j : Fin k, occupancyWeight n h k P j * dj h k P j ∧
(∀ j : Fin k, 3/8 ≤ dj h k P j ∧
|nu h k P j - theta P * dj h k P j| ≤ bias h k * dj h k P j) ∧
info n h k / 8 ≤ Wocc n h k P ∧ Wocc n h k P ≤ 9*info n h k/2 ∧
d0 n h k ≤ Dbar n h k Q ∧
|Nbar n h k Q / Dbar n h k Q - theta P| ≤ bias h k := by
let : IsProbabilityMeasure (Pobs P) :=
Measure.isProbabilityMeasure_map measurable_observe.aemeasurable
let : IsProbabilityMeasure (dataLaw n P) := by unfold dataLaw; infer_instance
let : IsProbabilityMeasure Q := hiid.symm ▸ inferInstance
have hconditional : ∀ j : Fin k,
(∫ D, numeratorCell n h k D j ∂Q) =
(∫ D, (if cellCount n h k D j < 2 then 0 else
(cellCount n h k D j : ℝ)) ∂Q) * nu h k P j ∧
(∫ D, denominatorCell n h k D j ∂Q) =
(∫ D, (if cellCount n h k D j < 2 then 0 else
(cellCount n h k D j : ℝ)) ∂Q) * dj h k P j := by
intro j
rw [hiid]
have hlabel (I : Finset (Fin n)) :=
conditional_cell_contribution_expectation n k h hk hh P hP j I
have hlocal (I : Finset (Fin n)) :
(∫ D in {D | cellRecords n h k D j = I}, numeratorCell n h k D j ∂dataLaw n P) =
(∫ D in {D | cellRecords n h k D j = I},
(if cellCount n h k D j < 2 then 0 else (cellCount n h k D j : ℝ))
∂dataLaw n P) * nu h k P j ∧
(∫ D in {D | cellRecords n h k D j = I}, denominatorCell n h k D j ∂dataLaw n P) =
(∫ D in {D | cellRecords n h k D j = I},
(if cellCount n h k D j < 2 then 0 else (cellCount n h k D j : ℝ))
∂dataLaw n P) * dj h k P j := by
simp only [cell_membership_setIntegral_normalize n k h hk hh P hP j I]
rw [(hlabel I).1, (hlabel I).2]
constructor <;> ring
rw [integral_cell_membership_partition n k h j _ _
(integrable_numeratorCell n k h j _),
integral_cell_membership_partition n k h j _ _
(integrable_denominatorCell n k h j _),
integral_cell_membership_partition n k h j _ _
(integrable_cell_partner_count n k h j _)]
constructor
· simp_rw [(hlocal _).1, Finset.sum_mul]
· simp_rw [(hlocal _).2, Finset.sum_mul]
have hweight (j : Fin k) :
(∫ D, (if cellCount n h k D j < 2 then 0 else
(cellCount n h k D j : ℝ)) ∂Q) = occupancyWeight n h k P j := by
rw [hiid]
exact iid_cell_partner_count_expectation n k h P j
have hpopulation :
Nbar n h k Q = ∑ j : Fin k, occupancyWeight n h k P j * nu h k P j ∧
Dbar n h k Q = ∑ j : Fin k, occupancyWeight n h k P j * dj h k P j := by
obtain ⟨hsN, hsD⟩ := population_cell_sum n k h Q
constructor
· rw [hsN]
apply Finset.sum_congr rfl
intro j hj
rw [(hconditional j).1, hweight]
· rw [hsD]
apply Finset.sum_congr rfl
intro j hj
rw [(hconditional j).2, hweight]
obtain ⟨hN, hD⟩ := hpopulation
have hc : ∀ j : Fin k, 3 / 8 ≤ dj h k P j ∧
|nu h k P j - theta P * dj h k P j| ≤ bias h k * dj h k P j :=
fun j => ⟨conditional_cell_denominator_lower k h (by omega) hh P hP j,
conditional_cell_bias_bound k h (by omega) hh P hP j⟩
obtain ⟨hWlo, hWhi⟩ := occupancy_comparisons n k h hn (by omega) hh P hP
obtain ⟨hfloor, hbias⟩ := occupancy_ratio_assembly n k h P Q
(by omega) (by omega) hh.1 hN hD hc hWlo
exact ⟨hN, hD, hc, hWlo, hWhi, hfloor, hbias⟩
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.ComponentAmplitude 27 declarations Free-amplitude conditional mark arrays and their derivative bounds for the second-order component contraction argument.
Free-amplitude conditional mark arrays and their derivative bounds for the second-order component contraction argument.
The Fourier mark mass with a free amplitude and fixed perturbation and bump.
The first amplitude derivative of the four conditional masses.
The second amplitude derivative of the four conditional masses.
A Bernoulli Fourier mark has absolute value one. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Proof (Lean source)
-- @node: abs_bit_sign
lemma abs_bit_sign (b : Bool) : |2*bit b-1| = 1 := by
cases b <;> norm_num [bit]
The finite Fourier triangle inequality bounds the sum of absolute mark masses. the theorem's stated inputs and assumptions, and the asserted conclusion follows.
Formal statement
Proof (Lean source)
-- @node: markFourier_abs_sum_le
lemma markFourier_abs_sum_le (a b c d : ℝ) :
(∑ z : Bool × Bool,
|(a + (2*bit z.1-1)*b + (2*bit z.2-1)*c +
(2*bit z.1-1)*(2*bit z.2-1)*d)/4|) ≤ |a|+|b|+|c|+|d| := by
have hp (z : Bool × Bool) :
|(a + (2*bit z.1-1)*b + (2*bit z.2-1)*c +
(2*bit z.1-1)*(2*bit z.2-1)*d)/4| ≤ (|a|+|b|+|c|+|d|)/4 := by
rw [abs_div, abs_of_pos (by norm_num : (0 : ℝ) < 4)]
apply div_le_div_of_nonneg_right _ (by norm_num)
calc
_ ≤ |a| + |(2*bit z.1-1)*b| + |(2*bit z.2-1)*c| +
|(2*bit z.1-1)*(2*bit z.2-1)*d| :=
by
have h1 := abs_add_le a ((2*bit z.1-1)*b)
have h2 := abs_add_le (a+(2*bit z.1-1)*b) ((2*bit z.2-1)*c)
have h3 := abs_add_le (a+(2*bit z.1-1)*b+(2*bit z.2-1)*c)
((2*bit z.1-1)*(2*bit z.2-1)*d)
linarith
_ = _ := by simp only [abs_mul, abs_bit_sign, one_mul]
have hs := Finset.sum_le_sum (fun z (_ : z ∈ univ) => hp z)
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_prod,
Fintype.card_bool, nsmul_eq_mul, Nat.cast_mul, Nat.cast_ofNat] at hs
linarith
theorem amplitudeMarkMass_factorization
The free-amplitude mass is the product of the treatment and outcome Bernoulli factors. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: amplitudeMarkMass_factorization
lemma amplitudeMarkMass_factorization (S q u : ℝ) (z : Bool × Bool) :
amplitudeMarkMass S q u z =
((1+(2*bit z.1-1)*u*S)/2) *
((1+(2*bit z.2-1)*(-u*S+(2*bit z.1-1)*u^2*q))/2) := by
rcases z with ⟨a, b⟩
cases a <;> cases b <;> norm_num [amplitudeMarkMass, bit] <;> ring
theorem amplitudeMarkMass_sum
Summing the four conditional masses gives one at every amplitude. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: amplitudeMarkMass_sum
lemma amplitudeMarkMass_sum (S q u : ℝ) :
(∑ z : Bool × Bool, amplitudeMarkMass S q u z) = 1 := by
simp [Fintype.sum_prod_type, amplitudeMarkMass, bit]
ring
theorem amplitudeMarkMass_zero
The mass at zero amplitude is the fair pair mass. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: amplitudeMarkMass_zero
lemma amplitudeMarkMass_zero (S q : ℝ) (z : Bool × Bool) :
amplitudeMarkMass S q 0 z = 1 / 4 := by
simp [amplitudeMarkMass]
theorem amplitudeMarkMass_actual
The actual alternative is the free-amplitude array evaluated at the square-root separation. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
= conditionalLikelihood hL lam x z.1 z.2 / 4
Proof (Lean source)
-- @node: amplitudeMarkMass_actual
lemma amplitudeMarkMass_actual (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(lam : SignVector hL) (x : Covariate) (z : Bool × Bool) :
amplitudeMarkMass (perturbation hL lam x) (bump hL x)
(sqrt (separation hL)) z = conditionalLikelihood hL lam x z.1 z.2 / 4 := by
rw [conditionalLikelihood_fourier hL hhL]
have hs := Real.sq_sqrt (separation_unit_range hL hhL).1
dsimp [amplitudeMarkMass]
rw [pow_succ, hs]
ring
theorem amplitudeMarkMass_hasDerivAt
The polynomial first derivative is valid at every real amplitude. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: amplitudeMarkMass_hasDerivAt
lemma amplitudeMarkMass_hasDerivAt (S q u : ℝ) (z : Bool × Bool) :
HasDerivAt (fun v => amplitudeMarkMass S q v z) (amplitudeMarkSlope S q u z) u := by
convert (((hasDerivAt_const u (1 : ℝ)).add
(((hasDerivAt_id u).const_mul (2*bit z.1-1)).mul_const S)).add
(((((hasDerivAt_id u).pow 3).mul_const q).sub (hasDerivAt_id u)).const_mul
(2*bit z.2-1) |>.mul_const S)).add
((((hasDerivAt_id u).pow 2).const_mul ((2*bit z.1-1)*(2*bit z.2-1))).mul_const
(q-S^2)) |>.div_const 4 using 1 <;>
first | rfl | (dsimp [amplitudeMarkMass, amplitudeMarkSlope]; ring)
theorem amplitudeMarkSlope_hasDerivAt
The polynomial second derivative is valid at every real amplitude. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: amplitudeMarkSlope_hasDerivAt
lemma amplitudeMarkSlope_hasDerivAt (S q u : ℝ) (z : Bool × Bool) :
HasDerivAt (fun v => amplitudeMarkSlope S q v z) (amplitudeMarkCurvature S q u z) u := by
convert (((hasDerivAt_const u ((2*bit z.1-1)*S)).add
((((((hasDerivAt_id u).pow 2).const_mul 3).mul_const q).sub
(hasDerivAt_const u (1 : ℝ))).const_mul (2*bit z.2-1) |>.mul_const S)).add
(((hasDerivAt_id u).const_mul ((2*bit z.1-1)*(2*bit z.2-1)*2)).mul_const
(q-S^2))) |>.div_const 4 using 1 <;>
first | rfl | (dsimp [amplitudeMarkSlope, amplitudeMarkCurvature]; ring)
theorem amplitudeMarkMass_pos
Legal free amplitudes keep both Bernoulli factors strictly positive. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
0 < amplitudeMarkMass S q u z
Proof (Lean source)
-- @node: amplitudeMarkMass_pos
lemma amplitudeMarkMass_pos (S q u : ℝ) (hS : |S| ≤ 3 / 2)
(hq : q ∈ Icc 0 1) (hu : u ∈ Icc 0 (1 / 10)) (z : Bool × Bool) :
0 < amplitudeMarkMass S q u z := by
have hlin : |u*S| ≤ 3 / 20 := by
rw [abs_mul, abs_of_nonneg hu.1]
have := mul_le_mul hu.2 hS (abs_nonneg S) (by norm_num : (0 : ℝ) ≤ 1 / 10)
norm_num at this
exact this
have hquad : |u^2*q| ≤ 1 / 100 := by
rw [abs_mul, abs_of_nonneg (sq_nonneg u), abs_of_nonneg hq.1]
have hu2 : u^2 ≤ 1 / 100 := by nlinarith [hu.1, hu.2]
nlinarith [mul_le_mul_of_nonneg_left hq.2 (sq_nonneg u)]
have ha : |(2*bit z.1-1)*u*S| ≤ 3 / 20 := by
simpa only [mul_assoc, abs_mul, abs_bit_sign, one_mul] using hlin
have hb : |(2*bit z.2-1)*(-u*S+(2*bit z.1-1)*u^2*q)| ≤ 4 / 25 := by
rw [abs_mul, abs_bit_sign, one_mul]
have htri := abs_add_le (-u*S) ((2*bit z.1-1)*u^2*q)
have hfirst : |-u*S| = |u*S| := by rw [neg_mul, abs_neg]
have hsecond : |(2*bit z.1-1)*u^2*q| = |u^2*q| := by
rw [mul_assoc, abs_mul, abs_bit_sign, one_mul]
rw [hfirst, hsecond] at htri
linarith
rw [amplitudeMarkMass_factorization]
exact mul_pos (by linarith [(abs_le.mp ha).1]) (by linarith [(abs_le.mp hb).1])
theorem amplitudeMarkMass_abs_sum
The four conditional masses have unit ℓ₁ norm on the free-amplitude path. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
S q u :
ℝ
hS :
|S| ≤ 3 / 2
hq :
q ∈ Icc 0 1
hu :
u ∈ Icc 0 (1 / 10)
conclusion
Proof (Lean source)
-- @node: amplitudeMarkMass_abs_sum
lemma amplitudeMarkMass_abs_sum (S q u : ℝ) (hS : |S| ≤ 3 / 2)
(hq : q ∈ Icc 0 1) (hu : u ∈ Icc 0 (1 / 10)) :
(∑ z : Bool × Bool, |amplitudeMarkMass S q u z|) = 1 := by
simp_rw [abs_of_pos (amplitudeMarkMass_pos S q u hS hq hu _)]
exact amplitudeMarkMass_sum S q u
theorem amplitudeMarkSlope_abs_sum_le
The Fourier coefficients bound the ℓ₁ norm of the first derivative. The displayed conclusion follows.
Formal statement
hypotheses
S q u :
ℝ
conclusion
≤ |S| + |3*u^2*q-1| * |S| + 2*|u| * |q - S^2|
Proof (Lean source)
-- @node: amplitudeMarkSlope_abs_sum_le
lemma amplitudeMarkSlope_abs_sum_le (S q u : ℝ) :
(∑ z : Bool × Bool, |amplitudeMarkSlope S q u z|) ≤
|S| + |3*u^2*q-1| * |S| + 2*|u| * |q - S^2| := by
have h := markFourier_abs_sum_le 0 S ((3*u^2*q-1)*S) (2*u*(q-S^2))
have he (z : Bool × Bool) :
amplitudeMarkSlope S q u z =
(0+(2*bit z.1-1)*S+(2*bit z.2-1)*((3*u^2*q-1)*S)+
(2*bit z.1-1)*(2*bit z.2-1)*(2*u*(q-S^2)))/4 := by
dsimp [amplitudeMarkSlope]
ring
simp_rw [he]
simpa only [abs_zero, zero_add, abs_mul,
abs_of_nonneg (by norm_num : (0 : ℝ) ≤ 2),
abs_of_nonneg (by norm_num : (0 : ℝ) ≤ 6)] using h
theorem amplitudeMarkCurvature_abs_sum_le
The Fourier coefficients bound the ℓ₁ norm of the second derivative. The displayed conclusion follows.
Formal statement
hypotheses
S q u :
ℝ
conclusion
Proof (Lean source)
-- @node: amplitudeMarkCurvature_abs_sum_le
lemma amplitudeMarkCurvature_abs_sum_le (S q u : ℝ) :
(∑ z : Bool × Bool, |amplitudeMarkCurvature S q u z|) ≤
6*|u| * |q| * |S| + 2*|q - S^2| := by
have h := markFourier_abs_sum_le 0 0 (6*u*q*S) (2*(q-S^2))
have he (z : Bool × Bool) :
amplitudeMarkCurvature S q u z =
(0+(2*bit z.1-1)*0+(2*bit z.2-1)*(6*u*q*S)+
(2*bit z.1-1)*(2*bit z.2-1)*(2*(q-S^2)))/4 := by
dsimp [amplitudeMarkCurvature]
ring
simp_rw [he]
simpa only [abs_zero, zero_add, abs_mul,
abs_of_nonneg (by norm_num : (0 : ℝ) ≤ 2),
abs_of_nonneg (by norm_num : (0 : ℝ) ≤ 6)] using h
theorem amplitudeMarkSlope_abs_sum_le_four
The first derivative has ℓ₁ norm at most four throughout the amplitude path. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
S q u :
ℝ
hS :
|S| ≤ 3 / 2
hq :
q ∈ Icc 0 1
hr :
|q - S^2| ≤ 1
hu :
u ∈ Icc 0 (1 / 10)
conclusion
Proof (Lean source)
-- @node: amplitudeMarkSlope_abs_sum_le_four
lemma amplitudeMarkSlope_abs_sum_le_four (S q u : ℝ) (hS : |S| ≤ 3 / 2)
(hq : q ∈ Icc 0 1) (hr : |q - S^2| ≤ 1) (hu : u ∈ Icc 0 (1 / 10)) :
(∑ z : Bool × Bool, |amplitudeMarkSlope S q u z|) ≤ 4 := by
have hu2 : u^2 ≤ 1 / 100 := by nlinarith [hu.1, hu.2]
have hcoef : |3*u^2*q-1| ≤ 1 := by
apply abs_le.mpr
constructor
· nlinarith [mul_nonneg (sq_nonneg u) hq.1]
· nlinarith [mul_le_mul_of_nonneg_left hq.2 (sq_nonneg u)]
have hmiddle : |3*u^2*q-1| * |S| ≤ 3 / 2 := by
have := mul_le_mul hcoef hS (abs_nonneg S) (by norm_num : (0 : ℝ) ≤ 1)
simpa using this
have hlast : 2*|u| * |q - S^2| ≤ 1 / 5 := by
rw [abs_of_nonneg hu.1]
nlinarith [mul_le_mul_of_nonneg_left hr hu.1]
exact (amplitudeMarkSlope_abs_sum_le S q u).trans (by linarith)
theorem amplitudeMarkCurvature_abs_sum_le_four
The second derivative has ℓ₁ norm at most four throughout the amplitude path. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
S q u :
ℝ
hS :
|S| ≤ 3 / 2
hq :
q ∈ Icc 0 1
hr :
|q - S^2| ≤ 1
hu :
u ∈ Icc 0 (1 / 10)
conclusion
Proof (Lean source)
-- @node: amplitudeMarkCurvature_abs_sum_le_four
lemma amplitudeMarkCurvature_abs_sum_le_four (S q u : ℝ) (hS : |S| ≤ 3 / 2)
(hq : q ∈ Icc 0 1) (hr : |q - S^2| ≤ 1) (hu : u ∈ Icc 0 (1 / 10)) :
(∑ z : Bool × Bool, |amplitudeMarkCurvature S q u z|) ≤ 4 := by
have hprod : u*q ≤ 1 / 10 := by
nlinarith [hu.2, mul_le_mul_of_nonneg_left hq.2 hu.1]
have hfirst : 6*u*q*|S| ≤ 9 / 10 := by
have := mul_le_mul hprod hS (abs_nonneg S) (by norm_num : (0 : ℝ) ≤ 1 / 10)
nlinarith
have h := amplitudeMarkCurvature_abs_sum_le S q u
rw [abs_of_nonneg hu.1, abs_of_nonneg hq.1] at h
linarith
The actual perturbation and bump satisfy all coefficient bounds used by the path. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
conclusion 1
|perturbation hL lam x| ≤ 3 / 2
conclusion 2
bump hL x ∈ Icc 0 1
conclusion 3
|bump hL x-(perturbation hL lam x)^2| ≤ 1
Proof (Lean source)
-- @node: amplitudeMark_coefficients_valid
lemma amplitudeMark_coefficients_valid (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(lam : SignVector hL) (x : Covariate) :
|perturbation hL lam x| ≤ 3 / 2 ∧ bump hL x ∈ Icc 0 1 ∧
|bump hL x-(perturbation hL lam x)^2| ≤ 1 := by
exact ⟨perturbation_abs_le_three_halves hL hhL lam x, bump_range hL x,
(perturbation_square_residual_abs_le_bump hL hhL lam x).trans (bump_range hL x).2⟩
The actual square-root separation belongs to the path's amplitude interval. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: sqrt_separation_amplitude_range
lemma sqrt_separation_amplitude_range (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4) :
sqrt (separation hL) ∈ Icc 0 (1 / 10) := by
have ht : separation hL ≤ 1 / 16384 := by
dsimp [separation, kappa]
linarith [hhL.2]
have hs := Real.sq_sqrt (separation_unit_range hL hhL).1
exact ⟨Real.sqrt_nonneg _, by nlinarith [Real.sqrt_nonneg (separation hL)]⟩
theorem amplitudeMarkSlope_zero
At zero amplitude each first derivative is linear in the latent perturbation. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: amplitudeMarkSlope_zero
lemma amplitudeMarkSlope_zero (S q : ℝ) (z : Bool × Bool) :
amplitudeMarkSlope S q 0 z = ((2*bit z.1-1)-(2*bit z.2-1))*S/4 := by
dsimp [amplitudeMarkSlope]
ring
theorem amplitudeMarkSlope_sign_sum_zero
Averaging the zero-amplitude derivative cancels the independent fair signs. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: amplitudeMarkSlope_sign_sum_zero
lemma amplitudeMarkSlope_sign_sum_zero (hL : ℝ) (x : Covariate) (z : Bool × Bool) :
(∑ lam : SignVector hL,
amplitudeMarkSlope (perturbation hL lam x) (bump hL x) 0 z) = 0 := by
simp_rw [amplitudeMarkSlope_zero]
rw [← Finset.sum_div, ← Finset.mul_sum, perturbation_sign_sum, mul_zero, zero_div]
theorem amplitudeMarkMass_sign_sum_exact
Every free-amplitude singleton mixture is exactly the fair mark law. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
= (card (SignVector hL) : ℝ)/4
Proof (Lean source)
-- @node: amplitudeMarkMass_sign_sum_exact
lemma amplitudeMarkMass_sign_sum_exact (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(x : Covariate) (u : ℝ) (z : Bool × Bool) :
(∑ lam : SignVector hL,
amplitudeMarkMass (perturbation hL lam x) (bump hL x) u z) =
(card (SignVector hL) : ℝ)/4 := by
have hquad : (∑ lam : SignVector hL, (perturbation hL lam x)^2) =
(card (SignVector hL) : ℝ)*bump hL x := by
rw [perturbation_sign_square_sum, weighted_frame_square_partition hL hhL]
simp only [amplitudeMarkMass, ← Finset.sum_div, Finset.sum_add_distrib,
← Finset.mul_sum, Finset.sum_sub_distrib,
perturbation_sign_sum, hquad, Finset.sum_const, Finset.card_univ, nsmul_eq_mul]
ring
The finite sign mixture of component mark masses along the amplitude path.
Definition (Lean source)
parameters
def
componentAmplitudeMass hL n x C z u :
ℝ
given by
(2 : ℝ)^(-((activeSigns hL).card : ℤ)) * ∑ lam : SignVector hL, ∏ i : C, amplitudeMarkMass (perturbation hL lam (x i)) (bump hL (x i)) u (z i)
theorem componentAmplitudeMass_actual
The amplitude path's endpoint is precisely the existing alternative component array. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: componentAmplitudeMass_actual
lemma componentAmplitudeMass_actual (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) (x : Fin n → Covariate) (C : Finset (Fin n)) (z : Marks C) :
componentAmplitudeMass hL n x C z (sqrt (separation hL)) =
alternativeMass hL n x C z := by
simp only [componentAmplitudeMass, amplitudeMarkMass_actual hL hhL, alternativeMass]
theorem componentAmplitudeMass_zero
At zero amplitude the component path equals the fair component array. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: componentAmplitudeMass_zero
lemma componentAmplitudeMass_zero (hL : ℝ) (n : ℕ) (x : Fin n → Covariate)
(C : Finset (Fin n)) (z : Marks C) :
componentAmplitudeMass hL n x C z 0 = nullMass n C z := by
simp only [componentAmplitudeMass, amplitudeMarkMass_zero, Finset.prod_const,
Finset.card_univ, Fintype.card_coe, Finset.sum_const, nsmul_eq_mul]
rw [← mul_assoc, sign_weight_real_normalization, one_mul]
simp [nullMass, zpow_neg, zpow_natCast, one_div]
theorem componentAmplitudeMass_hasDerivAt
Differentiation of the finite mixture uses the ordinary finite product rule. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
HasDerivAt (componentAmplitudeMass hL n x C z) ((2 : ℝ)^(-((activeSigns hL).card : ℤ)) * ∑ lam : SignVector hL, ∑ i : C, (∏ j ∈ (Finset.univ : Finset C).erase i, amplitudeMarkMass (perturbation hL lam (x j)) (bump hL (x j)) u (z j)) * amplitudeMarkSlope (perturbation hL lam (x i)) (bump hL (x i)) u (z i)) u
Proof (Lean source)
-- @node: componentAmplitudeMass_hasDerivAt
lemma componentAmplitudeMass_hasDerivAt (hL : ℝ) (n : ℕ) (x : Fin n → Covariate)
(C : Finset (Fin n)) (z : Marks C) (u : ℝ) :
HasDerivAt (componentAmplitudeMass hL n x C z)
((2 : ℝ)^(-((activeSigns hL).card : ℤ)) *
∑ lam : SignVector hL, ∑ i : C,
(∏ j ∈ (Finset.univ : Finset C).erase i,
amplitudeMarkMass (perturbation hL lam (x j)) (bump hL (x j)) u (z j)) *
amplitudeMarkSlope (perturbation hL lam (x i)) (bump hL (x i)) u (z i)) u := by
classical
have hp (lam : SignVector hL) := HasDerivAt.fun_finsetProd
(u := (Finset.univ : Finset C))
(fun i _ => amplitudeMarkMass_hasDerivAt (perturbation hL lam (x i))
(bump hL (x i)) u (z i))
have hs := (HasDerivAt.fun_sum (u := (Finset.univ : Finset (SignVector hL)))
(fun lam _ => hp lam)).const_mul ((2 : ℝ)^(-((activeSigns hL).card : ℤ)))
convert hs using 1 <;> rfl
theorem componentAmplitudeMass_hasDerivAt_zero
Independent fair signs cancel the entire component path's first derivative at zero. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
HasDerivAt (componentAmplitudeMass hL n x C z) 0 0
Proof (Lean source)
-- @node: componentAmplitudeMass_hasDerivAt_zero
lemma componentAmplitudeMass_hasDerivAt_zero (hL : ℝ) (n : ℕ) (x : Fin n → Covariate)
(C : Finset (Fin n)) (z : Marks C) :
HasDerivAt (componentAmplitudeMass hL n x C z) 0 0 := by
classical
have h := componentAmplitudeMass_hasDerivAt hL n x C z 0
have hz : (∑ lam : SignVector hL, ∑ i : C,
(∏ j ∈ (Finset.univ : Finset C).erase i,
amplitudeMarkMass (perturbation hL lam (x j)) (bump hL (x j)) 0 (z j)) *
amplitudeMarkSlope (perturbation hL lam (x i)) (bump hL (x i)) 0 (z i)) = 0 := by
simp_rw [amplitudeMarkMass_zero]
rw [Finset.sum_comm]
simp_rw [← Finset.mul_sum, amplitudeMarkSlope_sign_sum_zero, mul_zero]
simp
simpa only [hz, mul_zero] using h
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.ComponentContraction 1 declarations Assembly of the Borel measurable shared-sign coupling and its private contraction bound.
Assembly of the Borel measurable shared-sign coupling and its private contraction bound.
theorem measurable_component_contraction
Finite graph factorization yields measurable component couplings and, in the sparse regime, a Hamming bound and its private-output consequence. Both cited gates remain explicit inputs. the theorem's stated inputs and assumptions, and the asserted conclusion follows.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1 / 4
n :
ℕ
cayley_of_gate :
boundedDifferences_of_gate :
epsilon :
ℝ
he :
0 < epsilon ∧ epsilon ≤ 1
B :
M :
hM :
PrivateKernel n epsilon M
conclusion
conclusion 2
x i :
C :
componentVertices hL n x i
C.card = 1
alternativeComponentLaw hL n x C = nullComponentLaw n C
≤ 4*separation hL*(componentSize n C : ℝ)^2
conclusion 3
conclusion 4
Proof (Lean source)
-- @node: lem:measurable-component-contraction
lemma measurable_component_contraction
(cayley_of_gate : CayleyLabeledTreeCount)
(boundedDifferences_of_gate : BoundedDifferencesMGF)
(epsilon : ℝ) (he : 0 < epsilon ∧ epsilon ≤ 1)
{B : Type*} [MeasurableSpace B] [StandardBorelSpace B]
(M : Kernel (Dataset n) B) [IsMarkovKernel M] (hM : PrivateKernel n epsilon M) :
(∀ x z,
fullAlternativeMass hL n x z =
((orderedComponents hL n x).map (fun C =>
alternativeMass hL n x C (fun i => z i))).prod ∧
(4 : ℝ)^(-(n : ℤ)) =
((orderedComponents hL n x).map (fun C => nullMass n C (fun i => z i))).prod) ∧
(∀ x i,
let C := componentVertices hL n x i
(C.card = 1 → alternativeComponentLaw hL n x C = nullComponentLaw n C) ∧
TV (alternativeComponentLaw hL n x C) (nullComponentLaw n C) ≤
4*separation hL*(componentSize n C : ℝ)^2 ∧
IsCoupling (componentCoupling hL n x C)
(alternativeComponentLaw hL n x C) (nullComponentLaw n C)) ∧
Measurable (conditionalDatasetCoupling hL n) ∧
IsCoupling (commonMassCoupling hL n)
(signMixture hL hhL n) (dataLaw n fairNull) ∧
((n : ℝ)*deltaL hL ≤ 1/128 →
WH n (signMixture hL hhL n) (dataLaw n fairNull) ≤
ofReal (1024*separation hL*(n : ℝ)^2*hL*deltaL hL) ∧
TV (M ∘ₘ signMixture hL hhL n) (M ∘ₘ dataLaw n fairNull) ≤
2048*epsilon*separation hL*(n : ℝ)^2*hL*deltaL hL) := by
refine ⟨?_, ?_, measurable_conditionalDatasetCoupling hL n,
commonMassCoupling_isCoupling hL hhL n, ?_⟩
· intro x z
exact ⟨fullAlternativeMass_component_factorization hL n x z,
nullMass_component_factorization hL n x z⟩
· intro x i
exact ⟨alternativeComponentLaw_eq_null_of_card_one hL hhL n x _,
component_TV_le_quadratic hL hhL n x _, componentCoupling_isCoupling hL hhL n x _⟩
· intro hcap
have heps : 0 ≤ epsilon := he.1.le
have hradius : 0 ≤ hL := hhL.1.le
have hγ := commonMassCoupling_isCoupling hL hhL n
let := hγ.isProbabilityMeasure
have hcost : (∫⁻ zw, (dHam n zw.1 zw.2 : ℝ≥0∞) ∂commonMassCoupling hL n) ≤
ofReal (1024*separation hL*(n : ℝ)^2*hL*deltaL hL) := by
by_cases hn : 1 ≤ n
· exact commonMassCoupling_hamming_cost_le_sparse cayley_of_gate hL hhL n hn hcap
· have hn0 : n = 0 := by omega
subst n
simp [dHam, hammingDist]
have hWH : WH n (signMixture hL hhL n) (dataLaw n fairNull) ≤
ofReal (1024*separation hL*(n : ℝ)^2*hL*deltaL hL) := by
exact (iInf_le_of_le (commonMassCoupling hL n) (iInf_le_of_le hγ le_rfl)).trans hcost
refine ⟨hWH, ?_⟩
let : IsProbabilityMeasure (signMixture hL hhL n) := by
rw [← hγ.map_fst]
exact Measure.isProbabilityMeasure_map measurable_fst.aemeasurable
let : IsProbabilityMeasure (dataLaw n fairNull) := by
rw [← hγ.map_snd]
exact Measure.isProbabilityMeasure_map measurable_snd.aemeasurable
have hout := private_TV_le_coupling_cost n epsilon he M hM
(signMixture hL hhL n) (dataLaw n fairNull) (commonMassCoupling hL n) hγ
have hbound : ofReal (TV (M ∘ₘ signMixture hL hhL n)
(M ∘ₘ dataLaw n fairNull)) ≤
ofReal (2048*epsilon*separation hL*(n : ℝ)^2*hL*deltaL hL) := by
calc
_ ≤ ofReal (2*epsilon) *
ofReal (1024*separation hL*(n : ℝ)^2*hL*deltaL hL) :=
hout.trans (mul_le_mul' le_rfl hcost)
_ = _ := by
rw [← ENNReal.ofReal_mul (by positivity : 0 ≤ 2*epsilon)]
congr 1
ring
exact (ENNReal.ofReal_le_ofReal_iff (by
have hs := (separation_unit_range hL hhL).1
have hd : 0 ≤ deltaL hL := by unfold deltaL; positivity
positivity)).mp hbound
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.ComponentCounting 18 declarations Component-count expectations as sums over fixed vertex sets, and ordered-tree probability bounds on arbitrary distinct coordinates of the full sample.
Component-count expectations as sums over fixed vertex sets, and ordered-tree probability bounds on arbitrary distinct coordinates of the full sample.
Selecting distinct coordinates of an iid uniform sample preserves the uniform product law. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: uniform_covariate_injection_map
lemma uniform_covariate_injection_map (n s : ℕ) (e : Fin s → Fin n)
(he : Injective e) :
(Measure.pi (fun _ : Fin n => (volume : Measure Covariate))).map
(fun x i => x (e i)) =
Measure.pi (fun _ : Fin s => (volume : Measure Covariate)) := by
have hi : iIndepFun (fun i : Fin n => fun x : Fin n → Covariate => x i)
(Measure.pi (fun _ : Fin n => (volume : Measure Covariate))) :=
iIndepFun_pi (fun _ => measurable_id.aemeasurable)
rw [(hi.precomp he).map_fun_eq_pi_map (fun i => (measurable_pi_apply (e i)).aemeasurable)]
congr 1
funext i
exact (measurePreserving_eval
(fun _ : Fin n => (volume : Measure Covariate)) (e i)).map_eq
theorem uniform_covariate_injection_preimage Lemma uniform_covariate_injection_preimage in the paper ↗
A Borel event on distinct selected sample coordinates has exactly its smaller product probability. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: uniform_covariate_injection_preimage
lemma uniform_covariate_injection_preimage (n s : ℕ) (e : Fin s → Fin n)
(he : Injective e) (A : Set (Fin s → Covariate)) (hA : MeasurableSet A) :
(Measure.pi (fun _ : Fin n => (volume : Measure Covariate)))
((fun x i => x (e i)) ⁻¹' A) =
(Measure.pi (fun _ : Fin s => (volume : Measure Covariate))) A := by
rw [← uniform_covariate_injection_map n s e he, Measure.map_apply (by fun_prop) hA]
The event that a fixed vertex set is one of the selected graph components is Borel. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: measurableSet_fixed_component
lemma measurableSet_fixed_component (hL : ℝ) (n : ℕ) (C : Finset (Fin n)) :
MeasurableSet {x : Fin n → Covariate | C ∈ orderedComponents hL n x} := by
classical
have he : {x : Fin n → Covariate | C ∈ orderedComponents hL n x} =
⋃ G : SimpleGraph (Fin n),
(if C ∈ componentsOfGraph n G then {x | sharedGraph hL n x = G} else ∅) := by
ext x
simp only [Set.mem_ofPred_eq, Set.mem_iUnion]
constructor
· intro hx
refine ⟨sharedGraph hL n x, ?_⟩
have hc : C ∈ componentsOfGraph n (sharedGraph hL n x) := hx
simp [hc]
· rintro ⟨G, hx⟩
split_ifs at hx with hc
· have hg : sharedGraph hL n x = G := hx
rw [orderedComponents_eq_of_sharedGraph_eq hL n x G hg]
exact hc
· exact False.elim hx
rw [he]
apply MeasurableSet.iUnion
intro G
split_ifs
· exact measurableSet_sharedGraph_eq hL n G
· exact MeasurableSet.empty
theorem sizeComponentCount_eq_sum_indicators
Counting size-s components is exactly summing their indicators over all size-s vertex sets. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: sizeComponentCount_eq_sum_indicators
lemma sizeComponentCount_eq_sum_indicators (hL : ℝ) (n s : ℕ)
(x : Fin n → Covariate) :
(sizeComponentCount hL n x s : ℝ≥0∞) =
∑ C ∈ (Finset.univ : Finset (Fin n)).powersetCard s,
{x : Fin n → Covariate | C ∈ orderedComponents hL n x}.indicator
(fun _ => (1 : ℝ≥0∞)) x := by
classical
have he : ((orderedComponents hL n x).toFinset.filter (fun C => C.card = s)) =
((Finset.univ : Finset (Fin n)).powersetCard s).filter
(fun C => C ∈ orderedComponents hL n x) := by
ext C
simp only [mem_filter, List.mem_toFinset, Finset.mem_powersetCard,
Finset.subset_univ, true_and]
exact and_comm
rw [sizeComponentCount, he]
simp [indicator]
theorem lintegral_sizeComponentCount_eq_sum
Taking expectations of the exact indicator count gives a sum of fixed-component probabilities. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: lintegral_sizeComponentCount_eq_sum
lemma lintegral_sizeComponentCount_eq_sum (hL : ℝ) (n s : ℕ)
(μ : Measure (Fin n → Covariate)) :
(∫⁻ x, (sizeComponentCount hL n x s : ℝ≥0∞) ∂μ) =
∑ C ∈ (Finset.univ : Finset (Fin n)).powersetCard s,
μ {x | C ∈ orderedComponents hL n x} := by
classical
simp_rw [sizeComponentCount_eq_sum_indicators]
rw [lintegral_finsetSum _ (fun C _ =>
measurable_const.indicator (measurableSet_fixed_component hL n C))]
apply Finset.sum_congr rfl
intro C _
rw [lintegral_indicator_const (measurableSet_fixed_component hL n C), one_mul]
theorem lintegral_sizeComponentCount_le_choose_mul
A common probability bound for fixed size-s components gives the binomial component-count bound. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: lintegral_sizeComponentCount_le_choose_mul
lemma lintegral_sizeComponentCount_le_choose_mul (hL : ℝ) (n s : ℕ)
(μ : Measure (Fin n → Covariate)) (b : ℝ≥0∞)
(hb : ∀ C : Finset (Fin n), C.card = s →
μ {x | C ∈ orderedComponents hL n x} ≤ b) :
(∫⁻ x, (sizeComponentCount hL n x s : ℝ≥0∞) ∂μ) ≤
(n.choose s : ℝ≥0∞) * b := by
classical
rw [lintegral_sizeComponentCount_eq_sum]
calc
_ ≤ ∑ C ∈ (Finset.univ : Finset (Fin n)).powersetCard s, b := by
apply Finset.sum_le_sum
intro C hC
exact hb C (Finset.mem_powersetCard.mp hC).2
_ = _ := by simp [Finset.card_powersetCard, nsmul_eq_mul]
The component event for any specified two-set is bounded by the corresponding single-edge event. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: pair_component_uniform_volume_le
lemma pair_component_uniform_volume_le (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) (C : Finset (Fin n)) (hc : C.card = 2) :
(Measure.pi (fun _ : Fin n => (volume : Measure Covariate)))
{x | C ∈ orderedComponents hL n x} ≤ ofReal (2 * hL * (4 * deltaL hL)) := by
classical
obtain ⟨i, j, hij, rfl⟩ := Finset.card_eq_two.mp hc
exact (measure_mono (fun x hx => sharedAdj_of_pair_component hL n x i j hij hx)).trans
(sharedAdj_uniform_volume_le hL hhL n i j hij)
theorem lintegral_sizeComponentCount_two_le
The actual expected number of two-vertex components satisfies the roadmap's binomial labeled-tree bound, with no assumed probability estimate. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: lintegral_sizeComponentCount_two_le
lemma lintegral_sizeComponentCount_two_le (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) :
(∫⁻ x, (sizeComponentCount hL n x 2 : ℝ≥0∞)
∂(Measure.pi (fun _ : Fin n => (volume : Measure Covariate)))) ≤
ofReal ((n.choose 2 : ℝ) * (2 * hL * (4 * deltaL hL))) := by
have hb := lintegral_sizeComponentCount_le_choose_mul hL n 2
(Measure.pi (fun _ : Fin n => (volume : Measure Covariate))) _
(pair_component_uniform_volume_le hL hhL n)
rw [← ENNReal.ofReal_natCast, ← ENNReal.ofReal_mul (Nat.cast_nonneg _)] at hb
exact hb
Every vertex in a component with another vertex has a neighbor in that component. This uses the first step of a connecting walk, without assuming any graph geometry. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: component_vertex_has_neighbor
lemma component_vertex_has_neighbor (hL : ℝ) (n : ℕ) (x : Fin n → Covariate)
(C : Finset (Fin n)) (hC : C ∈ orderedComponents hL n x)
(i j : Fin n) (hi : i ∈ C) (hj : j ∈ C) (hij : i ≠ j) :
∃ v ∈ C, sharedAdj hL n x i v := by
obtain ⟨r, _, hr⟩ := (mem_orderedComponents hL n x C).mp hC
have hri : (sharedGraph hL n x).Reachable r i :=
(mem_componentVertices hL n x r i).mp (by rw [hr]; exact hi)
have hrj : (sharedGraph hL n x).Reachable r j :=
(mem_componentVertices hL n x r j).mp (by rw [hr]; exact hj)
obtain ⟨w⟩ := hri.symm.trans hrj
cases w with
| nil => exact False.elim (hij rfl)
| @cons u v w hedge tail =>
refine ⟨v, ?_, hedge⟩
rw [← hr]
exact (mem_componentVertices hL n x r v).mpr (hri.trans hedge.reachable)
A three-vertex component contains one of the three labeled spanning trees. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
Proof (Lean source)
-- @node: triple_component_contains_tree
lemma triple_component_contains_tree (hL : ℝ) (n : ℕ) (x : Fin n → Covariate)
(i j k : Fin n) (hij : i ≠ j) (hik : i ≠ k) (hjk : j ≠ k)
(hC : ({i, j, k} : Finset (Fin n)) ∈ orderedComponents hL n x) :
(sharedAdj hL n x i j ∧ sharedAdj hL n x i k) ∨
(sharedAdj hL n x j i ∧ sharedAdj hL n x j k) ∨
(sharedAdj hL n x k i ∧ sharedAdj hL n x k j) := by
classical
have hn (a b : Fin n) (ha : a ∈ ({i, j, k} : Finset (Fin n)))
(hb : b ∈ ({i, j, k} : Finset (Fin n))) (hab : a ≠ b) :=
component_vertex_has_neighbor hL n x {i, j, k} hC a b ha hb hab
obtain ⟨v, hv, hiv⟩ := hn i j (by simp) (by simp) hij
have hv' : v = j ∨ v = k := by
simpa [hiv.1.symm] using hv
rcases hv' with hvj | hvk
<;> subst v
· obtain ⟨v, hv, hkv⟩ := hn k i (by simp) (by simp) hik.symm
have hv' : v = i ∨ v = j := by simpa [hkv.1.symm] using hv
rcases hv' with hvi | hvj
<;> subst v
· exact inl ⟨hiv, SimpleGraph.Adj.symm (G := sharedGraph hL n x) hkv⟩
· exact inr (inl ⟨SimpleGraph.Adj.symm (G := sharedGraph hL n x) hiv,
SimpleGraph.Adj.symm (G := sharedGraph hL n x) hkv⟩)
· obtain ⟨v, hv, hjv⟩ := hn j i (by simp) (by simp) hij.symm
have hv' : v = i ∨ v = k := by simpa [hjv.1.symm] using hv
rcases hv' with hvi | hvk
<;> subst v
· exact inl ⟨SimpleGraph.Adj.symm (G := sharedGraph hL n x) hjv, hiv⟩
· exact inr (inr ⟨SimpleGraph.Adj.symm (G := sharedGraph hL n x) hiv,
SimpleGraph.Adj.symm (G := sharedGraph hL n x) hjv⟩)
A specified two-edge star on three distinct records obeys the tree envelope. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: triple_star_uniform_volume_le
lemma triple_star_uniform_volume_le (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) (i j k : Fin n) (hij : i ≠ j) (hik : i ≠ k) (hjk : j ≠ k) :
(Measure.pi (fun _ : Fin n => (volume : Measure Covariate)))
{x | sharedAdj hL n x i j ∧ sharedAdj hL n x i k} ≤
ofReal (2 * hL * (4 * deltaL hL) ^ 2) := by
let e : Fin 3 → Fin n := ![i, j, k]
have he : Injective e := by
intro a b hab
fin_cases a <;> fin_cases b <;> simp_all [e]
have ht := selected_orderedSharedTreeEvent_volume_le hL hhL n 2 (by omega)
e he (fun _ => 0)
have hsub : {x : Fin n → Covariate | sharedAdj hL n x i j ∧ sharedAdj hL n x i k} ⊆
{x | (fun a => x (e a)) ∈ orderedSharedTreeEvent hL (fun _ => 0) 2} := by
intro x hx
have hj := SimpleGraph.Adj.symm (G := sharedGraph hL n x) hx.1
have hk := SimpleGraph.Adj.symm (G := sharedGraph hL n x) hx.2
simpa [orderedSharedTreeEvent, sharedAdj, e] using intro hj.2 hk.2
exact (measure_mono hsub).trans ht
The probability that a specified triple is a component is at most the sum of its three labeled spanning-tree envelopes. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: triple_component_uniform_volume_le
lemma triple_component_uniform_volume_le (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) (i j k : Fin n) (hij : i ≠ j) (hik : i ≠ k) (hjk : j ≠ k) :
(Measure.pi (fun _ : Fin n => (volume : Measure Covariate)))
{x | ({i, j, k} : Finset (Fin n)) ∈ orderedComponents hL n x} ≤
ofReal (3 * (2 * hL * (4 * deltaL hL) ^ 2)) := by
classical
let μ := Measure.pi (fun _ : Fin n => (volume : Measure Covariate))
let A := {x | sharedAdj hL n x i j ∧ sharedAdj hL n x i k}
let B := {x | sharedAdj hL n x j i ∧ sharedAdj hL n x j k}
let D := {x | sharedAdj hL n x k i ∧ sharedAdj hL n x k j}
have hs : {x | ({i, j, k} : Finset (Fin n)) ∈ orderedComponents hL n x} ⊆
A ∪ (B ∪ D) := fun x hx =>
triple_component_contains_tree hL n x i j k hij hik hjk hx
have ha := triple_star_uniform_volume_le hL hhL n i j k hij hik hjk
have hb := triple_star_uniform_volume_le hL hhL n j i k hij.symm hjk hik
have hd := triple_star_uniform_volume_le hL hhL n k i j hik.symm hjk.symm hij
calc
_ ≤ μ (A ∪ (B ∪ D)) := measure_mono hs
_ ≤ μ A + (μ B + μ D) :=
(measure_union_le A (B ∪ D)).trans (add_le_add le_rfl (measure_union_le B D))
_ ≤ ofReal (2 * hL * (4 * deltaL hL) ^ 2) +
(ofReal (2 * hL * (4 * deltaL hL) ^ 2) +
ofReal (2 * hL * (4 * deltaL hL) ^ 2)) :=
add_le_add ha (add_le_add hb hd)
_ = _ := by
have hh : 0 < hL := hhL.1
have hn : 0 ≤ 2 * hL * (4 * deltaL hL) ^ 2 := by positivity
rw [← ENNReal.ofReal_add hn hn, ← ENNReal.ofReal_add hn (add_nonneg hn hn)]
congr 1
ring
theorem card_three_component_uniform_volume_le Lemma card_three_component_uniform_volume_le in the paper ↗
Every fixed size-three component obeys the Cayley envelope with three trees. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: card_three_component_uniform_volume_le
lemma card_three_component_uniform_volume_le (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) (C : Finset (Fin n)) (hc : C.card = 3) :
(Measure.pi (fun _ : Fin n => (volume : Measure Covariate)))
{x | C ∈ orderedComponents hL n x} ≤
ofReal (3 * (2 * hL * (4 * deltaL hL) ^ 2)) := by
classical
obtain ⟨i, j, k, hij, hik, hjk, rfl⟩ := Finset.card_eq_three.mp hc
exact triple_component_uniform_volume_le hL hhL n i j k hij hik hjk
theorem lintegral_sizeComponentCount_three_le
The expected number of size-three components satisfies the roadmap's binomial Cayley bound with no assumed component probability estimate. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: lintegral_sizeComponentCount_three_le
lemma lintegral_sizeComponentCount_three_le (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) :
(∫⁻ x, (sizeComponentCount hL n x 3 : ℝ≥0∞)
∂(Measure.pi (fun _ : Fin n => (volume : Measure Covariate)))) ≤
ofReal ((n.choose 3 : ℝ) * 2 * hL * 3 ^ (3 - 2 : ℕ) *
(4 * deltaL hL) ^ (3 - 1 : ℕ)) := by
have hb := lintegral_sizeComponentCount_le_choose_mul hL n 3
(Measure.pi (fun _ : Fin n => (volume : Measure Covariate))) _
(card_three_component_uniform_volume_le hL hhL n)
rw [← ENNReal.ofReal_natCast, ← ENNReal.ofReal_mul (Nat.cast_nonneg _)] at hb
convert hb using 1 <;> congr 1 <;> norm_num <;> ring
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.ComponentCoupling 52 declarations Shared-sign graphs, ordered finite components, explicit common-mass couplings and conditional sparse Hamming/private-output contraction.
Shared-sign graphs, ordered finite components, explicit common-mass couplings and conditional sparse Hamming/private-output contraction.
def componentVertices
A graph component contains exactly the vertices reachable from its selected vertex.
def componentRoots
Each component is represented by its least vertex.
def orderedComponents
Components are selected by increasing least vertex, so the construction is deterministic.
Definition (Lean source)
parameters
given by
def componentSize
Component size is its vertex cardinality.
def Marks
Component marks are binary treatment-outcome pairs indexed by the component vertices.
def nullMass
The fair component mark law gives every configuration mass four to the power minus component size.
def alternativeMass
The alternative component mark mass averages the conditional Bernoulli products over the latent signs.
Definition (Lean source)
parameters
def
alternativeMass hL n x C z :
ℝ
given by
(2 : ℝ)^(-((activeSigns hL).card : ℤ)) * ∑ lam : SignVector hL, ∏ i : C, (conditionalLikelihood hL lam (x i) (z i).1 (z i).2 / 4)
def commonMass
The common mass is the pointwise minimum of the alternative and null component masses.
Total overlap sums the common component masses.
def alternativeComponentLaw
The alternative component measure is the finite measure with the specified alternative mark masses.
def nullComponentLaw
The null component measure is the finite measure with the specified fair mark masses.
def componentCouplingMass
The common-mass coupling combines diagonal common mass with the normalized product of the two residual masses, omitting the residual when overlap is one.
Definition (Lean source)
parameters
def
componentCouplingMass hL n x C z w :
ℝ
given by
if overlapMass hL n x C < 1 then diagonal
def componentCoupling
The finite component coupling measure has the explicit common-mass joint array.
def jointMarkMass
The joint mark array is the product of the common-mass couplings across the ordered graph components.
Definition (Lean source)
parameters
def
jointMarkMass hL n x z w :
ℝ
given by
((orderedComponents hL n x).map (fun C => componentCouplingMass hL n x C (fun i => z i) (fun i => w i))).prod
def attachMarks
The same covariate vector is attached to a vector of treatment-outcome marks.
def conditionalDatasetCoupling
The conditional dataset coupling attaches the same covariates to both sides of the product component coupling.
Definition (Lean source)
parameters
given by
∑ zw : (Fin n → Bool × Bool) × (Fin n → Bool × Bool), ofReal (jointMarkMass hL n x zw.1 zw.2) • Measure.dirac (attachMarks n x zw.1, attachMarks n x zw.2)
The dataset coupling integrates the ordered product of explicit component couplings over shared uniform covariates.
def fullAlternativeMass
The full conditional mark mass averages the dataset Bernoulli likelihood product uniformly over all latent signs.
Definition (Lean source)
parameters
def
fullAlternativeMass hL n x z :
ℝ
given by
(2 : ℝ)^(-((activeSigns hL).card : ℤ)) * ∑ lam : SignVector hL, ∏ i : Fin n, conditionalLikelihood hL lam (x i) (z i).1 (z i).2 / 4
The four conditional mark masses sum to one at every covariate and sign vector. The displayed conclusion follows.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: conditionalLikelihood_mark_sum
lemma conditionalLikelihood_mark_sum (lam : SignVector hL) (x : Covariate) :
(∑ z : Bool × Bool, conditionalLikelihood hL lam x z.1 z.2 / 4) = 1 := by
simp only [Fintype.sum_prod_type, Fintype.sum_bool, conditionalLikelihood,
bernoulliMass, Bool.false_eq_true, if_true, if_false]
ring
Every alternative component mass is nonnegative. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: alternativeMass_nonneg
lemma alternativeMass_nonneg (x : Fin n → Covariate) (C : Finset (Fin n))
(z : Marks C) : 0 ≤ alternativeMass hL n x C z := by
apply mul_nonneg (le_of_lt (zpow_pos (by norm_num : (0 : ℝ) < 2) _))
apply Finset.sum_nonneg
intro lam _
apply Finset.prod_nonneg
intro i _
exact div_nonneg (conditionalLikelihood_nonneg hL hhL lam (x i) (z i).1 (z i).2)
(by norm_num)
Summing the finite component array gives total mass one, including the empty component. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: alternativeMass_sum
lemma alternativeMass_sum (x : Fin n → Covariate) (C : Finset (Fin n)) :
(∑ z : Marks C, alternativeMass hL n x C z) = 1 := by
classical
simp only [alternativeMass, ← Finset.mul_sum]
rw [Finset.sum_comm]
have hp (lam : SignVector hL) :
(∑ z : Marks C, ∏ i : C, conditionalLikelihood hL lam (x i) (z i).1 (z i).2 / 4) = 1 := by
rw [← Fintype.prod_sum (fun (i : C) (z : Bool × Bool) =>
conditionalLikelihood hL lam (x i) z.1 z.2 / 4)]
simp only [conditionalLikelihood_mark_sum, Finset.prod_const_one]
simp_rw [hp]
rw [Finset.sum_const, Finset.card_univ, nsmul_eq_mul, mul_one]
exact sign_weight_real_normalization hL
Every fair component mass is positive. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: nullMass_pos
lemma nullMass_pos (C : Finset (Fin n)) (z : Marks C) : 0 < nullMass n C z := by
exact zpow_pos (by norm_num : (0 : ℝ) < 4) _
theorem alternativeMass_eq_nullMass_of_card_one Lemma alternativeMass_eq_nullMass_of_card_one in the paper ↗
A singleton alternative array equals the fair array by the exact sign cancellations. the theorem's stated inputs and assumptions, and the asserted conclusion follows.
Formal statement
Proof (Lean source)
-- @node: alternativeMass_eq_nullMass_of_card_one
lemma alternativeMass_eq_nullMass_of_card_one (x : Fin n → Covariate)
(C : Finset (Fin n)) (hc : C.card = 1) (z : Marks C) :
alternativeMass hL n x C z = nullMass n C z := by
classical
rcases Finset.card_eq_one.mp hc with ⟨i, rfl⟩
let j : ({i} : Finset (Fin n)) := ⟨i, Finset.mem_singleton_self i⟩
let : Subsingleton ({i} : Finset (Fin n)) := ⟨by
intro a b
apply Subtype.ext
exact (Finset.mem_singleton.mp a.property).trans
(Finset.mem_singleton.mp b.property).symm⟩
have hp (lam : SignVector hL) :
(∏ a : ({i} : Finset (Fin n)), conditionalLikelihood hL lam (x a) (z a).1 (z a).2 / 4) =
conditionalLikelihood hL lam (x j) (z j).1 (z j).2 / 4 :=
Fintype.prod_subsingleton _ j
simp only [alternativeMass, hp, ← Finset.sum_div,
conditionalLikelihood_sign_sum_exact hL hhL, nullMass, Finset.card_singleton]
rw [← mul_div_assoc, sign_weight_real_normalization]
norm_num
theorem alternativeComponentLaw_eq_null_of_card_one Lemma alternativeComponentLaw_eq_null_of_card_one in the paper ↗
Singleton component measures coincide pointwise, including at support boundaries. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1 / 4
n :
ℕ
hc :
C.card = 1
conclusion
alternativeComponentLaw hL n x C = nullComponentLaw n C
Proof (Lean source)
-- @node: alternativeComponentLaw_eq_null_of_card_one
lemma alternativeComponentLaw_eq_null_of_card_one (x : Fin n → Covariate)
(C : Finset (Fin n)) (hc : C.card = 1) :
alternativeComponentLaw hL n x C = nullComponentLaw n C := by
simp only [alternativeComponentLaw, nullComponentLaw,
alternativeMass_eq_nullMass_of_card_one hL hhL n x C hc]
Common component mass lies between zero and each marginal array. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
conclusion 1
0 ≤ commonMass hL n x C z
conclusion 2
commonMass hL n x C z ≤ alternativeMass hL n x C z
conclusion 3
commonMass hL n x C z ≤ nullMass n C z
Proof (Lean source)
-- @node: commonMass_bounds
lemma commonMass_bounds (x : Fin n → Covariate) (C : Finset (Fin n)) (z : Marks C) :
0 ≤ commonMass hL n x C z ∧
commonMass hL n x C z ≤ alternativeMass hL n x C z ∧
commonMass hL n x C z ≤ nullMass n C z := by
exact ⟨le_min (alternativeMass_nonneg hL hhL n x C z) (nullMass_pos n C z).le,
min_le_left _ _, min_le_right _ _⟩
The total common component mass is a probability overlap between zero and one. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1 / 4
n :
ℕ
conclusion
overlapMass hL n x C ∈ Icc 0 1
Proof (Lean source)
-- @node: overlapMass_range
lemma overlapMass_range (x : Fin n → Covariate) (C : Finset (Fin n)) :
overlapMass hL n x C ∈ Icc 0 1 := by
constructor
· exact Finset.sum_nonneg (fun z _ => (commonMass_bounds hL hhL n x C z).1)
· calc
_ ≤ ∑ z : Marks C, alternativeMass hL n x C z :=
Finset.sum_le_sum (fun z _ => (commonMass_bounds hL hhL n x C z).2.1)
_ = 1 := alternativeMass_sum hL n x C
The fair component array sums to one for every finite component. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: nullMass_sum
lemma nullMass_sum (C : Finset (Fin n)) : (∑ z : Marks C, nullMass n C z) = 1 := by
classical
simp only [nullMass, Finset.sum_const, Finset.card_univ, nsmul_eq_mul,
Fintype.card_fun, Fintype.card_prod, Fintype.card_bool, Fintype.card_coe]
norm_num only [Nat.cast_pow, Nat.cast_mul, Nat.cast_ofNat]
rw [zpow_neg, zpow_natCast, mul_inv_cancel₀ (by positivity)]
The first residual array has mass one minus the overlap. the theorem's stated inputs and assumptions, and the asserted conclusion follows.
Formal statement
hypotheses
conclusion
= 1-overlapMass hL n x C
Proof (Lean source)
-- @node: alternativeResidual_sum
lemma alternativeResidual_sum (x : Fin n → Covariate) (C : Finset (Fin n)) :
(∑ z : Marks C, (alternativeMass hL n x C z-commonMass hL n x C z)) =
1-overlapMass hL n x C := by
rw [Finset.sum_sub_distrib, alternativeMass_sum]
rfl
The second residual array has the same mass one minus the overlap. The displayed conclusion follows.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: nullResidual_sum
lemma nullResidual_sum (x : Fin n → Covariate) (C : Finset (Fin n)) :
(∑ z : Marks C, (nullMass n C z-commonMass hL n x C z)) =
1-overlapMass hL n x C := by
rw [Finset.sum_sub_distrib, nullMass_sum]
rfl
theorem alternativeMass_eq_common_of_overlap_one Lemma alternativeMass_eq_common_of_overlap_one in the paper ↗
At full overlap the alternative array equals the common array. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1 / 4
n :
ℕ
ha :
overlapMass hL n x C = 1
z :
Marks C
conclusion
alternativeMass hL n x C z = commonMass hL n x C z
Proof (Lean source)
-- @node: alternativeMass_eq_common_of_overlap_one
lemma alternativeMass_eq_common_of_overlap_one (x : Fin n → Covariate)
(C : Finset (Fin n)) (ha : overlapMass hL n x C = 1) (z : Marks C) :
alternativeMass hL n x C z = commonMass hL n x C z := by
have hs : (∑ w : Marks C, (alternativeMass hL n x C w-commonMass hL n x C w)) = 0 := by
rw [alternativeResidual_sum, ha, sub_self]
have hz := (Finset.sum_eq_zero_iff_of_nonneg (fun w _ => sub_nonneg.mpr
(commonMass_bounds hL hhL n x C w).2.1)).mp hs z (Finset.mem_univ z)
exact sub_eq_zero.mp hz
At full overlap the null array equals the common array. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1 / 4
n :
ℕ
ha :
overlapMass hL n x C = 1
z :
Marks C
conclusion
nullMass n C z = commonMass hL n x C z
Proof (Lean source)
-- @node: nullMass_eq_common_of_overlap_one
lemma nullMass_eq_common_of_overlap_one (x : Fin n → Covariate)
(C : Finset (Fin n)) (ha : overlapMass hL n x C = 1) (z : Marks C) :
nullMass n C z = commonMass hL n x C z := by
have hs : (∑ w : Marks C, (nullMass n C w-commonMass hL n x C w)) = 0 := by
rw [nullResidual_sum, ha, sub_self]
have hz := (Finset.sum_eq_zero_iff_of_nonneg (fun w _ => sub_nonneg.mpr
(commonMass_bounds hL hhL n x C w).2.2)).mp hs z (Finset.mem_univ z)
exact sub_eq_zero.mp hz
The explicit common-mass joint array is nonnegative in both overlap branches. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: componentCouplingMass_nonneg
lemma componentCouplingMass_nonneg (x : Fin n → Covariate)
(C : Finset (Fin n)) (z w : Marks C) : 0 ≤ componentCouplingMass hL n x C z w := by
classical
have hz := commonMass_bounds hL hhL n x C z
have hw := commonMass_bounds hL hhL n x C w
dsimp [componentCouplingMass]
split_ifs with ha he he
· exact add_nonneg hz.1 (div_nonneg
(mul_nonneg (sub_nonneg.mpr hz.2.1) (sub_nonneg.mpr hw.2.2)) (by linarith))
· rw [zero_add]
exact div_nonneg
(mul_nonneg (sub_nonneg.mpr hz.2.1) (sub_nonneg.mpr hw.2.2)) (by linarith)
· exact hz.1
· exact le_rfl
Summing a row of the explicit coupling recovers the alternative component mass. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: componentCouplingMass_row_sum
lemma componentCouplingMass_row_sum (x : Fin n → Covariate)
(C : Finset (Fin n)) (z : Marks C) :
(∑ w : Marks C, componentCouplingMass hL n x C z w) = alternativeMass hL n x C z := by
classical
by_cases ha : overlapMass hL n x C < 1
· have hd : 1-overlapMass hL n x C ≠ 0 := by linarith
simp only [componentCouplingMass, if_pos ha, Finset.sum_add_distrib,
← Finset.sum_div, ← Finset.mul_sum]
rw [nullResidual_sum]
simp only [Finset.sum_ite_eq, Finset.mem_univ, if_true]
field_simp
ring
· have he : overlapMass hL n x C = 1 :=
le_antisymm (overlapMass_range hL hhL n x C).2 (le_of_not_gt ha)
simp only [componentCouplingMass, if_neg ha, Finset.sum_ite_eq,
Finset.mem_univ, if_true]
exact (alternativeMass_eq_common_of_overlap_one hL hhL n x C he z).symm
Summing a column of the explicit coupling recovers the null component mass. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: componentCouplingMass_column_sum
lemma componentCouplingMass_column_sum (x : Fin n → Covariate)
(C : Finset (Fin n)) (w : Marks C) :
(∑ z : Marks C, componentCouplingMass hL n x C z w) = nullMass n C w := by
classical
by_cases ha : overlapMass hL n x C < 1
· have hd : 1-overlapMass hL n x C ≠ 0 := by linarith
simp only [componentCouplingMass, if_pos ha, Finset.sum_add_distrib,
← Finset.sum_div, ← Finset.sum_mul]
rw [alternativeResidual_sum]
simp only [Finset.sum_ite_eq', Finset.mem_univ, if_true]
field_simp
ring
· have he : overlapMass hL n x C = 1 :=
le_antisymm (overlapMass_range hL hhL n x C).2 (le_of_not_gt ha)
simp only [componentCouplingMass, if_neg ha, Finset.sum_ite_eq',
Finset.mem_univ, if_true]
exact (nullMass_eq_common_of_overlap_one hL hhL n x C he w).symm
The explicit finite coupling has the alternative component law as first marginal. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1 / 4
n :
ℕ
conclusion
Proof (Lean source)
-- @node: componentCoupling_map_fst
lemma componentCoupling_map_fst (x : Fin n → Covariate) (C : Finset (Fin n)) :
(componentCoupling hL n x C).map fst = alternativeComponentLaw hL n x C := by
classical
have hr (z : Marks C) :
(∑ w : Marks C, ofReal (componentCouplingMass hL n x C z w)) =
ofReal (alternativeMass hL n x C z) := by
rw [← ENNReal.ofReal_sum_of_nonneg
(fun w _ => componentCouplingMass_nonneg hL hhL n x C z w),
componentCouplingMass_row_sum hL hhL]
ext S hS
rw [Measure.map_apply measurable_fst hS]
simp only [componentCoupling, alternativeComponentLaw, Measure.finsetSum_apply,
Measure.smul_apply, smul_eq_mul, Measure.dirac_apply, Fintype.sum_prod_type]
apply Finset.sum_congr rfl
intro z _
change (∑ w : Marks C, ofReal (componentCouplingMass hL n x C z w) *
S.indicator 1 z) = ofReal (alternativeMass hL n x C z) * S.indicator 1 z
rw [← Finset.sum_mul, hr]
The explicit finite coupling has the fair component law as second marginal. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1 / 4
n :
ℕ
conclusion
Proof (Lean source)
-- @node: componentCoupling_map_snd
lemma componentCoupling_map_snd (x : Fin n → Covariate) (C : Finset (Fin n)) :
(componentCoupling hL n x C).map snd = nullComponentLaw n C := by
classical
have hc (w : Marks C) :
(∑ z : Marks C, ofReal (componentCouplingMass hL n x C z w)) =
ofReal (nullMass n C w) := by
rw [← ENNReal.ofReal_sum_of_nonneg
(fun z _ => componentCouplingMass_nonneg hL hhL n x C z w),
componentCouplingMass_column_sum hL hhL]
ext S hS
rw [Measure.map_apply measurable_snd hS]
simp only [componentCoupling, nullComponentLaw, Measure.finsetSum_apply,
Measure.smul_apply, smul_eq_mul, Measure.dirac_apply, Fintype.sum_prod_type]
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro w _
change (∑ z : Marks C, ofReal (componentCouplingMass hL n x C z w) *
S.indicator 1 w) = ofReal (nullMass n C w) * S.indicator 1 w
rw [← Finset.sum_mul, hc]
theorem nullComponentLaw_isProbabilityMeasure Lemma nullComponentLaw_isProbabilityMeasure in the paper ↗
The fair finite component measure is a probability measure. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: nullComponentLaw_isProbabilityMeasure
lemma nullComponentLaw_isProbabilityMeasure (C : Finset (Fin n)) :
IsProbabilityMeasure (nullComponentLaw n C) := by
constructor
simp only [nullComponentLaw, Measure.finsetSum_apply, Measure.smul_apply,
Measure.dirac_apply_of_mem (Set.mem_univ _), smul_eq_mul, mul_one]
rw [← ENNReal.ofReal_sum_of_nonneg (fun z _ => (nullMass_pos n C z).le), nullMass_sum]
exact ENNReal.ofReal_one
The common-mass component construction is a probability coupling of its two mark laws. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1 / 4
n :
ℕ
conclusion
Proof (Lean source)
-- @node: componentCoupling_isCoupling
lemma componentCoupling_isCoupling (x : Fin n → Covariate) (C : Finset (Fin n)) :
IsCoupling (componentCoupling hL n x C)
(alternativeComponentLaw hL n x C) (nullComponentLaw n C) := by
have hs := componentCoupling_map_snd hL hhL n x C
have hprob : IsProbabilityMeasure (componentCoupling hL n x C) := by
constructor
have hm := congrArg (fun mu : Measure (Marks C) => mu univ) hs
rw [Measure.map_apply measurable_snd MeasurableSet.univ] at hm
let := nullComponentLaw_isProbabilityMeasure n C
simpa using hm
exact ⟨hprob, componentCoupling_map_fst hL hhL n x C, hs⟩
At any mark vector one of the two residual masses is zero. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
= 0
Proof (Lean source)
-- @node: componentResidual_diagonal_zero
lemma componentResidual_diagonal_zero (x : Fin n → Covariate)
(C : Finset (Fin n)) (z : Marks C) :
(alternativeMass hL n x C z-commonMass hL n x C z) *
(nullMass n C z-commonMass hL n x C z) = 0 := by
dsimp [commonMass]
rcases le_total (alternativeMass hL n x C z) (nullMass n C z) with h | h
· rw [min_eq_left h, sub_self, zero_mul]
· rw [min_eq_right h, sub_self, mul_zero]
The diagonal entry of the joint array is exactly the common mass in either branch. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
componentCouplingMass hL n x C z z = commonMass hL n x C z
Proof (Lean source)
-- @node: componentCouplingMass_diagonal
lemma componentCouplingMass_diagonal (x : Fin n → Covariate)
(C : Finset (Fin n)) (z : Marks C) :
componentCouplingMass hL n x C z z = commonMass hL n x C z := by
classical
simp only [componentCouplingMass, if_true, componentResidual_diagonal_zero,
zero_div, add_zero, ite_self]
The component coupling's agreement probability is precisely its common mass overlap. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1 / 4
n :
ℕ
conclusion
Proof (Lean source)
-- @node: componentCoupling_diagonal_mass
lemma componentCoupling_diagonal_mass (x : Fin n → Covariate) (C : Finset (Fin n)) :
componentCoupling hL n x C {zw | zw.1 = zw.2} = ofReal (overlapMass hL n x C) := by
classical
simp only [componentCoupling, Measure.finsetSum_apply, Measure.smul_apply,
smul_eq_mul, Measure.dirac_apply, Fintype.sum_prod_type, indicator,
Set.mem_ofPred_eq, Pi.one_apply, mul_ite, mul_one, mul_zero,
Finset.sum_ite_eq, Finset.mem_univ, if_true, componentCouplingMass_diagonal]
exact (ENNReal.ofReal_sum_of_nonneg
(fun z _ => (commonMass_bounds hL hhL n x C z).1)).symm
The absolute array difference is twice the mass outside their pointwise overlap. The displayed conclusion follows.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: componentMass_abs_sum
lemma componentMass_abs_sum (x : Fin n → Covariate) (C : Finset (Fin n)) :
(∑ z : Marks C, |alternativeMass hL n x C z-nullMass n C z|) =
2*(1-overlapMass hL n x C) := by
have hp (z : Marks C) :
|alternativeMass hL n x C z-nullMass n C z| =
alternativeMass hL n x C z + nullMass n C z - 2*commonMass hL n x C z := by
dsimp [commonMass]
rcases le_total (alternativeMass hL n x C z) (nullMass n C z) with h | h
· rw [min_eq_left h, abs_of_nonpos (sub_nonpos.mpr h)]; ring
· rw [min_eq_right h, abs_of_nonneg (sub_nonneg.mpr h)]; ring
simp_rw [hp]
rw [Finset.sum_sub_distrib, Finset.sum_add_distrib, ← Finset.mul_sum,
alternativeMass_sum, nullMass_sum]
dsimp [overlapMass]
ring
Singleton components have full common-mass overlap. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1 / 4
n :
ℕ
hc :
C.card = 1
conclusion
overlapMass hL n x C = 1
Proof (Lean source)
-- @node: overlapMass_eq_one_of_card_one
lemma overlapMass_eq_one_of_card_one (x : Fin n → Covariate)
(C : Finset (Fin n)) (hc : C.card = 1) : overlapMass hL n x C = 1 := by
simp only [overlapMass, commonMass,
alternativeMass_eq_nullMass_of_card_one hL hhL n x C hc, min_self]
exact nullMass_sum n C
theorem componentCoupling_disagreement_mass Lemma componentCoupling_disagreement_mass in the paper ↗
The explicit component coupling fails with probability exactly one minus its overlap. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1 / 4
n :
ℕ
conclusion
Proof (Lean source)
-- @node: componentCoupling_disagreement_mass
lemma componentCoupling_disagreement_mass (x : Fin n → Covariate)
(C : Finset (Fin n)) :
componentCoupling hL n x C {zw | zw.1 ≠ zw.2} =
ofReal (1-overlapMass hL n x C) := by
letI := (componentCoupling_isCoupling hL hhL n x C).isProbabilityMeasure
have hD : MeasurableSet {zw : Marks C × Marks C | zw.1 = zw.2} := measurableSet_eq_fun measurable_fst measurable_snd
change componentCoupling hL n x C ({zw | zw.1 = zw.2}ᶜ) = _
rw [measure_compl hD (measure_ne_top _ _), measure_univ,
componentCoupling_diagonal_mass hL hhL]
simpa only [ENNReal.ofReal_one] using
(ENNReal.ofReal_sub 1 (overlapMass_range hL hhL n x C).1).symm
theorem componentCoupling_disagreement_zero_of_card_one Lemma componentCoupling_disagreement_zero_of_card_one in the paper ↗
Singleton couplings agree almost surely, including support-boundary singletons. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1 / 4
n :
ℕ
hc :
C.card = 1
conclusion
componentCoupling hL n x C {zw | zw.1 ≠ zw.2} = 0
Proof (Lean source)
-- @node: componentCoupling_disagreement_zero_of_card_one
lemma componentCoupling_disagreement_zero_of_card_one (x : Fin n → Covariate)
(C : Finset (Fin n)) (hc : C.card = 1) :
componentCoupling hL n x C {zw | zw.1 ≠ zw.2} = 0 := by
rw [componentCoupling_disagreement_mass hL hhL,
overlapMass_eq_one_of_card_one hL hhL n x C hc, sub_self, ENNReal.ofReal_zero]
A component can change at most its cardinality on the coupling's failure event. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: componentCoupling_hamming_cost_le
lemma componentCoupling_hamming_cost_le (x : Fin n → Covariate)
(C : Finset (Fin n)) :
(∫⁻ zw, (hammingDist zw.1 zw.2 : ℝ≥0∞) ∂componentCoupling hL n x C) ≤
(C.card : ℝ≥0∞) * ofReal (1-overlapMass hL n x C) := by
classical
calc
_ ≤ ∫⁻ zw, ({zw : Marks C × Marks C | zw.1 ≠ zw.2}.indicator
(fun _ => (C.card : ℝ≥0∞))) zw ∂componentCoupling hL n x C := by
apply lintegral_mono
intro zw
by_cases heq : zw.1 = zw.2
· simp [heq]
· rw [Set.indicator_of_mem heq]
exact Nat.cast_le.mpr (by simpa using
(hammingDist_le_card_fintype (x := zw.1) (y := zw.2)))
_ = _ := by
rw [lintegral_indicator (show MeasurableSet {zw : Marks C × Marks C | zw.1 ≠ zw.2} from
(measurableSet_eq_fun measurable_fst measurable_snd).compl), lintegral_const,
Measure.restrict_apply_univ, componentCoupling_disagreement_mass hL hhL]
theorem componentCoupling_hamming_cost_le_abs_sum Lemma componentCoupling_hamming_cost_le_abs_sum in the paper ↗
The component transport cost is at most half its size times the absolute array difference. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: componentCoupling_hamming_cost_le_abs_sum
lemma componentCoupling_hamming_cost_le_abs_sum (x : Fin n → Covariate)
(C : Finset (Fin n)) :
(∫⁻ zw, (hammingDist zw.1 zw.2 : ℝ≥0∞) ∂componentCoupling hL n x C) ≤
ofReal ((C.card : ℝ)/2 *
∑ z : Marks C, |alternativeMass hL n x C z-nullMass n C z|) := by
rw [componentMass_abs_sum]
have heq : (C.card : ℝ)/2 * (2*(1-overlapMass hL n x C)) =
(C.card : ℝ)*(1-overlapMass hL n x C) := by ring
rw [heq, ENNReal.ofReal_mul (Nat.cast_nonneg _), ENNReal.ofReal_natCast]
exact componentCoupling_hamming_cost_le hL hhL n x C
theorem componentCoupling_hamming_cost_zero_of_card_one Lemma componentCoupling_hamming_cost_zero_of_card_one in the paper ↗
Singleton components contribute no expected changed marks to the dataset coupling. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: componentCoupling_hamming_cost_zero_of_card_one
lemma componentCoupling_hamming_cost_zero_of_card_one (x : Fin n → Covariate)
(C : Finset (Fin n)) (hc : C.card = 1) :
(∫⁻ zw, (hammingDist zw.1 zw.2 : ℝ≥0∞) ∂componentCoupling hL n x C) = 0 := by
apply le_antisymm _ bot_le
have hcost := componentCoupling_hamming_cost_le hL hhL n x C
rw [overlapMass_eq_one_of_card_one hL hhL n x C hc, sub_self,
ENNReal.ofReal_zero, mul_zero] at hcost
exact hcost
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.ComponentFactorization 12 declarations Pointwise products over the ordered shared-sign partition: fixed-sign conditional likelihoods and fair null masses factor at every covariate vector.
Pointwise products over the ordered shared-sign partition: fixed-sign conditional likelihoods and fair null masses factor at every covariate vector.
The finite union of the ordered components is the whole dataset index set. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
(orderedComponents hL n x).toFinset.biUnion id = univ
Proof (Lean source)
-- @node: orderedComponents_biUnion
lemma orderedComponents_biUnion (x : Fin n → Covariate) :
(orderedComponents hL n x).toFinset.biUnion id = univ := by
classical
ext i
simp only [Finset.mem_biUnion, List.mem_toFinset, id_eq, Finset.mem_univ, iff_true]
exact orderedComponents_cover hL n x i
The finset underlying the component order is a disjoint partition. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: orderedComponents_pairwiseDisjoint
lemma orderedComponents_pairwiseDisjoint (x : Fin n → Covariate) :
PairwiseDisjoint ((orderedComponents hL n x).toFinset : Set (Finset (Fin n))) id := by
classical
intro C hC D hD hne
exact orderedComponents_disjoint hL n x C D
(List.mem_toFinset.mp hC) (List.mem_toFinset.mp hD) hne
Multiplying vertex factors component by component equals the full dataset product. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: orderedComponents_prod
lemma orderedComponents_prod {α : Type*} [CommMonoid α]
(x : Fin n → Covariate) (f : Fin n → α) :
((orderedComponents hL n x).map (fun C : Finset (Fin n) => ∏ i : C, f i)).prod = ∏ i, f i := by
classical
simp_rw [Finset.prod_coe_sort]
rw [← List.prod_toFinset _ (orderedComponents_nodup hL n x)]
have hp := Finset.prod_biUnion (f := f) (orderedComponents_pairwiseDisjoint hL n x)
rw [orderedComponents_biUnion] at hp
simpa only [id_eq] using hp.symm
Component cardinalities sum to the number of dataset records. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: orderedComponents_card_sum
lemma orderedComponents_card_sum (x : Fin n → Covariate) :
((orderedComponents hL n x).map card).sum = n := by
classical
have hs := Finset.sum_biUnion (f := fun _ : Fin n => (1 : ℕ))
(orderedComponents_pairwiseDisjoint hL n x)
rw [orderedComponents_biUnion] at hs
simpa only [Finset.sum_const, Finset.card_univ, Fintype.card_fin,
smul_eq_mul, mul_one, id_eq, List.sum_toFinset _ (orderedComponents_nodup hL n x)] using hs.symm
Conditional independence given signs factors its mark likelihood over the graph partition. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
(∏ i : Fin n, conditionalLikelihood hL lam (x i) (z i).1 (z i).2 / 4)
= ((orderedComponents hL n x).map (fun C : Finset (Fin n) => ∏ i : C, conditionalLikelihood hL lam (x i) (z i).1 (z i).2 / 4)).prod
Proof (Lean source)
-- @node: fixedSign_component_factorization
lemma fixedSign_component_factorization (x : Fin n → Covariate)
(z : Fin n → Bool × Bool) (lam : SignVector hL) :
(∏ i : Fin n, conditionalLikelihood hL lam (x i) (z i).1 (z i).2 / 4) =
((orderedComponents hL n x).map (fun C : Finset (Fin n) =>
∏ i : C, conditionalLikelihood hL lam (x i) (z i).1 (z i).2 / 4)).prod := by
exact (orderedComponents_prod hL n x _).symm
A fair component mass is the product of the fair one-record masses. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: nullMass_eq_vertex_prod
lemma nullMass_eq_vertex_prod (C : Finset (Fin n)) (z : Marks C) :
nullMass n C z = ∏ _i : C, (1 / 4 : ℝ) := by
simp [nullMass, zpow_neg, zpow_natCast, one_div]
The fair null factors over components at every covariate vector. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: nullMass_component_factorization
lemma nullMass_component_factorization (x : Fin n → Covariate)
(z : Fin n → Bool × Bool) :
(4 : ℝ)^(-(n : ℤ)) =
((orderedComponents hL n x).map (fun C => nullMass n C (fun i => z i))).prod := by
simp_rw [nullMass_eq_vertex_prod]
rw [orderedComponents_prod hL n x (fun _ => (1 / 4 : ℝ))]
simp [zpow_neg, zpow_natCast, one_div]
A component uses precisely those active signs nonzero at one of its vertices.
theorem mem_componentSignIndices
The sign block membership test is the existence of a vertex using that coordinate. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
j ∈ componentSignIndices hL n x C ↔ ∃ i ∈ C, envelope hL (x i) * frame hL j (x i) ≠ 0
Proof (Lean source)
-- @node: mem_componentSignIndices
lemma mem_componentSignIndices (x : Fin n → Covariate) (C : Finset (Fin n))
(j : activeSigns hL) :
j ∈ componentSignIndices hL n x C ↔
∃ i ∈ C, envelope hL (x i) * frame hL j (x i) ≠ 0 := by
classical
simp [componentSignIndices]
theorem orderedComponents_signIndices_pairwiseDisjoint
Distinct ordered graph components use disjoint coordinate blocks of the latent signs. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
PairwiseDisjoint ((orderedComponents hL n x).toFinset : Set (Finset (Fin n))) (componentSignIndices hL n x)
Proof (Lean source)
-- @node: orderedComponents_signIndices_pairwiseDisjoint
lemma orderedComponents_signIndices_pairwiseDisjoint (x : Fin n → Covariate) :
PairwiseDisjoint ((orderedComponents hL n x).toFinset : Set (Finset (Fin n)))
(componentSignIndices hL n x) := by
classical
intro C hC D hD hne
apply Finset.disjoint_left.mpr
intro j hjC hjD
obtain ⟨i, hi, hij⟩ := (mem_componentSignIndices hL n x C j).mp hjC
obtain ⟨l, hl, hlj⟩ := (mem_componentSignIndices hL n x D j).mp hjD
obtain ⟨r, hr, rfl⟩ := (mem_orderedComponents hL n x C).mp (List.mem_toFinset.mp hC)
obtain ⟨s, hs, rfl⟩ := (mem_orderedComponents hL n x D).mp (List.mem_toFinset.mp hD)
have hrs : ¬(sharedGraph hL n x).Reachable r s := by
intro hrs
exact hne (componentVertices_eq_of_reachable hL n x r s hrs)
exact hlj (component_activeSigns_disjoint hL n x r s i l hrs hi hl j hij)
theorem componentLikelihood_dependsOn_signIndices
A component mark product is a function only of its selected sign block. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
n :
ℕ
z :
Marks C
lam lam' :
SignVector hL
hagree :
∀ j ∈ componentSignIndices hL n x C, lam j = lam' j
conclusion
(∏ i : C, conditionalLikelihood hL lam (x i) (z i).1 (z i).2 / 4)
= ∏ i : C, conditionalLikelihood hL lam' (x i) (z i).1 (z i).2 / 4
Proof (Lean source)
-- @node: componentLikelihood_dependsOn_signIndices
lemma componentLikelihood_dependsOn_signIndices (x : Fin n → Covariate)
(C : Finset (Fin n)) (z : Marks C) (lam lam' : SignVector hL)
(hagree : ∀ j ∈ componentSignIndices hL n x C, lam j = lam' j) :
(∏ i : C, conditionalLikelihood hL lam (x i) (z i).1 (z i).2 / 4) =
∏ i : C, conditionalLikelihood hL lam' (x i) (z i).1 (z i).2 / 4 := by
apply componentLikelihood_eq_of_activeSigns_agree hL n x C z lam lam'
intro i j hj
exact hagree j ((mem_componentSignIndices hL n x C j).mpr ⟨i, i.property, hj⟩)
theorem fullAlternativeMass_component_factorization Lemma fullAlternativeMass_component_factorization in the paper ↗
Mixing independent signs preserves the product over distinct shared-sign components. This identity holds pointwise in the covariates, including all support boundaries. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
fullAlternativeMass hL n x z
Proof (Lean source)
-- @node: fullAlternativeMass_component_factorization
lemma fullAlternativeMass_component_factorization (x : Fin n → Covariate)
(z : Fin n → Bool × Bool) :
fullAlternativeMass hL n x z =
((orderedComponents hL n x).map (fun C =>
alternativeMass hL n x C (fun i => z i))).prod := by
classical
let F : Finset (Fin n) → SignVector hL → ℝ := fun C lam =>
∏ i : C, conditionalLikelihood hL lam (x i) (z i).1 (z i).2 / 4
have hfactor := fairSignAverage_disjointBlocks_prod
(orderedComponents hL n x).toFinset (componentSignIndices hL n x) F
(orderedComponents_signIndices_pairwiseDisjoint hL n x)
(fun C _ lam lam' hagree =>
componentLikelihood_dependsOn_signIndices hL n x C (fun i => z i) lam lam' hagree)
have hweight : card (activeSigns hL) = (activeSigns hL).card :=
Fintype.card_coe _
rw [hweight] at hfactor
have hprod (lam : SignVector hL) :
(∏ C ∈ (orderedComponents hL n x).toFinset, F C lam) =
∏ i : Fin n, conditionalLikelihood hL lam (x i) (z i).1 (z i).2 / 4 := by
rw [List.prod_toFinset (fun C => F C lam) (orderedComponents_nodup hL n x)]
exact orderedComponents_prod hL n x
(fun i => conditionalLikelihood hL lam (x i) (z i).1 (z i).2 / 4)
simp_rw [hprod] at hfactor
change fullAlternativeMass hL n x z =
∏ C ∈ (orderedComponents hL n x).toFinset, alternativeMass hL n x C (fun i => z i) at hfactor
rw [List.prod_toFinset (fun C => alternativeMass hL n x C (fun i => z i))
(orderedComponents_nodup hL n x)] at hfactor
exact hfactor
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.ComponentJointLaw 30 declarations Finite partition sums, conditional marginals of the product common-mass coupling, and the quadratic component total-variation bound.
Finite partition sums, conditional marginals of the product common-mass coupling, and the quadratic component total-variation bound. The certificates hold at every covariate vector, including support boundaries.
Restriction to the blocks of a finite partition is a bijection of assignment spaces.
Definition (Lean source)
parameters
ι α :
hcover :
∀ i, ∃ C ∈ s, i ∈ C
hdisj :
def
given by
by classical apply Equiv.ofBijective (fun (z : ι → α) (C : s) (i : ↥(C : Finset ι)) => z i) constructor · intro z w h funext i obtain ⟨C, hC, hi⟩ := hcover i exact congrFun (congrFun h ⟨C, hC⟩) ⟨i, hi⟩ · intro w choose C hC hi using hcover refine ⟨fun i
=> w ⟨C i, hC i⟩ ⟨i, hi i⟩, ?_⟩ funext D i have he : C i = D := by by_contra hne exact Finset.disjoint_left.mp (hdisj (hC i) D.property hne) (hi i) i.property have hs : (⟨C i, hC i⟩ : s) = D := Subtype.ext he have key : ∀ (E D : s) (he : E = D) (a : ↥(E : Finset ι)) (b : ↥(D : Finset ι)), (a : ι) = (b : ι) → w E a = w D b := by intro E D he a b hab subst D exact congrArg (w E) (Subtype.ext hab) exact key _ _ hs _ _ rfl
Summing products of independent block arrays factors into their individual sums. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
(∑ z : ι → α, ∏ C ∈ s, F C (fun i => z i)) = ∏ C ∈ s, ∑ z : C → α, F C z
Proof (Lean source)
-- @node: sum_prod_partition
lemma sum_prod_partition {ι α : Type*} [Fintype ι] [DecidableEq ι] [Fintype α]
(s : Finset (Finset ι))
(hcover : ∀ i, ∃ C ∈ s, i ∈ C)
(hdisj : PairwiseDisjoint (s : Set (Finset ι)) id)
(F : ∀ C : Finset ι, (C → α) → ℝ) :
(∑ z : ι → α, ∏ C ∈ s, F C (fun i => z i)) =
∏ C ∈ s, ∑ z : C → α, F C z := by
classical
let e := partitionRestrictionEquiv (α := α) s hcover hdisj
calc
_ = ∑ w : ∀ C : s, ↥(C : Finset ι) → α, ∏ C : s, F C (w C) := by
apply Fintype.sum_equiv e
intro z
change (∏ C ∈ s, F C (fun i => z i)) = ∏ C : s, F C (fun i => z i)
exact (Finset.prod_coe_sort s (fun C => F C (fun i => z i))).symm
_ = _ := by
rw [← Fintype.prod_sum]
exact Finset.prod_coe_sort s (fun C => ∑ z : C → α, F C z)
The ordered shared-sign components form a finite partition for any mark alphabet. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: orderedComponents_sum_prod
lemma orderedComponents_sum_prod {α : Type*} [Fintype α]
(hL : ℝ) (n : ℕ) (x : Fin n → Covariate)
(F : ∀ C : Finset (Fin n), (C → α) → ℝ) :
(∑ z : Fin n → α, ((orderedComponents hL n x).map
(fun C => F C (fun i => z i))).prod) =
((orderedComponents hL n x).map (fun (C : Finset (Fin n)) => ∑ z : C → α, F C z)).prod := by
classical
have hc : ∀ i, ∃ C ∈ (orderedComponents hL n x).toFinset, i ∈ C := by
intro i
obtain ⟨C, hC, hi⟩ := orderedComponents_cover hL n x i
exact ⟨C, List.mem_toFinset.mpr hC, hi⟩
have hd : PairwiseDisjoint
((orderedComponents hL n x).toFinset : Set (Finset (Fin n))) id := by
intro C hC D hD hne
exact orderedComponents_disjoint hL n x C D
(List.mem_toFinset.mp hC) (List.mem_toFinset.mp hD) hne
simp_rw [← List.prod_toFinset _ (orderedComponents_nodup hL n x)]
exact sum_prod_partition _ hc hd F
Every product joint mass is nonnegative. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: jointMarkMass_nonneg
lemma jointMarkMass_nonneg (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) (x : Fin n → Covariate) (z w : Fin n → Bool × Bool) :
0 ≤ jointMarkMass hL n x z w := by
apply List.prod_nonneg
intro a ha
obtain ⟨C, _, rfl⟩ := List.mem_map.mp ha
exact componentCouplingMass_nonneg hL hhL n x C _ _
Summing a row of the joint product recovers the full conditional sign mixture. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
(∑ w, jointMarkMass hL n x z w) = fullAlternativeMass hL n x z
Proof (Lean source)
-- @node: jointMarkMass_row_sum
lemma jointMarkMass_row_sum (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) (x : Fin n → Covariate) (z : Fin n → Bool × Bool) :
(∑ w, jointMarkMass hL n x z w) = fullAlternativeMass hL n x z := by
unfold jointMarkMass
rw [orderedComponents_sum_prod]
simp_rw [componentCouplingMass_row_sum hL hhL]
exact (fullAlternativeMass_component_factorization hL n x z).symm
Summing a column of the joint product recovers the full fair mark array. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: jointMarkMass_column_sum
lemma jointMarkMass_column_sum (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) (x : Fin n → Covariate) (w : Fin n → Bool × Bool) :
(∑ z, jointMarkMass hL n x z w) = (4 : ℝ)^(-(n : ℤ)) := by
unfold jointMarkMass
rw [orderedComponents_sum_prod hL n x
(fun C z => componentCouplingMass hL n x C z (fun i => w i))]
simp_rw [componentCouplingMass_column_sum hL hhL]
exact (nullMass_component_factorization hL n x w).symm
The full conditional alternative array sums to one. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: fullAlternativeMass_sum
lemma fullAlternativeMass_sum (hL : ℝ) (n : ℕ) (x : Fin n → Covariate) :
(∑ z, fullAlternativeMass hL n x z) = 1 := by
simp_rw [fullAlternativeMass_component_factorization, orderedComponents_sum_prod,
alternativeMass_sum]
simp
The finite joint product array has unit total mass. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: jointMarkMass_sum
lemma jointMarkMass_sum (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) (x : Fin n → Covariate) :
(∑ zw : (Fin n → Bool × Bool) × (Fin n → Bool × Bool),
jointMarkMass hL n x zw.1 zw.2) = 1 := by
rw [Fintype.sum_prod_type]
simp_rw [jointMarkMass_row_sum hL hhL]
exact fullAlternativeMass_sum hL n x
The conditional alternative dataset law attaches fixed covariates to the mixed marks.
Definition (Lean source)
parameters
given by
∑ z : Fin n → Bool × Bool, ofReal (fullAlternativeMass hL n x z) • Measure.dirac (attachMarks n x z)
The conditional null dataset law attaches fixed covariates to independent fair marks.
theorem conditionalDatasetCoupling_map_fst
The conditional product coupling has the exact mixed dataset law as first marginal. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: conditionalDatasetCoupling_map_fst
lemma conditionalDatasetCoupling_map_fst (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) (x : Fin n → Covariate) :
(conditionalDatasetCoupling hL n x).map fst =
conditionalAlternativeDatasetLaw hL n x := by
classical
have hr (z : Fin n → Bool × Bool) :
(∑ w, ofReal (jointMarkMass hL n x z w)) =
ofReal (fullAlternativeMass hL n x z) := by
rw [← ENNReal.ofReal_sum_of_nonneg
(fun w _ => jointMarkMass_nonneg hL hhL n x z w), jointMarkMass_row_sum hL hhL]
ext S hS
rw [Measure.map_apply measurable_fst hS]
simp only [conditionalDatasetCoupling, conditionalAlternativeDatasetLaw,
Measure.finsetSum_apply, Measure.smul_apply, smul_eq_mul, Measure.dirac_apply,
Fintype.sum_prod_type]
apply Finset.sum_congr rfl
intro z _
change (∑ w, ofReal (jointMarkMass hL n x z w) *
S.indicator 1 (attachMarks n x z)) = _
rw [← Finset.sum_mul, hr]
theorem conditionalDatasetCoupling_map_snd
The conditional product coupling has the exact fair dataset law as second marginal. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: conditionalDatasetCoupling_map_snd
lemma conditionalDatasetCoupling_map_snd (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) (x : Fin n → Covariate) :
(conditionalDatasetCoupling hL n x).map snd = conditionalNullDatasetLaw n x := by
classical
have hc (w : Fin n → Bool × Bool) :
(∑ z, ofReal (jointMarkMass hL n x z w)) =
ofReal ((4 : ℝ)^(-(n : ℤ))) := by
rw [← ENNReal.ofReal_sum_of_nonneg
(fun z _ => jointMarkMass_nonneg hL hhL n x z w), jointMarkMass_column_sum hL hhL]
ext S hS
rw [Measure.map_apply measurable_snd hS]
simp only [conditionalDatasetCoupling, conditionalNullDatasetLaw,
Measure.finsetSum_apply, Measure.smul_apply, smul_eq_mul, Measure.dirac_apply,
Fintype.sum_prod_type]
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro w _
change (∑ z, ofReal (jointMarkMass hL n x z w) *
S.indicator 1 (attachMarks n x w)) = _
rw [← Finset.sum_mul, hc]
theorem conditionalDatasetCoupling_isProbabilityMeasure Lemma conditionalDatasetCoupling_isProbabilityMeasure in the paper ↗
The conditional product coupling is a probability measure at every covariate vector. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: conditionalDatasetCoupling_isProbabilityMeasure
lemma conditionalDatasetCoupling_isProbabilityMeasure (hL : ℝ)
(hhL : 0 < hL ∧ hL ≤ 1 / 4) (n : ℕ) (x : Fin n → Covariate) :
IsProbabilityMeasure (conditionalDatasetCoupling hL n x) := by
constructor
simp only [conditionalDatasetCoupling, Measure.finsetSum_apply, Measure.smul_apply,
Measure.dirac_apply_of_mem (Set.mem_univ _), smul_eq_mul, mul_one]
rw [← ENNReal.ofReal_sum_of_nonneg
(fun zw _ => jointMarkMass_nonneg hL hhL n x zw.1 zw.2), jointMarkMass_sum hL hhL]
exact ENNReal.ofReal_one
theorem conditionalDatasetCoupling_isCoupling
The explicit Borel conditional product couples the two full conditional dataset laws. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
IsCoupling (conditionalDatasetCoupling hL n x) (conditionalAlternativeDatasetLaw hL n x) (conditionalNullDatasetLaw n x)
Proof (Lean source)
-- @node: conditionalDatasetCoupling_isCoupling
lemma conditionalDatasetCoupling_isCoupling (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) (x : Fin n → Covariate) :
IsCoupling (conditionalDatasetCoupling hL n x)
(conditionalAlternativeDatasetLaw hL n x) (conditionalNullDatasetLaw n x) := by
exact ⟨conditionalDatasetCoupling_isProbabilityMeasure hL hhL n x,
conditionalDatasetCoupling_map_fst hL hhL n x,
conditionalDatasetCoupling_map_snd hL hhL n x⟩
theorem commonMassCoupling_isProbabilityMeasure
Integrating the pointwise conditional coupling preserves unit total mass. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1 / 4
n :
ℕ
conclusion
Proof (Lean source)
-- @node: commonMassCoupling_isProbabilityMeasure
lemma commonMassCoupling_isProbabilityMeasure (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) : IsProbabilityMeasure (commonMassCoupling hL n) := by
constructor
rw [commonMassCoupling, Measure.bind_apply MeasurableSet.univ
(measurable_conditionalDatasetCoupling hL n).aemeasurable]
have hp (x : Fin n → Covariate) : conditionalDatasetCoupling hL n x univ = 1 := by
let := conditionalDatasetCoupling_isProbabilityMeasure hL hhL n x
exact measure_univ
simp only [hp, lintegral_const, one_mul, measure_univ]
theorem measurable_conditionalAlternativeDatasetLaw
The conditional alternative dataset law is Borel in the covariates. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
Proof (Lean source)
-- @node: measurable_conditionalAlternativeDatasetLaw
@[fun_prop] lemma measurable_conditionalAlternativeDatasetLaw (hL : ℝ) (n : ℕ) :
Measurable (conditionalAlternativeDatasetLaw hL n) := by
apply Measure.measurable_of_measurable_coe
intro E hE
simp only [conditionalAlternativeDatasetLaw, Measure.finsetSum_apply,
Measure.smul_apply, smul_eq_mul]
apply Finset.measurable_sum
intro z _
apply mul ((measurable_fullAlternativeMass hL n z).ennreal_ofReal)
simp only [Measure.dirac_apply' _ hE]
exact measurable_const.indicator (hE.preimage (measurable_attachMarks n z))
theorem measurable_conditionalNullDatasetLaw
The conditional null dataset law is Borel in the covariates. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: measurable_conditionalNullDatasetLaw
@[fun_prop] lemma measurable_conditionalNullDatasetLaw (n : ℕ) :
Measurable (conditionalNullDatasetLaw n) := by
apply Measure.measurable_of_measurable_coe
intro E hE
simp only [conditionalNullDatasetLaw, Measure.finsetSum_apply,
Measure.smul_apply, smul_eq_mul]
apply Finset.measurable_sum
intro z _
apply mul measurable_const
simp only [Measure.dirac_apply' _ hE]
exact measurable_const.indicator (hE.preimage (measurable_attachMarks n z))
theorem commonMassCoupling_map_fst_bind
Integrating the conditional coupling recovers the integrated alternative marginal. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1 / 4
n :
ℕ
conclusion
= (Measure.pi (fun _ : Fin n => (volume : Measure Covariate))).bind (conditionalAlternativeDatasetLaw hL n)
Proof (Lean source)
-- @node: commonMassCoupling_map_fst_bind
lemma commonMassCoupling_map_fst_bind (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) :
(commonMassCoupling hL n).map fst =
(Measure.pi (fun _ : Fin n => (volume : Measure Covariate))).bind
(conditionalAlternativeDatasetLaw hL n) := by
ext E hE
rw [Measure.map_apply measurable_fst hE, commonMassCoupling,
Measure.bind_apply (hE.preimage measurable_fst)
(measurable_conditionalDatasetCoupling hL n).aemeasurable,
Measure.bind_apply hE (measurable_conditionalAlternativeDatasetLaw hL n).aemeasurable]
apply lintegral_congr
intro x
rw [← conditionalDatasetCoupling_map_fst hL hhL n x,
Measure.map_apply measurable_fst hE]
theorem commonMassCoupling_map_snd_bind
Integrating the conditional coupling recovers the integrated null marginal. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1 / 4
n :
ℕ
conclusion
Proof (Lean source)
-- @node: commonMassCoupling_map_snd_bind
lemma commonMassCoupling_map_snd_bind (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) :
(commonMassCoupling hL n).map snd =
(Measure.pi (fun _ : Fin n => (volume : Measure Covariate))).bind
(conditionalNullDatasetLaw n) := by
ext E hE
rw [Measure.map_apply measurable_snd hE, commonMassCoupling,
Measure.bind_apply (hE.preimage measurable_snd)
(measurable_conditionalDatasetCoupling hL n).aemeasurable,
Measure.bind_apply hE (measurable_conditionalNullDatasetLaw n).aemeasurable]
apply lintegral_congr
intro x
rw [← conditionalDatasetCoupling_map_snd hL hhL n x,
Measure.map_apply measurable_snd hE]
theorem commonMassCoupling_isCoupling_bind
The Borel product construction couples the integrated conditional dataset laws. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1 / 4
n :
ℕ
conclusion
IsCoupling (commonMassCoupling hL n) ((Measure.pi (fun _ : Fin n => (volume : Measure Covariate))).bind (conditionalAlternativeDatasetLaw hL n)) ((Measure.pi (fun _ : Fin n => (volume : Measure Covariate))).bind (conditionalNullDatasetLaw n))
Proof (Lean source)
-- @node: commonMassCoupling_isCoupling_bind
lemma commonMassCoupling_isCoupling_bind (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) :
IsCoupling (commonMassCoupling hL n)
((Measure.pi (fun _ : Fin n => (volume : Measure Covariate))).bind
(conditionalAlternativeDatasetLaw hL n))
((Measure.pi (fun _ : Fin n => (volume : Measure Covariate))).bind
(conditionalNullDatasetLaw n)) := by
exact ⟨commonMassCoupling_isProbabilityMeasure hL hhL n,
commonMassCoupling_map_fst_bind hL hhL n, commonMassCoupling_map_snd_bind hL hhL n⟩
A nonnegative finite atomic array evaluates an event by summing its selected entries. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
Proof (Lean source)
-- @node: finite_atomic_array_real
lemma finite_atomic_array_real {α : Type*} [Fintype α] [MeasurableSpace α]
(a : α → ℝ) (ha : ∀ z, 0 ≤ a z) (E : Set α) (hE : MeasurableSet E) :
(∑ z, ofReal (a z) • Measure.dirac z).real E =
∑ z, if z ∈ E then a z else 0 := by
classical
have hn (z : α) : 0 ≤ if z ∈ E then a z else 0 := by
split_ifs <;> first | exact ha z | exact le_rfl
have he (z : α) : ofReal (a z) * E.indicator 1 z =
ofReal (if z ∈ E then a z else 0) := by
by_cases hz : z ∈ E <;> simp [hz]
simp only [measureReal_def, Measure.finsetSum_apply, Measure.smul_apply,
Measure.dirac_apply' _ hE, smul_eq_mul, he]
rw [← ENNReal.ofReal_sum_of_nonneg (fun z _ => hn z),
ENNReal.toReal_ofReal (Finset.sum_nonneg (fun z _ => hn z))]
theorem finite_array_event_gap_le_overlap_deficit Lemma finite_array_event_gap_le_overlap_deficit in the paper ↗
Event gaps of two probability arrays are bounded by their mass outside the common part. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: finite_array_event_gap_le_overlap_deficit
lemma finite_array_event_gap_le_overlap_deficit {α : Type*} [Fintype α]
(a b : α → ℝ)
(hsa : ∑ z, a z = 1) (hsb : ∑ z, b z = 1) (E : Set α) :
|(∑ z, if z ∈ E then a z else 0) - (∑ z, if z ∈ E then b z else 0)| ≤
1 - ∑ z, min (a z) (b z) := by
classical
let c (z : α) := min (a z) (b z)
have bounds (f : α → ℝ) (hf : ∀ z, 0 ≤ f z) :
0 ≤ (∑ z, if z ∈ E then f z else 0) ∧
(∑ z, if z ∈ E then f z else 0) ≤ ∑ z, f z := by
constructor
· apply Finset.sum_nonneg; intro z _; split_ifs <;> first | exact hf z | exact le_rfl
· apply Finset.sum_le_sum; intro z _; split_ifs <;> first | exact le_rfl | exact hf z
have har := bounds (fun z => a z-c z) (fun z => sub_nonneg.mpr (min_le_left _ _))
have hbr := bounds (fun z => b z-c z) (fun z => sub_nonneg.mpr (min_le_right _ _))
rw [Finset.sum_sub_distrib, hsa] at har
rw [Finset.sum_sub_distrib, hsb] at hbr
have split_sum (f : α → ℝ) :
(∑ z, if z ∈ E then f z else 0) =
(∑ z, if z ∈ E then f z-c z else 0) + (∑ z, if z ∈ E then c z else 0) := by
rw [← Finset.sum_add_distrib]
apply Finset.sum_congr rfl
intro z _
split_ifs <;> ring
rw [split_sum a, split_sum b, abs_le]
change -(1 - ∑ z, c z) ≤ _ ∧ _ ≤ 1 - ∑ z, c z
constructor <;> linarith [har.1, har.2, hbr.1, hbr.2]
The TV of the finite component laws is bounded by the explicit coupling's failure mass. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: component_TV_le_overlap_deficit
lemma component_TV_le_overlap_deficit (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) (x : Fin n → Covariate) (C : Finset (Fin n)) :
TV (alternativeComponentLaw hL n x C) (nullComponentLaw n C) ≤
1-overlapMass hL n x C := by
classical
unfold TV tvDist
apply ciSup_le
intro E
rw [alternativeComponentLaw, nullComponentLaw,
finite_atomic_array_real _ (alternativeMass_nonneg hL hhL n x C) E.1 E.2,
finite_atomic_array_real _ (fun z => (nullMass_pos n C z).le) E.1 E.2]
exact finite_array_event_gap_le_overlap_deficit _ _
(alternativeMass_sum hL n x C) (nullMass_sum n C) E.1
The curvature certificate gives the uniform four-times-separation component TV bound. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
≤ 4*separation hL*(componentSize n C : ℝ)^2
Proof (Lean source)
-- @node: component_TV_le_quadratic
lemma component_TV_le_quadratic (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) (x : Fin n → Covariate) (C : Finset (Fin n)) :
TV (alternativeComponentLaw hL n x C) (nullComponentLaw n C) ≤
4*separation hL*(componentSize n C : ℝ)^2 := by
exact (component_TV_le_overlap_deficit hL hhL n x C).trans
(overlapMass_deficit_le hL hhL n x C)
Selecting entries where the first array is larger attains its overlap deficit. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: finite_array_positive_event_gap
lemma finite_array_positive_event_gap {α : Type*} [Fintype α]
(a b : α → ℝ) (hsa : ∑ z, a z = 1) :
(∑ z, if b z ≤ a z then a z else 0) -
(∑ z, if b z ≤ a z then b z else 0) =
1 - ∑ z, min (a z) (b z) := by
classical
rw [← Finset.sum_sub_distrib, ← hsa, ← Finset.sum_sub_distrib]
apply Finset.sum_congr rfl
intro z _
by_cases h : b z ≤ a z
· rw [if_pos h, if_pos h, min_eq_right h]
· rw [if_neg h, if_neg h, min_eq_left (le_of_not_ge h)]
ring
theorem alternativeComponentLaw_isProbabilityMeasure Lemma alternativeComponentLaw_isProbabilityMeasure in the paper ↗
Each nonnegative alternative component array defines a probability measure. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
IsProbabilityMeasure (alternativeComponentLaw hL n x C)
Proof (Lean source)
-- @node: alternativeComponentLaw_isProbabilityMeasure
lemma alternativeComponentLaw_isProbabilityMeasure (hL : ℝ)
(hhL : 0 < hL ∧ hL ≤ 1 / 4) (n : ℕ) (x : Fin n → Covariate)
(C : Finset (Fin n)) : IsProbabilityMeasure (alternativeComponentLaw hL n x C) := by
constructor
simp only [alternativeComponentLaw, Measure.finsetSum_apply, Measure.smul_apply,
Measure.dirac_apply_of_mem (Set.mem_univ _), smul_eq_mul, mul_one]
rw [← ENNReal.ofReal_sum_of_nonneg
(fun z _ => alternativeMass_nonneg hL hhL n x C z), alternativeMass_sum]
exact ENNReal.ofReal_one
The explicit finite common part gives the exact total variation of component laws. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: component_TV_eq_overlap_deficit
lemma component_TV_eq_overlap_deficit (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) (x : Fin n → Covariate) (C : Finset (Fin n)) :
TV (alternativeComponentLaw hL n x C) (nullComponentLaw n C) =
1-overlapMass hL n x C := by
classical
apply le_antisymm (component_TV_le_overlap_deficit hL hhL n x C)
let E : Set (Marks C) := {z | nullMass n C z ≤ alternativeMass hL n x C z}
have hE : MeasurableSet E := Set.to_countable E |>.measurableSet
let := alternativeComponentLaw_isProbabilityMeasure hL hhL n x C
let := nullComponentLaw_isProbabilityMeasure n C
have h := abs_measureReal_sub_le_tvDist
(μ := alternativeComponentLaw hL n x C) (ν := nullComponentLaw n C) hE
rw [alternativeComponentLaw, nullComponentLaw,
finite_atomic_array_real _ (alternativeMass_nonneg hL hhL n x C) E hE,
finite_atomic_array_real _ (fun z => (nullMass_pos n C z).le) E hE] at h
change |(∑ z, if nullMass n C z ≤ alternativeMass hL n x C z then
alternativeMass hL n x C z else 0) -
(∑ z, if nullMass n C z ≤ alternativeMass hL n x C z then nullMass n C z else 0)| ≤ _ at h
rw [finite_array_positive_event_gap _ _ (alternativeMass_sum hL n x C)] at h
change |1-overlapMass hL n x C| ≤ TV (alternativeComponentLaw hL n x C)
(nullComponentLaw n C) at h
rw [abs_of_nonneg (sub_nonneg.mpr (overlapMass_range hL hhL n x C).2)] at h
exact h
theorem componentCoupling_disagreement_eq_TV Lemma componentCoupling_disagreement_eq_TV in the paper ↗
The common-mass component coupling fails with probability exactly its marginal TV. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
componentCoupling hL n x C {zw | zw.1 ≠ zw.2}
Proof (Lean source)
-- @node: componentCoupling_disagreement_eq_TV
lemma componentCoupling_disagreement_eq_TV (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) (x : Fin n → Covariate) (C : Finset (Fin n)) :
componentCoupling hL n x C {zw | zw.1 ≠ zw.2} =
ofReal (TV (alternativeComponentLaw hL n x C) (nullComponentLaw n C)) := by
rw [component_TV_eq_overlap_deficit hL hhL,
componentCoupling_disagreement_mass hL hhL]
Component total variation is half the absolute difference of its finite arrays. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: component_TV_eq_half_abs_sum
lemma component_TV_eq_half_abs_sum (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) (x : Fin n → Covariate) (C : Finset (Fin n)) :
TV (alternativeComponentLaw hL n x C) (nullComponentLaw n C) =
1/2 * ∑ z : Marks C, |alternativeMass hL n x C z-nullMass n C z| := by
rw [component_TV_eq_overlap_deficit hL hhL, componentMass_abs_sum]
ring
theorem componentCoupling_hamming_cost_le_TV Lemma componentCoupling_hamming_cost_le_TV in the paper ↗
The explicit coupling changes at most the component size times its marginal TV. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
(∫⁻ zw, (hammingDist zw.1 zw.2 : ℝ≥0∞) ∂componentCoupling hL n x C)
Proof (Lean source)
-- @node: componentCoupling_hamming_cost_le_TV
lemma componentCoupling_hamming_cost_le_TV (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) (x : Fin n → Covariate) (C : Finset (Fin n)) :
(∫⁻ zw, (hammingDist zw.1 zw.2 : ℝ≥0∞) ∂componentCoupling hL n x C) ≤
(C.card : ℝ≥0∞) *
ofReal (TV (alternativeComponentLaw hL n x C) (nullComponentLaw n C)) := by
rw [component_TV_eq_overlap_deficit hL hhL]
exact componentCoupling_hamming_cost_le hL hhL n x C
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.ComponentMeasurability 16 declarations Borel regularity of finite common-mass arrays, fixed-component couplings and the full conditional dataset coupling, using finitely many labeled-graph strata.
Borel regularity of finite common-mass arrays, fixed-component couplings and the full conditional dataset coupling, using finitely many labeled-graph strata. These certificates include the overlap-one branch and require no radius restrictions.
theorem measurable_conditionalLikelihood_covariate Lemma measurable_conditionalLikelihood_covariate in the paper ↗
Fixing the two marks leaves a Borel conditional likelihood in the covariate. The displayed conclusion follows.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: measurable_conditionalLikelihood_covariate
@[fun_prop] lemma measurable_conditionalLikelihood_covariate (lam : SignVector hL)
(a y : Bool) : Measurable (fun x : Covariate => conditionalLikelihood hL lam x a y) := by
cases a <;> cases y <;> simp only [conditionalLikelihood, bernoulliMass,
Bool.false_eq_true, if_false, if_true] <;> fun_prop
Each fixed-component alternative mass is Borel in the full covariate vector. The displayed conclusion follows.
Formal statement
hypotheses
conclusion
Measurable (fun x => alternativeMass hL n x C z)
Proof (Lean source)
-- @node: measurable_alternativeMass
@[fun_prop] lemma measurable_alternativeMass (C : Finset (Fin n)) (z : Marks C) :
Measurable (fun x => alternativeMass hL n x C z) := by
unfold alternativeMass
fun_prop
Taking the pointwise common part preserves Borel regularity. The displayed conclusion follows.
Formal statement
hypotheses
conclusion
Measurable (fun x => commonMass hL n x C z)
Proof (Lean source)
-- @node: measurable_commonMass
@[fun_prop] lemma measurable_commonMass (C : Finset (Fin n)) (z : Marks C) :
Measurable (fun x => commonMass hL n x C z) := by
unfold commonMass
fun_prop
The finite sum of common masses is Borel in the covariates. The displayed conclusion follows.
Formal statement
hypotheses
conclusion
Measurable (fun x => overlapMass hL n x C)
Proof (Lean source)
-- @node: measurable_overlapMass
@[fun_prop] lemma measurable_overlapMass (C : Finset (Fin n)) :
Measurable (fun x => overlapMass hL n x C) := by
unfold overlapMass
fun_prop
The explicit joint mass is Borel, including the branch where overlap is one. The displayed conclusion follows.
Formal statement
hypotheses
conclusion
Measurable (fun x => componentCouplingMass hL n x C z w)
Proof (Lean source)
-- @node: measurable_componentCouplingMass
@[fun_prop] lemma measurable_componentCouplingMass (C : Finset (Fin n)) (z w : Marks C) :
Measurable (fun x => componentCouplingMass hL n x C z w) := by
classical
dsimp only [componentCouplingMass]
have hd : Measurable (fun x : Fin n → Covariate =>
if z = w then commonMass hL n x C z else 0) := by
by_cases hzw : z = w <;> simp only [hzw, if_true, if_false] <;> fun_prop
apply Measurable.ite
(measurableSet_lt (measurable_overlapMass hL n C) measurable_const) _ hd
exact hd.add (((measurable_alternativeMass hL n C z).sub
(measurable_commonMass hL n C z)).mul
(measurable_const.sub (measurable_commonMass hL n C w)) |>.div
(measurable_const.sub (measurable_overlapMass hL n C)))
The full sign-mixture conditional mass is Borel for each fixed dataset mark vector. The displayed conclusion follows.
Formal statement
hypotheses
conclusion
Measurable (fun x => fullAlternativeMass hL n x z)
Proof (Lean source)
-- @node: measurable_fullAlternativeMass
@[fun_prop] lemma measurable_fullAlternativeMass (z : Fin n → Bool × Bool) :
Measurable (fun x => fullAlternativeMass hL n x z) := by
unfold fullAlternativeMass
fun_prop
Attaching fixed marks to covariates is a Borel map into the dataset space. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: measurable_attachMarks
@[fun_prop] lemma measurable_attachMarks (z : Fin n → Bool × Bool) :
Measurable (fun x => attachMarks n x z) := by
unfold attachMarks
fun_prop
The alternative finite component measure varies measurably with the covariates. The displayed conclusion follows.
Formal statement
hypotheses
conclusion
Measurable (fun x => alternativeComponentLaw hL n x C)
Proof (Lean source)
-- @node: measurable_alternativeComponentLaw
@[fun_prop] lemma measurable_alternativeComponentLaw (C : Finset (Fin n)) :
Measurable (fun x => alternativeComponentLaw hL n x C) := by
unfold alternativeComponentLaw
fun_prop
The common-mass coupling of a fixed component varies measurably with the covariates. The displayed conclusion follows.
Formal statement
hypotheses
conclusion
Measurable (fun x => componentCoupling hL n x C)
Proof (Lean source)
-- @node: measurable_componentCoupling
@[fun_prop] lemma measurable_componentCoupling (C : Finset (Fin n)) :
Measurable (fun x => componentCoupling hL n x C) := by
unfold componentCoupling
fun_prop
The deterministic ordered components of a fixed graph use increasing least vertices.
A product over any fixed list of components is a Borel joint mark mass. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Measurable (fun x => (Cs.map (fun C => componentCouplingMass hL n x C (fun i => z i) (fun i => w i))).prod)
Proof (Lean source)
-- @node: measurable_fixedComponentProduct
lemma measurable_fixedComponentProduct (Cs : List (Finset (Fin n)))
(z w : Fin n → Bool × Bool) :
Measurable (fun x => (Cs.map (fun C =>
componentCouplingMass hL n x C (fun i => z i) (fun i => w i))).prod) := by
induction Cs with
| nil => simpa using (measurable_const : Measurable (fun _ : Fin n → Covariate => (1 : ℝ)))
| cons C Cs ih =>
convert (measurable_componentCouplingMass hL n C
(fun i => z i) (fun i => w i)).mul ih using 1
funext x
simp only [List.map_cons, List.prod_cons, Pi.mul_apply]
The varying-graph joint array is Borel by its finite decomposition into graph strata. The displayed conclusion follows.
Formal statement
hypotheses
conclusion
Measurable (fun x => jointMarkMass hL n x z w)
Proof (Lean source)
-- @node: measurable_jointMarkMass
@[fun_prop] lemma measurable_jointMarkMass (z w : Fin n → Bool × Bool) :
Measurable (fun x => jointMarkMass hL n x z w) := by
classical
have hm : Measurable (fun x => ∑ G : SimpleGraph (Fin n),
{x | sharedGraph hL n x = G}.indicator (fun x =>
((componentsOfGraph n G).map (fun C =>
componentCouplingMass hL n x C (fun i => z i) (fun i => w i))).prod) x) := by
apply Finset.measurable_sum
intro G _
exact (measurable_fixedComponentProduct hL n (componentsOfGraph n G) z w).indicator
(measurableSet_sharedGraph_eq hL n G)
convert hm using 1
funext x
simp only [indicator, Set.mem_ofPred_eq]
rw [Finset.sum_ite_eq, if_pos (Finset.mem_univ _)]
simp only [jointMarkMass,
orderedComponents_eq_of_sharedGraph_eq hL n x (sharedGraph hL n x) rfl]
theorem measurable_conditionalDatasetCoupling Lemma measurable_conditionalDatasetCoupling in the paper ↗
The explicit conditional dataset coupling is Borel on every covariate vector. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: measurable_conditionalDatasetCoupling
@[fun_prop] lemma measurable_conditionalDatasetCoupling :
Measurable (conditionalDatasetCoupling hL n) := by
apply Measure.measurable_of_measurable_coe
intro E hE
simp only [conditionalDatasetCoupling, Measure.finsetSum_apply, Measure.smul_apply,
smul_eq_mul]
apply Finset.measurable_sum
intro zw _
apply mul ((measurable_jointMarkMass hL n zw.1 zw.2).ennreal_ofReal)
simp only [Measure.dirac_apply' _ hE]
exact measurable_const.indicator (hE.preimage
((measurable_attachMarks n zw.1).prodMk (measurable_attachMarks n zw.2)))
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.ComponentPartition 18 declarations The ordered shared-sign components partition the dataset indices.
The ordered shared-sign components partition the dataset indices. Distinct components use disjoint active sign coordinates. Conditional likelihoods depend only on those coordinates.
Component membership is exactly reachability in the shared-sign graph. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: mem_componentVertices
lemma mem_componentVertices (x : Fin n → Covariate) (i l : Fin n) :
l ∈ componentVertices hL n x i ↔ (sharedGraph hL n x).Reachable i l := by
classical
simp [componentVertices]
Every vertex belongs to its own component. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
i ∈ componentVertices hL n x i
Proof (Lean source)
-- @node: self_mem_componentVertices
lemma self_mem_componentVertices (x : Fin n → Covariate) (i : Fin n) :
i ∈ componentVertices hL n x i :=
(mem_componentVertices hL n x i i).mpr .rfl
Reachable vertices select the same component. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
conclusion
componentVertices hL n x i = componentVertices hL n x l
Proof (Lean source)
-- @node: componentVertices_eq_of_reachable
lemma componentVertices_eq_of_reachable (x : Fin n → Covariate) (i l : Fin n)
(hil : (sharedGraph hL n x).Reachable i l) :
componentVertices hL n x i = componentVertices hL n x l := by
ext v
rw [mem_componentVertices, mem_componentVertices]
exact ⟨fun hiv => hil.symm.trans hiv, fun hlv => hil.trans hlv⟩
theorem componentVertices_disjoint_of_not_reachable Lemma componentVertices_disjoint_of_not_reachable in the paper ↗
Two components either coincide or are disjoint. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: componentVertices_disjoint_of_not_reachable
lemma componentVertices_disjoint_of_not_reachable (x : Fin n → Covariate) (i l : Fin n)
(hil : ¬(sharedGraph hL n x).Reachable i l) :
Disjoint (componentVertices hL n x i) (componentVertices hL n x l) := by
classical
apply Finset.disjoint_left.mpr
intro v hiv hlv
exact hil (((mem_componentVertices hL n x i v).mp hiv).trans
((mem_componentVertices hL n x l v).mp hlv).symm)
Component roots are precisely the least vertices of their reachable classes. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: mem_componentRoots
lemma mem_componentRoots (x : Fin n → Covariate) (i : Fin n) :
i ∈ componentRoots hL n x ↔ ∀ l, (sharedGraph hL n x).Reachable i l → i ≤ l := by
classical
simp [componentRoots]
Every vertex has a component root in its reachable class. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
∃ r ∈ componentRoots hL n x, i ∈ componentVertices hL n x r
Proof (Lean source)
-- @node: exists_componentRoot
lemma exists_componentRoot (x : Fin n → Covariate) (i : Fin n) :
∃ r ∈ componentRoots hL n x, i ∈ componentVertices hL n x r := by
classical
let C := componentVertices hL n x i
have hC : C.Nonempty := ⟨i, self_mem_componentVertices hL n x i⟩
let r := C.min' hC
have hir : (sharedGraph hL n x).Reachable i r :=
(mem_componentVertices hL n x i r).mp (C.min'_mem hC)
refine ⟨r, (mem_componentRoots hL n x r).mpr ?_,
(mem_componentVertices hL n x r i).mpr hir.symm⟩
intro l hrl
exact C.min'_le l ((mem_componentVertices hL n x i l).mpr (hir.trans hrl))
A reachable class has at most one selected least vertex. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
n :
ℕ
i l :
Fin n
hi :
i ∈ componentRoots hL n x
hl :
l ∈ componentRoots hL n x
hil :
(sharedGraph hL n x).Reachable i l
conclusion
i = l
Proof (Lean source)
-- @node: componentRoots_eq_of_reachable
lemma componentRoots_eq_of_reachable (x : Fin n → Covariate) (i l : Fin n)
(hi : i ∈ componentRoots hL n x) (hl : l ∈ componentRoots hL n x)
(hil : (sharedGraph hL n x).Reachable i l) : i = l := by
exact le_antisymm ((mem_componentRoots hL n x i).mp hi l hil)
((mem_componentRoots hL n x l).mp hl i hil.symm)
Distinct selected roots have disjoint component vertex sets. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
n :
ℕ
i l :
Fin n
hi :
i ∈ componentRoots hL n x
hl :
l ∈ componentRoots hL n x
hne :
i ≠ l
conclusion
Proof (Lean source)
-- @node: componentRoots_pairwise_disjoint
lemma componentRoots_pairwise_disjoint (x : Fin n → Covariate) (i l : Fin n)
(hi : i ∈ componentRoots hL n x) (hl : l ∈ componentRoots hL n x) (hne : i ≠ l) :
Disjoint (componentVertices hL n x i) (componentVertices hL n x l) := by
apply componentVertices_disjoint_of_not_reachable
intro hil
exact hne (componentRoots_eq_of_reachable hL n x i l hi hl hil)
The ordered list contains exactly the components of selected roots. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: mem_orderedComponents
lemma mem_orderedComponents (x : Fin n → Covariate) (C : Finset (Fin n)) :
C ∈ orderedComponents hL n x ↔
∃ r ∈ componentRoots hL n x, componentVertices hL n x r = C := by
classical
simp [orderedComponents]
The ordered components cover every dataset index. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
∃ C ∈ orderedComponents hL n x, i ∈ C
Proof (Lean source)
-- @node: orderedComponents_cover
lemma orderedComponents_cover (x : Fin n → Covariate) (i : Fin n) :
∃ C ∈ orderedComponents hL n x, i ∈ C := by
obtain ⟨r, hr, hi⟩ := exists_componentRoot hL n x i
exact ⟨componentVertices hL n x r,
(mem_orderedComponents hL n x _).mpr ⟨r, hr, rfl⟩, hi⟩
Distinct graph components cannot use the same active sign coordinate. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
n :
ℕ
r s i l :
Fin n
hrs :
¬(sharedGraph hL n x).Reachable r s
hi :
i ∈ componentVertices hL n x r
hl :
l ∈ componentVertices hL n x s
j :
activeSigns hL
hij :
envelope hL (x i) * frame hL j (x i) ≠ 0
conclusion
envelope hL (x l) * frame hL j (x l) = 0
Proof (Lean source)
-- @node: component_activeSigns_disjoint
lemma component_activeSigns_disjoint (x : Fin n → Covariate) (r s i l : Fin n)
(hrs : ¬(sharedGraph hL n x).Reachable r s)
(hi : i ∈ componentVertices hL n x r) (hl : l ∈ componentVertices hL n x s)
(j : activeSigns hL) (hij : envelope hL (x i) * frame hL j (x i) ≠ 0) :
envelope hL (x l) * frame hL j (x l) = 0 := by
by_contra hlj
exact hrs (((mem_componentVertices hL n x r i).mp hi).trans
((reachable_of_shared_activeSign hL n x i l j hij hlj).trans
((mem_componentVertices hL n x s l).mp hl).symm))
theorem perturbation_eq_of_activeSigns_agree Lemma perturbation_eq_of_activeSigns_agree in the paper ↗
The perturbation depends only on signs with a nonzero envelope-weighted frame. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
lam lam' :
SignVector hL
x :
hagree :
∀ j : activeSigns hL
if
envelope hL x * frame hL j x ≠ 0
then
lam j = lam' j
conclusion
perturbation hL lam x = perturbation hL lam' x
Proof (Lean source)
-- @node: perturbation_eq_of_activeSigns_agree
lemma perturbation_eq_of_activeSigns_agree (lam lam' : SignVector hL) (x : Covariate)
(hagree : ∀ j : activeSigns hL,
envelope hL x * frame hL j x ≠ 0 → lam j = lam' j) :
perturbation hL lam x = perturbation hL lam' x := by
classical
simp only [perturbation, Finset.mul_sum]
apply Finset.sum_congr rfl
intro j _
by_cases hj : envelope hL x * frame hL j x = 0
· calc
_ = (2*bit (lam j)-1)*(envelope hL x * frame hL j x) := by ring
_ = (2*bit (lam' j)-1)*(envelope hL x * frame hL j x) := by rw [hj]; ring
_ = _ := by ring
· rw [hagree j hj]
theorem conditionalLikelihood_eq_of_activeSigns_agree Lemma conditionalLikelihood_eq_of_activeSigns_agree in the paper ↗
The entire conditional likelihood, including its quadratic term, uses only active signs. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
lam lam' :
SignVector hL
x :
a y :
hagree :
∀ j : activeSigns hL
if
envelope hL x * frame hL j x ≠ 0
then
lam j = lam' j
conclusion
conditionalLikelihood hL lam x a y = conditionalLikelihood hL lam' x a y
Proof (Lean source)
-- @node: conditionalLikelihood_eq_of_activeSigns_agree
lemma conditionalLikelihood_eq_of_activeSigns_agree (lam lam' : SignVector hL)
(x : Covariate) (a y : Bool)
(hagree : ∀ j : activeSigns hL,
envelope hL x * frame hL j x ≠ 0 → lam j = lam' j) :
conditionalLikelihood hL lam x a y = conditionalLikelihood hL lam' x a y := by
simp only [conditionalLikelihood, altE, altMu0, altMu1,
perturbation_eq_of_activeSigns_agree hL lam lam' x hagree]
Every component in the ordered list contains its selected root. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: orderedComponents_nonempty
lemma orderedComponents_nonempty (x : Fin n → Covariate) (C : Finset (Fin n))
(hC : C ∈ orderedComponents hL n x) : C.Nonempty := by
obtain ⟨r, _, rfl⟩ := (mem_orderedComponents hL n x C).mp hC
exact ⟨r, self_mem_componentVertices hL n x r⟩
Distinct members of the ordered component list are disjoint. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
n :
ℕ
hC :
C ∈ orderedComponents hL n x
hD :
D ∈ orderedComponents hL n x
hne :
C ≠ D
conclusion
Disjoint C D
Proof (Lean source)
-- @node: orderedComponents_disjoint
lemma orderedComponents_disjoint (x : Fin n → Covariate) (C D : Finset (Fin n))
(hC : C ∈ orderedComponents hL n x) (hD : D ∈ orderedComponents hL n x)
(hne : C ≠ D) : Disjoint C D := by
obtain ⟨r, hr, rfl⟩ := (mem_orderedComponents hL n x C).mp hC
obtain ⟨s, hs, rfl⟩ := (mem_orderedComponents hL n x D).mp hD
apply componentRoots_pairwise_disjoint hL n x r s hr hs
intro heq
exact hne (congrArg (componentVertices hL n x) heq)
The deterministic component ordering never repeats a component. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
(orderedComponents hL n x).Nodup
Proof (Lean source)
-- @node: orderedComponents_nodup
lemma orderedComponents_nodup (x : Fin n → Covariate) :
(orderedComponents hL n x).Nodup := by
classical
apply List.Nodup.map_on _ (Finset.sort_nodup _ _)
intro r hr s hs heq
have hr' : r ∈ componentRoots hL n x := (Finset.mem_sort _).mp hr
have hs' : s ∈ componentRoots hL n x := (Finset.mem_sort _).mp hs
apply componentRoots_eq_of_reachable hL n x r s hr' hs'
apply (mem_componentVertices hL n x r s).mp
rw [heq]
exact self_mem_componentVertices hL n x s
theorem componentLikelihood_eq_of_activeSigns_agree Lemma componentLikelihood_eq_of_activeSigns_agree in the paper ↗
Agreement on a component's active signs preserves its entire conditional mark product. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
n :
ℕ
z :
Marks C
lam lam' :
SignVector hL
hagree :
∀ i : C, ∀ j : activeSigns hL, envelope hL (x i) * frame hL j (x i) ≠ 0 → lam j = lam' j
conclusion
(∏ i : C, conditionalLikelihood hL lam (x i) (z i).1 (z i).2 / 4)
= ∏ i : C, conditionalLikelihood hL lam' (x i) (z i).1 (z i).2 / 4
Proof (Lean source)
-- @node: componentLikelihood_eq_of_activeSigns_agree
lemma componentLikelihood_eq_of_activeSigns_agree (x : Fin n → Covariate)
(C : Finset (Fin n)) (z : Marks C) (lam lam' : SignVector hL)
(hagree : ∀ i : C, ∀ j : activeSigns hL,
envelope hL (x i) * frame hL j (x i) ≠ 0 → lam j = lam' j) :
(∏ i : C, conditionalLikelihood hL lam (x i) (z i).1 (z i).2 / 4) =
∏ i : C, conditionalLikelihood hL lam' (x i) (z i).1 (z i).2 / 4 := by
classical
apply Finset.prod_congr rfl
intro i _
rw [conditionalLikelihood_eq_of_activeSigns_agree hL lam lam' (x i)
(z i).1 (z i).2 (hagree i)]
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.ComponentProduct 22 declarations Finite tensor product differentiation and second-order integration for the component amplitude path, retaining the sharp finite-array absolute-mass bound.
Finite tensor product differentiation and second-order integration for the component amplitude path, retaining the sharp finite-array absolute-mass bound.
The first product-rule array for a finite collection of scalar paths.
Definition (Lean source)
parameters
ι :
p q :
ι → ℝ
def
finiteProductSlope p q :
ℝ
The second product-rule array, including every ordered pair of distinct factors.
theorem finiteProductSlope_hasDerivAt
The first product-rule array differentiates to the second product-rule array. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
ι :
p q :
ι → ℝ → ℝ
r :
ι → ℝ
u :
ℝ
hp :
∀ i, HasDerivAt (p i) (q i u) u
hq :
∀ i, HasDerivAt (q i) (r i) u
conclusion
HasDerivAt (fun v => finiteProductSlope (fun i => p i v) (fun i => q i v)) (finiteProductCurvature (fun i => p i u) (fun i => q i u) r) u
Proof (Lean source)
-- @node: finiteProductSlope_hasDerivAt
lemma finiteProductSlope_hasDerivAt {ι : Type*} [Fintype ι] [DecidableEq ι]
(p q : ι → ℝ → ℝ) (r : ι → ℝ) (u : ℝ)
(hp : ∀ i, HasDerivAt (p i) (q i u) u)
(hq : ∀ i, HasDerivAt (q i) (r i) u) :
HasDerivAt (fun v => finiteProductSlope (fun i => p i v) (fun i => q i v))
(finiteProductCurvature (fun i => p i u) (fun i => q i u) r) u := by
exact HasDerivAt.fun_sum (fun i _ =>
(HasDerivAt.fun_finsetProd (fun j _ => hp j)).mul (hq i))
The absolute mass of a tensor array factors as the product of its absolute masses. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
ι α :
p :
ι → α → ℝ
conclusion
(∑ z : ι → α, |∏ i, p i (z i)|) = ∏ i, ∑ a, |p i a|
Proof (Lean source)
-- @node: finiteTensor_abs_sum
lemma finiteTensor_abs_sum {ι α : Type*} [Fintype ι] [DecidableEq ι] [Fintype α]
(p : ι → α → ℝ) :
(∑ z : ι → α, |∏ i, p i (z i)|) = ∏ i, ∑ a, |p i a| := by
simp_rw [Finset.abs_prod]
exact (Fintype.prod_sum (fun i a => |p i a|)).symm
Replacing one tensor factor gives its absolute mass times those of the remaining factors. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: finiteTensor_replace_abs_sum
lemma finiteTensor_replace_abs_sum {ι α : Type*} [Fintype ι] [DecidableEq ι] [Fintype α]
(p : ι → α → ℝ) (q : α → ℝ) (i : ι) :
(∑ z : ι → α, |(∏ j ∈ (Finset.univ : Finset ι).erase i, p j (z j)) * q (z i)|) =
(∏ j ∈ (Finset.univ : Finset ι).erase i, ∑ a, |p j a|) * (∑ a, |q a|) := by
classical
let f : ι → α → ℝ := update p i q
have hprod (z : ι → α) : (∏ j, f j (z j)) =
(∏ j ∈ (Finset.univ : Finset ι).erase i, p j (z j)) * q (z i) := by
rw [← Finset.prod_erase_mul _ _ (Finset.mem_univ i)]
congr 1
· apply Finset.prod_congr rfl
intro j hj
simp only [f, Function.update_of_ne (Finset.ne_of_mem_erase hj)]
· simp [f]
simp_rw [← hprod]
rw [finiteTensor_abs_sum]
rw [← Finset.prod_erase_mul _ _ (Finset.mem_univ i)]
congr 1
· apply Finset.prod_congr rfl
intro j hj
simp only [f, Function.update_of_ne (Finset.ne_of_mem_erase hj)]
· simp [f]
Unit-mass factors make a one-factor replacement's absolute mass exactly that factor's mass. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
ι α :
p :
ι → α → ℝ
hp :
∀ i, (∑ a, |p i a|) = 1
q :
α → ℝ
i :
ι
Proof (Lean source)
-- @node: finiteTensor_replace_abs_sum_unit
lemma finiteTensor_replace_abs_sum_unit {ι α : Type*} [Fintype ι] [DecidableEq ι] [Fintype α]
(p : ι → α → ℝ) (hp : ∀ i, (∑ a, |p i a|) = 1) (q : α → ℝ) (i : ι) :
(∑ z : ι → α, |(∏ j ∈ (Finset.univ : Finset ι).erase i, p j (z j)) * q (z i)|) =
∑ a, |q a| := by
rw [finiteTensor_replace_abs_sum]
simp_rw [hp]
simp
theorem finiteTensor_double_replace_abs_sum_unit Lemma finiteTensor_double_replace_abs_sum_unit in the paper ↗
Unit-mass factors make two distinct replacements' absolute mass the product of their masses. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: finiteTensor_double_replace_abs_sum_unit
lemma finiteTensor_double_replace_abs_sum_unit {ι α : Type*}
[Fintype ι] [DecidableEq ι] [Fintype α]
(p : ι → α → ℝ) (hp : ∀ i, (∑ a, |p i a|) = 1)
(q r : α → ℝ) (i j : ι) (hji : j ≠ i) :
(∑ z : ι → α,
|((∏ k ∈ ((Finset.univ : Finset ι).erase i).erase j, p k (z k)) *
r (z j)) * q (z i)|) = (∑ a, |r a|) * (∑ a, |q a|) := by
classical
let f : ι → α → ℝ := update p j r
have hprod (z : ι → α) :
(∏ k ∈ (Finset.univ : Finset ι).erase i, f k (z k)) =
(∏ k ∈ ((Finset.univ : Finset ι).erase i).erase j, p k (z k)) * r (z j) := by
rw [← Finset.prod_erase_mul _ _ (Finset.mem_erase.mpr ⟨hji, Finset.mem_univ j⟩)]
congr 1
· apply Finset.prod_congr rfl
intro k hk
simp only [f, Function.update_of_ne (Finset.ne_of_mem_erase hk)]
· simp [f]
simp_rw [← hprod]
rw [finiteTensor_replace_abs_sum]
rw [← Finset.prod_erase_mul _ _ (Finset.mem_erase.mpr ⟨hji, Finset.mem_univ j⟩)]
have hone : (∏ k ∈ ((Finset.univ : Finset ι).erase i).erase j, ∑ a, |f k a|) = 1 := by
apply Finset.prod_eq_one
intro k hk
simp only [f, Function.update_of_ne (Finset.ne_of_mem_erase hk), hp]
simp [hone, f]
theorem finiteProductCurvature_abs_sum_le
The second product derivative has quadratic absolute mass when individual derivatives have absolute masses at most four and the base factors have unit absolute mass. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
ι α :
p q r :
ι → α → ℝ
hp :
∀ i, (∑ a, |p i a|) = 1
hq :
∀ i, (∑ a, |q i a|) ≤ 4
hr :
∀ i, (∑ a, |r i a|) ≤ 4
conclusion
(∑ z : ι → α, |finiteProductCurvature (fun i => p i (z i)) (fun i => q i (z i)) (fun i => r i (z i))|)
≤ 16 * (Fintype.card ι : ℝ)^2
Proof (Lean source)
-- @node: finiteProductCurvature_abs_sum_le
lemma finiteProductCurvature_abs_sum_le {ι α : Type*}
[Fintype ι] [DecidableEq ι] [Fintype α]
(p q r : ι → α → ℝ) (hp : ∀ i, (∑ a, |p i a|) = 1)
(hq : ∀ i, (∑ a, |q i a|) ≤ 4) (hr : ∀ i, (∑ a, |r i a|) ≤ 4) :
(∑ z : ι → α,
|finiteProductCurvature (fun i => p i (z i))
(fun i => q i (z i)) (fun i => r i (z i))|) ≤
16 * (Fintype.card ι : ℝ)^2 := by
classical
have hpoint (z : ι → α) :
|finiteProductCurvature (fun i => p i (z i))
(fun i => q i (z i)) (fun i => r i (z i))| ≤
∑ i, ((∑ j ∈ (Finset.univ : Finset ι).erase i,
|((∏ k ∈ ((Finset.univ : Finset ι).erase i).erase j, p k (z k)) *
q j (z j)) * q i (z i)|) +
|(∏ j ∈ (Finset.univ : Finset ι).erase i, p j (z j)) * r i (z i)|) := by
dsimp [finiteProductCurvature]
simp_rw [Finset.sum_mul]
apply (Finset.abs_sum_le_sum_abs _ _).trans
apply Finset.sum_le_sum
intro i _
exact (abs_add_le _ _).trans (add_le_add (Finset.abs_sum_le_sum_abs _ _) le_rfl)
have hi (i : ι) :
(∑ z : ι → α, ((∑ j ∈ (Finset.univ : Finset ι).erase i,
|((∏ k ∈ ((Finset.univ : Finset ι).erase i).erase j, p k (z k)) *
q j (z j)) * q i (z i)|) +
|(∏ j ∈ (Finset.univ : Finset ι).erase i, p j (z j)) * r i (z i)|)) ≤
16 * ((Fintype.card ι : ℝ)-1) + 4 := by
rw [Finset.sum_add_distrib, Finset.sum_comm,
finiteTensor_replace_abs_sum_unit p hp]
apply add_le_add _ (hr i)
calc
_ ≤ ∑ j ∈ (Finset.univ : Finset ι).erase i, (16 : ℝ) := by
apply Finset.sum_le_sum
intro j hj
rw [finiteTensor_double_replace_abs_sum_unit p hp _ _ i j
(Finset.ne_of_mem_erase hj)]
exact (mul_le_mul (hq j) (hq i) (Finset.sum_nonneg (fun _ _ => abs_nonneg _))
(by norm_num : (0 : ℝ) ≤ 4)).trans_eq (by norm_num)
_ = _ := by
simp only [Finset.sum_const, nsmul_eq_mul, Finset.card_erase_of_mem (Finset.mem_univ i),
Finset.card_univ, Nat.cast_sub (Nat.succ_le_iff.mpr
(Fintype.card_pos_iff.mpr ⟨i⟩)), Nat.cast_one]
ring
calc
_ ≤ ∑ z : ι → α, ∑ i, ((∑ j ∈ (Finset.univ : Finset ι).erase i,
|((∏ k ∈ ((Finset.univ : Finset ι).erase i).erase j, p k (z k)) *
q j (z j)) * q i (z i)|) +
|(∏ j ∈ (Finset.univ : Finset ι).erase i, p j (z j)) * r i (z i)|) :=
Finset.sum_le_sum (fun z _ => hpoint z)
_ ≤ ∑ i : ι, (16 * ((Fintype.card ι : ℝ)-1) + 4) := by
rw [Finset.sum_comm]
exact Finset.sum_le_sum (fun i _ => hi i)
_ ≤ 16 * (Fintype.card ι : ℝ)^2 := by
simp only [Finset.sum_const, Finset.card_univ, nsmul_eq_mul]
nlinarith [Nat.cast_nonneg (α := ℝ) (Fintype.card ι)]
theorem componentAmplitudeMass_sum
Free-amplitude component masses sum to one at every real amplitude. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: componentAmplitudeMass_sum
lemma componentAmplitudeMass_sum (hL : ℝ) (n : ℕ) (x : Fin n → Covariate)
(C : Finset (Fin n)) (u : ℝ) :
(∑ z : Marks C, componentAmplitudeMass hL n x C z u) = 1 := by
classical
simp only [componentAmplitudeMass, ← Finset.mul_sum]
rw [Finset.sum_comm]
simp_rw [← Fintype.prod_sum, amplitudeMarkMass_sum]
simp only [Finset.prod_const_one, Finset.sum_const, Finset.card_univ, nsmul_eq_mul,
mul_one]
exact sign_weight_real_normalization hL
theorem componentAmplitudeMass_nonneg
On the legal path every component mark mass is nonnegative. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: componentAmplitudeMass_nonneg
lemma componentAmplitudeMass_nonneg (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) (x : Fin n → Covariate) (C : Finset (Fin n)) (z : Marks C)
(u : ℝ) (hu : u ∈ Icc 0 (1 / 10)) :
0 ≤ componentAmplitudeMass hL n x C z u := by
apply mul_nonneg (by positivity)
apply Finset.sum_nonneg
intro lam _
apply Finset.prod_nonneg
intro i _
obtain ⟨hS, hq, _⟩ := amplitudeMark_coefficients_valid hL hhL lam (x i)
exact (amplitudeMarkMass_pos _ _ u hS hq hu (z i)).le
theorem componentAmplitudeMass_abs_sum
The component probability array has unit absolute mass on the legal path. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: componentAmplitudeMass_abs_sum
lemma componentAmplitudeMass_abs_sum (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) (x : Fin n → Covariate) (C : Finset (Fin n))
(u : ℝ) (hu : u ∈ Icc 0 (1 / 10)) :
(∑ z : Marks C, |componentAmplitudeMass hL n x C z u|) = 1 := by
simp_rw [abs_of_nonneg (componentAmplitudeMass_nonneg hL hhL n x C _ u hu)]
exact componentAmplitudeMass_sum hL n x C u
The averaged first product-rule array along the component path.
Definition (Lean source)
parameters
def
componentAmplitudeSlope hL n x C z u :
ℝ
given by
(2 : ℝ)^(-((activeSigns hL).card : ℤ)) * ∑ lam : SignVector hL, finiteProductSlope (fun i : C => amplitudeMarkMass (perturbation hL lam (x i)) (bump hL (x i)) u (z i)) (fun i : C => amplitudeMarkSlope (perturbation hL lam (x i)) (bump hL (x i)) u (z i))
The averaged second product-rule array along the component path.
Definition (Lean source)
parameters
def
componentAmplitudeCurvature hL n x C z u :
ℝ
given by
(2 : ℝ)^(-((activeSigns hL).card : ℤ)) * ∑ lam : SignVector hL, finiteProductCurvature (fun i : C => amplitudeMarkMass (perturbation hL lam (x i)) (bump hL (x i)) u (z i)) (fun i : C => amplitudeMarkSlope (perturbation hL lam (x i)) (bump hL (x i)) u (z i)) (fun i : C => amplitudeMarkCurvature (perturbation hL lam (x i)) (bump hL (x i)) u (z i))
theorem componentAmplitudeMass_hasDerivAt_slope
The named slope is the derivative of the component mass path. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: componentAmplitudeMass_hasDerivAt_slope
lemma componentAmplitudeMass_hasDerivAt_slope (hL : ℝ) (n : ℕ)
(x : Fin n → Covariate) (C : Finset (Fin n)) (z : Marks C) (u : ℝ) :
HasDerivAt (componentAmplitudeMass hL n x C z)
(componentAmplitudeSlope hL n x C z u) u :=
componentAmplitudeMass_hasDerivAt hL n x C z u
theorem componentAmplitudeSlope_hasDerivAt
The named curvature is the derivative of the component slope path. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: componentAmplitudeSlope_hasDerivAt
lemma componentAmplitudeSlope_hasDerivAt (hL : ℝ) (n : ℕ)
(x : Fin n → Covariate) (C : Finset (Fin n)) (z : Marks C) (u : ℝ) :
HasDerivAt (componentAmplitudeSlope hL n x C z)
(componentAmplitudeCurvature hL n x C z u) u := by
classical
exact (HasDerivAt.fun_sum (fun lam _ => finiteProductSlope_hasDerivAt _ _ _ u
(fun i => amplitudeMarkMass_hasDerivAt _ _ u (z i))
(fun i => amplitudeMarkSlope_hasDerivAt _ _ u (z i)))).const_mul _
theorem componentAmplitudeSlope_zero
The named component slope vanishes at zero amplitude by fair-sign cancellation. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: componentAmplitudeSlope_zero
lemma componentAmplitudeSlope_zero (hL : ℝ) (n : ℕ)
(x : Fin n → Covariate) (C : Finset (Fin n)) (z : Marks C) :
componentAmplitudeSlope hL n x C z 0 = 0 := by
exact (componentAmplitudeMass_hasDerivAt_slope hL n x C z 0).unique
(componentAmplitudeMass_hasDerivAt_zero hL n x C z)
theorem componentAmplitudeCurvature_abs_sum_le
Averaging over signs preserves the quadratic bound on the component second derivative. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: componentAmplitudeCurvature_abs_sum_le
lemma componentAmplitudeCurvature_abs_sum_le (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) (x : Fin n → Covariate) (C : Finset (Fin n))
(u : ℝ) (hu : u ∈ Icc 0 (1 / 10)) :
(∑ z : Marks C, |componentAmplitudeCurvature hL n x C z u|) ≤
16 * (C.card : ℝ)^2 := by
classical
have hw : 0 ≤ (2 : ℝ)^(-((activeSigns hL).card : ℤ)) := by
positivity
have hbound (lam : SignVector hL) :
(∑ z : Marks C, |finiteProductCurvature
(fun i : C => amplitudeMarkMass (perturbation hL lam (x i)) (bump hL (x i)) u (z i))
(fun i : C => amplitudeMarkSlope (perturbation hL lam (x i)) (bump hL (x i)) u (z i))
(fun i : C => amplitudeMarkCurvature
(perturbation hL lam (x i)) (bump hL (x i)) u (z i))|) ≤
16 * (C.card : ℝ)^2 := by
simpa only [Fintype.card_coe] using finiteProductCurvature_abs_sum_le
(fun i : C => fun a => amplitudeMarkMass (perturbation hL lam (x i)) (bump hL (x i)) u a)
(fun i : C => fun a => amplitudeMarkSlope (perturbation hL lam (x i)) (bump hL (x i)) u a)
(fun i : C => fun a => amplitudeMarkCurvature (perturbation hL lam (x i)) (bump hL (x i)) u a)
(by
intro i
obtain ⟨hS, hq, hr⟩ := amplitudeMark_coefficients_valid hL hhL lam (x i)
exact amplitudeMarkMass_abs_sum _ _ u hS hq hu)
(by
intro i
obtain ⟨hS, hq, hr⟩ := amplitudeMark_coefficients_valid hL hhL lam (x i)
exact amplitudeMarkSlope_abs_sum_le_four _ _ u hS hq hr hu)
(by
intro i
obtain ⟨hS, hq, hr⟩ := amplitudeMark_coefficients_valid hL hhL lam (x i)
exact amplitudeMarkCurvature_abs_sum_le_four _ _ u hS hq hr hu)
unfold componentAmplitudeCurvature
simp_rw [abs_mul, abs_of_nonneg hw]
apply (Finset.sum_le_sum (fun z _ => mul_le_mul_of_nonneg_left
(Finset.abs_sum_le_sum_abs _ _) hw)).trans
rw [← Finset.mul_sum, Finset.sum_comm]
apply (mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun lam _ => hbound lam)) hw).trans_eq
simp only [Finset.sum_const, Finset.card_univ, nsmul_eq_mul]
rw [← mul_assoc, sign_weight_real_normalization, one_mul]
theorem scalar_second_derivative_remainder_le Lemma scalar_second_derivative_remainder_le in the paper ↗
A vanishing initial slope and a bounded second derivative give the quadratic remainder by the mean-value inequality followed by integration of the first derivative. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
f g r :
ℝ → ℝ
K u :
ℝ
hu :
0 ≤ u
hf :
∀ v, HasDerivAt f (g v) v
hg :
∀ v, HasDerivAt g (r v) v
hzero :
g 0 = 0
hbound :
∀ v ∈ Icc 0 u, |r v| ≤ K
conclusion
|f u-f 0| ≤ K*u^2/2
Proof (Lean source)
-- @node: scalar_second_derivative_remainder_le
lemma scalar_second_derivative_remainder_le (f g r : ℝ → ℝ) (K u : ℝ)
(hu : 0 ≤ u) (hf : ∀ v, HasDerivAt f (g v) v)
(hg : ∀ v, HasDerivAt g (r v) v) (hzero : g 0 = 0)
(hbound : ∀ v ∈ Icc 0 u, |r v| ≤ K) :
|f u-f 0| ≤ K*u^2/2 := by
have hgc : Continuous g := continuous_iff_continuousAt.mpr
(fun v => (hg v).continuousAt)
have hgi : IntervalIntegrable g volume 0 u := hgc.intervalIntegrable _ _
have hslope (v : ℝ) (hv : v ∈ Icc 0 u) : |g v| ≤ K*v := by
have hm := norm_image_sub_le_of_norm_deriv_le_segment'
(a := (0 : ℝ)) (b := u) (f := g) (f' := r)
(fun v _ => (hg v).hasDerivWithinAt)
(fun v hv => by
simpa only [Real.norm_eq_abs] using (hbound v ⟨hv.1, hv.2.le⟩)) v hv
simpa only [hzero, sub_zero, Real.norm_eq_abs] using hm
rw [← intervalIntegral.integral_eq_sub_of_hasDerivAt (fun v _ => hf v) hgi]
calc
_ ≤ ∫ v in (0 : ℝ)..u, |g v| := intervalIntegral.abs_integral_le_integral_abs hu
_ ≤ ∫ v in (0 : ℝ)..u, K*v :=
intervalIntegral.integral_mono_on hu hgi.abs
((continuous_const.mul continuous_id).intervalIntegrable _ _) hslope
_ = _ := by rw [intervalIntegral.integral_const_mul, integral_id]; simp; ring
theorem finite_array_second_derivative_remainder_le Lemma finite_array_second_derivative_remainder_le in the paper ↗
Applying the scalar second-order bound to a signed projection controls the whole finite array without multiplying the bound by the number of its coordinates. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
ι :
Type*
Fintype ι
f g r :
ι → ℝ → ℝ
K u :
ℝ
hu :
0 ≤ u
hf :
∀ i v, HasDerivAt (f i) (g i v) v
hg :
∀ i v, HasDerivAt (g i) (r i v) v
hzero :
∀ i, g i 0 = 0
hbound :
∀ v ∈ Icc 0 u, (∑ i, |r i v|) ≤ K
conclusion
(∑ i, |f i u-f i 0|) ≤ K*u^2/2
Proof (Lean source)
-- @node: finite_array_second_derivative_remainder_le
lemma finite_array_second_derivative_remainder_le {ι : Type*} [Fintype ι]
(f g r : ι → ℝ → ℝ) (K u : ℝ) (hu : 0 ≤ u)
(hf : ∀ i v, HasDerivAt (f i) (g i v) v)
(hg : ∀ i v, HasDerivAt (g i) (r i v) v)
(hzero : ∀ i, g i 0 = 0)
(hbound : ∀ v ∈ Icc 0 u, (∑ i, |r i v|) ≤ K) :
(∑ i, |f i u-f i 0|) ≤ K*u^2/2 := by
classical
let a (i : ι) : ℝ := if 0 ≤ f i u-f i 0 then 1 else -1
have ha (i : ι) : |a i| = 1 := by dsimp [a]; split_ifs <;> norm_num
have hd (i : ι) : a i*(f i u-f i 0) = |f i u-f i 0| := by
dsimp [a]
split_ifs with h
· rw [one_mul, abs_of_nonneg h]
· rw [neg_one_mul, abs_of_neg (lt_of_not_ge h)]
have hs := scalar_second_derivative_remainder_le
(fun v => ∑ i, a i*f i v) (fun v => ∑ i, a i*g i v)
(fun v => ∑ i, a i*r i v) K u hu
(fun v => HasDerivAt.fun_sum (fun i _ => (hf i v).const_mul (a i)))
(fun v => HasDerivAt.fun_sum (fun i _ => (hg i v).const_mul (a i)))
(by simp only [hzero, mul_zero, Finset.sum_const_zero]) (by
intro v hv
apply (Finset.abs_sum_le_sum_abs _ _).trans
simpa only [abs_mul, ha, one_mul] using hbound v hv)
have he : (∑ i, a i*f i u)-(∑ i, a i*f i 0) = ∑ i, |f i u-f i 0| := by
rw [← Finset.sum_sub_distrib]
simp_rw [← mul_sub, hd]
rw [he, abs_of_nonneg (Finset.sum_nonneg (fun _ _ => abs_nonneg _))] at hs
exact hs
theorem alternativeMass_abs_sub_nullMass_sum_le Lemma alternativeMass_abs_sub_nullMass_sum_le in the paper ↗
Integrating the component curvature twice gives the sharp second-order array bound. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
≤ 8*separation hL*(C.card : ℝ)^2
Proof (Lean source)
-- @node: alternativeMass_abs_sub_nullMass_sum_le
lemma alternativeMass_abs_sub_nullMass_sum_le (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) (x : Fin n → Covariate) (C : Finset (Fin n)) :
(∑ z : Marks C, |alternativeMass hL n x C z-nullMass n C z|) ≤
8*separation hL*(C.card : ℝ)^2 := by
have hu := sqrt_separation_amplitude_range hL hhL
have h := finite_array_second_derivative_remainder_le
(componentAmplitudeMass hL n x C) (componentAmplitudeSlope hL n x C)
(componentAmplitudeCurvature hL n x C) (16*(C.card : ℝ)^2)
(sqrt (separation hL)) hu.1
(componentAmplitudeMass_hasDerivAt_slope hL n x C)
(componentAmplitudeSlope_hasDerivAt hL n x C)
(componentAmplitudeSlope_zero hL n x C) (by
intro v hv
exact componentAmplitudeCurvature_abs_sum_le hL hhL n x C v ⟨hv.1, hv.2.trans hu.2⟩)
simp_rw [componentAmplitudeMass_actual hL hhL, componentAmplitudeMass_zero] at h
rw [Real.sq_sqrt (separation_unit_range hL hhL).1] at h
convert h using 1
ring
The absolute-array bound gives the component coupling's failure-probability bound. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
1-overlapMass hL n x C ≤ 4*separation hL*(C.card : ℝ)^2
Proof (Lean source)
-- @node: overlapMass_deficit_le
lemma overlapMass_deficit_le (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) (x : Fin n → Covariate) (C : Finset (Fin n)) :
1-overlapMass hL n x C ≤ 4*separation hL*(C.card : ℝ)^2 := by
have h := alternativeMass_abs_sub_nullMass_sum_le hL hhL n x C
rw [componentMass_abs_sum] at h
linarith
theorem componentCoupling_hamming_cost_le_cubic Lemma componentCoupling_hamming_cost_le_cubic in the paper ↗
A nonsingleton component's expected number of changed marks is at most four times separation times the cube of its size, as required by the component-counting argument. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: componentCoupling_hamming_cost_le_cubic
lemma componentCoupling_hamming_cost_le_cubic (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) (x : Fin n → Covariate) (C : Finset (Fin n)) :
(∫⁻ zw, (hammingDist zw.1 zw.2 : ℝ≥0∞) ∂componentCoupling hL n x C) ≤
ofReal (4*separation hL*(C.card : ℝ)^3) := by
apply (componentCoupling_hamming_cost_le hL hhL n x C).trans
rw [← ENNReal.ofReal_natCast C.card, ← ENNReal.ofReal_mul (Nat.cast_nonneg C.card)]
apply ENNReal.ofReal_le_ofReal
have h := mul_le_mul_of_nonneg_left (overlapMass_deficit_le hL hhL n x C)
(Nat.cast_nonneg (α := ℝ) C.card)
calc
_ ≤ (C.card : ℝ)*(4*separation hL*(C.card : ℝ)^2) := h
_ = _ := by ring
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.ComponentSpanning 11 declarations General labeled spanning-tree assembly for the shared-sign component count.
General labeled spanning-tree assembly for the shared-sign component count.
A component induces a connected graph. Enumerating its vertices identifies a spanning tree
on Fin s; a finite union bound then applies the published Cayley count.
The induced graph on any selected component is connected, with no size restriction. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: orderedComponent_induce_connected
lemma orderedComponent_induce_connected (hL : ℝ) (n : ℕ) (x : Fin n → Covariate)
(C : Finset (Fin n)) (hC : C ∈ orderedComponents hL n x) :
((sharedGraph hL n x).induce (C : Set (Fin n))).Connected := by
classical
obtain ⟨r, _, hr⟩ := (mem_orderedComponents hL n x C).mp hC
have hs : ((sharedGraph hL n x).connectedComponentMk r).supp = (C : Set (Fin n)) := by
ext i
rw [← hr]
simp only [Finset.mem_coe, mem_componentVertices,
SimpleGraph.ConnectedComponent.mem_supp_iff,
SimpleGraph.ConnectedComponent.eq, SimpleGraph.reachable_comm]
have h := ((sharedGraph hL n x).connectedComponentMk r).connected_toSimpleGraph
change ((sharedGraph hL n x).induce
((sharedGraph hL n x).connectedComponentMk r).supp).Connected at h
rw [hs] at h
exact h
theorem orderedComponent_contains_labeled_tree
For any fixed enumeration of a component, a labeled spanning tree exists in that component's graph. The enumeration is fixed before taking probabilities. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
∃ T : {T : SimpleGraph (Fin s) // T.IsTree},
x ∈ labeledSharedTreeEvent hL n s (fun i => (e i).val) T.val
Proof (Lean source)
-- @node: orderedComponent_contains_labeled_tree
lemma orderedComponent_contains_labeled_tree (hL : ℝ) (n s : ℕ)
(C : Finset (Fin n)) (e : Fin s ≃ C) (x : Fin n → Covariate)
(hC : C ∈ orderedComponents hL n x) :
∃ T : {T : SimpleGraph (Fin s) // T.IsTree},
x ∈ labeledSharedTreeEvent hL n s (fun i => (e i).val) T.val := by
classical
let G := (sharedGraph hL n x).induce (C : Set (Fin n))
have hG : G.Connected := orderedComponent_induce_connected hL n x C hC
have hGe : (G.comap e).Connected := (SimpleGraph.Iso.comap e G).connected_iff.mpr hG
obtain ⟨T, hle, ht⟩ := hGe.exists_isTree_le
refine ⟨⟨T, ht⟩, ?_⟩
intro i j hij
exact hle hij
theorem fixed_component_subset_labeled_tree_union
The component event is contained in the union over all labeled trees on its fixed vertex enumeration. This inclusion also covers singleton components. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
{x : Fin n → Covariate | C ∈ orderedComponents hL n x} ⊆ ⋃ T : {T : SimpleGraph (Fin s) // T.IsTree}, labeledSharedTreeEvent hL n s (fun i => (e i).val) T.val
Proof (Lean source)
-- @node: fixed_component_subset_labeled_tree_union
lemma fixed_component_subset_labeled_tree_union (hL : ℝ) (n s : ℕ)
(C : Finset (Fin n)) (e : Fin s ≃ C) :
{x : Fin n → Covariate | C ∈ orderedComponents hL n x} ⊆
⋃ T : {T : SimpleGraph (Fin s) // T.IsTree},
labeledSharedTreeEvent hL n s (fun i => (e i).val) T.val := by
intro x hx
obtain ⟨T, ht⟩ := orderedComponent_contains_labeled_tree hL n s C e x hx
exact Set.mem_iUnion.mpr ⟨T, ht⟩
theorem fixed_component_measure_le_cayley_mul
A common probability envelope for specified labeled trees gives the component probability bound by the finite union bound and the explicit Cayley gate. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
cayley :
hL :
ℝ
n s :
ℕ
hs :
2 ≤ s
e :
Fin s ≃ C
b :
ℝ≥0∞
hb :
∀ T : {T : SimpleGraph (Fin s) // T.IsTree},
μ (labeledSharedTreeEvent hL n s (fun i => (e i).val) T.val) ≤ b
conclusion
μ {x | C ∈ orderedComponents hL n x} ≤ (s ^ (s - 2) : ℕ) * b
Proof (Lean source)
-- @node: fixed_component_measure_le_cayley_mul
lemma fixed_component_measure_le_cayley_mul (cayley : CayleyLabeledTreeCount)
(hL : ℝ) (n s : ℕ) (hs : 2 ≤ s) (C : Finset (Fin n)) (e : Fin s ≃ C)
(μ : Measure (Fin n → Covariate)) (b : ℝ≥0∞)
(hb : ∀ T : {T : SimpleGraph (Fin s) // T.IsTree},
μ (labeledSharedTreeEvent hL n s (fun i => (e i).val) T.val) ≤ b) :
μ {x | C ∈ orderedComponents hL n x} ≤ (s ^ (s - 2) : ℕ) * b := by
classical
let := ofFinite {T : SimpleGraph (Fin s) // T.IsTree}
calc
_ ≤ μ (⋃ T : {T : SimpleGraph (Fin s) // T.IsTree},
labeledSharedTreeEvent hL n s (fun i => (e i).val) T.val) :=
measure_mono (fixed_component_subset_labeled_tree_union hL n s C e)
_ ≤ ∑' T : {T : SimpleGraph (Fin s) // T.IsTree},
μ (labeledSharedTreeEvent hL n s (fun i => (e i).val) T.val) :=
measure_iUnion_le _
_ ≤ ∑' _T : {T : SimpleGraph (Fin s) // T.IsTree}, b := ENNReal.tsum_le_tsum hb
_ = _ := by
rw [tsum_fintype]
simp only [Finset.sum_const, Finset.card_univ, nsmul_eq_mul]
rw [← Nat.card_eq_fintype_card, cayley s hs]
theorem lintegral_sizeComponentCount_le_of_labeled_tree_bound
The general size-s count assembly: a specified-tree envelope, on arbitrary distinct sample coordinates, implies the binomial Cayley estimate for the actual component count. The tree probability estimate remains an explicit input here, rather than a paper-theorem premise. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
cayley :
hL :
ℝ
n s :
ℕ
hs :
2 ≤ s
b :
ℝ≥0∞
hb :
if
then
conclusion
(∫⁻ x, (sizeComponentCount hL n x s : ℝ≥0∞) ∂μ)
≤ (n.choose s : ℝ≥0∞) * (s ^ (s - 2) : ℕ) * b
Proof (Lean source)
-- @node: lintegral_sizeComponentCount_le_of_labeled_tree_bound
lemma lintegral_sizeComponentCount_le_of_labeled_tree_bound
(cayley : CayleyLabeledTreeCount) (hL : ℝ) (n s : ℕ) (hs : 2 ≤ s)
(μ : Measure (Fin n → Covariate)) (b : ℝ≥0∞)
(hb : ∀ (e : Fin s → Fin n), Function.Injective e →
∀ T : {T : SimpleGraph (Fin s) // T.IsTree},
μ (labeledSharedTreeEvent hL n s e T.val) ≤ b) :
(∫⁻ x, (sizeComponentCount hL n x s : ℝ≥0∞) ∂μ) ≤
(n.choose s : ℝ≥0∞) * (s ^ (s - 2) : ℕ) * b := by
classical
rw [mul_assoc]
apply lintegral_sizeComponentCount_le_choose_mul
intro C hC
let e : Fin s ≃ C := (Finset.equivFinOfCardEq hC).symm
apply fixed_component_measure_le_cayley_mul cayley hL n s hs C e μ b
apply hb
intro i j hij
exact e.injective (Subtype.ext hij)
theorem lintegral_sizeComponentCount_le_of_uniform_labeled_tree_bound
The roadmap's general component-count bound follows from specified-tree probabilities on their own vertex spaces. Cayley and coordinate selection supply all the assembly factors. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
cayley :
hL :
ℝ
n s :
ℕ
hs :
2 ≤ s
htree :
∀ T : {T : SimpleGraph (Fin s) // T.IsTree},
(Measure.pi (fun _ : Fin s => (volume : Measure Covariate))) (labeledSharedTreeEvent hL s s id T.val)
Proof (Lean source)
-- @node: lintegral_sizeComponentCount_le_of_uniform_labeled_tree_bound
lemma lintegral_sizeComponentCount_le_of_uniform_labeled_tree_bound
(cayley : CayleyLabeledTreeCount) (hL : ℝ)
(n s : ℕ) (hs : 2 ≤ s)
(htree : ∀ T : {T : SimpleGraph (Fin s) // T.IsTree},
(Measure.pi (fun _ : Fin s => (volume : Measure Covariate)))
(labeledSharedTreeEvent hL s s id T.val) ≤
ofReal (2 * hL * (4 * deltaL hL) ^ (s - 1))) :
(∫⁻ x, (sizeComponentCount hL n x s : ℝ≥0∞)
∂(Measure.pi (fun _ : Fin n => (volume : Measure Covariate)))) ≤
ofReal ((n.choose s : ℝ) * 2 * hL * (s : ℝ) ^ (s - 2) *
(4 * deltaL hL) ^ (s - 1)) := by
have hb := lintegral_sizeComponentCount_le_of_labeled_tree_bound cayley hL n s hs
(Measure.pi (fun _ : Fin n => (volume : Measure Covariate)))
(ofReal (2 * hL * (4 * deltaL hL) ^ (s - 1)))
(fun e he T => by rw [labeledSharedTreeEvent_uniform_volume_eq hL n s e he]; exact htree T)
rw [← ENNReal.ofReal_natCast, ← ENNReal.ofReal_natCast,
← ENNReal.ofReal_mul (Nat.cast_nonneg _),
← ENNReal.ofReal_mul (mul_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _))] at hb
convert hb using 1
congr 1
push_cast
ring
theorem commonMassCoupling_hamming_cost_le_of_labeled_tree_bound
The actual sparse dataset coupling obeys the Hamming-cost bound once each specified labeled tree has its geometric probability envelope. All component-count assembly is proved. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
cayley :
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1 / 4
n :
ℕ
hn :
1 ≤ n
hcap :
(n : ℝ) * deltaL hL ≤ 1 / 128
htree :
∀ s : ℕ
if
2 ≤ s
and
s ≤ n
then
∀ T : {T : SimpleGraph (Fin s) // T.IsTree},
(Measure.pi (fun _ : Fin s => (volume : Measure Covariate))) (labeledSharedTreeEvent hL s s id T.val)
conclusion
(∫⁻ zw, (dHam n zw.1 zw.2 : ℝ≥0∞) ∂commonMassCoupling hL n)
Proof (Lean source)
-- @node: commonMassCoupling_hamming_cost_le_of_labeled_tree_bound
lemma commonMassCoupling_hamming_cost_le_of_labeled_tree_bound
(cayley : CayleyLabeledTreeCount) (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) (hn : 1 ≤ n) (hcap : (n : ℝ) * deltaL hL ≤ 1 / 128)
(htree : ∀ s : ℕ, 2 ≤ s → s ≤ n →
∀ T : {T : SimpleGraph (Fin s) // T.IsTree},
(Measure.pi (fun _ : Fin s => (volume : Measure Covariate)))
(labeledSharedTreeEvent hL s s id T.val) ≤
ofReal (2 * hL * (4 * deltaL hL) ^ (s - 1))) :
(∫⁻ zw, (dHam n zw.1 zw.2 : ℝ≥0∞) ∂commonMassCoupling hL n) ≤
ofReal (1024 * separation hL * (n : ℝ) ^ 2 * hL * deltaL hL) := by
apply commonMassCoupling_hamming_cost_le_of_tree_counts hL hhL n hn hcap
intro i hi
have hi' := Finset.mem_range.mp hi
have hb := lintegral_sizeComponentCount_le_of_uniform_labeled_tree_bound
cayley hL n (i + 2) (by omega) (htree (i + 2) (by omega) (by omega))
convert hb using 1
congr 1
simp
ring
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.ComponentTransport 15 declarations Additive transport costs for the finite product of component couplings.
Additive transport costs for the finite product of component couplings.
Pairing two assignments coordinatewise preserves all finite assignments.
Partition restriction factors array sums over any commutative semiring. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
(∑ z : ι → α, ∏ C ∈ s, F C (fun i => z i)) = ∏ C ∈ s, ∑ z : C → α, F C z
Proof (Lean source)
-- @node: sum_prod_partition_semiring
lemma sum_prod_partition_semiring {ι α R : Type*} [Fintype ι] [DecidableEq ι]
[Fintype α] [CommSemiring R]
(s : Finset (Finset ι)) (hcover : ∀ i, ∃ C ∈ s, i ∈ C)
(hdisj : PairwiseDisjoint (s : Set (Finset ι)) id)
(F : ∀ C : Finset ι, (C → α) → R) :
(∑ z : ι → α, ∏ C ∈ s, F C (fun i => z i)) =
∏ C ∈ s, ∑ z : C → α, F C z := by
classical
let e := partitionRestrictionEquiv (α := α) s hcover hdisj
calc
_ = ∑ w : ∀ C : s, ↥(C : Finset ι) → α, ∏ C : s, F C (w C) := by
apply Fintype.sum_equiv e
intro z
exact (Finset.prod_coe_sort s (fun C => F C (fun i => z i))).symm
_ = _ := by
rw [← Fintype.prod_sum]
exact Finset.prod_coe_sort s (fun C => ∑ z : C → α, F C z)
A normalized partition product preserves the expectation of any one block observable. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: sum_prod_partition_observable
lemma sum_prod_partition_observable {ι α R : Type*} [Fintype ι] [DecidableEq ι]
[Fintype α] [CommSemiring R]
(s : Finset (Finset ι)) (hcover : ∀ i, ∃ C ∈ s, i ∈ C)
(hdisj : PairwiseDisjoint (s : Set (Finset ι)) id)
(F : ∀ C : Finset ι, (C → α) → R)
(hF : ∀ C ∈ s, ∑ z : C → α, F C z = 1)
(D : Finset ι) (hD : D ∈ s) (g : (D → α) → R) :
(∑ z : ι → α, (∏ C ∈ s, F C (fun i => z i)) * g (fun i => z i)) =
∑ z : D → α, F D z * g z := by
classical
let G (C : Finset ι) (z : C → α) : R :=
if h : C = D then F D (fun i => z ⟨i, h.symm ▸ i.property⟩) *
g (fun i => z ⟨i, h.symm ▸ i.property⟩) else F C z
have hp (z : ι → α) :
(∏ C ∈ s, G C (fun i => z i)) =
(∏ C ∈ s, F C (fun i => z i)) * g (fun i => z i) := by
rw [← Finset.prod_erase_mul _ _ hD, ← Finset.prod_erase_mul _ _ hD]
have he : (∏ C ∈ s.erase D, G C (fun i => z i)) =
∏ C ∈ s.erase D, F C (fun i => z i) := by
apply Finset.prod_congr rfl
intro C hC
simp [G, Finset.ne_of_mem_erase hC]
rw [he]
simp [G, mul_assoc]
simp_rw [← hp]
rw [sum_prod_partition_semiring s hcover hdisj G, ← Finset.prod_erase_mul _ _ hD]
have he : (∏ C ∈ s.erase D, ∑ z : C → α, G C z) = 1 := by
apply Finset.prod_eq_one
intro C hC
simpa [G, Finset.ne_of_mem_erase hC] using hF C (mem_of_mem_erase hC)
rw [he, one_mul]
simp [G]
The shared-sign component list partitions any sum over dataset records. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
(∑ i, f i) = ((orderedComponents hL n x).map (fun C => ∑ i ∈ C, f i)).sum
Proof (Lean source)
-- @node: orderedComponents_sum_records
lemma orderedComponents_sum_records {M : Type*} [AddCommMonoid M]
(hL : ℝ) (n : ℕ) (x : Fin n → Covariate) (f : Fin n → M) :
(∑ i, f i) = ((orderedComponents hL n x).map
(fun C => ∑ i ∈ C, f i)).sum := by
classical
let s := (orderedComponents hL n x).toFinset
have hc : s.biUnion id = univ := by
ext i
simp only [Finset.mem_biUnion, Finset.mem_univ, iff_true]
obtain ⟨C, hC, hi⟩ := orderedComponents_cover hL n x i
exact ⟨C, List.mem_toFinset.mpr hC, hi⟩
have hd : PairwiseDisjoint (s : Set (Finset (Fin n))) id := by
intro C hC D hD hne
exact orderedComponents_disjoint hL n x C D
(List.mem_toFinset.mp hC) (List.mem_toFinset.mp hD) hne
rw [← hc, Finset.sum_biUnion hd]
exact List.sum_toFinset _ (orderedComponents_nodup hL n x)
Identical attached covariates make dataset replacements exactly mark replacements. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: attachMarks_hammingDist
lemma attachMarks_hammingDist (n : ℕ) (x : Fin n → Covariate)
(z w : Fin n → Bool × Bool) :
dHam n (attachMarks n x z) (attachMarks n x w) = hammingDist z w := by
classical
unfold dHam hammingDist
congr 1
ext i
simp [attachMarks, Prod.ext_iff]
Hamming cost decomposes over the disjoint ordered components. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
hammingDist z w
= ((orderedComponents hL n x).map (fun (C : Finset (Fin n)) => hammingDist (fun i : C => z i) (fun i : C => w i))).sum
Proof (Lean source)
-- @node: hammingDist_orderedComponents
lemma hammingDist_orderedComponents (hL : ℝ) (n : ℕ) (x : Fin n → Covariate)
(z w : Fin n → Bool × Bool) :
hammingDist z w = ((orderedComponents hL n x).map
(fun (C : Finset (Fin n)) => hammingDist (fun i : C => z i) (fun i : C => w i))).sum := by
classical
have hc (C : Finset (Fin n)) :
hammingDist (fun i : C => z i) (fun i : C => w i) =
∑ i ∈ C, if z i ≠ w i then 1 else 0 := by
simp only [hammingDist, Finset.card_eq_sum_ones, Finset.sum_filter]
exact Finset.sum_coe_sort C (fun i => if z i ≠ w i then 1 else 0)
simp_rw [hc]
simpa only [hammingDist, Finset.card_eq_sum_ones, Finset.sum_filter] using
orderedComponents_sum_records hL n x (fun i => if z i ≠ w i then 1 else 0 : Fin n → ℕ)
Summation over pairs of assignments equals summation over paired coordinates. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
ι α M :
f :
(ι → α) × (ι → α) → M
conclusion
(∑ zw, f zw) = ∑ v : ι → α × α, f (fun i => (v i).1, fun i => (v i).2)
Proof (Lean source)
-- @node: sum_pairedAssignments
lemma sum_pairedAssignments {ι α M : Type*} [Fintype ι] [DecidableEq ι] [Fintype α]
[AddCommMonoid M]
(f : (ι → α) × (ι → α) → M) :
(∑ zw, f zw) = ∑ v : ι → α × α, f (fun i => (v i).1, fun i => (v i).2) := by
apply Fintype.sum_equiv (pairedAssignmentEquiv ι α)
intro zw
rfl
Each component's paired-coordinate coupling array has total mass one. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: componentCouplingMass_paired_sum
lemma componentCouplingMass_paired_sum (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) (x : Fin n → Covariate) (C : Finset (Fin n)) :
(∑ v : C → (Bool × Bool) × (Bool × Bool),
ofReal (componentCouplingMass hL n x C
(fun i => (v i).1) (fun i => (v i).2))) = 1 := by
rw [← sum_pairedAssignments (fun zw : Marks C × Marks C =>
ofReal (componentCouplingMass hL n x C zw.1 zw.2)), Fintype.sum_prod_type]
have hr (z : Marks C) :
(∑ w, ofReal (componentCouplingMass hL n x C z w)) =
ofReal (alternativeMass hL n x C z) := by
rw [← ENNReal.ofReal_sum_of_nonneg
(fun w _ => componentCouplingMass_nonneg hL hhL n x C z w),
componentCouplingMass_row_sum hL hhL]
simp_rw [hr]
rw [← ENNReal.ofReal_sum_of_nonneg
(fun z _ => alternativeMass_nonneg hL hhL n x C z), alternativeMass_sum]
exact ENNReal.ofReal_one
Converting the joint mass to an extended real preserves its component product. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
ofReal (jointMarkMass hL n x z w)
= ∏ C ∈ (orderedComponents hL n x).toFinset, ofReal (componentCouplingMass hL n x C (fun i => z i) (fun i => w i))
Proof (Lean source)
-- @node: jointMarkMass_ofReal_prod
lemma jointMarkMass_ofReal_prod (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) (x : Fin n → Covariate) (z w : Fin n → Bool × Bool) :
ofReal (jointMarkMass hL n x z w) =
∏ C ∈ (orderedComponents hL n x).toFinset,
ofReal (componentCouplingMass hL n x C (fun i => z i) (fun i => w i)) := by
unfold jointMarkMass
rw [← List.prod_toFinset _ (orderedComponents_nodup hL n x)]
exact ENNReal.ofReal_prod_of_nonneg (fun C _ => componentCouplingMass_nonneg hL hhL n x C _ _)
Any component observable has its prescribed marginal under the full product coupling array. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
Proof (Lean source)
-- @node: jointMarkMass_component_observable
lemma jointMarkMass_component_observable (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) (x : Fin n → Covariate) (C : Finset (Fin n))
(hC : C ∈ orderedComponents hL n x) (g : Marks C × Marks C → ℝ≥0∞) :
(∑ zw : (Fin n → Bool × Bool) × (Fin n → Bool × Bool),
ofReal (jointMarkMass hL n x zw.1 zw.2) *
g (fun i => zw.1 i, fun i => zw.2 i)) =
∑ zw : Marks C × Marks C,
ofReal (componentCouplingMass hL n x C zw.1 zw.2) * g zw := by
classical
conv_lhs => rw [sum_pairedAssignments]
conv_rhs => rw [sum_pairedAssignments]
simp_rw [jointMarkMass_ofReal_prod hL hhL]
apply sum_prod_partition_observable (α := (Bool × Bool) × (Bool × Bool))
(s := (orderedComponents hL n x).toFinset)
(D := C)
(F := fun D v => ofReal (componentCouplingMass hL n x D
(fun i => (v i).1) (fun i => (v i).2)))
(g := fun v => g (fun i => (v i).1, fun i => (v i).2))
· intro i
obtain ⟨D, hD, hi⟩ := orderedComponents_cover hL n x i
exact ⟨D, List.mem_toFinset.mpr hD, hi⟩
· intro D hD E hE hne
exact orderedComponents_disjoint hL n x D E
(List.mem_toFinset.mp hD) (List.mem_toFinset.mp hE) hne
· intro D _
exact componentCouplingMass_paired_sum hL hhL n x D
· exact List.mem_toFinset.mpr hC
theorem conditionalDatasetCoupling_hamming_cost_eq Lemma conditionalDatasetCoupling_hamming_cost_eq in the paper ↗
The conditional dataset coupling's expected Hamming cost is the sum of component costs. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
(∫⁻ zw, (dHam n zw.1 zw.2 : ℝ≥0∞) ∂conditionalDatasetCoupling hL n x)
= ∑ C ∈ (orderedComponents hL n x).toFinset, ∫⁻ zw, (hammingDist zw.1 zw.2 : ℝ≥0∞) ∂componentCoupling hL n x C
Proof (Lean source)
-- @node: conditionalDatasetCoupling_hamming_cost_eq
lemma conditionalDatasetCoupling_hamming_cost_eq (hL : ℝ)
(hhL : 0 < hL ∧ hL ≤ 1 / 4) (n : ℕ) (x : Fin n → Covariate) :
(∫⁻ zw, (dHam n zw.1 zw.2 : ℝ≥0∞) ∂conditionalDatasetCoupling hL n x) =
∑ C ∈ (orderedComponents hL n x).toFinset,
∫⁻ zw, (hammingDist zw.1 zw.2 : ℝ≥0∞) ∂componentCoupling hL n x C := by
classical
have hd (z w : Fin n → Bool × Bool) :
(dHam n (attachMarks n x z) (attachMarks n x w) : ℝ≥0∞) =
∑ C ∈ (orderedComponents hL n x).toFinset,
(hammingDist (fun i : C => z i) (fun i : C => w i) : ℝ≥0∞) := by
rw [attachMarks_hammingDist, hammingDist_orderedComponents hL n x]
rw [← List.sum_toFinset _ (orderedComponents_nodup hL n x), Nat.cast_sum]
simp only [conditionalDatasetCoupling, lintegral_finsetSum_measure,
lintegral_smul_measure, lintegral_dirac, smul_eq_mul]
simp_rw [hd, Finset.mul_sum]
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro C hC
rw [jointMarkMass_component_observable hL hhL n x C (List.mem_toFinset.mp hC)
(fun zw => (hammingDist zw.1 zw.2 : ℝ≥0∞))]
simp only [componentCoupling, lintegral_finsetSum_measure,
lintegral_smul_measure, lintegral_dirac, smul_eq_mul]
theorem conditionalDatasetCoupling_hamming_cost_le_cubic_sum Lemma conditionalDatasetCoupling_hamming_cost_le_cubic_sum in the paper ↗
The product coupling costs at most the sum of the nonsingleton cubic certificates. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
(∫⁻ zw, (dHam n zw.1 zw.2 : ℝ≥0∞) ∂conditionalDatasetCoupling hL n x)
≤ ∑ C ∈ (orderedComponents hL n x).toFinset, if C.card = 1 then 0 else ofReal (4*separation hL*(C.card : ℝ)^3)
Proof (Lean source)
-- @node: conditionalDatasetCoupling_hamming_cost_le_cubic_sum
lemma conditionalDatasetCoupling_hamming_cost_le_cubic_sum (hL : ℝ)
(hhL : 0 < hL ∧ hL ≤ 1 / 4) (n : ℕ) (x : Fin n → Covariate) :
(∫⁻ zw, (dHam n zw.1 zw.2 : ℝ≥0∞) ∂conditionalDatasetCoupling hL n x) ≤
∑ C ∈ (orderedComponents hL n x).toFinset,
if C.card = 1 then 0 else ofReal (4*separation hL*(C.card : ℝ)^3) := by
classical
rw [conditionalDatasetCoupling_hamming_cost_eq hL hhL]
apply Finset.sum_le_sum
intro C _
by_cases hc : C.card = 1
· rw [if_pos hc, componentCoupling_hamming_cost_zero_of_card_one hL hhL n x C hc]
· rw [if_neg hc]
exact componentCoupling_hamming_cost_le_cubic hL hhL n x C
Replacement Hamming cost is a Borel function of the two datasets. The displayed conclusion follows.
Formal statement
hypotheses
n :
ℕ
conclusion
Proof (Lean source)
-- @node: measurable_datasetHammingCost
@[fun_prop] lemma measurable_datasetHammingCost (n : ℕ) :
Measurable (fun zw : Dataset n × Dataset n => (dHam n zw.1 zw.2 : ℝ≥0∞)) := by
classical
simp only [dHam, hammingDist, Finset.card_eq_sum_ones, Finset.sum_filter,
Nat.cast_sum, Nat.cast_ite, Nat.cast_one, Nat.cast_zero]
apply Finset.measurable_sum
intro i _
apply Measurable.ite
· exact (measurableSet_eq_fun ((measurable_pi_apply i).comp measurable_fst)
((measurable_pi_apply i).comp measurable_snd)).compl
· exact measurable_const
· exact measurable_const
theorem commonMassCoupling_hamming_cost_eq
Integrating the Borel conditional kernel preserves the exact additive transport cost. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1 / 4
n :
ℕ
conclusion
(∫⁻ zw, (dHam n zw.1 zw.2 : ℝ≥0∞) ∂commonMassCoupling hL n)
= ∫⁻ x, ∑ C ∈ (orderedComponents hL n x).toFinset, ∫⁻ zw, (hammingDist zw.1 zw.2 : ℝ≥0∞) ∂componentCoupling hL n x C ∂(Measure.pi (fun _ : Fin n => (volume : Measure Covariate)))
Proof (Lean source)
-- @node: commonMassCoupling_hamming_cost_eq
lemma commonMassCoupling_hamming_cost_eq (hL : ℝ)
(hhL : 0 < hL ∧ hL ≤ 1 / 4) (n : ℕ) :
(∫⁻ zw, (dHam n zw.1 zw.2 : ℝ≥0∞) ∂commonMassCoupling hL n) =
∫⁻ x, ∑ C ∈ (orderedComponents hL n x).toFinset,
∫⁻ zw, (hammingDist zw.1 zw.2 : ℝ≥0∞) ∂componentCoupling hL n x C
∂(Measure.pi (fun _ : Fin n => (volume : Measure Covariate))) := by
rw [commonMassCoupling, Measure.lintegral_bind
(measurable_conditionalDatasetCoupling hL n).aemeasurable
(measurable_datasetHammingCost n).aemeasurable]
apply lintegral_congr
intro x
exact conditionalDatasetCoupling_hamming_cost_eq hL hhL n x
theorem commonMassCoupling_hamming_cost_le_cubic_sum
The integrated dataset cost is bounded by the expected nonsingleton cubic component sum. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1 / 4
n :
ℕ
conclusion
(∫⁻ zw, (dHam n zw.1 zw.2 : ℝ≥0∞) ∂commonMassCoupling hL n)
≤ ∫⁻ x, ∑ C ∈ (orderedComponents hL n x).toFinset, if C.card = 1 then 0 else ofReal (4*separation hL*(C.card : ℝ)^3) ∂(Measure.pi (fun _ : Fin n => (volume : Measure Covariate)))
Proof (Lean source)
-- @node: commonMassCoupling_hamming_cost_le_cubic_sum
lemma commonMassCoupling_hamming_cost_le_cubic_sum (hL : ℝ)
(hhL : 0 < hL ∧ hL ≤ 1 / 4) (n : ℕ) :
(∫⁻ zw, (dHam n zw.1 zw.2 : ℝ≥0∞) ∂commonMassCoupling hL n) ≤
∫⁻ x, ∑ C ∈ (orderedComponents hL n x).toFinset,
if C.card = 1 then 0 else ofReal (4*separation hL*(C.card : ℝ)^3)
∂(Measure.pi (fun _ : Fin n => (volume : Measure Covariate))) := by
rw [commonMassCoupling, Measure.lintegral_bind
(measurable_conditionalDatasetCoupling hL n).aemeasurable
(measurable_datasetHammingCost n).aemeasurable]
exact lintegral_mono (fun x =>
conditionalDatasetCoupling_hamming_cost_le_cubic_sum hL hhL n x)
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.Construction 54 declarations Public cells, count-weighted pair statistics and the single Laplace release; scalar, interval, dyadic tuning and inversion postprocessings are defined concretely.
Public cells, count-weighted pair statistics and the single Laplace release; scalar, interval, dyadic tuning and inversion postprocessings are defined concretely.
def cellWidth
The equal cell width is twice the public radius divided by the public cell count.
Definition (Lean source)
parameters
h :
ℝ
k :
ℕ
def
cellWidth h k :
ℝ
given by
2*h/k
def cell
Public cells partition the localization window with left endpoints included and only the final right endpoint included.
def cellRecords
A cell selects the dataset indices whose covariates belong to that public cell.
def cellCount
Cell occupancy is the cardinality of the selected dataset indices.
Definition (Lean source)
def KN
The numerator pair kernel multiplies treatment difference by outcome difference.
def KD
The denominator pair kernel squares the treatment difference.
def pairContribution
The count-weighted pair sum is zero below occupancy two, and otherwise weights the sum over unordered pairs by occupancy divided by the number of pairs.
def cellSum
A cell moment sum adds a record function across the selected dataset indices.
Definition (Lean source)
def numeratorCell
The numerator cell contribution uses the displayed closed treatment-outcome moment formula, with zero below occupancy two.
Definition (Lean source)
parameters
def
numeratorCell n h k D j :
ℝ
def denominatorCell
The denominator cell contribution uses the displayed closed treatment moment formula, with zero below occupancy two.
def SN
The numerator statistic sums the closed cell contributions.
Definition (Lean source)
def SD
The denominator statistic sums the closed cell contributions.
Definition (Lean source)
The sum of pairwise difference products equals the centered finite cross moment. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: pair_difference_sum
lemma pair_difference_sum (s : ℕ) (a b : Fin s → ℝ) :
(∑ i : Fin s, ∑ j : Fin s, if i < j then (a i-a j)*(b i-b j) else 0) =
(s : ℝ) * ∑ i, a i*b i - (∑ i, a i)*(∑ i, b i) := by
classical
let K := fun i j => (a i-a j)*(b i-b j)
have hsymm : ∀ i j, K i j = K j i := by intros; dsimp [K]; ring
have hsplit : ∀ i j, K i j = (if i < j then K i j else 0) +
(if j < i then K i j else 0) := by
intro i j
rcases lt_trichotomy i j with hij | hij | hij
· simp [hij, not_lt.mpr hij.le]
· subst j; simp [K]
· simp [hij, not_lt.mpr hij.le]
have htwice : (∑ i : Fin s, ∑ j : Fin s, K i j) =
2 * ∑ i : Fin s, ∑ j : Fin s, if i < j then K i j else 0 := by
calc
_ = ∑ i : Fin s, ∑ j : Fin s, ((if i < j then K i j else 0) +
(if j < i then K i j else 0)) := by
apply Finset.sum_congr rfl
intro i hi
apply Finset.sum_congr rfl
intro j hj
exact hsplit i j
_ = _ := by
simp only [Finset.sum_add_distrib]
rw [Finset.sum_comm (f := fun i j => if j < i then K i j else 0)]
conv_lhs => rhs; arg 2; ext i; arg 2; ext j; rw [hsymm j i]
ring
have hfull : (∑ i : Fin s, ∑ j : Fin s, K i j) =
2 * ((s : ℝ) * ∑ i, a i*b i - (∑ i, a i)*(∑ i, b i)) := by
dsimp [K]
simp_rw [sub_mul, mul_sub, Finset.sum_sub_distrib]
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul,
← Finset.mul_sum, ← Finset.sum_mul]
ring
dsimp [K] at htwice
linarith
The pair kernels give the displayed count-weighted moment formulas, including s < 2. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: pair_moment_formulas
lemma pair_moment_formulas (s : ℕ) (z : Fin s → O) :
pairContribution KN z = (if s < 2 then 0 else
2 * ((s : ℝ) * ∑ i, bit (z i).2.1 * bit (z i).2.2 -
(∑ i, bit (z i).2.1) * (∑ i, bit (z i).2.2)) / ((s : ℝ) - 1)) ∧
pairContribution KD z = (if s < 2 then 0 else
2 * ((s : ℝ) * ∑ i, bit (z i).2.1 - (∑ i, bit (z i).2.1)^2) /
((s : ℝ) - 1)) := by
classical
by_cases hs : s < 2
· simp [pairContribution, hs]
have hs0 : (s : ℝ) ≠ 0 := by
have : 0 < s := by omega
exact_mod_cast this.ne'
have hs1 : (s : ℝ) - 1 ≠ 0 := by
have : (2 : ℝ) ≤ s := by exact_mod_cast (show 2 ≤ s by omega)
linarith
have hN := pair_difference_sum s (fun i => bit (z i).2.1) (fun i => bit (z i).2.2)
have hD := pair_difference_sum s (fun i => bit (z i).2.1) (fun i => bit (z i).2.1)
have hbit : ∀ a : Bool, bit a * bit a = bit a := by
intro a; cases a <;> norm_num [bit]
simp only [hbit] at hD
constructor
· simp only [pairContribution, hs, if_false, KN]
rw [hN, Nat.cast_choose_two ℝ]
field_simp [hs0, hs1]
<;> ring
· simp only [pairContribution, hs, if_false, KD, pow_two]
rw [hD, Nat.cast_choose_two ℝ]
field_simp [hs0, hs1]
<;> ring
Each public cell is a Borel covariate set, including the final endpoint. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: measurableSet_cell
lemma measurableSet_cell (j : Fin k) : MeasurableSet (cell h k j) := by
unfold cell
measurability
Counting cell members is measurable on the full dataset space. The displayed conclusion follows.
Formal statement
hypotheses
conclusion
Measurable (fun D => cellCount n h k D j)
Proof (Lean source)
-- @node: measurable_cellCount
@[fun_prop] lemma measurable_cellCount (j : Fin k) : Measurable (fun D => cellCount n h k D j) := by
classical
have heq : (fun D => cellCount n h k D j) =
(fun D => ∑ i : Fin n, if (D i).1 ∈ cell h k j then (1 : ℕ) else 0) := by
funext D
simp [cellCount, cellRecords, Finset.sum_boole]
rw [heq]
apply Finset.measurable_fun_sum
intro i hi
exact measurable_const.ite ((measurableSet_cell h k j).preimage (by fun_prop))
measurable_const
Summing a measurable record function over a public cell is measurable. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
conclusion
Measurable (fun D => cellSum n h k D j v)
Proof (Lean source)
-- @node: measurable_cellSum
@[fun_prop] lemma measurable_cellSum (j : Fin k) (v : O → ℝ) (hv : Measurable v) :
Measurable (fun D => cellSum n h k D j v) := by
classical
simp only [cellSum, cellRecords, Finset.sum_filter]
apply Finset.measurable_fun_sum
intro i hi
exact (hv.comp (measurable_pi_apply i)).ite
((measurableSet_cell h k j).preimage (by fun_prop)) measurable_const
theorem measurable_bit
Numerically encoding a binary mark is measurable. The displayed conclusion follows.
Formal statement
conclusion
Proof (Lean source)
@[fun_prop] lemma measurable_bit : Measurable bit := by
fun_prop
Each closed numerator cell contribution is measurable, including occupancies zero and one. The displayed conclusion follows.
Formal statement
hypotheses
conclusion
Measurable (fun D => numeratorCell n h k D j)
Proof (Lean source)
-- @node: measurable_numeratorCell
@[fun_prop] lemma measurable_numeratorCell (j : Fin k) :
Measurable (fun D => numeratorCell n h k D j) := by
unfold numeratorCell
dsimp only
refine measurable_const.ite (measurableSet_lt (measurable_cellCount n h k j) measurable_const) ?_
fun_prop
Each closed denominator cell contribution is measurable, including occupancies zero and one. The displayed conclusion follows.
Formal statement
hypotheses
conclusion
Measurable (fun D => denominatorCell n h k D j)
Proof (Lean source)
-- @node: measurable_denominatorCell
@[fun_prop] lemma measurable_denominatorCell (j : Fin k) :
Measurable (fun D => denominatorCell n h k D j) := by
unfold denominatorCell
dsimp only
refine measurable_const.ite (measurableSet_lt (measurable_cellCount n h k j) measurable_const) ?_
fun_prop
The numerator statistic is measurable. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: measurable_SN
@[fun_prop] lemma measurable_SN : Measurable (SN n h k) := by
unfold SN
fun_prop
The denominator statistic is measurable. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: measurable_SD
@[fun_prop] lemma measurable_SD : Measurable (SD n h k) := by
unfold SD
fun_prop
def info
The public occupancy scale is sample size times radius times the truncated expected cell occupancy.
Definition (Lean source)
parameters
n :
ℕ
h :
ℝ
k :
ℕ
def
info n h k :
ℝ
given by
n*h*min 1 (n*cellWidth h k)
def d0
The public denominator floor is three sixty-fourths of the occupancy scale.
Definition (Lean source)
def bias
The public bias envelope retains the localization and rough-nuisance cell terms.
Definition (Lean source)
parameters
h :
ℝ
k :
ℕ
def
bias h k :
ℝ
given by
3*h + 24*(cellWidth h k)^(1/5 : ℝ)
The public squared-error envelope retains localization, cell bias, occupancy and the Laplace second-moment terms with the specified multiplier.
def pairQuery
The joint query consists of the numerator and denominator statistics.
Both coordinates of the joint query are measurable functions of the dataset. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: measurable_pairQuery
@[fun_prop] lemma measurable_pairQuery : Measurable (pairQuery n h k) := by
unfold pairQuery
fun_prop
The joint Laplace release is measurable. The displayed conclusion follows.
Formal statement
hypotheses
n :
ℕ
epsilon :
ℝ
h :
ℝ
k :
ℕ
conclusion
Proof (Lean source)
-- @node: measurable_pairRelease
@[fun_prop] lemma measurable_pairRelease :
Measurable (Causalean.Stat.Privacy.laplaceMechPi (12/epsilon) (pairQuery n h k)) := by
classical
letI : SigmaFinite (Causalean.Stat.Privacy.laplaceMeasure (12/epsilon)) := by
unfold Causalean.Stat.Privacy.laplaceMeasure
infer_instance
unfold Causalean.Stat.Privacy.laplaceMechPi
apply Measure.measurable_of_measurable_coe
intro E hE
have hshift : Measurable (fun p : Dataset n × (Fin 2 → ℝ) => p.2 + pairQuery n h k p.1) := by
fun_prop
have hm : Measurable (fun p : Dataset n × (Fin 2 → ℝ) =>
E.indicator (fun _ => (1 : ℝ≥0∞)) (p.2 + pairQuery n h k p.1)) :=
(measurable_const.indicator hE).comp hshift
have hmeas := hm.lintegral_prod_right' (ν := Measure.pi
(fun _ : Fin 2 => Causalean.Stat.Privacy.laplaceMeasure (12/epsilon)))
convert hmeas using 1
funext D
rw [Measure.map_apply (by fun_prop) hE]
simpa only [indicator, preimage, Set.mem_setOf_eq, Pi.one_apply] using
(lintegral_indicator_one (μ := Measure.pi
(fun _ : Fin 2 => Causalean.Stat.Privacy.laplaceMeasure (12/epsilon)))
(hE.preimage (show Measurable (fun z : Fin 2 → ℝ => z + pairQuery n h k D) by
fun_prop))).symm
The single joint Laplace release adds the specified independent noise to the two count- weighted statistics.
Definition (Lean source)
parameters
n :
ℕ
epsilon :
ℝ
h :
ℝ
k :
ℕ
given by
clause 1
toFun := laplaceMechPi (12/epsilon) (pairQuery n h k)
clause 2
measurable' := measurable_pairRelease n epsilon h k
def clip
Clipping truncates a real number to the specified lower and upper endpoints.
def ratioMap
The scalar postprocessing floors the noisy denominator and clips the ratio to the target range.
def closedInterval
Finite closed endpoints give an interval code with both endpoints included.
Definition (Lean source)
def intervalMap
The conservative interval centers the public radius at the clipped ratio and truncates it to the target range.
Definition (Lean source)
The floored clipped ratio is measurable. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: measurable_ratioMap
@[fun_prop] lemma measurable_ratioMap : Measurable (ratioMap n h k) := by
unfold ratioMap clip
fun_prop
The conservative interval postprocessing is measurable. The displayed conclusion follows.
Formal statement
hypotheses
n :
ℕ
epsilon :
ℝ
h :
ℝ
k :
ℕ
conclusion
Measurable (intervalMap n epsilon h k)
Proof (Lean source)
-- @node: measurable_intervalMap
@[fun_prop] lemma measurable_intervalMap : Measurable (intervalMap n epsilon h k) := by
unfold intervalMap closedInterval
fun_prop
The scalar release is measurable postprocessing of the single joint noisy release.
Definition (Lean source)
parameters
n :
ℕ
epsilon :
ℝ
h :
ℝ
k :
ℕ
given by
The interval release is measurable postprocessing of that same single joint noisy release.
Definition (Lean source)
parameters
n :
ℕ
epsilon :
ℝ
h :
ℝ
k :
ℕ
def
given by
(privateRatioRelease n epsilon h k).mapOfMeasurable (intervalMap n epsilon h k) (measurable_intervalMap n epsilon h k)
The numerical benchmark is the truncated maximum of the sampling, sparse-privacy and dense- privacy terms.
def tunedH
The public radius is the specified dyadic rounding of the benchmark scale.
def tunedK
The public integer cell count is twice the inverse fourth power of the dyadic radius.
Definition (Lean source)
parameters
n :
ℕ
epsilon :
ℝ
def
tunedK n epsilon :
ℕ
given by
⌊2*(tunedH n epsilon)^(-4 : ℤ)⌋₊
theorem rate_pos
The benchmark is strictly positive at every positive sample size. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
theorem tunedH_bounds
Dyadic rounding brackets the benchmark between the radius and twice the radius. the theorem's stated inputs and assumptions, and the asserted conclusion follows.
Formal statement
Proof (Lean source)
-- @node: tunedH_bounds
lemma tunedH_bounds (hn : 0 < n) :
0 < tunedH n epsilon ∧ rate n epsilon / 2 ≤ tunedH n epsilon ∧
tunedH n epsilon ≤ rate n epsilon := by
have hr := rate_pos n epsilon hn
let c := ceil (Real.logb 2 (1 / rate n epsilon))
have hc : Real.logb 2 (1 / rate n epsilon) ≤ (c : ℝ) := Int.le_ceil _
have hc' : (c : ℝ) ≤ Real.logb 2 (1 / rate n epsilon) + 1 :=
le_of_lt (Int.ceil_lt_add_one _)
have hp : 0 < (2 : ℝ) ^ c := zpow_pos (by norm_num) _
have hl : 1 / rate n epsilon ≤ (2 : ℝ) ^ c := by
rw [← Real.rpow_intCast]
exact (Real.logb_le_iff_le_rpow (by norm_num) (by positivity)).mp hc
have hu : (2 : ℝ) ^ c ≤ 2 / rate n epsilon := by
rw [← Real.rpow_intCast]
calc
(2 : ℝ) ^ (c : ℝ) ≤ 2 ^ (Real.logb 2 (1 / rate n epsilon) + 1) :=
Real.rpow_le_rpow_of_exponent_le (by norm_num) hc'
_ = 2 / rate n epsilon := by
rw [Real.rpow_add (by norm_num), Real.rpow_logb (by norm_num)
(by norm_num) (by positivity), Real.rpow_one]
ring
change 0 < (2 : ℝ) ^ (-c) ∧ rate n epsilon / 2 ≤ (2 : ℝ) ^ (-c) ∧
(2 : ℝ) ^ (-c) ≤ rate n epsilon
rw [zpow_neg]
refine ⟨inv_pos.mpr hp, ?_, ?_⟩
· rw [← one_div]
apply (le_div_iff₀ hp).mpr
have hu' := (le_div_iff₀ hr).mp hu
nlinarith
· apply (inv_le_iff_one_le_mul₀ hp).mpr
have hl' := (div_le_iff₀ hr).mp hl
nlinarith
theorem public_tuning_parameters
The integer conversion is exact on the dyadic branch; its radius, cell count and cell width obey the public construction constraints. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
n :
ℕ
epsilon :
ℝ
hn :
2 ≤ n
he :
0 < epsilon ∧ epsilon ≤ 1
hr :
rate n epsilon < 1 / 8
Proof (Lean source)
-- @node: public_tuning_parameters
lemma public_tuning_parameters (hn : 2 ≤ n) (he : 0 < epsilon ∧ epsilon ≤ 1)
(hr : rate n epsilon < 1 / 8) :
0 < tunedH n epsilon ∧ tunedH n epsilon ≤ 1 / 4 ∧ 2 ≤ tunedK n epsilon ∧
(tunedK n epsilon : ℝ) = 2*(tunedH n epsilon)^(-4 : ℤ) ∧
cellWidth (tunedH n epsilon) (tunedK n epsilon) = (tunedH n epsilon)^5 := by
have hb := tunedH_bounds n epsilon (by omega)
let c := ceil (Real.logb 2 (1 / rate n epsilon))
have hc : 0 ≤ c := by
apply Int.ceil_nonneg
apply Real.logb_nonneg (by norm_num)
apply (le_div_iff₀ (rate_pos n epsilon (by omega))).mpr
linarith
have hex : 2 * (tunedH n epsilon) ^ (-4 : ℤ) =
((2 * 2 ^ (c.toNat * 4) : ℕ) : ℝ) := by
unfold tunedH
change 2 * ((2 : ℝ) ^ (-c)) ^ (-4 : ℤ) = _
rw [← zpow_mul]
have hm : -c * (-4 : ℤ) = ((c.toNat * 4 : ℕ) : ℤ) := by
rw [Nat.cast_mul, Int.toNat_of_nonneg hc]
ring
rw [hm, zpow_natCast]
push_cast
rfl
have hk : tunedK n epsilon = 2 * 2 ^ (c.toNat * 4) := by
unfold tunedK
rw [hex, Nat.floor_natCast]
have hkcast : (tunedK n epsilon : ℝ) = 2*(tunedH n epsilon)^(-4 : ℤ) := by
rw [hk, ← hex]
refine ⟨hb.1, by linarith [hb.2.2], ?_, hkcast, ?_⟩
· rw [hk]
have hp : 1 ≤ (2 : ℕ) ^ (c.toNat * 4) := Nat.one_le_pow _ _ (by omega)
omega
· unfold cellWidth
rw [hkcast]
rw [zpow_neg, zpow_ofNat]
field_simp
<;> ring
Public tuning returns zero in the constant branch and otherwise uses the specified dyadic pair-ratio release.
Public tuning returns the entire target range in the constant branch and otherwise uses the conservative pair-ratio interval.
Definition (Lean source)
parameters
n :
ℕ
epsilon :
ℝ
def
given by
if 1/8
≤ rate n epsilon then Kernel.const _ (Measure.dirac (closedInterval (-1) 1)) else Ihk n epsilon (tunedH n epsilon) (tunedK n epsilon)
def cellMass
The population cell mass is the real measure of the cell under the design marginal.
def occupancyWeight
The population weight is expected occupancy multiplied by the probability of another record in the same cell.
def Wocc
The total occupancy weight sums the population cell weights.
Definition (Lean source)
def conditionalCellLaw
The conditional record law is the observed marginal restricted to the cell and normalized by its mass.
def nu
The conditional numerator moment integrates the pair kernel under two independent draws from the conditional cell law.
Definition (Lean source)
parameters
def
nu h k P j :
ℝ
given by
def dj
The conditional denominator moment integrates its pair kernel under those independent draws.
Definition (Lean source)
parameters
def
dj h k P j :
ℝ
given by
def Nbar
The population numerator is the expectation of the count-weighted numerator under the dataset law.
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.CosineCertificate 24 declarations Assembly of complete sign-family membership and the fixed-sample chi-square certificate.
Assembly of complete sign-family membership and the fixed-sample chi-square certificate.
The micro radius is positive and no larger than the macro radius. the asserted conclusion follows.
Formal statement
Proof (Lean source)
-- @node: deltaL_pos_le_macro
lemma deltaL_pos_le_macro : 0 < deltaL hL ∧ deltaL hL ≤ hL := by
have hp : 0 < hL^5 := pow_pos hhL.1 5
have h1 : hL ≤ 1 := hhL.2.trans (by norm_num)
have h4 : hL^4 ≤ 1 := pow_le_one₀ (le_of_lt hhL.1) h1
constructor
· exact hp
· dsimp [deltaL]
calc
hL^5 = hL * hL^4 := by ring
_ ≤ hL * 1 := mul_le_mul_of_nonneg_left h4 (le_of_lt hhL.1)
_ = hL := mul_one _
theorem activeSigns_subset_endpoint_interval Lemma activeSigns_subset_endpoint_interval in the paper ↗
Active centers lie in the integer interval between the two macro endpoint roundings. the asserted conclusion follows.
Formal statement
Proof (Lean source)
-- @node: activeSigns_subset_endpoint_interval
lemma activeSigns_subset_endpoint_interval :
activeSigns hL ⊆ Icc ⌊(x0-hL)/deltaL hL⌋ ⌈(x0+hL)/deltaL hL⌉ := by
intro j hj
have hd := (deltaL_pos_le_macro hL hhL).1
have hm := (mem_activeSigns hL hhL j).mp hj
have hl : (x0-hL)/deltaL hL < (j : ℝ)+1 := by
apply (div_lt_iff₀ hd).mpr
nlinarith [hm.2]
have hu : (j : ℝ)-1 < (x0+hL)/deltaL hL := by
apply (lt_div_iff₀ hd).mpr
nlinarith [hm.1]
have hf := Int.floor_le ((x0-hL)/deltaL hL)
have hc := Int.le_ceil ((x0+hL)/deltaL hL)
have hf' : ⌊(x0-hL)/deltaL hL⌋ < j+1 := by exact_mod_cast (hf.trans_lt hl)
have hc' : j-1 < ⌈(x0+hL)/deltaL hL⌉ := by exact_mod_cast (hu.trans_le hc)
rw [Finset.mem_Icc]
omega
Counting the active centers costs at most the macro interval length plus three micro cells. the asserted conclusion follows.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1 / 4
conclusion
((activeSigns hL).card : ℝ) ≤ 2*hL/deltaL hL+3
Proof (Lean source)
-- @node: activeSigns_card_le_length
lemma activeSigns_card_le_length :
((activeSigns hL).card : ℝ) ≤ 2*hL/deltaL hL+3 := by
have hd := (deltaL_pos_le_macro hL hhL).1
have he : (x0-hL)/deltaL hL ≤ (x0+hL)/deltaL hL := by
apply div_le_div_of_nonneg_right _ (le_of_lt hd)
linarith [hhL.1]
have hl := Int.floor_le ((x0-hL)/deltaL hL)
have hu := Int.le_ceil ((x0+hL)/deltaL hL)
have hround : ⌊(x0-hL)/deltaL hL⌋ ≤ ⌈(x0+hL)/deltaL hL⌉ := by
exact_mod_cast (hl.trans (he.trans hu))
have hc := Finset.card_le_card (activeSigns_subset_endpoint_interval hL hhL)
have hcount := Int.card_Icc_of_le ⌊(x0-hL)/deltaL hL⌋ ⌈(x0+hL)/deltaL hL⌉ (by omega)
have hcR : ((activeSigns hL).card : ℝ) ≤
(⌈(x0+hL)/deltaL hL⌉ : ℝ)+1-(⌊(x0-hL)/deltaL hL⌋ : ℝ) := by
exact_mod_cast (show ((activeSigns hL).card : ℤ) ≤
⌈(x0+hL)/deltaL hL⌉+1-⌊(x0-hL)/deltaL hL⌋ by
rw [← hcount]; exact_mod_cast hc)
have hf := Int.lt_floor_add_one ((x0-hL)/deltaL hL)
have hce := Int.ceil_lt_add_one ((x0+hL)/deltaL hL)
have hdiff : (x0+hL)/deltaL hL-(x0-hL)/deltaL hL = 2*hL/deltaL hL := by ring
linarith
The number of active signs is at most five times the macro-to-micro radius ratio. the asserted conclusion follows.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1 / 4
conclusion
((activeSigns hL).card : ℝ) ≤ 5*hL/deltaL hL
Proof (Lean source)
-- @node: activeSigns_card_le_five_ratio
lemma activeSigns_card_le_five_ratio :
((activeSigns hL).card : ℝ) ≤ 5*hL/deltaL hL := by
have hd := deltaL_pos_le_macro hL hhL
have hr : 1 ≤ hL/deltaL hL := (le_div_iff₀ hd.1).mpr (by simpa using hd.2)
have hc := activeSigns_card_le_length hL hhL
have he : 5*hL/deltaL hL = 5*(hL/deltaL hL) := by ring
have he' : 2*hL/deltaL hL = 2*(hL/deltaL hL) := by ring
rw [he]
rw [he'] at hc
linarith
Updating one latent sign changes only its localized frame contribution. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: perturbation_update_difference
lemma perturbation_update_difference (lam : SignVector hL) (j : activeSigns hL)
(b : Bool) (x : Covariate) :
perturbation hL lam x - perturbation hL (update lam j b) x =
envelope hL x * (2*bit (lam j)-2*bit b) * frame hL j x := by
classical
have hs : (∑ i : activeSigns hL,
((2*bit (lam i)-1)*frame hL i x -
(2*bit (update lam j b i)-1)*frame hL i x)) =
(2*bit (lam j)-2*bit b)*frame hL j x := by
rw [Finset.sum_eq_single j]
· simp only [Function.update_self]
ring
· intro i _ hij
simp [Function.update_of_ne hij]
· simp
simp only [perturbation]
rw [← mul_sub, ← Finset.sum_sub_distrib, hs]
ring
A sign update has pointwise oscillation at most twice its envelope-weighted frame. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
≤ 2*envelope hL x*|frame hL j x|
Proof (Lean source)
-- @node: perturbation_update_abs_le
lemma perturbation_update_abs_le (lam : SignVector hL) (j : activeSigns hL)
(b : Bool) (x : Covariate) :
|perturbation hL lam x - perturbation hL (update lam j b) x| ≤
2*envelope hL x*|frame hL j x| := by
rw [perturbation_update_difference hL hhL, abs_mul, abs_mul,
abs_of_nonneg (envelope_range hL x).1]
have hb : |2*bit (lam j)-2*bit b| ≤ 2 := by
cases h : lam j <;> cases b <;> norm_num [bit, h]
simpa only [mul_comm (envelope hL x) 2] using mul_le_mul_of_nonneg_right
(mul_le_mul_of_nonneg_left hb (envelope_range hL x).1) (abs_nonneg (frame hL j x))
Outside the updated frame's support the perturbation is unchanged. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: perturbation_update_eq_off_support
lemma perturbation_update_eq_off_support (lam : SignVector hL) (j : activeSigns hL)
(b : Bool) (x : Covariate) (hx : deltaL hL ≤ |(x : ℝ)-(j : ℤ)*deltaL hL|) :
perturbation hL lam x = perturbation hL (update lam j b) x := by
have he := perturbation_update_difference hL hhL lam j b x
rw [frame_zero_of_radius_le hL hhL j x hx, mul_zero] at he
exact sub_eq_zero.mp he
The roadmap's coordinate oscillation constants have the required total squared budget. The displayed conclusion follows.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1 / 4
conclusion
≤ 1280*(separation hL)^2*hL*deltaL hL
Proof (Lean source)
-- @node: sign_coordinate_squared_budget
lemma sign_coordinate_squared_budget :
(∑ j : activeSigns hL, (16*separation hL*deltaL hL)^2) ≤
1280*(separation hL)^2*hL*deltaL hL := by
classical
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_coe, nsmul_eq_mul]
have hd := (deltaL_pos_le_macro hL hhL).1
have hc := activeSigns_card_le_five_ratio hL hhL
have hm := mul_le_mul_of_nonneg_right hc (sq_nonneg (16*separation hL*deltaL hL))
calc
_ ≤ (5*hL/deltaL hL)*(16*separation hL*deltaL hL)^2 := hm
_ = _ := by field_simp; ring
theorem signOverlapIntegrand_update_eq_off_support
The Fourier overlap integrand is unchanged outside a replaced sign's frame support. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
signOverlapIntegrand hL lam lam' x
= signOverlapIntegrand hL (update lam j b) lam' x
Proof (Lean source)
-- @node: signOverlapIntegrand_update_eq_off_support
lemma signOverlapIntegrand_update_eq_off_support (lam lam' : SignVector hL)
(j : activeSigns hL) (b : Bool) (x : Covariate)
(hx : deltaL hL ≤ |(x : ℝ)-(j : ℤ)*deltaL hL|) :
signOverlapIntegrand hL lam lam' x =
signOverlapIntegrand hL (update lam j b) lam' x := by
simp only [signOverlapIntegrand, perturbation_update_eq_off_support hL hhL lam j b x hx]
theorem signOverlap_range
The centered overlap stays in a fixed bounded interval after covariate integration. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
signOverlap hL lam lam' ∈ Icc (-1) 15
Proof (Lean source)
-- @node: signOverlap_range
lemma signOverlap_range (lam lam' : SignVector hL) : signOverlap hL lam lam' ∈ Icc (-1) 15 := by
have hi := integrable_signOverlapIntegrand hL hhL lam lam'
constructor
· have hm := integral_mono (integrable_const (-1 : ℝ)) hi
(fun x => (signOverlapIntegrand_range hL hhL lam lam' x).1)
simpa [signOverlap] using hm
· have hm := integral_mono hi (integrable_const (15 : ℝ))
(fun x => (signOverlapIntegrand_range hL hhL lam lam' x).2)
simpa [signOverlap] using hm
theorem signOverlap_abs_le_fifteen
The boundedness hypothesis of the bounded-differences gate holds for each fixed second sign. the theorem's stated inputs and assumptions, and the asserted conclusion follows.
Formal statement
hypotheses
conclusion
|signOverlap hL lam lam'| ≤ 15
Proof (Lean source)
-- @node: signOverlap_abs_le_fifteen
lemma signOverlap_abs_le_fifteen (lam lam' : SignVector hL) :
|signOverlap hL lam lam'| ≤ 15 := by
have hr := signOverlap_range hL hhL lam lam'
exact abs_le.mpr ⟨by linarith [hr.1], hr.2⟩
theorem signOverlap_update_difference_integral
A coordinate replacement in the overlap is exactly the integral of its localized change. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
= ∫ x : Covariate, (signOverlapIntegrand hL lam lam' x - signOverlapIntegrand hL (update lam j b) lam' x)
Proof (Lean source)
-- @node: signOverlap_update_difference_integral
lemma signOverlap_update_difference_integral (lam lam' : SignVector hL)
(j : activeSigns hL) (b : Bool) :
signOverlap hL lam lam' - signOverlap hL (update lam j b) lam' =
∫ x : Covariate, (signOverlapIntegrand hL lam lam' x -
signOverlapIntegrand hL (update lam j b) lam' x) := by
exact (integral_sub (integrable_signOverlapIntegrand hL hhL lam lam')
(integrable_signOverlapIntegrand hL hhL (update lam j b) lam')).symm
Only the two neighboring lattice frames can be nonzero, including at grid boundaries. The displayed conclusion follows.
Formal statement
hypotheses
conclusion
(Finset.univ.filter (fun j : activeSigns hL => frame hL j x ≠ 0)).card ≤ 2
Proof (Lean source)
-- @node: frame_nonzero_card_le_two
lemma frame_nonzero_card_le_two (x : Covariate) :
(Finset.univ.filter (fun j : activeSigns hL => frame hL j x ≠ 0)).card ≤ 2 := by
classical
let q : ℤ := ⌊(x : ℝ) / deltaL hL⌋
have hd := (deltaL_pos_le_macro hL hhL).1
have hl : (q : ℝ) * deltaL hL ≤ x := (le_div_iff₀ hd).mp (Int.floor_le _)
have hu : (x : ℝ) ≤ (q + 1 : ℤ) * deltaL hL := by
have hu := (div_lt_iff₀ hd).mp (Int.lt_floor_add_one ((x : ℝ) / deltaL hL))
simpa [q] using hu.le
have hc := Finset.card_le_card_of_injOn (fun j : activeSigns hL => (j : ℤ))
(s := Finset.univ.filter (fun j : activeSigns hL => frame hL j x ≠ 0))
(t := {q, q+1}) (by
intro j hj
have hn := (Finset.mem_filter.mp hj).2
by_cases hj0 : (j : ℤ) = q
· simp [hj0]
· by_cases hj1 : (j : ℤ) = q+1
· simp [hj1]
· exact (hn (frame_zero_off_neighbors hL hhL q j x hl hu hj0 hj1)).elim)
(by intro i hi j hj hij; exact Subtype.ext hij)
exact hc.trans (Finset.card_le_two)
The two-frame Cauchy-Schwarz bound keeps every squared perturbation below twice the bump. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
(perturbation hL lam x)^2 ≤ 2*bump hL x
Proof (Lean source)
-- @node: perturbation_square_le_two_bump
lemma perturbation_square_le_two_bump (lam : SignVector hL) (x : Covariate) :
(perturbation hL lam x)^2 ≤ 2*bump hL x := by
classical
let s := Finset.univ.filter (fun j : activeSigns hL => frame hL j x ≠ 0)
have hsum : (∑ j ∈ s, (2*bit (lam j)-1)*frame hL j x) =
∑ j : activeSigns hL, (2*bit (lam j)-1)*frame hL j x := by
apply Finset.sum_subset (filter_subset _ _)
intro j hj hn
have hz : frame hL j x = 0 := by simpa [s, hj] using hn
simp [hz]
have hsquares : (∑ j ∈ s, (frame hL j x)^2) =
∑ j : activeSigns hL, (frame hL j x)^2 := by
apply Finset.sum_subset (filter_subset _ _)
intro j hj hn
have hz : frame hL j x = 0 := by simpa [s, hj] using hn
simp [hz]
have hc : (s.card : ℝ) ≤ 2 := by exact_mod_cast frame_nonzero_card_le_two hL hhL x
have hcs := Finset.sum_mul_sq_le_sq_mul_sq s
(fun j => 2*bit (lam j)-1) (fun j => frame hL j x)
have hbitSq (b : Bool) : (2 * bit b - 1) ^ 2 = (1 : ℝ) := by
cases b <;> norm_num [bit]
simp only [hbitSq, Finset.sum_const, nsmul_eq_mul, mul_one] at hcs
rw [hsum, hsquares] at hcs
have ht := hcs.trans (mul_le_mul_of_nonneg_right hc
(Finset.sum_nonneg (fun j _ => sq_nonneg (frame hL j x))))
have hw := mul_le_mul_of_nonneg_left ht (sq_nonneg (envelope hL x))
calc
(perturbation hL lam x)^2 = bump hL x *
(∑ j : activeSigns hL, (2*bit (lam j)-1)*frame hL j x)^2 := by
simp only [perturbation, bump, mul_pow]
_ ≤ bump hL x * (2 * ∑ j : activeSigns hL, (frame hL j x)^2) := hw
_ = 2*bump hL x := by
rw [← mul_assoc, mul_comm (bump hL x) 2, mul_assoc,
weighted_frame_square_partition hL hhL]
theorem perturbation_square_residual_abs_le_bump Lemma perturbation_square_residual_abs_le_bump in the paper ↗
The centered quadratic Fourier coefficient is bounded by the local bump. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: perturbation_square_residual_abs_le_bump
lemma perturbation_square_residual_abs_le_bump (lam : SignVector hL) (x : Covariate) :
|bump hL x-(perturbation hL lam x)^2| ≤ bump hL x := by
have hs := perturbation_square_le_two_bump hL hhL lam x
exact abs_le.mpr ⟨by linarith, by linarith [sq_nonneg (perturbation hL lam x)]⟩
Replacing one sign changes the squared perturbation by at most twice the bump. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: perturbation_update_square_abs_le
lemma perturbation_update_square_abs_le (lam : SignVector hL) (j : activeSigns hL)
(b : Bool) (x : Covariate) :
|(perturbation hL lam x)^2-(perturbation hL (update lam j b) x)^2| ≤
2*bump hL x := by
have hs := perturbation_square_le_two_bump hL hhL lam x
have hu := perturbation_square_le_two_bump hL hhL (update lam j b) x
exact abs_le.mpr ⟨by linarith [sq_nonneg (perturbation hL lam x)],
by linarith [sq_nonneg (perturbation hL (update lam j b) x)]⟩
The sharp square bound gives a convenient rational sup-norm envelope. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
|perturbation hL lam x| ≤ 3/2
Proof (Lean source)
-- @node: perturbation_abs_le_three_halves
lemma perturbation_abs_le_three_halves (lam : SignVector hL) (x : Covariate) :
|perturbation hL lam x| ≤ 3/2 := by
have hs := perturbation_square_le_two_bump hL hhL lam x
have hq := (bump_range hL x).2
have ha := sq_abs (perturbation hL lam x)
nlinarith [abs_nonneg (perturbation hL lam x)]
The effect amplitude lies between zero and one for every legal macro radius. the asserted conclusion follows.
Formal statement
Proof (Lean source)
-- @node: separation_unit_range
lemma separation_unit_range : separation hL ∈ Icc 0 1 := by
norm_num [separation, kappa] at *
constructor <;> linarith [hhL.1, hhL.2]
theorem signOverlapIntegrand_update_abs_le
A coordinate replacement changes the Fourier overlap integrand by at most eight amplitudes. the theorem's stated inputs and assumptions, and the asserted conclusion follows.
Formal statement
hypotheses
conclusion
|signOverlapIntegrand hL lam lam' x
- signOverlapIntegrand hL (update lam j b) lam' x|
≤ 8*separation hL
Proof (Lean source)
-- @node: signOverlapIntegrand_update_abs_le
lemma signOverlapIntegrand_update_abs_le (lam lam' : SignVector hL)
(j : activeSigns hL) (b : Bool) (x : Covariate) :
|signOverlapIntegrand hL lam lam' x -
signOverlapIntegrand hL (update lam j b) lam' x| ≤ 8*separation hL := by
have ht := separation_unit_range hL hhL
have hq := bump_range hL x
have hcoef : 0 ≤ 1+(separation hL*bump hL x-1)^2 ∧
1+(separation hL*bump hL x-1)^2 ≤ 2 := by
have hp := mul_nonneg ht.1 hq.1
have hu := mul_le_one₀ ht.2 hq.1 hq.2
constructor
· positivity
· nlinarith [mul_nonneg hp (sub_nonneg.mpr hu)]
have hframe : |frame hL j x| ≤ 1 := by
exact (frame_abs_le hL j x).trans (by split_ifs <;> norm_num)
have hdiff : |perturbation hL lam x-perturbation hL (update lam j b) x| ≤ 2 := by
have hd := perturbation_update_abs_le hL hhL lam j b x
have hf := mul_le_mul (envelope_range hL x).2 hframe
(abs_nonneg (frame hL j x)) (by norm_num : (0 : ℝ) ≤ 1)
nlinarith
have hfirst : |separation hL*(1+(separation hL*bump hL x-1)^2)*
(perturbation hL lam x-perturbation hL (update lam j b) x)*
perturbation hL lam' x| ≤ 6*separation hL := by
rw [abs_mul, abs_mul, abs_mul, abs_of_nonneg ht.1, abs_of_nonneg hcoef.1]
calc
_ ≤ separation hL*2*2*(3/2) := by
have ht0 := ht.1
gcongr <;> first
| positivity
| exact hcoef.2
| exact hdiff
| exact perturbation_abs_le_three_halves hL hhL lam' x
_ = _ := by ring
have hsecond : |(separation hL)^2*
((perturbation hL (update lam j b) x)^2-(perturbation hL lam x)^2)*
(bump hL x-(perturbation hL lam' x)^2)| ≤ 2*(separation hL)^2 := by
rw [abs_mul, abs_mul, abs_of_nonneg (sq_nonneg _), abs_sub_comm
((perturbation hL (update lam j b) x)^2)]
have hd := (perturbation_update_square_abs_le hL hhL lam j b x).trans
(mul_le_mul_of_nonneg_left hq.2 (by norm_num : (0 : ℝ) ≤ 2))
have hr := (perturbation_square_residual_abs_le_bump hL hhL lam' x).trans hq.2
calc
_ ≤ (separation hL)^2*2*1 := by
gcongr
simpa using hd
_ = _ := by ring
have he : signOverlapIntegrand hL lam lam' x -
signOverlapIntegrand hL (update lam j b) lam' x =
separation hL*(1+(separation hL*bump hL x-1)^2)*
(perturbation hL lam x-perturbation hL (update lam j b) x)*
perturbation hL lam' x +
(separation hL)^2*
((perturbation hL (update lam j b) x)^2-(perturbation hL lam x)^2)*
(bump hL x-(perturbation hL lam' x)^2) := by
dsimp [signOverlapIntegrand]
ring
rw [he]
calc
_ ≤ _ := abs_add_le _ _
_ ≤ 6*separation hL+2*(separation hL)^2 := add_le_add hfirst hsecond
_ ≤ 8*separation hL := by nlinarith [mul_nonneg ht.1 (sub_nonneg.mpr ht.2)]
Restricting a micro support to the unit interval can only decrease its length. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: frame_open_support_volume_le
lemma frame_open_support_volume_le (j : ℤ) :
(volume : Measure Covariate).real
{x | |(x : ℝ)-j*deltaL hL| < deltaL hL} ≤ 2*deltaL hL := by
have hd := (deltaL_pos_le_macro hL hhL).1
have hsub : val '' {x : Covariate | |(x : ℝ)-j*deltaL hL| < deltaL hL} ⊆
Ioo ((j : ℝ)*deltaL hL-deltaL hL) ((j : ℝ)*deltaL hL+deltaL hL) := by
rintro y ⟨x, hx, rfl⟩
change |(x : ℝ)-j*deltaL hL| < deltaL hL at hx
have hx' := abs_lt.mp hx
constructor <;> linarith
have hm : (volume : Measure ℝ) (val '' {x : Covariate |
|(x : ℝ)-j*deltaL hL| < deltaL hL}) ≤
volume (Ioo ((j : ℝ)*deltaL hL-deltaL hL) ((j : ℝ)*deltaL hL+deltaL hL)) :=
measure_mono hsub
rw [← unitInterval.volume_apply, Real.volume_Ioo,
show (j : ℝ)*deltaL hL+deltaL hL-((j : ℝ)*deltaL hL-deltaL hL) =
2*deltaL hL by ring] at hm
have hr := ENNReal.toReal_mono ENNReal.ofReal_ne_top hm
simpa only [Measure.real, ENNReal.toReal_ofReal (by positivity : 0 ≤ 2*deltaL hL)] using hr
theorem signOverlap_update_abs_le
Integrating the localized pointwise change gives the roadmap's sixteen-amplitude coordinate bound. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
≤ 16*separation hL*deltaL hL
Proof (Lean source)
-- @node: signOverlap_update_abs_le
lemma signOverlap_update_abs_le (lam lam' : SignVector hL)
(j : activeSigns hL) (b : Bool) :
|signOverlap hL lam lam'-signOverlap hL (update lam j b) lam'| ≤
16*separation hL*deltaL hL := by
classical
let s : Set Covariate := {x | |(x : ℝ)-(j : ℤ)*deltaL hL| < deltaL hL}
have hs : MeasurableSet s := by
apply measurableSet_lt
· fun_prop
· fun_prop
have hi := (integrable_signOverlapIntegrand hL hhL lam lam').sub
(integrable_signOverlapIntegrand hL hhL (update lam j b) lam')
have hbound (x : Covariate) :
|signOverlapIntegrand hL lam lam' x-
signOverlapIntegrand hL (update lam j b) lam' x| ≤
s.indicator (fun _ => 8*separation hL) x := by
by_cases hx : x ∈ s
· rw [Set.indicator_of_mem hx]
exact signOverlapIntegrand_update_abs_le hL hhL lam lam' j b x
· rw [Set.indicator_of_notMem hx]
have hx' : deltaL hL ≤ |(x : ℝ)-(j : ℤ)*deltaL hL| := by
exact le_of_not_gt hx
rw [signOverlapIntegrand_update_eq_off_support hL hhL lam lam' j b x hx']
simp
rw [signOverlap_update_difference_integral hL hhL]
calc
_ ≤ ∫ x : Covariate, |signOverlapIntegrand hL lam lam' x-
signOverlapIntegrand hL (update lam j b) lam' x| :=
abs_integral_le_integral_abs
_ ≤ ∫ x : Covariate, s.indicator (fun _ => 8*separation hL) x :=
integral_mono hi.abs ((integrable_const _).indicator hs) hbound
_ = (volume : Measure Covariate).real s * (8*separation hL) := by
rw [integral_indicator_const _ hs, smul_eq_mul]
_ ≤ (2*deltaL hL)*(8*separation hL) :=
mul_le_mul_of_nonneg_right (frame_open_support_volume_le hL hhL j)
(mul_nonneg (by norm_num) (separation_unit_range hL hhL).1)
_ = _ := by ring
theorem signOverlap_exp_average_le
For a fixed second sign vector, the centered overlap obeys the roadmap's mgf bound. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: signOverlap_exp_average_le
lemma signOverlap_exp_average_le (gate : BoundedDifferencesMGF)
(n : ℕ) (lam' : SignVector hL) :
(∑ lam : SignVector hL, exp ((n : ℝ)*signOverlap hL lam lam')) /
(card (SignVector hL) : ℝ) ≤
exp (160*(n : ℝ)^2*(separation hL)^2*hL*deltaL hL) := by
classical
have hm := finite_sign_mgf_average gate (fun lam => signOverlap hL lam lam')
(signOverlap_sign_sum hL hhL lam')
⟨15, fun lam => signOverlap_abs_le_fifteen hL hhL lam lam'⟩
(fun _ => 16*separation hL*deltaL hL)
(fun _ => mul_nonneg
(mul_nonneg (by norm_num) (separation_unit_range hL hhL).1)
(deltaL_pos_le_macro hL hhL).1.le)
(fun j lam b => signOverlap_update_abs_le hL hhL lam lam' j b)
(n : ℝ) (Nat.cast_nonneg n)
apply hm.trans
apply Real.exp_le_exp.mpr
calc
_ ≤ (n : ℝ)^2 / 8 * (1280*(separation hL)^2*hL*deltaL hL) :=
mul_le_mul_of_nonneg_left (sign_coordinate_squared_budget hL hhL) (by positivity)
_ = _ := by ring
theorem signOverlap_joint_exp_average_le
Averaging the fixed-second-sign mgf estimate gives the joint exponential bound. The displayed conclusion follows.
Formal statement
hypotheses
conclusion
(∑ lam : SignVector hL, ∑ lam' : SignVector hL, exp ((n : ℝ)*signOverlap hL lam lam')) / (card (SignVector hL) : ℝ)^2
Proof (Lean source)
-- @node: signOverlap_joint_exp_average_le
lemma signOverlap_joint_exp_average_le (gate : BoundedDifferencesMGF) (n : ℕ) :
(∑ lam : SignVector hL, ∑ lam' : SignVector hL,
exp ((n : ℝ)*signOverlap hL lam lam')) /
(card (SignVector hL) : ℝ)^2 ≤
exp (160*(n : ℝ)^2*(separation hL)^2*hL*deltaL hL) := by
classical
have hc : 0 < (card (SignVector hL) : ℝ) := by
exact_mod_cast Fintype.card_pos
have hb (lam' : SignVector hL) :=
(div_le_iff₀ hc).mp (signOverlap_exp_average_le hL hhL gate n lam')
have hs := Finset.sum_le_sum (fun lam' (_ : lam' ∈ univ) => hb lam')
rw [Finset.sum_comm] at hs
simp only [Finset.sum_const, Finset.card_univ, nsmul_eq_mul] at hs
apply (div_le_iff₀ (sq_pos_of_pos hc)).mpr
calc
_ ≤ _ := hs
_ = _ := by ring
theorem positive_family_certificate
Frame moments, model membership, the Fourier likelihood and fixed-sample chi-square certificate are conditional on the published bounded-differences mgf gate. The certificate records absolute continuity and square-density integrability explicitly, so its real-valued chi-square integral represents the finite divergence used in testing. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1 / 4
boundedDifferences_of_gate :
n :
ℕ
conclusion
conclusion 1
conclusion 2
theta fairNull = 0
conclusion 3
lam :
SignVector hL
CompleteModel (cosineFamily hL hhL lam)
theta (cosineFamily hL hhL lam) = separation hL
conclusion 5
conclusion 6
conclusion 7
conclusion 8
Integrable (fun z => (((signMixture hL hhL n).rnDeriv (dataLaw n fairNull) z).toReal - 1)^2) (dataLaw n fairNull)
Proof (Lean source)
-- @node: lem:positive-family-certificate
lemma positive_family_certificate (boundedDifferences_of_gate : BoundedDifferencesMGF)
(n : ℕ) :
CompleteModel fairNull ∧ theta fairNull = 0 ∧
(∀ lam : SignVector hL, CompleteModel (cosineFamily hL hhL lam) ∧
theta (cosineFamily hL hhL lam) = separation hL) ∧
(∀ lam x av yv,
conditionalLikelihood hL lam x av yv =
1 + (2*bit av-1)*sqrt (separation hL)*perturbation hL lam x +
(2*bit yv-1)*sqrt (separation hL)*(separation hL*bump hL x-1)*
perturbation hL lam x +
(2*bit av-1)*(2*bit yv-1)*separation hL*
(bump hL x-(perturbation hL lam x)^2)) ∧
oneRecordMixture hL hhL = Pobs fairNull ∧
chiSqDiv (signMixture hL hhL n) (dataLaw n fairNull) ≤
exp (160*(n : ℝ)^2*(separation hL)^2*hL*deltaL hL)-1 ∧
signMixture hL hhL n ≪ dataLaw n fairNull ∧
Integrable (fun z => (((signMixture hL hhL n).rnDeriv
(dataLaw n fairNull) z).toReal - 1)^2) (dataLaw n fairNull) := by
have hchi :
chiSqDiv (signMixture hL hhL n) (dataLaw n fairNull) ≤
exp (160*(n : ℝ)^2*(separation hL)^2*hL*deltaL hL)-1 := by
have hmgf :
(∑ lam : SignVector hL, ∑ lam' : SignVector hL,
exp ((n : ℝ)*signOverlap hL lam lam')) /
(card (SignVector hL) : ℝ)^2 ≤
exp (160*(n : ℝ)^2*(separation hL)^2*hL*deltaL hL) := by
exact signOverlap_joint_exp_average_le hL hhL boundedDifferences_of_gate n
have hmoment := (signMixture_chiSq_le_fourier_exp_average hL hhL n).trans hmgf
linarith
have halt : ∀ lam : SignVector hL, CompleteModel (cosineFamily hL hhL lam) :=
cosineFamily_completeModel hL hhL
exact ⟨fairNull_completeModel, fairNull_target,
fun lam => ⟨halt lam, cosineFamily_target hL hhL lam⟩,
conditionalLikelihood_fourier hL hhL, oneRecordMixture_eq_fairNull hL hhL, hchi,
signMixture_absolutelyContinuous hL hhL n,
signMixture_square_density_integrable hL hhL n⟩
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.CosineFamily 53 declarations Explicit uniform-design Bernoulli causal laws, overlapping cosine signs, and the conditional published-mgf certificate for the fixed-sample alternative mixture.
Explicit uniform-design Bernoulli causal laws, overlapping cosine signs, and the conditional published-mgf certificate for the fixed-sample alternative mixture.
def signLaw
The latent sign law is uniform on all sign vectors.
Definition (Lean source)
parameters
hL :
ℝ
def
given by
def perturbation
The nuisance perturbation multiplies the signed frame sum by the macro envelope.
Definition (Lean source)
parameters
def
perturbation hL lam x :
ℝ
given by
A finite signed sum of Borel frames times the envelope is Borel measurable. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: measurable_perturbation
@[fun_prop] lemma measurable_perturbation (lam : SignVector hL) :
Measurable (perturbation hL lam) := by
unfold perturbation
fun_prop
Fair independent signs cancel the nuisance perturbation pointwise, including boundaries. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: perturbation_sign_sum
lemma perturbation_sign_sum (x : Covariate) :
(∑ lam : SignVector hL, perturbation hL lam x) = 0 := by
classical
have hsign (b : Bool) : 2 * bit b - 1 = if b then (1 : ℝ) else -1 := by
cases b <;> norm_num [bit]
simp only [perturbation, ← Finset.mul_sum]
simp_rw [hsign]
rw [show (∑ lam : SignVector hL,
∑ j : activeSigns hL, (if lam j then (1 : ℝ) else -1) * frame hL j x) = 0 by
exact signLinear_sum (fun j : activeSigns hL => frame hL j x), mul_zero]
Off-diagonal frame terms cancel in the exact finite-sign second moment. The remaining geometric sum of squared frames is kept explicit. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: perturbation_sign_square_sum
lemma perturbation_sign_square_sum (x : Covariate) :
(∑ lam : SignVector hL, (perturbation hL lam x)^2) =
(card (SignVector hL) : ℝ) *
(bump hL x * ∑ j : activeSigns hL, (frame hL j x)^2) := by
classical
have hsign (b : Bool) : 2 * bit b - 1 = if b then (1 : ℝ) else -1 := by
cases b <;> norm_num [bit]
simp only [perturbation, mul_pow, ← Finset.mul_sum]
simp_rw [hsign]
rw [show (∑ lam : SignVector hL,
(∑ j : activeSigns hL, (if lam j then (1 : ℝ) else -1) * frame hL j x) ^ 2) =
(card (SignVector hL) : ℝ) *
∑ j : activeSigns hL, (frame hL j x) ^ 2 by
exact signLinear_sq_sum (fun j : activeSigns hL => frame hL j x)]
unfold bump
ring
def altE
The alternative propensity is one half plus the specified shared-sign perturbation.
Definition (Lean source)
parameters
def
altE hL lam x :
ℝ
given by
def altMu0
The alternative control mean has the negative shared-sign term and negative effect bump.
Definition (Lean source)
parameters
def
altMu0 hL lam x :
ℝ
given by
def altMu1
The alternative treated mean has the negative shared-sign term and positive effect bump.
Definition (Lean source)
parameters
def
altMu1 hL lam x :
ℝ
given by
The shared-sign propensity is measurable. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: measurable_altE
@[fun_prop] lemma measurable_altE (lam : SignVector hL) : Measurable (altE hL lam) := by
unfold altE
fun_prop
The shared-sign control mean is measurable. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: measurable_altMu0
@[fun_prop] lemma measurable_altMu0 (lam : SignVector hL) : Measurable (altMu0 hL lam) := by
unfold altMu0
fun_prop
The shared-sign treated mean is measurable. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: measurable_altMu1
@[fun_prop] lemma measurable_altMu1 (lam : SignVector hL) : Measurable (altMu1 hL lam) := by
unfold altMu1
fun_prop
A localized frame has absolute value at most one, and is zero outside its support. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: frame_abs_le
lemma frame_abs_le (j : ℤ) (x : Covariate) :
|frame hL j x| ≤ if |(x : ℝ) - j * deltaL hL| ≤ deltaL hL then 1 else 0 := by
classical
unfold frame
split
· exact Real.abs_cos_le_one _
· simp
At most three closed frame supports cover a covariate, including support endpoints. The displayed conclusion follows.
Formal statement
hypotheses
conclusion
≤ 3
Proof (Lean source)
-- @node: frame_support_card_le
lemma frame_support_card_le (x : Covariate) :
(Finset.univ.filter (fun j : activeSigns hL =>
|(x : ℝ) - (j : ℤ) * deltaL hL| ≤ deltaL hL)).card ≤ 3 := by
classical
let q : ℤ := ⌊(x : ℝ) / deltaL hL⌋
have hd : 0 < deltaL hL := pow_pos hhL.1 5
have hq := Int.floor_le ((x : ℝ) / deltaL hL)
have hq' := Int.lt_floor_add_one ((x : ℝ) / deltaL hL)
have hc := Finset.card_le_card_of_injOn (fun j : activeSigns hL => (j : ℤ))
(s := Finset.univ.filter (fun j : activeSigns hL =>
|(x : ℝ) - (j : ℤ) * deltaL hL| ≤ deltaL hL))
(t := Icc (q - 1) (q + 1)) (by
intro j hj
have hj' := abs_le.mp (Finset.mem_filter.mp hj).2
rw [Finset.mem_coe, Finset.mem_Icc]
have hl : (q : ℝ) - 1 ≤ (j : ℤ) := by
have h := (le_div_iff₀ hd).mp hq
change (q : ℝ) * deltaL hL ≤ (x : ℝ) at h
nlinarith [hj'.2]
have hu : (j : ℤ) < (q : ℝ) + 2 := by
have h := (div_lt_iff₀ hd).mp hq'
change (x : ℝ) < ((q : ℝ) + 1) * deltaL hL at h
nlinarith [hj'.1]
constructor
· exact_mod_cast hl
· change (j : ℤ) ≤ q + 1
have : (j : ℤ) < q + 2 := by exact_mod_cast hu
omega) (by
intro i hi j hj hij
exact Subtype.ext hij)
have hcard : (Icc (q - 1) (q + 1)).card = 3 := by
rw [Int.card_Icc]
omega
exact hc.trans (by rw [hcard])
Closed-support counting gives a uniform bound on the signed perturbation, also at boundaries. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
|perturbation hL lam x| ≤ 3
Proof (Lean source)
-- @node: perturbation_abs_le_three
lemma perturbation_abs_le_three (lam : SignVector hL) (x : Covariate) :
|perturbation hL lam x| ≤ 3 := by
classical
have hg := envelope_range hL x
have hs : |∑ j : activeSigns hL, (2 * bit (lam j) - 1) * frame hL j x| ≤ 3 := by
calc
_ ≤ ∑ j : activeSigns hL, |(2 * bit (lam j) - 1) * frame hL j x| :=
Finset.abs_sum_le_sum_abs _ _
_ ≤ ∑ j : activeSigns hL,
if |(x : ℝ) - (j : ℤ) * deltaL hL| ≤ deltaL hL then (1 : ℝ) else 0 := by
apply Finset.sum_le_sum
intro j hj
have hb : |2 * bit (lam j) - 1| = 1 := by
cases h : lam j <;> norm_num [bit, h]
rw [abs_mul, hb, one_mul]
exact frame_abs_le hL j x
_ = ((Finset.univ.filter (fun j : activeSigns hL =>
|(x : ℝ) - (j : ℤ) * deltaL hL| ≤ deltaL hL)).card : ℝ) := by
simp [← Finset.sum_filter]
_ ≤ 3 := by exact_mod_cast frame_support_card_le hL hhL x
unfold perturbation
rw [abs_mul, abs_of_nonneg hg.1]
calc
_ ≤ envelope hL x * 3 := mul_le_mul_of_nonneg_left hs hg.1
_ ≤ 3 := by nlinarith [hg.2]
Every legal macro radius gives valid binary probabilities for every sign vector. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
Proof (Lean source)
-- @node: alt_parameters_range
lemma alt_parameters_range (lam : SignVector hL) :
∀ x, altE hL lam x ∈ Icc 0 1 ∧ altMu0 hL lam x ∈ Icc 0 1 ∧
altMu1 hL lam x ∈ Icc 0 1 := by
intro x
have hq := bump_range hL x
have hs := abs_le.mp (perturbation_abs_le_three hL hhL lam x)
have ht : 0 ≤ separation hL ∧ separation hL ≤ 1 / 16384 := by
unfold separation kappa
norm_num
constructor <;> nlinarith [hhL.1, hhL.2]
have hroot := Real.sqrt_nonneg (separation hL)
have hsq := Real.sq_sqrt ht.1
have hsmall : sqrt (separation hL) ≤ 1 / 128 := by nlinarith
have hprodlo := mul_le_mul_of_nonneg_left hs.1 hroot
have hprodhi := mul_le_mul_of_nonneg_left hs.2 hroot
have hbumplo := mul_nonneg ht.1 hq.1
have hbumphi := mul_le_mul_of_nonneg_left hq.2 ht.1
dsimp [altE, altMu0, altMu1]
refine ⟨⟨?_, ?_⟩, ⟨?_, ?_⟩, ⟨?_, ?_⟩⟩ <;> nlinarith
The small public amplitude keeps every sign alternative inside the required overlap interval, uniformly over covariates and signs. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
Proof (Lean source)
-- @node: altE_overlap
lemma altE_overlap (lam : SignVector hL) (x : Covariate) :
1 / 4 ≤ altE hL lam x ∧ altE hL lam x ≤ 3 / 4 := by
have hs := abs_le.mp (perturbation_abs_le_three hL hhL lam x)
have ht : 0 ≤ separation hL ∧ separation hL ≤ 1 / 16384 := by
unfold separation kappa
norm_num
constructor <;> nlinarith [hhL.1, hhL.2]
have hroot := Real.sqrt_nonneg (separation hL)
have hsq := Real.sq_sqrt ht.1
have hsmall : sqrt (separation hL) ≤ 1 / 128 := by nlinarith
have hlo := mul_le_mul_of_nonneg_left hs.1 hroot
have hhi := mul_le_mul_of_nonneg_left hs.2 hroot
dsimp [altE]
constructor <;> nlinarith
def cosineFamily
The concrete causal alternative has the specified shared-sign propensity and arm means, independent binary potential outcomes given the covariate, and consistency.
Definition (Lean source)
parameters
def
given by
binaryCausalLaw (altE hL lam) (altMu0 hL lam) (altMu1 hL lam) (measurable_altE hL lam) (measurable_altMu0 hL lam) (measurable_altMu1 hL lam) (alt_parameters_range hL hhL lam)
The fair Bernoulli parameters are valid probabilities. The displayed conclusion follows.
Formal statement
hypotheses
x :
↑CausalSmith.Stat.PrivateCateRoughdesign.Covariate
x :
conclusion
conclusion 1
(1/2 : ℝ) ∈ Icc 0 1
conclusion 2
(1/2 : ℝ) ∈ Icc 0 1
conclusion 3
(1/2 : ℝ) ∈ Icc 0 1
Proof (Lean source)
-- @node: fair_parameters_range
lemma fair_parameters_range : ∀ x : Covariate,
(1/2 : ℝ) ∈ Icc 0 1 ∧ (1/2 : ℝ) ∈ Icc 0 1 ∧ (1/2 : ℝ) ∈ Icc 0 1 := by
intro x
norm_num
def fairNull
The full-support null uses a uniform covariate and three conditionally independent fair binary variables, followed by consistency.
Definition (Lean source)
def
given by
binaryCausalLaw (fun _ => 1/2) (fun _ => 1/2) (fun _ => 1/2) measurable_const measurable_const measurable_const fair_parameters_range
def signMixture
The fixed-sample alternative mixture averages the observed product laws uniformly over all sign vectors.
Definition (Lean source)
parameters
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1 / 4
n :
ℕ
given by
(ofReal ((2 : ℝ)^(-(activeSigns hL).card : ℤ))) • ∑ lam : SignVector hL, dataLaw n (cosineFamily hL hhL lam)
def oneRecordMixture
The one-record mixture averages the observed marginals uniformly over all sign vectors.
Definition (Lean source)
parameters
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1 / 4
def
given by
(ofReal ((2 : ℝ)^(-(activeSigns hL).card : ℤ))) • ∑ lam : SignVector hL, Pobs (cosineFamily hL hhL lam)
def U
The treatment Fourier mark maps the binary treatment to minus one or one.
Definition (Lean source)
def conditionalLikelihood
The conditional mark likelihood is the actual Bernoulli cell mass divided by the fair cell mass.
Definition (Lean source)
parameters
def
conditionalLikelihood hL lam x av yv :
ℝ
given by
4 * bernoulliMass (altE hL lam x) av * bernoulliMass (if av then altMu1 hL lam x else altMu0 hL lam x) yv
The effect bump equals one at the target covariate. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Proof (Lean source)
-- @node: bump_at_target
lemma bump_at_target (hh : 0 ≤ hL) : bump hL x0 = 1 := by
simp [bump, envelope, hh]
theorem cosineFamily_contrast
The shared nuisance terms cancel exactly in the alternative contrast. the theorem's stated inputs and assumptions, and the asserted conclusion follows.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: cosineFamily_contrast
lemma cosineFamily_contrast (lam : SignVector hL) (x : Covariate) :
tau (cosineFamily hL hhL lam) x = separation hL * bump hL x := by
change altMu1 hL lam x - altMu0 hL lam x = _
dsimp [altMu1, altMu0]
ring
theorem cosineFamily_target
Every sign alternative has the prescribed point target. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
theta (cosineFamily hL hhL lam) = separation hL
Proof (Lean source)
-- @node: cosineFamily_target
lemma cosineFamily_target (lam : SignVector hL) :
theta (cosineFamily hL hhL lam) = separation hL := by
rw [theta, cosineFamily_contrast, bump_at_target hL hhL.1.le, mul_one]
theorem fairNull_target
The fair null has zero contrast at the target. The displayed conclusion follows.
Formal statement
conclusion
theta fairNull = 0
Proof (Lean source)
-- @node: fairNull_target
lemma fairNull_target : theta fairNull = 0 := by
change (1 / 2 : ℝ) - 1 / 2 = 0
ring
theorem fairNull_exchangeability
The fair null's treatment is independent of its potential outcomes given the covariate. the asserted conclusion follows.
Formal statement
conclusion
Proof (Lean source)
-- @node: fairNull_exchangeability
lemma fairNull_exchangeability : Exchangeability fairNull :=
binaryCausalLaw_exchangeability _ _ _ measurable_const measurable_const measurable_const
fair_parameters_range
theorem cosineFamily_exchangeability
Each shared-sign law has conditionally independent treatment and potential outcomes. The displayed conclusion follows.
Formal statement
hypotheses
conclusion
Exchangeability (cosineFamily hL hhL lam)
Proof (Lean source)
-- @node: cosineFamily_exchangeability
lemma cosineFamily_exchangeability (lam : SignVector hL) :
Exchangeability (cosineFamily hL hhL lam) :=
binaryCausalLaw_exchangeability _ _ _
(measurable_altE hL lam) (measurable_altMu0 hL lam) (measurable_altMu1 hL lam)
(alt_parameters_range hL hhL lam)
theorem fairNull_completeModel
The fair null satisfies all conditions of the complete causal model. The displayed conclusion follows.
Formal statement
conclusion
Proof (Lean source)
-- @node: fairNull_completeModel
lemma fairNull_completeModel : CompleteModel fairNull := by
apply binaryCausalLaw_completeModel (fun _ => 1/2) (fun _ => 1/2) (fun _ => 1/2)
measurable_const measurable_const measurable_const fair_parameters_range fairNull_exchangeability
· intro x
norm_num
· intro x y
simp only [sub_self, abs_zero]
exact mul_nonneg (by norm_num [L]) (Real.rpow_nonneg (abs_nonneg _) _)
· intro x y
simp only [sub_self, abs_zero]
exact mul_nonneg (by norm_num [L]) (Real.rpow_nonneg (abs_nonneg _) _)
· intro x y
simp only [sub_self, abs_zero]
exact mul_nonneg (by norm_num [L]) (abs_nonneg _)
Expanding the four Bernoulli cells gives the Fourier likelihood relative to fair marks. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
conditionalLikelihood hL lam x av yv
= 1
Proof (Lean source)
-- @node: conditionalLikelihood_fourier
lemma conditionalLikelihood_fourier (lam : SignVector hL) (x : Covariate)
(av yv : Bool) :
conditionalLikelihood hL lam x av yv =
1 + (2*bit av-1)*sqrt (separation hL)*perturbation hL lam x +
(2*bit yv-1)*sqrt (separation hL)*(separation hL*bump hL x-1)*
perturbation hL lam x +
(2*bit av-1)*(2*bit yv-1)*separation hL*
(bump hL x-(perturbation hL lam x)^2) := by
have ht : 0 ≤ separation hL := by
unfold separation kappa
exact mul_nonneg (by norm_num) hhL.1.le
have hs := Real.sq_sqrt ht
cases av <;> cases yv <;>
simp only [conditionalLikelihood, bernoulliMass, bit, Bool.false_eq_true,
if_false, if_true, altE, altMu0, altMu1] <;>
nlinarith [hs]
Averaging the actual Fourier likelihood cancels both linear sign terms; its only remaining deviation from the fair likelihood is the squared-frame defect. The displayed conclusion follows.
Formal statement
hypotheses
conclusion
(∑ lam : SignVector hL, conditionalLikelihood hL lam x av yv)
= (card (SignVector hL) : ℝ) * (1 + (2 * bit av - 1) * (2 * bit yv - 1) * separation hL * (bump hL x - bump hL x * ∑ j : activeSigns hL, (frame hL j x)^2))
Proof (Lean source)
-- @node: conditionalLikelihood_sign_sum
lemma conditionalLikelihood_sign_sum (x : Covariate) (av yv : Bool) :
(∑ lam : SignVector hL, conditionalLikelihood hL lam x av yv) =
(card (SignVector hL) : ℝ) *
(1 + (2 * bit av - 1) * (2 * bit yv - 1) * separation hL *
(bump hL x - bump hL x * ∑ j : activeSigns hL, (frame hL j x)^2)) := by
classical
simp_rw [conditionalLikelihood_fourier hL hhL]
simp only [Finset.sum_add_distrib, Finset.sum_sub_distrib, ← Finset.mul_sum,
perturbation_sign_sum, perturbation_sign_square_sum, mul_zero,
Finset.sum_const, Finset.card_univ, nsmul_eq_mul, add_zero]
ring
The uniform sign weight times the number of binary sign vectors is one. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
hL :
ℝ
conclusion
= 1
Proof (Lean source)
-- @node: sign_weight_normalization
lemma sign_weight_normalization (hL : ℝ) :
ofReal ((2 : ℝ)^(-((activeSigns hL).card : ℤ))) *
(card (SignVector hL) : ℝ≥0∞) = 1 := by
classical
have hcard : card (SignVector hL) = 2 ^ (activeSigns hL).card := by
simp [SignVector, Fintype.card_fun]
rw [hcard, zpow_neg, zpow_natCast, ENNReal.ofReal_inv_of_pos (by positivity),
ENNReal.ofReal_pow (by norm_num), ENNReal.ofReal_ofNat]
norm_cast
exact ENNReal.inv_mul_cancel (by positivity) (by finiteness)
The real-valued uniform sign weight normalizes the finite number of vectors. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
hL :
ℝ
conclusion
Proof (Lean source)
-- @node: sign_weight_real_normalization
lemma sign_weight_real_normalization (hL : ℝ) :
(2 : ℝ)^(-((activeSigns hL).card : ℤ)) *
(card (SignVector hL) : ℝ) = 1 := by
classical
have h := congrArg toReal (sign_weight_normalization hL)
simpa only [ENNReal.toReal_mul, ENNReal.toReal_ofReal (le_of_lt (zpow_pos (by norm_num : (0 : ℝ) < 2) _)),
ENNReal.toReal_natCast, ENNReal.toReal_one] using h
Integrating any real function against the sign law is its finite uniform average. The displayed conclusion follows.
Formal statement
hypotheses
hL :
ℝ
F :
SignVector hL → ℝ
conclusion
(∫ lam, F lam ∂signLaw hL) = (2 : ℝ)^(-((activeSigns hL).card : ℤ)) * ∑ lam, F lam
Proof (Lean source)
-- @node: integral_signLaw
lemma integral_signLaw (F : SignVector hL → ℝ) :
(∫ lam, F lam ∂signLaw hL) =
(2 : ℝ)^(-((activeSigns hL).card : ℤ)) * ∑ lam, F lam := by
classical
rw [signLaw, integral_smul_measure, integral_finsetSum_measure]
· simp only [integral_dirac, ENNReal.toReal_ofReal (le_of_lt (zpow_pos (by norm_num : (0 : ℝ) < 2) _)), smul_eq_mul]
· intro lam _
exact integrable_dirac (by finiteness)
The nuisance perturbation has expectation zero under the actual sign probability law. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: perturbation_sign_mean
lemma perturbation_sign_mean (x : Covariate) :
(∫ lam, perturbation hL lam x ∂signLaw hL) = 0 := by
rw [integral_signLaw, perturbation_sign_sum, mul_zero]
The actual sign-law second moment leaves only the squared-frame geometric factor. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: perturbation_sign_second_moment
lemma perturbation_sign_second_moment (x : Covariate) :
(∫ lam, (perturbation hL lam x)^2 ∂signLaw hL) =
bump hL x * ∑ j : activeSigns hL, (frame hL j x)^2 := by
classical
rw [integral_signLaw, perturbation_sign_square_sum, ← mul_assoc,
sign_weight_real_normalization, one_mul]
Under the actual sign law, the averaged likelihood has exactly the squared-frame defect. The displayed conclusion follows.
Formal statement
hypotheses
conclusion
(∫ lam, conditionalLikelihood hL lam x av yv ∂signLaw hL)
= 1
+ (2 * bit av - 1) * (2 * bit yv - 1) * separation hL * (bump hL x - bump hL x * ∑ j : activeSigns hL, (frame hL j x)^2)
Proof (Lean source)
-- @node: conditionalLikelihood_sign_mean
lemma conditionalLikelihood_sign_mean (x : Covariate) (av yv : Bool) :
(∫ lam, conditionalLikelihood hL lam x av yv ∂signLaw hL) =
1 + (2 * bit av - 1) * (2 * bit yv - 1) * separation hL *
(bump hL x - bump hL x * ∑ j : activeSigns hL, (frame hL j x)^2) := by
classical
rw [integral_signLaw, conditionalLikelihood_sign_sum hL hhL,
← mul_assoc, sign_weight_real_normalization, one_mul]
theorem perturbation_sign_second_moment_exact Lemma perturbation_sign_second_moment_exact in the paper ↗
The exact localized frame partition makes the sign second moment equal to the effect bump. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
(∫ lam, (perturbation hL lam x)^2 ∂signLaw hL) = bump hL x
Proof (Lean source)
-- @node: perturbation_sign_second_moment_exact
lemma perturbation_sign_second_moment_exact (x : Covariate) :
(∫ lam, (perturbation hL lam x)^2 ∂signLaw hL) = bump hL x := by
rw [perturbation_sign_second_moment, weighted_frame_square_partition hL hhL]
theorem conditionalLikelihood_sign_mean_one Lemma conditionalLikelihood_sign_mean_one in the paper ↗
The actual conditional likelihood has sign average one at every covariate and mark. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
(∫ lam, conditionalLikelihood hL lam x av yv ∂signLaw hL) = 1
Proof (Lean source)
-- @node: conditionalLikelihood_sign_mean_one
lemma conditionalLikelihood_sign_mean_one (x : Covariate) (av yv : Bool) :
(∫ lam, conditionalLikelihood hL lam x av yv ∂signLaw hL) = 1 := by
rw [conditionalLikelihood_sign_mean hL hhL, weighted_frame_square_partition hL hhL]
ring
theorem conditionalLikelihood_sign_sum_exact Lemma conditionalLikelihood_sign_sum_exact in the paper ↗
Summing the four-cell likelihood over all signs gives exactly the number of sign vectors. The displayed conclusion follows.
Formal statement
hypotheses
conclusion
(∑ lam : SignVector hL, conditionalLikelihood hL lam x av yv)
= (card (SignVector hL) : ℝ)
Proof (Lean source)
-- @node: conditionalLikelihood_sign_sum_exact
lemma conditionalLikelihood_sign_sum_exact (x : Covariate) (av yv : Bool) :
(∑ lam : SignVector hL, conditionalLikelihood hL lam x av yv) =
(card (SignVector hL) : ℝ) := by
rw [conditionalLikelihood_sign_sum hL hhL, weighted_frame_square_partition hL hhL]
ring
The uniform average of every actual conditional Bernoulli mark mass is the fair mass. The displayed conclusion follows.
Formal statement
hypotheses
conclusion
ofReal ((2 : ℝ)^(-((activeSigns hL).card : ℤ))) * ∑ lam : SignVector hL, ofReal (bernoulliMass (altE hL lam x) av * bernoulliMass (if av then altMu1 hL lam x else altMu0 hL lam x) yv)
= ofReal (1 / 4 : ℝ)
Proof (Lean source)
-- @node: conditional_mark_mass_sign_average
lemma conditional_mark_mass_sign_average (x : Covariate) (av yv : Bool) :
ofReal ((2 : ℝ)^(-((activeSigns hL).card : ℤ))) *
∑ lam : SignVector hL,
ofReal (bernoulliMass (altE hL lam x) av *
bernoulliMass (if av then altMu1 hL lam x else altMu0 hL lam x) yv) =
ofReal (1 / 4 : ℝ) := by
classical
let m : SignVector hL → ℝ := fun lam => bernoulliMass (altE hL lam x) av *
bernoulliMass (if av then altMu1 hL lam x else altMu0 hL lam x) yv
have hm (lam : SignVector hL) : 0 ≤ m lam := by
apply mul_nonneg (bernoulliMass_nonneg (alt_parameters_range hL hhL lam x).1 av)
cases av
· exact bernoulliMass_nonneg (alt_parameters_range hL hhL lam x).2.1 yv
· exact bernoulliMass_nonneg (alt_parameters_range hL hhL lam x).2.2 yv
have hsum : (2 : ℝ)^(-((activeSigns hL).card : ℤ)) * ∑ lam, m lam = 1 / 4 := by
have hs := conditionalLikelihood_sign_sum_exact hL hhL x av yv
simp only [conditionalLikelihood, mul_assoc] at hs
change (∑ lam, 4 * (m lam)) = (card (SignVector hL) : ℝ) at hs
rw [← Finset.mul_sum] at hs
have hn := sign_weight_real_normalization hL
calc
_ = (2 : ℝ)^(-((activeSigns hL).card : ℤ)) *
((card (SignVector hL) : ℝ) / 4) := by rw [← hs]; ring
_ = 1 / 4 := by rw [← mul_div_assoc, hn]
change ofReal ((2 : ℝ)^(-((activeSigns hL).card : ℤ))) *
∑ lam, ofReal (m lam) = _
rw [← ENNReal.ofReal_sum_of_nonneg (fun lam _ => hm lam),
← ENNReal.ofReal_mul (le_of_lt (zpow_pos (by norm_num : (0 : ℝ) < 2) _)), hsum]
Averaging the observed one-record laws gives the fair null measure exactly. the asserted conclusion follows.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1 / 4
conclusion
Proof (Lean source)
-- @node: oneRecordMixture_eq_fairNull
lemma oneRecordMixture_eq_fairNull : oneRecordMixture hL hhL = Pobs fairNull := by
classical
ext s hs
let w := ofReal ((2 : ℝ)^(-((activeSigns hL).card : ℤ)))
let f := fun (lam : SignVector hL) (x : Covariate) =>
∑ z : Bool × Bool,
ofReal (bernoulliMass (altE hL lam x) z.1 *
bernoulliMass (if z.1 then altMu1 hL lam x else altMu0 hL lam x) z.2) *
s.indicator (fun _ => (1 : ℝ≥0∞)) (x, z.1, z.2)
have hfm (lam : SignVector hL) : Measurable (f lam) := by
apply Finset.measurable_fun_sum
intro z hz
apply mul
· cases z.1 <;> cases z.2 <;> unfold bernoulliMass <;> simp only [if_true,
Bool.false_eq_true, if_false] <;> fun_prop
· exact (measurable_const.indicator hs).comp (by fun_prop)
have hobs (lam : SignVector hL) : Pobs (cosineFamily hL hhL lam) s = ∫⁻ x, f lam x :=
binaryCausalLaw_observed_apply _ _ _ (measurable_altE hL lam)
(measurable_altMu0 hL lam) (measurable_altMu1 hL lam)
(alt_parameters_range hL hhL lam) s hs
have hnull : Pobs fairNull s = ∫⁻ x : Covariate, ∑ z : Bool × Bool,
ofReal (1 / 4 : ℝ) * s.indicator (fun _ => (1 : ℝ≥0∞)) (x, z.1, z.2) := by
rw [fairNull, binaryCausalLaw_observed_apply _ _ _ measurable_const measurable_const
measurable_const fair_parameters_range s hs]
apply lintegral_congr
intro x
apply Finset.sum_congr rfl
intro z hz
cases z.1 <;> cases z.2 <;> norm_num [bernoulliMass]
simp only [oneRecordMixture, Measure.smul_apply, Measure.finsetSum_apply, smul_eq_mul]
simp_rw [hobs]
change w * (∑ lam, ∫⁻ x, f lam x) = _
rw [← lintegral_finsetSum _ (fun lam _ => hfm lam),
← lintegral_const_mul' w _ ENNReal.ofReal_ne_top, hnull]
apply lintegral_congr
intro x
dsimp [f]
rw [Finset.sum_comm, Finset.mul_sum]
apply Finset.sum_congr rfl
intro z hz
rw [← Finset.sum_mul, ← mul_assoc]
exact congrArg (fun a => a * s.indicator (fun _ => (1 : ℝ≥0∞)) (x, z.1, z.2))
(conditional_mark_mass_sign_average hL hhL x z.1 z.2)
theorem cosineFamily_observed_domination
Each observed sign alternative is dominated by eight times the fair observed null. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: cosineFamily_observed_domination
lemma cosineFamily_observed_domination (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1/4)
(lam : SignVector hL) : Pobs (cosineFamily hL hhL lam) ≤ (8 : ℝ≥0∞) • Pobs fairNull := by
have hm : Measurable observe := by unfold observe A Y; fun_prop
have h := Measure.map_mono (binaryLaw_fair_domination
(altE hL lam) (altMu0 hL lam) (altMu1 hL lam)
(measurable_altE hL lam) (measurable_altMu0 hL lam) (measurable_altMu1 hL lam)
(alt_parameters_range hL hhL lam)) hm
simpa only [Pobs, cosineFamily, fairNull, binaryCausalLaw, Measure.map_smul] using h
theorem signMixture_domination
The finite sign mixture is dominated by eight to the sample-size power times the null dataset law. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1/4
n :
ℕ
conclusion
Proof (Lean source)
-- @node: signMixture_domination
lemma signMixture_domination (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1/4) (n : ℕ) :
signMixture hL hhL n ≤ (8 : ℝ≥0∞)^n • dataLaw n fairNull := by
classical
have hm : Measurable observe := by unfold observe A Y; fun_prop
letI : IsProbabilityMeasure (Pobs fairNull) := Measure.isProbabilityMeasure_map hm.aemeasurable
letI : IsFiniteMeasure ((8 : ℝ≥0∞) • Pobs fairNull) := ⟨by simp⟩
have hd (lam : SignVector hL) : dataLaw n (cosineFamily hL hhL lam) ≤
(8 : ℝ≥0∞)^n • dataLaw n fairNull := by
exact finite_product_measure_domination n _ _ 8 (cosineFamily_observed_domination hL hhL lam)
apply Measure.le_iff.mpr
intro s hs
simp only [signMixture, Measure.smul_apply, Measure.finsetSum_apply, smul_eq_mul]
calc
_ ≤ ofReal ((2 : ℝ)^(-((activeSigns hL).card : ℤ))) *
∑ lam : SignVector hL, ((8 : ℝ≥0∞)^n • dataLaw n fairNull) s :=
mul_le_mul' le_rfl (Finset.sum_le_sum (fun lam _ => hd lam s))
_ = _ := by
simp only [Finset.sum_const, Finset.card_univ, nsmul_eq_mul, ← mul_assoc]
rw [sign_weight_normalization, one_mul, Measure.smul_apply, smul_eq_mul]
The uniform finite sign law is a probability measure. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: signLaw_probability
lemma signLaw_probability (hL : ℝ) : IsProbabilityMeasure (signLaw hL) := by
classical
rw [isProbabilityMeasure_iff, signLaw]
simp only [Measure.smul_apply, Measure.finsetSum_apply, Measure.dirac_apply_of_mem
(Set.mem_univ _), smul_eq_mul, Finset.sum_const, Finset.card_univ, nsmul_eq_mul, mul_one]
exact sign_weight_normalization hL
The sign mixture of observed product laws is a probability measure. the theorem's stated inputs and assumptions, and the asserted conclusion follows.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1/4
n :
ℕ
conclusion
IsProbabilityMeasure (signMixture hL hhL n)
Proof (Lean source)
-- @node: signMixture_probability
lemma signMixture_probability (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1/4) (n : ℕ) :
IsProbabilityMeasure (signMixture hL hhL n) := by
classical
have hm : Measurable observe := by unfold observe A Y; fun_prop
letI (P : CausalLaw) : IsProbabilityMeasure (Pobs P) :=
Measure.isProbabilityMeasure_map hm.aemeasurable
letI (P : CausalLaw) : IsProbabilityMeasure (dataLaw n P) := by unfold dataLaw; infer_instance
rw [isProbabilityMeasure_iff, signMixture]
simp only [Measure.smul_apply, Measure.finsetSum_apply, measure_univ, smul_eq_mul,
Finset.sum_const, Finset.card_univ, nsmul_eq_mul, mul_one]
exact sign_weight_normalization hL
theorem signMixture_absolutelyContinuous
The dataset sign mixture is absolutely continuous with respect to the fair null experiment. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1/4
n :
ℕ
conclusion
Proof (Lean source)
-- @node: signMixture_absolutelyContinuous
lemma signMixture_absolutelyContinuous (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1/4) (n : ℕ) :
signMixture hL hhL n ≪ dataLaw n fairNull :=
Measure.absolutelyContinuous_of_le_smul (signMixture_domination hL hhL n)
theorem signMixture_square_density_integrable
The squared density deviation of the sign mixture is integrable under the fair null experiment. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1/4
n :
ℕ
conclusion
Integrable (fun z => (((signMixture hL hhL n).rnDeriv (dataLaw n fairNull) z).toReal - 1)^2) (dataLaw n fairNull)
Proof (Lean source)
-- @node: signMixture_square_density_integrable
lemma signMixture_square_density_integrable (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1/4) (n : ℕ) :
Integrable (fun z => (((signMixture hL hhL n).rnDeriv
(dataLaw n fairNull) z).toReal - 1)^2) (dataLaw n fairNull) := by
have hm : Measurable observe := by unfold observe A Y; fun_prop
letI : IsProbabilityMeasure (Pobs fairNull) := Measure.isProbabilityMeasure_map hm.aemeasurable
letI : IsProbabilityMeasure (dataLaw n fairNull) := by unfold dataLaw; infer_instance
exact square_rnDeriv_integrable_of_domination _ _ ((8 : ℝ≥0∞)^n)
(by positivity) (by finiteness) (signMixture_domination hL hhL n)
def directMu1
The direct treated mean is one half plus the localized effect bump.
Definition (Lean source)
The direct treated mean is measurable. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: measurable_directMu1
@[fun_prop] lemma measurable_directMu1 : Measurable (directMu1 hL) := by
unfold directMu1
fun_prop
The direct alternative has valid binary parameters at every legal macro radius. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: direct_parameters_range
lemma direct_parameters_range : ∀ x : Covariate,
(1/2 : ℝ) ∈ Icc 0 1 ∧ (1/2 : ℝ) ∈ Icc 0 1 ∧ directMu1 hL x ∈ Icc 0 1 := by
intro x
have hq := bump_range hL x
have ht : 0 ≤ separation hL ∧ separation hL ≤ 1 / 2 := by
unfold separation kappa
norm_num at *
constructor <;> nlinarith [hhL.1, hhL.2]
have hp := mul_nonneg ht.1 hq.1
have hu := mul_le_mul_of_nonneg_left hq.2 ht.1
dsimp [directMu1]
constructor
· norm_num
constructor
· norm_num
· constructor <;> nlinarith
def directAlternative
The direct localized causal law has fair treatment and control potential outcome and the specified perturbed treated potential outcome, followed by consistency.
Definition (Lean source)
parameters
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1 / 4
def
given by
binaryCausalLaw (fun _ => 1/2) (fun _ => 1/2) (directMu1 hL) measurable_const measurable_const (measurable_directMu1 hL) (direct_parameters_range hL hhL)
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.CosineRegularity 14 declarations Global Lipschitz and Holder certificates for the shared-sign causal alternatives.
Global Lipschitz and Holder certificates for the shared-sign causal alternatives.
The micro radius is no larger than the macro radius in the published range. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: deltaL_le_macro
lemma deltaL_le_macro (hL : ℝ) (hh : 0 < hL ∧ hL ≤ 1/4) : deltaL hL ≤ hL := by
have h1 : hL ≤ 1 := by linarith [hh.2]
simpa only [deltaL, pow_one] using pow_le_pow_of_le_one hh.1.le h1 (by norm_num : 1 ≤ 5)
Only frames supported at one of the two points contribute to the signed-sum difference. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: signed_frame_difference_bound
lemma signed_frame_difference_bound (hL : ℝ) (hh : 0 < hL ∧ hL ≤ 1/4)
(lam : SignVector hL) (x y : Covariate) :
|(∑ j : activeSigns hL, (2*bit (lam j)-1)*frame hL j x) -
(∑ j : activeSigns hL, (2*bit (lam j)-1)*frame hL j y)| ≤
(3*pi/deltaL hL)*|(x : ℝ)-y| := by
classical
have hd : 0 < deltaL hL := pow_pos hh.1 5
let sx := Finset.univ.filter (fun j : activeSigns hL =>
|(x : ℝ)-(j : ℤ)*deltaL hL| ≤ deltaL hL)
let sy := Finset.univ.filter (fun j : activeSigns hL =>
|(y : ℝ)-(j : ℤ)*deltaL hL| ≤ deltaL hL)
let v (j : activeSigns hL) := (2*bit (lam j)-1)*(frame hL j x-frame hL j y)
have he : (∑ j : activeSigns hL, v j) = ∑ j ∈ sx ∪ sy, v j := by
symm
apply Finset.sum_subset (Finset.subset_univ _)
intro j hj hnot
have hn : j ∉ sx ∧ j ∉ sy := by simpa using hnot
have hx : ¬ |(x : ℝ)-(j : ℤ)*deltaL hL| ≤ deltaL hL := by
simpa [sx] using hn.1
have hy : ¬ |(y : ℝ)-(j : ℤ)*deltaL hL| ≤ deltaL hL := by
simpa [sy] using hn.2
simp [v, frame, hx, hy]
have hcard : (sx ∪ sy).card ≤ 6 := by
have hx := frame_support_card_le hL hh x
have hy := frame_support_card_le hL hh y
change sx.card ≤ 3 at hx
change sy.card ≤ 3 at hy
exact (Finset.card_union_le sx sy).trans (by omega)
have hv (j : activeSigns hL) : |v j| ≤
(pi/(2*deltaL hL))*|(x : ℝ)-y| := by
have hb : |2*bit (lam j)-1| = 1 := by cases h : lam j <;> norm_num [bit, h]
simpa only [v, abs_mul, hb, one_mul] using frame_difference_bound hL hh.1 j x y
rw [← Finset.sum_sub_distrib]
simp only [← mul_sub]
change |∑ j : activeSigns hL, v j| ≤ _
rw [he]
calc
_ ≤ ∑ j ∈ sx ∪ sy, |v j| := Finset.abs_sum_le_sum_abs _ _
_ ≤ ∑ j ∈ sx ∪ sy, (pi/(2*deltaL hL))*|(x : ℝ)-y| :=
Finset.sum_le_sum (fun j hj => hv j)
_ = ((sx ∪ sy).card : ℝ)*((pi/(2*deltaL hL))*|(x : ℝ)-y|) := by simp
_ ≤ 6*((pi/(2*deltaL hL))*|(x : ℝ)-y|) := by
apply mul_le_mul_of_nonneg_right (by exact_mod_cast hcard) (by positivity)
_ = _ := by ring
The same support counting bounds the unlocalized signed sum by three. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: signed_frame_abs_le_three
lemma signed_frame_abs_le_three (hL : ℝ) (hh : 0 < hL ∧ hL ≤ 1/4)
(lam : SignVector hL) (x : Covariate) :
|∑ j : activeSigns hL, (2*bit (lam j)-1)*frame hL j x| ≤ 3 := by
classical
calc
_ ≤ ∑ j : activeSigns hL, |(2*bit (lam j)-1)*frame hL j x| :=
Finset.abs_sum_le_sum_abs _ _
_ ≤ ∑ j : activeSigns hL,
if |(x : ℝ)-(j : ℤ)*deltaL hL| ≤ deltaL hL then (1 : ℝ) else 0 := by
apply Finset.sum_le_sum
intro j hj
have hb : |2*bit (lam j)-1| = 1 := by cases h : lam j <;> norm_num [bit, h]
simpa only [abs_mul, hb, one_mul] using frame_abs_le hL j x
_ = ((Finset.univ.filter (fun j : activeSigns hL =>
|(x : ℝ)-(j : ℤ)*deltaL hL| ≤ deltaL hL)).card : ℝ) := by
simp [← Finset.sum_filter]
_ ≤ 3 := by exact_mod_cast frame_support_card_le hL hh x
Multiplication by the macro envelope preserves a uniform micro-scale Lipschitz modulus. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: perturbation_difference_bound
lemma perturbation_difference_bound (hL : ℝ) (hh : 0 < hL ∧ hL ≤ 1/4)
(lam : SignVector hL) (x y : Covariate) :
|perturbation hL lam x-perturbation hL lam y| ≤
(24/deltaL hL)*|(x : ℝ)-y| := by
let s (z : Covariate) := ∑ j : activeSigns hL, (2*bit (lam j)-1)*frame hL j z
have hs := signed_frame_difference_bound hL hh lam x y
have hsy := signed_frame_abs_le_three hL hh lam y
have hg := envelope_difference_bound hL hh.1 x y
have hx := envelope_range hL x
have hd : 0 < deltaL hL := pow_pos hh.1 5
have hinv : pi/hL ≤ pi/deltaL hL :=
div_le_div_of_nonneg_left Real.pi_pos.le hd (deltaL_le_macro hL hh)
have he : perturbation hL lam x-perturbation hL lam y =
envelope hL x*(s x-s y)+(envelope hL x-envelope hL y)*s y := by
dsimp [perturbation, s]; ring
rw [he]
calc
_ ≤ |envelope hL x*(s x-s y)|+|(envelope hL x-envelope hL y)*s y| := abs_add_le _ _
_ = envelope hL x*|s x-s y|+|envelope hL x-envelope hL y| *|s y| := by
rw [abs_mul, abs_mul, abs_of_nonneg hx.1]
_ ≤ 1*((3*pi/deltaL hL)*|(x : ℝ)-y|)+
((pi/hL)*|(x : ℝ)-y|)*3 :=
add_le_add (mul_le_mul hx.2 hs (abs_nonneg _) (by norm_num))
(mul_le_mul hg hsy (abs_nonneg _) (mul_nonneg (div_nonneg Real.pi_pos.le hh.1.le) (abs_nonneg _)))
_ ≤ (6*pi/deltaL hL)*|(x : ℝ)-y| := by
have := mul_le_mul_of_nonneg_right hinv (abs_nonneg ((x : ℝ)-y))
have heq : (6*pi/deltaL hL)*|(x : ℝ)-y| =
3*((pi/deltaL hL)*|(x : ℝ)-y|)+
3*((pi/deltaL hL)*|(x : ℝ)-y|) := by ring
rw [heq]
have heq2 : 1*((3*pi/deltaL hL)*|(x : ℝ)-y|) =
3*((pi/deltaL hL)*|(x : ℝ)-y|) := by ring
rw [heq2]
linarith
_ ≤ _ := by
apply mul_le_mul_of_nonneg_right _ (abs_nonneg _)
apply div_le_div_of_nonneg_right _ hd.le
linarith [Real.pi_lt_four]
The effect bump satisfies the same coarse micro-scale modulus as the perturbation. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: bump_micro_difference_bound
lemma bump_micro_difference_bound (hL : ℝ) (hh : 0 < hL ∧ hL ≤ 1/4)
(x y : Covariate) :
|bump hL x-bump hL y| ≤ (24/deltaL hL)*|(x : ℝ)-y| := by
have hd : 0 < deltaL hL := pow_pos hh.1 5
have hi := div_le_div_of_nonneg_left (by positivity : 0 ≤ 2*pi) hd (deltaL_le_macro hL hh)
apply (bump_difference_bound hL hh.1 x y).trans
apply mul_le_mul_of_nonneg_right _ (abs_nonneg _)
apply hi.trans
apply div_le_div_of_nonneg_right _ hd.le
linarith [Real.pi_lt_four]
The micro radius to the one-tenth power is exactly the square root of the macro radius. The result uses the stated assumptions and establishes the displayed conclusion.
Proof (Lean source)
-- @node: deltaL_tenth_power
lemma deltaL_tenth_power (hL : ℝ) (hh : 0 < hL) :
(deltaL hL)^(1/10 : ℝ) = sqrt hL := by
rw [deltaL, ← Real.rpow_natCast_mul hh.le, Real.sqrt_eq_rpow]
norm_num
The square-root amplitude cancels the micro-scale Holder loss with a small constant. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
Proof (Lean source)
-- @node: separation_sqrt_scale
lemma separation_sqrt_scale (hL : ℝ) (hh : 0 < hL) :
sqrt (separation hL) / (deltaL hL)^(1/10 : ℝ) = 1/64 := by
have hk : kappa = (1/64 : ℝ)^2 := by norm_num [kappa]
rw [deltaL_tenth_power hL hh, separation, hk, Real.sqrt_mul (sq_nonneg _),
Real.sqrt_sq (by norm_num : (0 : ℝ) ≤ 1/64)]
exact mul_div_cancel_right₀ _ (Real.sqrt_pos.mpr hh).ne'
The small effect amplitude is bounded by the square-root amplitude. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
hh :
0 < hL ∧ hL ≤ 1/4
Proof (Lean source)
-- @node: separation_le_sqrt
lemma separation_le_sqrt (hL : ℝ) (hh : 0 < hL ∧ hL ≤ 1/4) :
0 ≤ separation hL ∧ separation hL ≤ sqrt (separation hL) := by
have ht : 0 ≤ separation hL ∧ separation hL ≤ 1 := by
norm_num [separation, kappa]
constructor <;> linarith [hh.1, hh.2]
refine ⟨ht.1, ?_⟩
nlinarith [Real.sq_sqrt ht.1, Real.sqrt_nonneg (separation hL),
mul_nonneg ht.1 (sub_nonneg.mpr ht.2)]
The perturbation's two-scale bounds imply the one-tenth Holder modulus. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
|perturbation hL lam x-perturbation hL lam y|
≤ 24*|(x : ℝ)-y|^(1/10 : ℝ)/(deltaL hL)^(1/10 : ℝ)
Proof (Lean source)
-- @node: perturbation_holder_bound
lemma perturbation_holder_bound (hL : ℝ) (hh : 0 < hL ∧ hL ≤ 1/4)
(lam : SignVector hL) (x y : Covariate) :
|perturbation hL lam x-perturbation hL lam y| ≤
24*|(x : ℝ)-y|^(1/10 : ℝ)/(deltaL hL)^(1/10 : ℝ) := by
apply holder_modulus_of_two_bounds _ _ _ _ (pow_pos hh.1 5) (abs_nonneg _) (by norm_num)
· have hx := perturbation_abs_le_three hL hh lam x
have hy := perturbation_abs_le_three hL hh lam y
linarith [abs_sub (perturbation hL lam x) (perturbation hL lam y)]
· convert perturbation_difference_bound hL hh lam x y using 1 <;> dsimp [deltaL] <;> ring
The bounded effect bump has the analogous one-tenth Holder modulus. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
Proof (Lean source)
-- @node: bump_holder_bound
lemma bump_holder_bound (hL : ℝ) (hh : 0 < hL ∧ hL ≤ 1/4) (x y : Covariate) :
|bump hL x-bump hL y| ≤
24*|(x : ℝ)-y|^(1/10 : ℝ)/(deltaL hL)^(1/10 : ℝ) := by
apply holder_modulus_of_two_bounds _ _ _ _ (pow_pos hh.1 5) (abs_nonneg _) (by norm_num)
· have hx := bump_range hL x
have hy := bump_range hL y
rw [abs_le]
constructor <;> linarith [hx.1, hx.2, hy.1, hy.2]
· convert bump_micro_difference_bound hL hh x y using 1 <;> dsimp [deltaL] <;> ring
theorem cosineFamily_propensity_holder
The actual propensity satisfies the model's radius-three Holder requirement. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
PropensityHolder (cosineFamily hL hh lam)
Proof (Lean source)
-- @node: cosineFamily_propensity_holder
lemma cosineFamily_propensity_holder (hL : ℝ) (hh : 0 < hL ∧ hL ≤ 1/4)
(lam : SignVector hL) : PropensityHolder (cosineFamily hL hh lam) := by
intro x y
change |altE hL lam x-altE hL lam y| ≤ _
have he : altE hL lam x-altE hL lam y =
sqrt (separation hL)*(perturbation hL lam x-perturbation hL lam y)/2 := by
dsimp [altE]; ring
rw [he, abs_div, abs_mul, abs_of_nonneg (Real.sqrt_nonneg _)]
rw [abs_of_pos (by norm_num : (0 : ℝ) < 2)]
have hb := mul_le_mul_of_nonneg_left (perturbation_holder_bound hL hh lam x y)
(Real.sqrt_nonneg (separation hL))
have hs := separation_sqrt_scale hL hh.1
have hid : sqrt (separation hL)*(24*|(x : ℝ)-y|^(1/10 : ℝ)/
(deltaL hL)^(1/10 : ℝ)) = (3/8)*|(x : ℝ)-y|^(1/10 : ℝ) := by
calc
_ = 24*|(x : ℝ)-y|^(1/10 : ℝ)*
(sqrt (separation hL)/(deltaL hL)^(1/10 : ℝ)) := by ring
_ = _ := by rw [hs]; ring
rw [hid] at hb
dsimp [L]
nlinarith [Real.rpow_nonneg (abs_nonneg ((x : ℝ)-y)) (1/10 : ℝ)]
theorem cosineFamily_control_holder
The actual control mean satisfies the model's radius-three Holder requirement. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
ControlHolder (cosineFamily hL hh lam)
Proof (Lean source)
-- @node: cosineFamily_control_holder
lemma cosineFamily_control_holder (hL : ℝ) (hh : 0 < hL ∧ hL ≤ 1/4)
(lam : SignVector hL) : ControlHolder (cosineFamily hL hh lam) := by
intro x y
change |altMu0 hL lam x-altMu0 hL lam y| ≤ _
have he : altMu0 hL lam x-altMu0 hL lam y =
-(sqrt (separation hL)*(perturbation hL lam x-perturbation hL lam y)+
separation hL*(bump hL x-bump hL y))/2 := by
dsimp [altMu0]; ring
have ht := separation_le_sqrt hL hh
have hb := perturbation_holder_bound hL hh lam x y
have hq := bump_holder_bound hL hh x y
let B := 24*|(x : ℝ)-y|^(1/10 : ℝ)/(deltaL hL)^(1/10 : ℝ)
have hB : 0 ≤ B := div_nonneg (mul_nonneg (by norm_num)
(Real.rpow_nonneg (abs_nonneg _) _)) (Real.rpow_nonneg (pow_pos hh.1 5).le _)
have hs : sqrt (separation hL)*B = (3/8)*|(x : ℝ)-y|^(1/10 : ℝ) := by
dsimp [B]
calc
_ = 24*|(x : ℝ)-y|^(1/10 : ℝ)*
(sqrt (separation hL)/(deltaL hL)^(1/10 : ℝ)) := by ring
_ = _ := by rw [separation_sqrt_scale hL hh.1]; ring
have hp := mul_le_mul_of_nonneg_left hb (Real.sqrt_nonneg (separation hL))
have hqb := (mul_le_mul_of_nonneg_left hq ht.1).trans
(mul_le_mul_of_nonneg_right ht.2 hB)
change sqrt (separation hL)*|perturbation hL lam x-perturbation hL lam y| ≤
sqrt (separation hL)*B at hp
rw [hs] at hp hqb
have ha := abs_add_le
(sqrt (separation hL)*(perturbation hL lam x-perturbation hL lam y))
(separation hL*(bump hL x-bump hL y))
simp only [abs_mul, abs_of_nonneg (Real.sqrt_nonneg _), abs_of_nonneg ht.1] at ha
rw [he, abs_div, abs_neg]
rw [abs_of_pos (by norm_num : (0 : ℝ) < 2)]
dsimp [L]
nlinarith [Real.rpow_nonneg (abs_nonneg ((x : ℝ)-y)) (1/10 : ℝ)]
theorem cosineFamily_contrast_lipschitz
Nuisance cancellation and the effect-bump modulus give the required contrast regularity. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
ContrastLipschitz (cosineFamily hL hh lam)
Proof (Lean source)
-- @node: cosineFamily_contrast_lipschitz
lemma cosineFamily_contrast_lipschitz (hL : ℝ) (hh : 0 < hL ∧ hL ≤ 1/4)
(lam : SignVector hL) : ContrastLipschitz (cosineFamily hL hh lam) := by
intro x y
rw [cosineFamily_contrast, cosineFamily_contrast, ← mul_sub, abs_mul]
have ht : 0 ≤ separation hL := (separation_le_sqrt hL hh).1
rw [abs_of_nonneg ht]
calc
_ ≤ separation hL*((2*pi/hL)*|(x : ℝ)-y|) :=
mul_le_mul_of_nonneg_left (bump_difference_bound hL hh.1 x y) ht
_ = (2*pi*kappa)*|(x : ℝ)-y| := by
unfold separation; field_simp [hh.1.ne']
_ ≤ L*|(x : ℝ)-y| := by
apply mul_le_mul_of_nonneg_right _ (abs_nonneg _)
norm_num [kappa, L]
linarith [Real.pi_lt_four]
theorem cosineFamily_completeModel
Every shared-sign alternative belongs to the complete model, with all three regularity requirements derived from its explicit cosine construction. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
CompleteModel (cosineFamily hL hh lam)
Proof (Lean source)
-- @node: cosineFamily_completeModel
lemma cosineFamily_completeModel (hL : ℝ) (hh : 0 < hL ∧ hL ≤ 1/4)
(lam : SignVector hL) : CompleteModel (cosineFamily hL hh lam) := by
exact binaryCausalLaw_completeModel _ _ _
(measurable_altE hL lam) (measurable_altMu0 hL lam) (measurable_altMu1 hL lam)
(alt_parameters_range hL hh lam) (cosineFamily_exchangeability hL hh lam)
(altE_overlap hL hh lam) (cosineFamily_propensity_holder hL hh lam)
(cosineFamily_control_holder hL hh lam) (cosineFamily_contrast_lipschitz hL hh lam)
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.DatasetMarginals 20 declarations Finite atomic product kernels identify the integrated conditional dataset laws with the sign mixture and null product experiment, certifying the coupling marginals.
Finite atomic product kernels identify the integrated conditional dataset laws with the sign mixture and null product experiment, certifying the coupling marginals.
Independent finite atomic laws expand into the array of products of their weights. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
ι α β :
w :
ι → α → ℝ≥0∞
f :
ι → α → β
hw :
∀ i a, w i a ≠ ⊤
conclusion
Measure.pi (fun i => ∑ a, w i a • Measure.dirac (f i a))
= ∑ z : ι → α, (∏ i, w i (z i)) • Measure.dirac (fun i => f i (z i))
Proof (Lean source)
-- @node: pi_finite_atomic
lemma pi_finite_atomic {ι α β : Type*} [Fintype ι] [DecidableEq ι] [Fintype α]
[MeasurableSpace β] (w : ι → α → ℝ≥0∞) (f : ι → α → β)
(hw : ∀ i a, w i a ≠ ⊤) :
Measure.pi (fun i => ∑ a, w i a • Measure.dirac (f i a)) =
∑ z : ι → α, (∏ i, w i (z i)) • Measure.dirac (fun i => f i (z i)) := by
classical
let (i : ι) : IsFiniteMeasure (∑ a, w i a • Measure.dirac (f i a)) := by
constructor
simpa [Measure.finsetSum_apply, Measure.smul_apply, smul_eq_mul] using
(ENNReal.sum_lt_top.mpr (fun a (_ : a ∈ (Finset.univ : Finset α)) => (hw i a).lt_top))
apply Measure.pi_eq
intro s hs
simp only [Measure.finsetSum_apply, Measure.smul_apply, smul_eq_mul,
Measure.dirac_apply' _ (MeasurableSet.univ_pi hs)]
simp only [Measure.dirac_apply' _ (hs _)]
rw [Fintype.prod_sum]
apply Finset.sum_congr rfl
intro z _
by_cases hz : ∀ i, f i (z i) ∈ s i
· simp [Set.mem_pi, hz]
· obtain ⟨i, hi⟩ := not_forall.mp hz
have hprod : (∏ j, w j (z j) * (s j).indicator (1 : β → ℝ≥0∞) (f j (z j))) = 0 := by
apply Finset.prod_eq_zero (Finset.mem_univ i)
simp [hi]
simpa [Set.mem_pi, hz] using hprod.symm
The conditional observed law for a fixed sign vector at one covariate.
Definition (Lean source)
parameters
def
given by
∑ z : Bool × Bool, ofReal (conditionalLikelihood hL lam x z.1 z.2 / 4) • Measure.dirac (x, z.1, z.2)
theorem measurable_conditionalSignRecordLaw
The fixed-sign conditional record law varies measurably with the covariate. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
Proof (Lean source)
-- @node: measurable_conditionalSignRecordLaw
@[fun_prop] lemma measurable_conditionalSignRecordLaw (hL : ℝ) (lam : SignVector hL) :
Measurable (conditionalSignRecordLaw hL lam) := by
apply Measure.measurable_of_measurable_coe
intro E hE
simp only [conditionalSignRecordLaw, Measure.finsetSum_apply, Measure.smul_apply,
Measure.dirac_apply' _ hE, smul_eq_mul]
apply Finset.measurable_sum
intro z _
apply mul
· fun_prop
· exact (measurable_const.indicator hE).comp (by fun_prop)
theorem conditionalSignRecordLaw_isProbabilityMeasure
Each fixed-sign conditional record law is a probability measure. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
IsProbabilityMeasure (conditionalSignRecordLaw hL lam x)
Proof (Lean source)
-- @node: conditionalSignRecordLaw_isProbabilityMeasure
lemma conditionalSignRecordLaw_isProbabilityMeasure (hL : ℝ)
(hhL : 0 < hL ∧ hL ≤ 1 / 4) (lam : SignVector hL) (x : Covariate) :
IsProbabilityMeasure (conditionalSignRecordLaw hL lam x) := by
constructor
simp only [conditionalSignRecordLaw, Measure.finsetSum_apply, Measure.smul_apply,
Measure.dirac_apply_of_mem (Set.mem_univ _), smul_eq_mul, mul_one]
rw [← ENNReal.ofReal_sum_of_nonneg (fun z _ =>
div_nonneg (conditionalLikelihood_nonneg hL hhL lam x z.1 z.2) (by norm_num)),
conditionalLikelihood_mark_sum, ENNReal.ofReal_one]
The fixed-sign observation kernel attaches the covariate to its conditional marks.
Definition (Lean source)
parameters
hL :
ℝ
lam :
SignVector hL
given by
clause 1
toFun := conditionalSignRecordLaw hL lam
clause 2
measurable' := measurable_conditionalSignRecordLaw hL lam
theorem signRecordKernel_isMarkovKernel
The fixed-sign observation kernel is everywhere probability-valued. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
IsMarkovKernel (signRecordKernel hL lam)
Proof (Lean source)
-- @node: signRecordKernel_isMarkovKernel
lemma signRecordKernel_isMarkovKernel (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(lam : SignVector hL) : IsMarkovKernel (signRecordKernel hL lam) :=
⟨conditionalSignRecordLaw_isProbabilityMeasure hL hhL lam⟩
theorem signRecordKernel_comp_volume
Integrating a fixed-sign conditional record reproduces the observed causal law. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: signRecordKernel_comp_volume
lemma signRecordKernel_comp_volume (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(lam : SignVector hL) :
signRecordKernel hL lam ∘ₘ (volume : Measure Covariate) =
Pobs (cosineFamily hL hhL lam) := by
ext E hE
change ((volume : Measure Covariate).bind (conditionalSignRecordLaw hL lam)) E = _
rw [Measure.bind_apply hE (measurable_conditionalSignRecordLaw hL lam).aemeasurable]
rw [cosineFamily, binaryCausalLaw_observed_apply _ _ _ (measurable_altE hL lam)
(measurable_altMu0 hL lam) (measurable_altMu1 hL lam)
(alt_parameters_range hL hhL lam) E hE]
apply lintegral_congr
intro x
simp only [conditionalSignRecordLaw, Measure.finsetSum_apply,
Measure.smul_apply, Measure.dirac_apply' _ hE, smul_eq_mul]
apply Finset.sum_congr rfl
intro z _
congr 2
dsimp [conditionalLikelihood]
ring
Given fixed signs, the conditional dataset law has independent mark weights.
Definition (Lean source)
parameters
given by
∑ z : Fin n → Bool × Bool, ofReal (∏ i, conditionalLikelihood hL lam (x i) (z i).1 (z i).2 / 4) • Measure.dirac (attachMarks n x z)
theorem conditionalSignDatasetLaw_eq_product
The conditional fixed-sign dataset array equals the coordinatewise product kernel. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
conditionalSignDatasetLaw hL n lam x
= finProductKernel n (signRecordKernel hL lam) x
Proof (Lean source)
-- @node: conditionalSignDatasetLaw_eq_product
lemma conditionalSignDatasetLaw_eq_product (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) (lam : SignVector hL) (x : Fin n → Covariate) :
conditionalSignDatasetLaw hL n lam x =
finProductKernel n (signRecordKernel hL lam) x := by
let := signRecordKernel_isMarkovKernel hL hhL lam
rw [finProductKernel_apply]
change _ = Measure.pi (fun i => conditionalSignRecordLaw hL lam (x i))
dsimp only [conditionalSignRecordLaw, conditionalSignDatasetLaw]
rw [pi_finite_atomic _ _ (fun _ _ => ENNReal.ofReal_ne_top)]
apply Finset.sum_congr rfl
intro z _
rw [ENNReal.ofReal_prod_of_nonneg (fun i _ =>
div_nonneg (conditionalLikelihood_nonneg hL hhL lam (x i) (z i).1 (z i).2)
(by norm_num))]
rfl
theorem measurable_conditionalSignDatasetLaw
The fixed-sign conditional dataset array is Borel measurable. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
conclusion
Measurable (conditionalSignDatasetLaw hL n lam)
Proof (Lean source)
-- @node: measurable_conditionalSignDatasetLaw
@[fun_prop] lemma measurable_conditionalSignDatasetLaw (hL : ℝ)
(hhL : 0 < hL ∧ hL ≤ 1 / 4) (n : ℕ) (lam : SignVector hL) :
Measurable (conditionalSignDatasetLaw hL n lam) := by
have heq : conditionalSignDatasetLaw hL n lam =
finProductKernel n (signRecordKernel hL lam) := by
funext x
exact conditionalSignDatasetLaw_eq_product hL hhL n lam x
rw [heq]
exact Kernel.measurable _
theorem conditionalSignDatasetLaw_bind_eq_dataLaw
Integrating a fixed-sign dataset kernel yields the observed product law. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
(Measure.pi (fun _ : Fin n => (volume : Measure Covariate))).bind (conditionalSignDatasetLaw hL n lam)
= dataLaw n (cosineFamily hL hhL lam)
Proof (Lean source)
-- @node: conditionalSignDatasetLaw_bind_eq_dataLaw
lemma conditionalSignDatasetLaw_bind_eq_dataLaw (hL : ℝ)
(hhL : 0 < hL ∧ hL ≤ 1 / 4) (n : ℕ) (lam : SignVector hL) :
(Measure.pi (fun _ : Fin n => (volume : Measure Covariate))).bind
(conditionalSignDatasetLaw hL n lam) = dataLaw n (cosineFamily hL hhL lam) := by
let := signRecordKernel_isMarkovKernel hL hhL lam
have heq : conditionalSignDatasetLaw hL n lam =
finProductKernel n (signRecordKernel hL lam) := by
funext x
exact conditionalSignDatasetLaw_eq_product hL hhL n lam x
rw [heq, finProductKernel_comp_pi, signRecordKernel_comp_volume hL hhL]
rfl
theorem conditionalAlternativeDatasetLaw_eq_sign_sum
The conditional alternative law is the uniform mixture of the fixed-sign dataset kernels. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
= ofReal ((2 : ℝ)^(-((activeSigns hL).card : ℤ))) • ∑ lam : SignVector hL, conditionalSignDatasetLaw hL n lam x
Proof (Lean source)
-- @node: conditionalAlternativeDatasetLaw_eq_sign_sum
lemma conditionalAlternativeDatasetLaw_eq_sign_sum (hL : ℝ)
(hhL : 0 < hL ∧ hL ≤ 1 / 4) (n : ℕ) (x : Fin n → Covariate) :
conditionalAlternativeDatasetLaw hL n x =
ofReal ((2 : ℝ)^(-((activeSigns hL).card : ℤ))) •
∑ lam : SignVector hL, conditionalSignDatasetLaw hL n lam x := by
classical
have hp (z : Fin n → Bool × Bool) (lam : SignVector hL) :
0 ≤ ∏ i, conditionalLikelihood hL lam (x i) (z i).1 (z i).2 / 4 := by
apply Finset.prod_nonneg
intro i _
exact div_nonneg (conditionalLikelihood_nonneg hL hhL lam (x i) _ _) (by norm_num)
simp only [conditionalAlternativeDatasetLaw, fullAlternativeMass,
ENNReal.ofReal_mul (le_of_lt (zpow_pos (by norm_num : (0 : ℝ) < 2) _)),
ENNReal.ofReal_sum_of_nonneg (fun lam _ => hp _ lam),
mul_smul, Finset.sum_smul, ← Finset.smul_sum, conditionalSignDatasetLaw]
rw [Finset.sum_comm]
theorem conditionalAlternativeDatasetLaw_bind_eq_signMixture
Integrating the alternative conditional law gives the specified mixture of product laws. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1 / 4
n :
ℕ
conclusion
(Measure.pi (fun _ : Fin n => (volume : Measure Covariate))).bind (conditionalAlternativeDatasetLaw hL n)
= signMixture hL hhL n
Proof (Lean source)
-- @node: conditionalAlternativeDatasetLaw_bind_eq_signMixture
lemma conditionalAlternativeDatasetLaw_bind_eq_signMixture (hL : ℝ)
(hhL : 0 < hL ∧ hL ≤ 1 / 4) (n : ℕ) :
(Measure.pi (fun _ : Fin n => (volume : Measure Covariate))).bind
(conditionalAlternativeDatasetLaw hL n) = signMixture hL hhL n := by
classical
ext E hE
rw [Measure.bind_apply hE (measurable_conditionalAlternativeDatasetLaw hL n).aemeasurable]
simp_rw [conditionalAlternativeDatasetLaw_eq_sign_sum hL hhL n]
simp only [Measure.smul_apply, Measure.finsetSum_apply, smul_eq_mul]
rw [lintegral_const_mul' _ _ ENNReal.ofReal_ne_top,
lintegral_finsetSum (Finset.univ : Finset (SignVector hL)) (fun lam _ =>
(show Measurable (fun x => (conditionalSignDatasetLaw hL n lam x) E) from
(Measure.measurable_coe hE).comp (measurable_conditionalSignDatasetLaw hL hhL n lam)))]
simp only [signMixture, Measure.smul_apply, Measure.finsetSum_apply, smul_eq_mul]
congr 1
apply Finset.sum_congr rfl
intro lam _
rw [← conditionalSignDatasetLaw_bind_eq_dataLaw hL hhL n lam,
Measure.bind_apply hE (measurable_conditionalSignDatasetLaw hL hhL n lam).aemeasurable]
theorem commonMassCoupling_map_fst_signMixture
The constructed dataset coupling's first marginal is the actual sign-mixture experiment. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1 / 4
n :
ℕ
conclusion
Proof (Lean source)
-- @node: commonMassCoupling_map_fst_signMixture
lemma commonMassCoupling_map_fst_signMixture (hL : ℝ)
(hhL : 0 < hL ∧ hL ≤ 1 / 4) (n : ℕ) :
(commonMassCoupling hL n).map fst = signMixture hL hhL n := by
rw [commonMassCoupling_map_fst_bind hL hhL,
conditionalAlternativeDatasetLaw_bind_eq_signMixture hL hhL]
The fair observation kernel attaches independent uniform binary marks to its covariate.
Definition (Lean source)
given by
instance fairRecordKernel_isMarkovKernel
The fair observation kernel is a Markov kernel.
Definition (Lean source)
-- @node: fairRecordKernel_isMarkovKernel
instance fairRecordKernel_isMarkovKernel : IsMarkovKernel fairRecordKernel where
isProbabilityMeasure x := by
constructor
change (∑ z : Bool × Bool, ofReal (1/4 : ℝ) • Measure.dirac (x, z.1, z.2)) univ = 1
simp only [Measure.finsetSum_apply, Measure.smul_apply, smul_eq_mul,
Measure.dirac_apply_of_mem (Set.mem_univ _), mul_one]
rw [← ENNReal.ofReal_sum_of_nonneg (fun _ _ => by norm_num)]
norm_num [Fintype.sum_prod_type, Fintype.sum_bool]
theorem fairRecordKernel_comp_volume
The fair conditional observation integrates to the observed null law. The displayed conclusion follows.
Formal statement
conclusion
Proof (Lean source)
-- @node: fairRecordKernel_comp_volume
lemma fairRecordKernel_comp_volume :
fairRecordKernel ∘ₘ (volume : Measure Covariate) = Pobs fairNull := by
ext E hE
rw [Measure.bind_apply hE (Kernel.measurable fairRecordKernel).aemeasurable,
fairNull, binaryCausalLaw_observed_apply _ _ _ measurable_const measurable_const
measurable_const fair_parameters_range E hE]
apply lintegral_congr
intro x
change (∑ z : Bool × Bool, ofReal (1/4 : ℝ) • Measure.dirac (x, z.1, z.2)) E = _
simp only [Measure.finsetSum_apply, Measure.smul_apply,
Measure.dirac_apply' _ hE, smul_eq_mul]
apply Finset.sum_congr rfl
intro z _
cases z.1 <;> cases z.2 <;> norm_num [bernoulliMass, indicator, Pi.one_apply]
theorem conditionalNullDatasetLaw_eq_product
The fair conditional dataset array equals the coordinatewise fair observation kernel. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: conditionalNullDatasetLaw_eq_product
lemma conditionalNullDatasetLaw_eq_product (n : ℕ) (x : Fin n → Covariate) :
conditionalNullDatasetLaw n x = finProductKernel n fairRecordKernel x := by
rw [finProductKernel_apply]
change _ = Measure.pi (fun i => ∑ z : Bool × Bool,
ofReal (1/4 : ℝ) • Measure.dirac (x i, z.1, z.2))
rw [pi_finite_atomic _ _ (fun _ _ => ENNReal.ofReal_ne_top)]
dsimp only [conditionalNullDatasetLaw]
apply Finset.sum_congr rfl
intro z _
congr 1
rw [Finset.prod_const, Finset.card_univ, Fintype.card_fin,
← ENNReal.ofReal_pow (by norm_num : (0 : ℝ) ≤ 1/4)]
congr 1
simp [zpow_neg, zpow_natCast, one_div]
theorem conditionalNullDatasetLaw_bind_eq_dataLaw
Integrating the fair conditional dataset law yields the null observed product law. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: conditionalNullDatasetLaw_bind_eq_dataLaw
lemma conditionalNullDatasetLaw_bind_eq_dataLaw (n : ℕ) :
(Measure.pi (fun _ : Fin n => (volume : Measure Covariate))).bind
(conditionalNullDatasetLaw n) = dataLaw n fairNull := by
have heq : conditionalNullDatasetLaw n = finProductKernel n fairRecordKernel := by
funext x
exact conditionalNullDatasetLaw_eq_product n x
rw [heq, finProductKernel_comp_pi, fairRecordKernel_comp_volume]
rfl
theorem commonMassCoupling_isCoupling
The Borel common-mass construction couples the two specified dataset experiments. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1 / 4
n :
ℕ
conclusion
Proof (Lean source)
-- @node: commonMassCoupling_isCoupling
lemma commonMassCoupling_isCoupling (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4) (n : ℕ) :
IsCoupling (commonMassCoupling hL n)
(signMixture hL hhL n) (dataLaw n fairNull) := by
have h := commonMassCoupling_isCoupling_bind hL hhL n
rw [conditionalAlternativeDatasetLaw_bind_eq_signMixture hL hhL,
conditionalNullDatasetLaw_bind_eq_dataLaw] at h
exact h
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.DesignPairMoments 10 declarations Actual conditional design laws and cross-treatment pair moments.
Actual conditional design laws and cross-treatment pair moments.
def conditionalDesignLaw
The actual design distribution conditioned on a public cell.
theorem conditionalDesignLaw_probability
Every admissible public cell has a normalized conditional design law. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
IsProbabilityMeasure (conditionalDesignLaw h k P j)
Proof (Lean source)
-- @node: conditionalDesignLaw_probability
lemma conditionalDesignLaw_probability (k : ℕ) (h : ℝ) (hk : 0 < k)
(hh : 0 < h ∧ h ≤ 1 / 4) (P : CausalLaw) (hP : CompleteModel P) (j : Fin k) :
IsProbabilityMeasure (conditionalDesignLaw h k P j) := by
let : IsProbabilityMeasure (PX P) := Measure.isProbabilityMeasure_map measurable_X.aemeasurable
have hp : 0 < cellMass h k P j := lt_of_lt_of_le
(half_pos (cell_endpoint_bounds k h hk hh j).2.2.2)
(cellMass_density_bounds k h hk hh P hP j).1
have hz : (PX P) (cell h k j) ≠ 0 := by
intro hz
simp [cellMass, measureReal_def, hz] at hp
refine ⟨?_⟩
simp only [conditionalDesignLaw, Measure.smul_apply,
Measure.restrict_apply MeasurableSet.univ, Set.univ_inter, smul_eq_mul]
exact ENNReal.inv_mul_cancel hz (measure_ne_top _ _)
theorem conditionalDesignLaw_integral
Conditional design integrals retain the actual within-cell density. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: conditionalDesignLaw_integral
lemma conditionalDesignLaw_integral (k : ℕ) (h : ℝ) (P : CausalLaw) (j : Fin k)
(f : Covariate → ℝ) :
(∫ x, f x ∂conditionalDesignLaw h k P j) =
(∫ x in cell h k j, f x ∂PX P) / cellMass h k P j := by
rw [conditionalDesignLaw, integral_smul_measure, ENNReal.toReal_inv]
simp only [cellMass, measureReal_def, smul_eq_mul, div_eq_mul_inv, mul_comm]
theorem conditionalDesignLaw_pair_mem
Two independent conditional covariates belong to their conditioning cell almost surely. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
∀ᵐ z ∂(conditionalDesignLaw h k P j).prod (conditionalDesignLaw h k P j), z.1 ∈ cell h k j ∧ z.2 ∈ cell h k j
Proof (Lean source)
-- @node: conditionalDesignLaw_pair_mem
lemma conditionalDesignLaw_pair_mem (k : ℕ) (h : ℝ) (hk : 0 < k)
(hh : 0 < h ∧ h ≤ 1 / 4) (P : CausalLaw) (hP : CompleteModel P) (j : Fin k) :
∀ᵐ z ∂(conditionalDesignLaw h k P j).prod (conditionalDesignLaw h k P j),
z.1 ∈ cell h k j ∧ z.2 ∈ cell h k j := by
let μ := conditionalDesignLaw h k P j
let : IsProbabilityMeasure μ := conditionalDesignLaw_probability k h hk hh P hP j
have hm : ∀ᵐ x ∂μ, x ∈ cell h k j :=
Measure.ae_smul_measure (ae_restrict_mem (measurableSet_cell h k j)) _
apply (Measure.ae_prod_iff_ae_ae
((measurableSet_cell h k j).preimage measurable_fst |>.inter
((measurableSet_cell h k j).preimage measurable_snd))).mpr
filter_upwards [hm] with x hx
filter_upwards [hm] with y hy
exact ⟨hx, hy⟩
def designPairNumerator
Cross-treatment numerator at two selected covariates.
def designPairDenominator
Cross-treatment probability at two selected covariates.
theorem design_regressions_integrable
The propensity, arm means, and arm-response regressions are integrable under any finite covariate measure, because the complete model gives everywhere binary-mean bounds. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
conclusion 1
Integrable P.e μ
conclusion 2
Integrable (fun x => 1 - P.e x) μ
conclusion 3
Integrable (fun x => P.e x * P.mu1 x) μ
conclusion 4
Integrable (fun x => (1 - P.e x) * P.mu0 x) μ
Proof (Lean source)
-- @node: design_regressions_integrable
lemma design_regressions_integrable (P : CausalLaw) (hP : CompleteModel P)
(μ : Measure Covariate) [IsFiniteMeasure μ] :
Integrable P.e μ ∧ Integrable (fun x => 1 - P.e x) μ ∧
Integrable (fun x => P.e x * P.mu1 x) μ ∧
Integrable (fun x => (1 - P.e x) * P.mu0 x) μ := by
have hme := P.e_measurable
have hmm0 := P.mu0_measurable
have hmm1 := P.mu1_measurable
have he : ∀ x, P.e x ∈ Icc (0 : ℝ) 1 := by
intro x
constructor <;> linarith [(hP.overlap x).1, (hP.overlap x).2]
have hm0 (x) : P.mu0 x ∈ Icc (0 : ℝ) 1 := arm_mean_mem_Icc P hP false x
have hm1 (x) : P.mu1 x ∈ Icc (0 : ℝ) 1 := arm_mean_mem_Icc P hP true x
have hi : Integrable P.e μ := Integrable.of_mem_Icc 0 1 (by fun_prop) (ae_of_all _ he)
refine ⟨hi, (integrable_const _).sub hi, ?_, ?_⟩
· apply Integrable.of_mem_Icc 0 1 (by fun_prop)
exact ae_of_all _ (fun x => ⟨mul_nonneg (he x).1 (hm1 x).1,
mul_le_one₀ (he x).2 (hm1 x).1 (hm1 x).2⟩)
· apply Integrable.of_mem_Icc 0 1 (by fun_prop)
exact ae_of_all _ (fun x => ⟨mul_nonneg (by linarith [(he x).2]) (hm0 x).1,
mul_le_one₀ (by linarith [(he x).1]) (hm0 x).1 (hm0 x).2⟩)
theorem design_pair_numerator_moment
Factorization of independent design pairs gives the exact numerator formula. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
conclusion 1
Integrable (designPairNumerator P) (μ.prod μ)
conclusion 2
(∫ z, designPairNumerator P z ∂μ.prod μ)
= 2 * ((∫ x, P.e x * P.mu1 x ∂μ) - (∫ x, P.e x ∂μ) * ((∫ x, (1 - P.e x) * P.mu0 x ∂μ) + (∫ x, P.e x * P.mu1 x ∂μ)))
Proof (Lean source)
-- @node: design_pair_numerator_moment
lemma design_pair_numerator_moment (P : CausalLaw) (hP : CompleteModel P)
(μ : Measure Covariate) [IsProbabilityMeasure μ] :
Integrable (designPairNumerator P) (μ.prod μ) ∧
(∫ z, designPairNumerator P z ∂μ.prod μ) =
2 * ((∫ x, P.e x * P.mu1 x ∂μ) - (∫ x, P.e x ∂μ) *
((∫ x, (1 - P.e x) * P.mu0 x ∂μ) + (∫ x, P.e x * P.mu1 x ∂μ))) := by
obtain ⟨he, hc, ht, hu⟩ := design_regressions_integrable P hP μ
have hfun : designPairNumerator P = (fun z =>
(P.e z.1 * P.mu1 z.1) * (1 - P.e z.2) -
P.e z.1 * ((1 - P.e z.2) * P.mu0 z.2) +
(1 - P.e z.1) * (P.e z.2 * P.mu1 z.2) -
((1 - P.e z.1) * P.mu0 z.1) * P.e z.2) := by
funext z
unfold designPairNumerator
ring
rw [hfun]
dsimp only
refine ⟨((ht.mul_prod hc).sub (he.mul_prod hu)).add (hc.mul_prod ht) |>.sub
(hu.mul_prod he), ?_⟩
have hsub := integral_sub (((ht.mul_prod hc).sub (he.mul_prod hu)).add (hc.mul_prod ht))
(hu.mul_prod he)
have hadd := integral_add ((ht.mul_prod hc).sub (he.mul_prod hu)) (hc.mul_prod ht)
have hsub' := integral_sub (ht.mul_prod hc) (he.mul_prod hu)
simp only [Pi.add_apply, Pi.sub_apply] at hsub hadd hsub'
rw [hsub, hadd, hsub',
integral_prod_mul (fun x => P.e x * P.mu1 x) (fun x => 1 - P.e x),
integral_prod_mul P.e (fun x => (1 - P.e x) * P.mu0 x),
integral_prod_mul (fun x => 1 - P.e x) (fun x => P.e x * P.mu1 x),
integral_prod_mul (fun x => (1 - P.e x) * P.mu0 x) P.e]
rw [integral_sub (integrable_const _) he]
simp only [integral_const, probReal_univ, smul_eq_mul, one_mul]
ring
theorem design_pair_denominator_moment
Factorization of independent design pairs gives the exact denominator formula. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
conclusion 1
Integrable (designPairDenominator P) (μ.prod μ)
conclusion 2
(∫ z, designPairDenominator P z ∂μ.prod μ) = 2 * (∫ x, P.e x ∂μ) * (1 - ∫ x, P.e x ∂μ)
Proof (Lean source)
-- @node: design_pair_denominator_moment
lemma design_pair_denominator_moment (P : CausalLaw) (hP : CompleteModel P)
(μ : Measure Covariate) [IsProbabilityMeasure μ] :
Integrable (designPairDenominator P) (μ.prod μ) ∧
(∫ z, designPairDenominator P z ∂μ.prod μ) =
2 * (∫ x, P.e x ∂μ) * (1 - ∫ x, P.e x ∂μ) := by
obtain ⟨he, hc, _, _⟩ := design_regressions_integrable P hP μ
have hfun : designPairDenominator P =
(fun z => P.e z.1 * (1 - P.e z.2) + (1 - P.e z.1) * P.e z.2) := by
funext z
unfold designPairDenominator
ring
rw [hfun]
dsimp only
refine ⟨(he.mul_prod hc).add (hc.mul_prod he), ?_⟩
rw [integral_add (he.mul_prod hc) (hc.mul_prod he)]
rw [integral_prod_mul P.e (fun x => 1 - P.e x),
integral_prod_mul (fun x => 1 - P.e x) P.e]
rw [integral_sub (integrable_const _) he]
simp only [integral_const, probReal_univ, smul_eq_mul, one_mul]
ring
theorem conditional_design_pair_bias
Integration of the pointwise causal certificate preserves the density-weighted cell bias. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
- theta P * (∫ z, designPairDenominator P z ∂(conditionalDesignLaw h k P j).prod (conditionalDesignLaw h k P j))|
≤ bias h k * (∫ z, designPairDenominator P z ∂(conditionalDesignLaw h k P j).prod (conditionalDesignLaw h k P j))
Proof (Lean source)
-- @node: conditional_design_pair_bias
lemma conditional_design_pair_bias (k : ℕ) (h : ℝ) (hk : 0 < k)
(hh : 0 < h ∧ h ≤ 1 / 4) (P : CausalLaw) (hP : CompleteModel P) (j : Fin k) :
|(∫ z, designPairNumerator P z
∂(conditionalDesignLaw h k P j).prod (conditionalDesignLaw h k P j)) -
theta P * (∫ z, designPairDenominator P z
∂(conditionalDesignLaw h k P j).prod (conditionalDesignLaw h k P j))| ≤
bias h k * (∫ z, designPairDenominator P z
∂(conditionalDesignLaw h k P j).prod (conditionalDesignLaw h k P j)) := by
let μ := conditionalDesignLaw h k P j
let : IsProbabilityMeasure μ := conditionalDesignLaw_probability k h hk hh P hP j
have hN := (design_pair_numerator_moment P hP μ).1
have hD := (design_pair_denominator_moment P hP μ).1
have hb : ∀ᵐ z ∂μ.prod μ,
|designPairNumerator P z - theta P * designPairDenominator P z| ≤
bias h k * designPairDenominator P z := by
filter_upwards [conditionalDesignLaw_pair_mem k h hk hh P hP j] with z hz
exact (cell_cross_treatment_certificate k h hk hh P hP j z.1 z.2 hz.1 hz.2).2
calc
_ = |∫ z, designPairNumerator P z - theta P * designPairDenominator P z
∂μ.prod μ| := by rw [integral_sub hN (hD.const_mul _), integral_const_mul]
_ ≤ ∫ z, |designPairNumerator P z - theta P * designPairDenominator P z|
∂μ.prod μ := abs_integral_le_integral_abs
_ ≤ ∫ z, bias h k * designPairDenominator P z ∂μ.prod μ :=
integral_mono_ae (hN.sub (hD.const_mul _)).abs (hD.const_mul _) hb
_ = _ := integral_const_mul _ _
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.DirectCoupling 17 declarations The shared-threshold direct experiment coupling and its independent-record transport cost.
The shared-threshold direct experiment coupling and its independent-record transport cost.
The localized bump has integrated mass at most the length of its macro support. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
Proof (Lean source)
-- @node: bump_lintegral_le_macro_length
lemma bump_lintegral_le_macro_length (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4) :
(∫⁻ x : Covariate, ofReal (bump hL x)) ≤ ofReal (2*hL) := by
classical
let W : Set Covariate := {x | |(x : ℝ)-x0| < hL}
have hW : MeasurableSet W := measurableSet_lt (by fun_prop) measurable_const
calc
_ ≤ ∫⁻ x : Covariate, W.indicator (fun _ => (1 : ℝ≥0∞)) x := by
apply lintegral_mono
intro x
by_cases hx : x ∈ W
· simp only [Set.indicator_of_mem hx]
exact (ENNReal.ofReal_le_ofReal (bump_range hL x).2).trans_eq (by simp)
· have hg : envelope hL x = 0 :=
envelope_zero_of_radius_le hL hhL x (le_of_not_gt hx)
simp [Set.indicator_of_notMem hx, bump, hg]
_ = (volume : Measure Covariate) W := by
rw [lintegral_indicator_const hW]
simp
_ ≤ _ := volume_covariate_neighborhood_le x0 hL
The crossing probability includes the fair treatment probability.
Definition (Lean source)
theorem directCrossingMass_range
The crossing mass fits within the fair treated-zero atom. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
Proof (Lean source)
-- @node: directCrossingMass_range
lemma directCrossingMass_range (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1/4) (x : Covariate) :
0 ≤ directCrossingMass hL x ∧ directCrossingMass hL x ≤ 1/4 := by
have ht := separation_le_sqrt hL hhL
have hb := bump_range hL x
have hs : separation hL ≤ 1/4 := by
dsimp [separation, kappa]
linarith [hhL.2]
dsimp [directCrossingMass]
constructor
· exact div_nonneg (mul_nonneg ht.1 hb.1) (by norm_num)
· nlinarith [mul_le_mul_of_nonneg_left hb.2 ht.1]
Integrating the shared thresholds leaves four common atoms and one crossing atom.
Definition (Lean source)
parameters
hL :
ℝ
x :
def
given by
theorem measurable_directConditionalCoupling
The finite shared-threshold law is a measurable conditional measure. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
Proof (Lean source)
-- @node: measurable_directConditionalCoupling
@[fun_prop] lemma measurable_directConditionalCoupling (hL : ℝ) :
Measurable (directConditionalCoupling hL) := by
apply Measure.measurable_of_measurable_coe
intro E hE
simp only [directConditionalCoupling, Measure.add_apply, Measure.smul_apply,
Measure.dirac_apply' _ hE, smul_eq_mul]
unfold directCrossingMass
have hdir (a b c d : Bool) :
Measurable (fun x : Covariate => E.indicator (1 : O × O → ℝ≥0∞)
((x,a,b),(x,c,d))) :=
(measurable_const.indicator hE).comp (by fun_prop)
fun_prop
theorem directConditionalCoupling_probability
Conditional shared thresholds have total mass one. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: directConditionalCoupling_probability
lemma directConditionalCoupling_probability (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1/4)
(x : Covariate) : IsProbabilityMeasure (directConditionalCoupling hL x) := by
constructor
have hw := directCrossingMass_range hL hhL x
simp only [directConditionalCoupling, Measure.add_apply, Measure.smul_apply,
Measure.dirac_apply_of_mem (Set.mem_univ _), smul_eq_mul, mul_one]
rw [← ENNReal.ofReal_add (by norm_num : (0 : ℝ) ≤ 1/4) (by norm_num : (0 : ℝ) ≤ 1/4),
← ENNReal.ofReal_add (by norm_num : (0 : ℝ) ≤ 1/4+1/4) (sub_nonneg.mpr hw.2),
← ENNReal.ofReal_add (by linarith : 0 ≤ 1/4+1/4+(1/4-directCrossingMass hL x))
(by norm_num : (0 : ℝ) ≤ 1/4),
← ENNReal.ofReal_add (by linarith : 0 ≤ 1/4+1/4+(1/4-directCrossingMass hL x)+1/4) hw.1]
rw [show (1/4+1/4+(1/4-directCrossingMass hL x)+1/4+directCrossingMass hL x : ℝ) = 1 by ring]
exact ENNReal.ofReal_one
theorem directConditionalCoupling_cost
Only the crossing atom contributes to the conditional disagreement cost. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
hL :
ℝ
x :
conclusion
(∫⁻ z : O × O, if z.1 = z.2 then (0 : ℝ≥0∞) else 1 ∂directConditionalCoupling hL x)
= ofReal (directCrossingMass hL x)
Proof (Lean source)
-- @node: directConditionalCoupling_cost
lemma directConditionalCoupling_cost (hL : ℝ) (x : Covariate) :
(∫⁻ z : O × O, if z.1 = z.2 then (0 : ℝ≥0∞) else 1
∂directConditionalCoupling hL x) = ofReal (directCrossingMass hL x) := by
have hm : Measurable (fun z : O × O => if z.1 = z.2 then (0 : ℝ≥0∞) else 1) := by
exact Measurable.ite (measurableSet_eq_fun measurable_fst measurable_snd)
measurable_const measurable_const
simp [directConditionalCoupling, lintegral_add_measure, lintegral_smul_measure,
lintegral_dirac' _ hm]
Integrate the conditional shared-threshold coupling over the common uniform covariate.
Definition (Lean source)
parameters
hL :
ℝ
def
given by
theorem directRecordCoupling_probability
The common covariate and conditional shared thresholds give a probability coupling law. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: directRecordCoupling_probability
lemma directRecordCoupling_probability (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1/4) :
IsProbabilityMeasure (directRecordCoupling hL) := by
constructor
rw [directRecordCoupling, Measure.bind_apply MeasurableSet.univ
(measurable_directConditionalCoupling hL).aemeasurable]
have hu (x : Covariate) : directConditionalCoupling hL x univ = 1 := by
let := directConditionalCoupling_probability hL hhL x
exact measure_univ
simp_rw [hu]
simp
theorem directConditionalCoupling_fst
The fair marginal of the shared-threshold conditional coupling. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
directConditionalCoupling hL x (fst ⁻¹' E)
Proof (Lean source)
-- @node: directConditionalCoupling_fst
lemma directConditionalCoupling_fst (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1/4)
(x : Covariate) (E : Set O) (hE : MeasurableSet E) :
directConditionalCoupling hL x (fst ⁻¹' E) =
∑ z : Bool × Bool, ofReal (1/4 : ℝ) *
E.indicator (1 : O → ℝ≥0∞) (x,z.1,z.2) := by
have hw := directCrossingMass_range hL hhL x
have ha : ofReal (1/4-directCrossingMass hL x) +
ofReal (directCrossingMass hL x) = ofReal (1/4 : ℝ) := by
rw [← ENNReal.ofReal_add (sub_nonneg.mpr hw.2) hw.1]
congr 1
ring
have hind (z : O × O) : (fst ⁻¹' E).indicator (1 : O × O → ℝ≥0∞) z =
E.indicator (1 : O → ℝ≥0∞) z.1 := by rfl
simp only [directConditionalCoupling, Measure.add_apply, Measure.smul_apply,
Measure.dirac_apply' _ (measurable_fst hE), smul_eq_mul]
simp_rw [hind]
change _ = _
simp only [Set.indicator_preimage, Function.comp_apply, Fintype.sum_prod_type,
Fintype.sum_bool]
calc
_ = ofReal (1/4 : ℝ) * E.indicator 1 (x,false,false) +
ofReal (1/4 : ℝ) * E.indicator 1 (x,false,true) +
(ofReal (1/4-directCrossingMass hL x) +
ofReal (directCrossingMass hL x)) * E.indicator 1 (x,true,false) +
ofReal (1/4 : ℝ) * E.indicator 1 (x,true,true) := by
simp only [indicator, preimage, Set.mem_setOf_eq, Pi.one_apply, fst, snd]
ring
_ = _ := by rw [ha]; ring
theorem directConditionalCoupling_snd
The treated marginal of the shared-threshold conditional coupling. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
directConditionalCoupling hL x (snd ⁻¹' E)
= ∑ z : Bool × Bool, ofReal (bernoulliMass (1/2) z.1 * bernoulliMass (if z.1 then directMu1 hL x else 1/2) z.2) * E.indicator (1 : O → ℝ≥0∞) (x,z.1,z.2)
Proof (Lean source)
-- @node: directConditionalCoupling_snd
lemma directConditionalCoupling_snd (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1/4)
(x : Covariate) (E : Set O) (hE : MeasurableSet E) :
directConditionalCoupling hL x (snd ⁻¹' E) =
∑ z : Bool × Bool,
ofReal (bernoulliMass (1/2) z.1 *
bernoulliMass (if z.1 then directMu1 hL x else 1/2) z.2) *
E.indicator (1 : O → ℝ≥0∞) (x,z.1,z.2) := by
have hw := directCrossingMass_range hL hhL x
have ha : ofReal (1/4 : ℝ) + ofReal (directCrossingMass hL x) =
ofReal (1/4+directCrossingMass hL x) :=
(ENNReal.ofReal_add (by norm_num) hw.1).symm
have hind (z : O × O) : (snd ⁻¹' E).indicator (1 : O × O → ℝ≥0∞) z =
E.indicator (1 : O → ℝ≥0∞) z.2 := by rfl
simp only [directConditionalCoupling, Measure.add_apply, Measure.smul_apply,
Measure.dirac_apply' _ (measurable_snd hE), smul_eq_mul,
Set.indicator_preimage, Function.comp_apply, Fintype.sum_prod_type,
Fintype.sum_bool, bernoulliMass, Bool.false_eq_true, if_false, if_true]
simp_rw [hind]
have hplus : (1/2 : ℝ) * directMu1 hL x = 1/4 + directCrossingMass hL x := by
dsimp [directMu1, directCrossingMass]; ring
have hminus : (1/2 : ℝ) * (1-directMu1 hL x) = 1/4 - directCrossingMass hL x := by
dsimp [directMu1, directCrossingMass]; ring
norm_num only at *
rw [hplus, hminus]
calc
_ = ofReal (1/4 : ℝ) * E.indicator 1 (x,false,false) +
ofReal (1/4 : ℝ) * E.indicator 1 (x,false,true) +
ofReal (1/4-directCrossingMass hL x) * E.indicator 1 (x,true,false) +
(ofReal (1/4 : ℝ) + ofReal (directCrossingMass hL x)) *
E.indicator 1 (x,true,true) := by
simp only [indicator, preimage, Set.mem_setOf_eq, Pi.one_apply, fst, snd]
ring
_ = _ := by rw [ha]; ring
theorem directRecordCoupling_map_fst
The first integrated marginal is the observed fair null law. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1/4
conclusion
Proof (Lean source)
-- @node: directRecordCoupling_map_fst
lemma directRecordCoupling_map_fst (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1/4) :
(directRecordCoupling hL).map fst = Pobs fairNull := by
ext E hE
rw [Measure.map_apply measurable_fst hE, directRecordCoupling,
Measure.bind_apply (measurable_fst hE) (measurable_directConditionalCoupling hL).aemeasurable]
simp_rw [directConditionalCoupling_fst hL hhL _ E hE]
rw [fairNull, binaryCausalLaw_observed_apply _ _ _ measurable_const measurable_const
measurable_const fair_parameters_range E hE]
apply lintegral_congr
intro x
apply Finset.sum_congr rfl
intro z _
cases z.1 <;> cases z.2 <;> norm_num [bernoulliMass, indicator, Pi.one_apply]
theorem directRecordCoupling_map_snd
The second integrated marginal is the observed direct alternative law. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1/4
conclusion
Proof (Lean source)
-- @node: directRecordCoupling_map_snd
lemma directRecordCoupling_map_snd (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1/4) :
(directRecordCoupling hL).map snd = Pobs (directAlternative hL hhL) := by
ext E hE
rw [Measure.map_apply measurable_snd hE, directRecordCoupling,
Measure.bind_apply (measurable_snd hE) (measurable_directConditionalCoupling hL).aemeasurable,
directAlternative, binaryCausalLaw_observed_apply _ _ _ measurable_const measurable_const
(measurable_directMu1 hL) (direct_parameters_range hL hhL) E hE]
exact lintegral_congr (fun x => directConditionalCoupling_snd hL hhL x E hE)
theorem directRecordCoupling_cost_le
The integrated disagreement cost is bounded by the crossing probability and macro support. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1/4
conclusion
(∫⁻ z : O × O, if z.1 = z.2 then (0 : ℝ≥0∞) else 1 ∂directRecordCoupling hL)
≤ ofReal (separation hL*hL)
Proof (Lean source)
-- @node: directRecordCoupling_cost_le
lemma directRecordCoupling_cost_le (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1/4) :
(∫⁻ z : O × O, if z.1 = z.2 then (0 : ℝ≥0∞) else 1
∂directRecordCoupling hL) ≤ ofReal (separation hL*hL) := by
have hm : Measurable (fun z : O × O => if z.1 = z.2 then (0 : ℝ≥0∞) else 1) :=
Measurable.ite (measurableSet_eq_fun measurable_fst measurable_snd)
measurable_const measurable_const
have ht := (separation_le_sqrt hL hhL).1
rw [directRecordCoupling, Measure.lintegral_bind
(measurable_directConditionalCoupling hL).aemeasurable hm.aemeasurable]
simp_rw [directConditionalCoupling_cost, directCrossingMass,
show ∀ x : Covariate, separation hL * bump hL x / 2 =
(separation hL/2)*bump hL x by intro x; ring,
ENNReal.ofReal_mul (show 0 ≤ separation hL/2 from div_nonneg ht (by norm_num))]
rw [lintegral_const_mul' _ _ ENNReal.ofReal_ne_top]
calc
_ ≤ ofReal (separation hL/2)*ofReal (2*hL) :=
mul_le_mul' le_rfl (bump_lintegral_le_macro_length hL hhL)
_ = _ := by
rw [← ENNReal.ofReal_mul (show 0 ≤ separation hL/2 from div_nonneg ht (by norm_num))]
congr 1
ring
Independent copies of the shared-threshold record pair are regrouped into two datasets.
theorem directDatasetCoupling_isCoupling
The independent-record construction has the two prescribed product marginals. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1/4
n :
ℕ
conclusion
Proof (Lean source)
-- @node: directDatasetCoupling_isCoupling
lemma directDatasetCoupling_isCoupling (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1/4) (n : ℕ) :
IsCoupling (directDatasetCoupling hL n)
(dataLaw n fairNull) (dataLaw n (directAlternative hL hhL)) := by
let := directRecordCoupling_probability hL hhL
have hu : Measurable (fun z : Fin n → O × O =>
(fun i => (z i).1, fun i => (z i).2)) := by fun_prop
refine ⟨Measure.isProbabilityMeasure_map hu.aemeasurable, ?_, ?_⟩
· rw [directDatasetCoupling, Measure.map_map measurable_fst hu]
change (Measure.pi (fun _ : Fin n => directRecordCoupling hL)).map
(fun z i => (z i).1) = _
rw [Measure.pi_map_pi (fun _ => measurable_fst.aemeasurable)]
simp_rw [directRecordCoupling_map_fst hL hhL]
rfl
· rw [directDatasetCoupling, Measure.map_map measurable_snd hu]
change (Measure.pi (fun _ : Fin n => directRecordCoupling hL)).map
(fun z i => (z i).2) = _
rw [Measure.pi_map_pi (fun _ => measurable_snd.aemeasurable)]
simp_rw [directRecordCoupling_map_snd hL hhL]
rfl
theorem directDatasetCoupling_cost_le
Hamming cost adds the identical one-record crossing costs over all independent records. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1/4
n :
ℕ
conclusion
(∫⁻ z, (dHam n z.1 z.2 : ℝ≥0∞) ∂directDatasetCoupling hL n)
≤ ofReal ((n : ℝ)*separation hL*hL)
Proof (Lean source)
-- @node: directDatasetCoupling_cost_le
lemma directDatasetCoupling_cost_le (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1/4) (n : ℕ) :
(∫⁻ z, (dHam n z.1 z.2 : ℝ≥0∞) ∂directDatasetCoupling hL n) ≤
ofReal ((n : ℝ)*separation hL*hL) := by
classical
let := directRecordCoupling_probability hL hhL
have hm : Measurable (fun z : O × O => if z.1 = z.2 then (0 : ℝ≥0∞) else 1) :=
Measurable.ite (measurableSet_eq_fun measurable_fst measurable_snd)
measurable_const measurable_const
rw [directDatasetCoupling, lintegral_map (measurable_datasetHammingCost n) (by fun_prop)]
simp only [dHam, hammingDist, Finset.card_eq_sum_ones, Finset.sum_filter,
Nat.cast_sum, Nat.cast_ite, Nat.cast_one, Nat.cast_zero]
rw [lintegral_finsetSum _ (fun i _ => by
simpa only [ite_not, Function.comp_def] using hm.comp (measurable_pi_apply i))]
have hi (i : Fin n) :
(∫⁻ z : Fin n → O × O, if (z i).1 ≠ (z i).2 then (1 : ℝ≥0∞) else 0
∂Measure.pi (fun _ : Fin n => directRecordCoupling hL)) =
∫⁻ z : O × O, if z.1 = z.2 then (0 : ℝ≥0∞) else 1 ∂directRecordCoupling hL := by
simpa only [ite_not, Function.comp_def] using
(measurePreserving_eval (fun _ : Fin n => directRecordCoupling hL) i).lintegral_comp hm
simp_rw [hi]
calc
_ ≤ ∑ _ : Fin n, ofReal (separation hL*hL) :=
Finset.sum_le_sum (fun _ _ => directRecordCoupling_cost_le hL hhL)
_ = _ := by
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul]
rw [← ENNReal.ofReal_natCast, ← ENNReal.ofReal_mul (Nat.cast_nonneg n)]
congr 1
ring
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.FrameGeometry 20 declarations Localized cosine frames and their exact squared partition identity, including lattice and macro support boundaries.
Localized cosine frames and their exact squared partition identity, including lattice and macro support boundaries.
def deltaL
The lower-family micro radius is the fifth power of its macro radius.
Definition (Lean source)
parameters
hL :
ℝ
def
deltaL hL :
ℝ
given by
hL^5
def kappa
The amplitude constant is two to the power minus twelve.
Definition (Lean source)
def
kappa :
ℝ
given by
(2 : ℝ)^(-12 : ℤ)
The target separation is the amplitude constant times the macro radius.
Definition (Lean source)
def envelope
The macro envelope is the squared cosine inside the localization window and zero outside.
def bump
The effect bump squares the macro envelope.
Definition (Lean source)
def frame
Each overlapping cosine frame element is supported within one micro radius of its integer center.
The localized macro envelope is Borel measurable, including its boundary convention. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: measurable_envelope
@[fun_prop] lemma measurable_envelope : Measurable (envelope hL) := by
unfold envelope
apply Measurable.ite (measurableSet_le (by fun_prop) measurable_const) <;> fun_prop
Squaring the macro envelope preserves measurability. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: measurable_bump
@[fun_prop] lemma measurable_bump : Measurable (bump hL) := by
unfold bump
fun_prop
Each localized cosine frame is Borel measurable. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: measurable_frame
@[fun_prop] lemma measurable_frame (j : ℤ) : Measurable (frame hL j) := by
unfold frame
apply Measurable.ite (measurableSet_le (by fun_prop) measurable_const) <;> fun_prop
The squared-cosine envelope takes values between zero and one. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: envelope_range
lemma envelope_range (x : Covariate) : envelope hL x ∈ Icc 0 1 := by
unfold envelope
split
· exact ⟨sq_nonneg _, Real.cos_sq_le_one _⟩
· exact ⟨le_rfl, zero_le_one⟩
The effect bump also takes values between zero and one. the theorem's stated inputs and assumptions, and the asserted conclusion follows.
Proof (Lean source)
-- @node: bump_range
lemma bump_range (x : Covariate) : bump hL x ∈ Icc 0 1 := by
have hg := envelope_range hL x
unfold bump
constructor
· exact sq_nonneg _
· nlinarith [mul_nonneg hg.1 (sub_nonneg.mpr hg.2)]
def activeSigns
The finite active integer indices have frame interiors intersecting the interior of the macro window.
The latent signs are binary encodings of minus one and one on the active index set.
Definition (Lean source)
The finite index construction selects exactly the strict support intersections for every legal macro radius. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: mem_activeSigns
lemma mem_activeSigns (j : ℤ) : j ∈ activeSigns hL ↔
((j : ℝ)*deltaL hL-deltaL hL < (x0 : ℝ)+hL ∧
(x0 : ℝ)-hL < (j : ℝ)*deltaL hL+deltaL hL) := by
classical
unfold activeSigns
rw [mem_filter]
constructor
· exact fun hj => hj.2
· intro hj
refine ⟨?_, hj⟩
rw [Finset.mem_Icc]
have hd : 0 < deltaL hL := pow_pos hhL.1 5
have hlo := Int.floor_le (((x0 : ℝ) - hL) / deltaL hL)
have hhi := Int.le_ceil (((x0 : ℝ) + hL) / deltaL hL)
have hl : (((x0 : ℝ) - hL) / deltaL hL) < (j : ℝ) + 1 := by
apply (div_lt_iff₀ hd).mpr
nlinarith [hj.2]
have hu : (j : ℝ) - 1 < (((x0 : ℝ) + hL) / deltaL hL) := by
apply (lt_div_iff₀ hd).mpr
nlinarith [hj.1]
constructor
· have : (⌊((x0 : ℝ) - hL) / deltaL hL⌋ : ℝ) - 1 ≤ (j : ℝ) := by
linarith
exact_mod_cast this
· have : (j : ℝ) ≤ (⌈((x0 : ℝ) + hL) / deltaL hL⌉ : ℝ) + 1 := by
linarith
exact_mod_cast this
A frame vanishes at either support endpoint and beyond it. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: frame_zero_of_radius_le
lemma frame_zero_of_radius_le (j : ℤ) (x : Covariate)
(hx : deltaL hL ≤ |(x : ℝ) - j * deltaL hL|) : frame hL j x = 0 := by
have hd : 0 < deltaL hL := pow_pos hhL.1 5
unfold frame
split
· rename_i hin
have he := le_antisymm hin hx
rcases (abs_eq (le_of_lt hd)).mp he with he | he
· rw [he]
have : pi * deltaL hL / (2 * deltaL hL) = pi / 2 := by
field_simp <;> ring
rw [this, Real.cos_pi_div_two]
· rw [he]
have : pi * (-deltaL hL) / (2 * deltaL hL) = -(pi / 2) := by
field_simp <;> ring
rw [this, Real.cos_neg, Real.cos_pi_div_two]
· rfl
The macro envelope vanishes at its endpoints as well as outside its support. the theorem's stated inputs and assumptions, and the asserted conclusion follows.
Formal statement
Proof (Lean source)
-- @node: envelope_zero_of_radius_le
lemma envelope_zero_of_radius_le (x : Covariate)
(hx : hL ≤ |(x : ℝ) - x0|) : envelope hL x = 0 := by
unfold envelope
split
· rename_i hin
have he := le_antisymm hin hx
rcases (abs_eq hhL.1.le).mp he with he | he
· rw [he]
have : pi * hL / (2 * hL) = pi / 2 := by field_simp [hhL.1.ne'] <;> ring
rw [this, Real.cos_pi_div_two]; norm_num
· rw [he]
have : pi * (-hL) / (2 * hL) = -(pi / 2) := by field_simp [hhL.1.ne'] <;> ring
rw [this, Real.cos_neg, Real.cos_pi_div_two]; norm_num
· rfl
An index omitted from the active frame set contributes zero after localization. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
Proof (Lean source)
-- @node: inactive_weighted_frame_zero
lemma inactive_weighted_frame_zero (j : ℤ) (x : Covariate)
(hj : j ∉ activeSigns hL) : bump hL x * (frame hL j x)^2 = 0 := by
by_cases hg : hL ≤ |(x : ℝ) - x0|
· rw [bump, envelope_zero_of_radius_le hL hhL x hg]; ring
· have hd : 0 < deltaL hL := pow_pos hhL.1 5
have hx := abs_lt.mp (lt_of_not_ge hg)
have hnj := mt (mem_activeSigns hL hhL j).mpr hj
push_neg at hnj
have hf : deltaL hL ≤ |(x : ℝ) - j * deltaL hL| := by
by_cases hl : (j : ℝ) * deltaL hL - deltaL hL < (x0 : ℝ) + hL
· have hu := hnj hl
rw [abs_of_nonneg (by linarith : 0 ≤ (x : ℝ) - j * deltaL hL)]
linarith [hx.1]
· rw [abs_of_nonpos (by linarith : (x : ℝ) - j * deltaL hL ≤ 0)]
linarith [hx.2]
rw [frame_zero_of_radius_le hL hhL j x hf]; ring
On a lattice interval the two neighboring frame squares sum to one. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
Proof (Lean source)
-- @node: neighboring_frame_squares
lemma neighboring_frame_squares (q : ℤ) (x : Covariate)
(hl : (q : ℝ) * deltaL hL ≤ x)
(hu : (x : ℝ) ≤ (q + 1 : ℤ) * deltaL hL) :
(frame hL q x)^2 + (frame hL (q + 1) x)^2 = 1 := by
have hd : 0 < deltaL hL := pow_pos hhL.1 5
have hqc : ((q + 1 : ℤ) : ℝ) = (q : ℝ) + 1 := by push_cast; rfl
have h0 : |(x : ℝ) - q * deltaL hL| ≤ deltaL hL := by
rw [abs_of_nonneg (by linarith)]
rw [hqc] at hu; nlinarith
have h1 : |(x : ℝ) - (q + 1 : ℤ) * deltaL hL| ≤ deltaL hL := by
rw [abs_of_nonpos (by linarith)]
rw [hqc]; nlinarith
simp only [frame, h0, h1, if_true]
have hang : pi * ((x : ℝ) - (q + 1 : ℤ) * deltaL hL) / (2 * deltaL hL) =
pi * ((x : ℝ) - q * deltaL hL) / (2 * deltaL hL) - pi / 2 := by
rw [hqc]; field_simp <;> ring
rw [hang, Real.cos_sub_pi_div_two, Real.cos_sq_add_sin_sq]
All frames except the two neighboring indices vanish on a lattice interval. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
Proof (Lean source)
-- @node: frame_zero_off_neighbors
lemma frame_zero_off_neighbors (q j : ℤ) (x : Covariate)
(hl : (q : ℝ) * deltaL hL ≤ x)
(hu : (x : ℝ) ≤ (q + 1 : ℤ) * deltaL hL)
(hj0 : j ≠ q) (hj1 : j ≠ q + 1) : frame hL j x = 0 := by
have hd : 0 < deltaL hL := pow_pos hhL.1 5
apply frame_zero_of_radius_le hL hhL
by_cases hj : j < q
· have hgap : (j : ℝ) + 1 ≤ q := by exact_mod_cast (show j + 1 ≤ q by omega)
rw [abs_of_nonneg (by nlinarith : 0 ≤ (x : ℝ) - j * deltaL hL)]
nlinarith
· have hgap : (q : ℝ) + 2 ≤ j := by exact_mod_cast (show q + 2 ≤ j by omega)
have huc : (x : ℝ) ≤ ((q : ℝ) + 1) * deltaL hL := by simpa using hu
rw [abs_of_nonpos (by nlinarith : (x : ℝ) - j * deltaL hL ≤ 0)]
nlinarith
Localizing the finite frame square sum gives exactly the effect bump everywhere. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: weighted_frame_square_partition
lemma weighted_frame_square_partition (x : Covariate) :
bump hL x * ∑ j : activeSigns hL, (frame hL j x)^2 = bump hL x := by
classical
let q : ℤ := ⌊(x : ℝ) / deltaL hL⌋
have hd : 0 < deltaL hL := pow_pos hhL.1 5
have hl : (q : ℝ) * deltaL hL ≤ x := (le_div_iff₀ hd).mp (Int.floor_le _)
have hu : (x : ℝ) ≤ (q + 1 : ℤ) * deltaL hL := by
have hu := (div_lt_iff₀ hd).mp (Int.lt_floor_add_one ((x : ℝ) / deltaL hL))
simpa [q] using hu.le
let f : ℤ → ℝ := fun j => bump hL x * (frame hL j x)^2
have hfinite : (∑ j ∈ activeSigns hL, f j) = ∑ j ∈ activeSigns hL ∪ {q, q+1}, f j := by
apply Finset.sum_subset (subset_union_left)
intro j hj hnot
exact inactive_weighted_frame_zero hL hhL j x hnot
have hpair : (∑ j ∈ ({q, q+1} : Finset ℤ), f j) =
∑ j ∈ activeSigns hL ∪ {q, q+1}, f j := by
apply Finset.sum_subset (subset_union_right)
intro j hj hnot
have hj0 : j ≠ q := by intro he; subst j; simp at hnot
have hj1 : j ≠ q+1 := by intro he; subst j; simp at hnot
dsimp [f]
rw [frame_zero_off_neighbors hL hhL q j x hl hu hj0 hj1]; ring
rw [Finset.mul_sum]
change (∑ j ∈ (activeSigns hL).attach, f j) = _
rw [Finset.sum_attach, hfinite, ← hpair]
simp only [Finset.sum_insert (by simp : q ∉ ({q+1} : Finset ℤ)), Finset.sum_singleton]
dsimp [f]
rw [← mul_add, neighboring_frame_squares hL hhL q x hl hu, mul_one]
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.FrameRegularity 7 declarations Global moduli of the localized cosine envelope and effect bump, including support boundaries.
Global moduli of the localized cosine envelope and effect bump, including support boundaries.
Clamping the displacement to the support interval realizes a zero-extended cosine exactly. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: localized_cosine_eq_clamped
lemma localized_cosine_eq_clamped (r c x : ℝ) (hr : 0 < r) :
(if |x - c| ≤ r then cos (pi * (x - c) / (2*r)) else 0) =
cos (pi * (projIcc (-r) r (by linarith) (x-c) : ℝ) / (2*r)) := by
classical
by_cases hx : |x-c| ≤ r
· rw [if_pos hx, projIcc_of_mem _ (abs_le.mp hx)]
· rw [if_neg hx]
have hout : x-c < -r ∨ r < x-c := by
simpa only [not_and_or, not_le] using (show ¬ (-r ≤ x-c ∧ x-c ≤ r) by
simpa only [← abs_le] using hx)
rcases hout with hlo | hhi
· rw [projIcc_of_le_left _ hlo.le]
have he : pi * (-r) / (2*r) = -(pi/2) := by field_simp
rw [he, Real.cos_neg, Real.cos_pi_div_two]
· rw [projIcc_of_right_le _ hhi.le]
have he : pi * r / (2*r) = pi/2 := by field_simp
rw [he, Real.cos_pi_div_two]
A zero-extended cosine has its global cosine Lipschitz bound across both support endpoints. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: localized_cosine_difference
lemma localized_cosine_difference (r c x y : ℝ) (hr : 0 < r) :
|(if |x-c| ≤ r then cos (pi*(x-c)/(2*r)) else 0) -
(if |y-c| ≤ r then cos (pi*(y-c)/(2*r)) else 0)| ≤
pi/(2*r) * |x-y| := by
rw [localized_cosine_eq_clamped r c x hr, localized_cosine_eq_clamped r c y hr]
have hc := (LipschitzWith.projIcc (show -r ≤ r by linarith)).dist_le_mul (x-c) (y-c)
simp only [Subtype.dist_eq, Real.dist_eq, NNReal.coe_one, one_mul] at hc
have ha : |pi * (projIcc (-r) r (by linarith) (x-c) : ℝ) / (2*r) -
pi * (projIcc (-r) r (by linarith) (y-c) : ℝ) / (2*r)| =
pi/(2*r) * |(projIcc (-r) r (by linarith) (x-c) : ℝ) -
(projIcc (-r) r (by linarith) (y-c) : ℝ)| := by
rw [← sub_div, ← mul_sub, abs_div, abs_mul, abs_of_pos Real.pi_pos,
abs_of_pos (by positivity : 0 < 2*r)]
ring
calc
_ ≤ _ := Real.abs_cos_sub_cos_le _ _
_ = _ := ha
_ ≤ _ := by
apply mul_le_mul_of_nonneg_left _ (by positivity)
simpa only [sub_sub_sub_cancel_right] using hc
Squaring values in the unit interval at most doubles their absolute difference. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
u v :
ℝ
hu :
|u| ≤ 1
hv :
|v| ≤ 1
conclusion
|u^2-v^2| ≤ 2*|u-v|
Proof (Lean source)
-- @node: square_difference_le_twice
lemma square_difference_le_twice (u v : ℝ) (hu : |u| ≤ 1) (hv : |v| ≤ 1) :
|u^2-v^2| ≤ 2*|u-v| := by
have he : u^2-v^2 = (u-v)*(u+v) := by ring
rw [he, abs_mul]
have hs : |u+v| ≤ 2 := (abs_add_le u v).trans (by linarith)
nlinarith [abs_nonneg (u-v)]
The macro envelope has a global Lipschitz modulus, with no exception at support boundaries. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: envelope_difference_bound
lemma envelope_difference_bound (hL : ℝ) (hh : 0 < hL) (x y : Covariate) :
|envelope hL x - envelope hL y| ≤ (pi/hL)*|(x : ℝ)-y| := by
classical
let f (z : Covariate) : ℝ :=
if |(z : ℝ)-x0| ≤ hL then cos (pi*((z : ℝ)-x0)/(2*hL)) else 0
have hf (z : Covariate) : |f z| ≤ 1 := by
dsimp [f]
split_ifs <;> first | exact Real.abs_cos_le_one _ | norm_num
have he (z : Covariate) : envelope hL z = (f z)^2 := by
dsimp [envelope, f]
split_ifs <;> ring
rw [he x, he y]
calc
_ ≤ 2*|f x-f y| := square_difference_le_twice _ _ (hf x) (hf y)
_ ≤ 2*(pi/(2*hL)*|(x : ℝ)-y|) :=
mul_le_mul_of_nonneg_left (localized_cosine_difference hL x0 x y hh) (by norm_num)
_ = _ := by field_simp
The squared envelope has a global Lipschitz modulus inherited from the envelope. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: bump_difference_bound
lemma bump_difference_bound (hL : ℝ) (hh : 0 < hL) (x y : Covariate) :
|bump hL x - bump hL y| ≤ (2*pi/hL)*|(x : ℝ)-y| := by
have hx := envelope_range hL x
have hy := envelope_range hL y
calc
_ ≤ 2*|envelope hL x-envelope hL y| := square_difference_le_twice _ _
(by simpa only [abs_of_nonneg hx.1] using hx.2)
(by simpa only [abs_of_nonneg hy.1] using hy.2)
_ ≤ 2*((pi/hL)*|(x : ℝ)-y|) :=
mul_le_mul_of_nonneg_left (envelope_difference_bound hL hh x y) (by norm_num)
_ = _ := by ring
Each micro-frame has the global modulus of its zero-extended cosine. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: frame_difference_bound
lemma frame_difference_bound (hL : ℝ) (hh : 0 < hL) (j : ℤ) (x y : Covariate) :
|frame hL j x-frame hL j y| ≤ (pi/(2*deltaL hL))*|(x : ℝ)-y| := by
exact localized_cosine_difference (deltaL hL) (j*deltaL hL) x y (pow_pos hh 5)
A bounded oscillation and a Lipschitz modulus give the one-tenth Holder modulus by splitting distances at the public radius. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
r z v C :
ℝ
hr :
0 < r
hz :
0 ≤ z
hC :
0 ≤ C
hbound :
v ≤ C
hlip :
v ≤ C*(z/r)
conclusion
v ≤ C*z^(1/10 : ℝ)/r^(1/10 : ℝ)
Proof (Lean source)
-- @node: holder_modulus_of_two_bounds
lemma holder_modulus_of_two_bounds (r z v C : ℝ) (hr : 0 < r) (hz : 0 ≤ z)
(hC : 0 ≤ C) (hbound : v ≤ C) (hlip : v ≤ C*(z/r)) :
v ≤ C*z^(1/10 : ℝ)/r^(1/10 : ℝ) := by
by_cases hzero : z = 0
· subst z
simpa using hlip
have hzpos : 0 < z := lt_of_le_of_ne hz (Ne.symm hzero)
have hu : 0 ≤ z/r := div_nonneg hz hr.le
have hid : (z/r)^(1/10 : ℝ) = z^(1/10 : ℝ)/r^(1/10 : ℝ) :=
Real.div_rpow hz hr.le _
rw [div_eq_mul_inv, mul_assoc, ← div_eq_mul_inv, ← hid]
by_cases hs : z/r ≤ 1
· have he : z/r ≤ (z/r)^(1/10 : ℝ) := by
have h := Real.rpow_le_rpow_of_exponent_ge (div_pos hzpos hr) hs (by norm_num : (1/10 : ℝ) ≤ 1)
simpa using h
exact hlip.trans (mul_le_mul_of_nonneg_left he hC)
· have he : 1 ≤ (z/r)^(1/10 : ℝ) := Real.one_le_rpow (le_of_not_ge hs) (by norm_num)
exact hbound.trans (by nlinarith)
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.IntervalEvents 1 declarations Borel point-containment events for the shared interval representation.
Borel point-containment events for the shared interval representation.
Point containment is Borel for every empty, finite, or unbounded interval code. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: measurableSet_interval_contains
lemma measurableSet_interval_contains (x : ℝ) :
MeasurableSet {c : IntervalCode | x ∈ c.toSet} := by
apply measurableSet_sum_iff.mpr
constructor
· simp [IntervalCode.toSet]
· change MeasurableSet {p : EReal × EReal × Bool × Bool |
(if p.2.2.1 then p.1 ≤ (x : EReal) else p.1 < (x : EReal)) ∧
(if p.2.2.2 then (x : EReal) ≤ p.2.1 else (x : EReal) < p.2.1)}
have hl : Measurable (fun p : EReal × EReal × Bool × Bool => p.1) := by fun_prop
have hu : Measurable (fun p : EReal × EReal × Bool × Bool => p.2.1) := by fun_prop
have hc0 : MeasurableSet {p : EReal × EReal × Bool × Bool | p.2.2.1 = true} := by
exact measurableSet_eq_fun (by fun_prop) measurable_const
have hc1 : MeasurableSet {p : EReal × EReal × Bool × Bool | p.2.2.2 = true} := by
exact measurableSet_eq_fun (by fun_prop) measurable_const
convert
(MeasurableSet.ite hc0 (measurableSet_le hl (measurable_const (a := (x : EReal))))
(measurableSet_lt hl (measurable_const (a := (x : EReal))))).inter
(MeasurableSet.ite hc1 (measurableSet_le (measurable_const (a := (x : EReal))) hu)
(measurableSet_lt (measurable_const (a := (x : EReal))) hu)) using 1
ext p
simp [Set.ite, ite_prop_iff_or, and_comm]
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.Inversion 11 declarations Explicit endpoints and measurability for the affine acceptance-set inversion.
Explicit endpoints and measurability for the affine acceptance-set inversion.
Positive denominators turn the affine acceptance inequality into two explicit endpoints. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
Proof (Lean source)
-- @node: affine_acceptance_positive
lemma affine_acceptance_positive (b z r : ℝ) (hb : 0 < b) (hr : 0 ≤ r) :
{x : ℝ | x ∈ Icc (-1) 1 ∧ |clip (-b) b z - x*b| ≤ r*b} =
Icc (max (-1) (clip (-b) b z / b - r))
(min 1 (clip (-b) b z / b + r)) := by
ext x
simp only [mem_setOf_eq, mem_Icc, max_le_iff, le_min_iff, abs_le]
constructor
· rintro ⟨⟨hx0, hx1⟩, h0, h1⟩
refine ⟨⟨hx0, ?_⟩, hx1, ?_⟩
· have : clip (-b) b z / b ≤ x + r :=
(div_le_iff₀ hb).mpr (by nlinarith)
linarith
· have : x - r ≤ clip (-b) b z / b :=
(le_div_iff₀ hb).mpr (by nlinarith)
linarith
· rintro ⟨⟨hx0, hxlo⟩, hx1, hxhi⟩
have hlo : clip (-b) b z ≤ (x+r)*b :=
(div_le_iff₀ hb).mp (by linarith)
have hhi : (x-r)*b ≤ clip (-b) b z :=
(le_div_iff₀ hb).mp (by linarith)
exact ⟨⟨hx0, hx1⟩, by nlinarith, by nlinarith⟩
theorem affine_acceptance_endpoints_ordered Lemma affine_acceptance_endpoints_ordered in the paper ↗
Clipping places the center inside the target range, so the truncated endpoints are ordered. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: affine_acceptance_endpoints_ordered
lemma affine_acceptance_endpoints_ordered (b z r : ℝ) (hb : 0 < b) (hr : 0 ≤ r) :
max (-1) (clip (-b) b z / b - r) ≤ min 1 (clip (-b) b z / b + r) := by
have hlo : -b ≤ clip (-b) b z := le_max_left _ _
have hhi : clip (-b) b z ≤ b := by
unfold clip
exact max_le (by linarith) (min_le_left _ _)
have hc0 : -1 ≤ clip (-b) b z / b := (le_div_iff₀ hb).mpr (by linarith)
have hc1 : clip (-b) b z / b ≤ 1 := (div_le_iff₀ hb).mpr (by linarith)
simp only [max_le_iff, le_min_iff]
constructor <;> constructor <;> linarith
A nonpositive denominator gives the full range at zero, a singleton at zero radius, or an empty acceptance set. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
b z r :
ℝ
hb :
b ≤ 0
hr :
0 ≤ r
conclusion
{x : ℝ | x ∈ Icc (-1) 1 ∧ |clip (-b) b z - x*b| ≤ r*b}
= if b = 0 then Icc (-1) 1 else if r = 0 then Icc (-1) (-1) else ∅
Proof (Lean source)
-- @node: affine_acceptance_nonpositive
lemma affine_acceptance_nonpositive (b z r : ℝ) (hb : b ≤ 0) (hr : 0 ≤ r) :
{x : ℝ | x ∈ Icc (-1) 1 ∧ |clip (-b) b z - x*b| ≤ r*b} =
if b = 0 then Icc (-1) 1 else if r = 0 then Icc (-1) (-1) else ∅ := by
have hc : clip (-b) b z = -b := by
unfold clip
exact max_eq_left (by have := min_le_left b z; linarith)
rw [hc]
split_ifs with hb0 hr0
· subst b
ext x
simp
· have hbneg : b < 0 := lt_of_le_of_ne hb hb0
subst r
ext x
simp only [mem_setOf_eq, mem_Icc, zero_mul, abs_le]
constructor
· rintro ⟨⟨hx0, hx1⟩, he⟩
have : x = -1 := by nlinarith [he.1, he.2]
simp [this]
· rintro ⟨hx0, hx1⟩
have : x = -1 := by linarith
subst x
constructor
· constructor <;> norm_num
· simp
· have hbneg : b < 0 := lt_of_le_of_ne hb hb0
have hrpos : 0 < r := lt_of_le_of_ne hr (Ne.symm hr0)
ext x
simp only [mem_setOf_eq, mem_empty_iff_false, iff_false, not_and]
intro hx ha
have := abs_nonneg (-b - x*b)
have := mul_neg_of_pos_of_neg hrpos hbneg
linarith
def acceptedSet
The inversion accepts exactly the target-range candidates satisfying the specified floored and clipped affine inequality.
def inversionMap
The inversion returns the closed interval hull of the accepted set, with the prescribed full- range fallback.
Definition (Lean source)
parameters
def
given by
if (acceptedSet n epsilon h k v).Nonempty then closedInterval (sInf (acceptedSet n epsilon h k v)) (sSup (acceptedSet n epsilon h k v)) else closedInterval (-1) 1
theorem inversionMap_eq_endpoints
The interval hull inversion has explicit endpoints on every denominator branch. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
inversionMap n epsilon h k v
= if 0 < max (d0 n h k) (v 1) then closedInterval (max (-1) (clip (-(max (d0 n h k) (v 1))) (max (d0 n h k) (v 1)) (v 0) / max (d0 n h k) (v 1) - sqrt (10*Vbound n epsilon h k))) (min 1 (clip (-(max (d0 n h k) (v 1))) (max (d0 n h k) (v 1)) (v 0) / max (d0 n h k) (v 1) + sqrt (10*Vbound n epsilon h k))) else if max (d0 n h k) (v 1) = 0 then closedInterval (-1) 1 else if sqrt (10*Vbound n epsilon h k) = 0 then closedInterval (-1) (-1) else closedInterval (-1) 1
Proof (Lean source)
-- @node: inversionMap_eq_endpoints
lemma inversionMap_eq_endpoints (v : Fin 2 → ℝ) :
inversionMap n epsilon h k v =
if 0 < max (d0 n h k) (v 1) then
closedInterval
(max (-1) (clip (-(max (d0 n h k) (v 1))) (max (d0 n h k) (v 1)) (v 0) /
max (d0 n h k) (v 1) - sqrt (10*Vbound n epsilon h k)))
(min 1 (clip (-(max (d0 n h k) (v 1))) (max (d0 n h k) (v 1)) (v 0) /
max (d0 n h k) (v 1) + sqrt (10*Vbound n epsilon h k)))
else if max (d0 n h k) (v 1) = 0 then closedInterval (-1) 1
else if sqrt (10*Vbound n epsilon h k) = 0 then closedInterval (-1) (-1)
else closedInterval (-1) 1 := by
classical
let b := max (d0 n h k) (v 1)
let r := sqrt (10*Vbound n epsilon h k)
have hr : 0 ≤ r := Real.sqrt_nonneg _
change inversionMap n epsilon h k v = if 0 < b then _ else _
by_cases hb : 0 < b
· rw [if_pos hb]
have hs : acceptedSet n epsilon h k v =
Icc (max (-1) (clip (-b) b (v 0) / b - r))
(min 1 (clip (-b) b (v 0) / b + r)) :=
affine_acceptance_positive b (v 0) r hb hr
have ho := affine_acceptance_endpoints_ordered b (v 0) r hb hr
simp only [inversionMap, hs, nonempty_Icc.mpr ho, ↓reduceIte, csInf_Icc ho, csSup_Icc ho]
rfl
· rw [if_neg hb]
have hs := affine_acceptance_nonpositive b (v 0) r (le_of_not_gt hb) hr
change acceptedSet n epsilon h k v = _ at hs
by_cases hb0 : b = 0
· rw [if_pos hb0]
have hs' : acceptedSet n epsilon h k v = Icc (-1) 1 := by simpa [hb0] using hs
simp [inversionMap, hs', csInf_Icc, csSup_Icc]
· rw [if_neg hb0]
by_cases hr0 : r = 0
· rw [if_pos hr0]
have hs' : acceptedSet n epsilon h k v = Icc (-1) (-1) := by
simpa [hb0, hr0] using hs
simp [inversionMap, hs', csInf_Icc, csSup_Icc]
· rw [if_neg hr0]
have hs' : acceptedSet n epsilon h k v = ∅ := by simpa [hb0, hr0] using hs
simp [inversionMap, hs']
The explicit inversion is measurable, including the zero and negative denominator cases. The displayed conclusion follows.
Formal statement
hypotheses
n :
ℕ
epsilon :
ℝ
h :
ℝ
k :
ℕ
conclusion
Measurable (inversionMap n epsilon h k)
Proof (Lean source)
-- @node: measurable_inversionMap
@[fun_prop] lemma measurable_inversionMap :
Measurable (inversionMap n epsilon h k) := by
have heq := funext (inversionMap_eq_endpoints n epsilon h k)
rw [heq]
have hb : Measurable (fun v : Fin 2 → ℝ => max (d0 n h k) (v 1)) := by fun_prop
apply Measurable.ite
· exact measurableSet_lt measurable_const hb
· unfold closedInterval clip
fun_prop
· apply Measurable.ite
· exact measurableSet_eq_fun hb measurable_const
· exact measurable_const
· by_cases hr : sqrt (10*Vbound n epsilon h k) = 0 <;>
simp only [hr, ↓reduceIte] <;> exact measurable_const
The publicly tuned inversion uses the same single joint release and returns the full target range in the constant branch.
Definition (Lean source)
parameters
n :
ℕ
epsilon :
ℝ
def
given by
if 1/8
≤ rate n epsilon then Kernel.const _ (Measure.dirac (closedInterval (-1) 1)) else (privateRatioRelease n epsilon (tunedH n epsilon) (tunedK n epsilon)).mapOfMeasurable (inversionMap n epsilon (tunedH n epsilon) (tunedK n epsilon)) (measurable_inversionMap n epsilon (tunedH n epsilon) (tunedK n epsilon))
The numerical interval scale output is the benchmark.
Definition (Lean source)
The fixed interval lower constant is two to the power minus twenty.
Definition (Lean source)
def
cI :
ℝ
given by
(2 : ℝ)^(-20 : ℤ)
The fixed interval upper constant is two to the power twenty-two.
Definition (Lean source)
def
CI :
ℝ
given by
2^22
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.MixtureOverlap 31 declarations Exact iid likelihood-overlap expansion of the shared-sign mixture and its reduction to a finite exponential average.
Exact iid likelihood-overlap expansion of the shared-sign mixture and its reduction to a finite exponential average. All domination and integrability conditions are derived.
theorem observed_probability
Every observed causal law is a probability measure. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: observed_probability
lemma observed_probability (P : CausalLaw) : IsProbabilityMeasure (Pobs P) := by
have hm : Measurable observe := by unfold observe X A Y; fun_prop
exact Measure.isProbabilityMeasure_map hm.aemeasurable
The finite sign weight is the reciprocal of the number of sign vectors. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
hL :
ℝ
conclusion
ofReal ((2 : ℝ)^(-((activeSigns hL).card : ℤ)))
= (card (SignVector hL) : ℝ≥0∞)⁻¹
Proof (Lean source)
-- @node: sign_weight_eq_inv_card
lemma sign_weight_eq_inv_card (hL : ℝ) :
ofReal ((2 : ℝ)^(-((activeSigns hL).card : ℤ))) =
(card (SignVector hL) : ℝ≥0∞)⁻¹ := by
classical
have hc : (card (SignVector hL) : ℝ≥0∞) ≠ 0 := by
exact_mod_cast Fintype.card_ne_zero
calc
_ = ofReal ((2 : ℝ)^(-((activeSigns hL).card : ℤ))) *
((card (SignVector hL) : ℝ≥0∞) *
(card (SignVector hL) : ℝ≥0∞)⁻¹) := by
rw [ENNReal.mul_inv_cancel hc (by simp), mul_one]
_ = _ := by rw [← mul_assoc, sign_weight_normalization, one_mul]
theorem signMixture_eq_uniformMixture
The dataset sign mixture is exactly the library's uniform finite iid mixture. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1 / 4
n :
ℕ
conclusion
signMixture hL hhL n
= uniformMixture (fun lam : SignVector hL => Measure.pi (fun _ : Fin n => Pobs (cosineFamily hL hhL lam)))
Proof (Lean source)
-- @node: signMixture_eq_uniformMixture
lemma signMixture_eq_uniformMixture (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) :
signMixture hL hhL n = uniformMixture
(fun lam : SignVector hL => Measure.pi
(fun _ : Fin n => Pobs (cosineFamily hL hhL lam))) := by
classical
simp only [signMixture, sign_weight_eq_inv_card, dataLaw,
uniformMixture, mixture,
Finset.smul_sum]
Finite domination bounds the real density under the reference measure. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
α :
Type*
μ ν :
c :
ℝ≥0∞
hc0 :
c ≠ 0
hct :
c ≠ ∞
hle :
μ ≤ c • ν
conclusion
∀ᵐ z ∂ν, (μ.rnDeriv ν z).toReal ≤ c.toReal
Proof (Lean source)
-- @node: rnDeriv_toReal_le_of_domination
lemma rnDeriv_toReal_le_of_domination {α : Type*} [MeasurableSpace α]
(μ ν : Measure α) [IsFiniteMeasure ν] (c : ℝ≥0∞)
(hc0 : c ≠ 0) (hct : c ≠ ∞) (hle : μ ≤ c • ν) :
∀ᵐ z ∂ν, (μ.rnDeriv ν z).toReal ≤ c.toReal := by
letI : IsFiniteMeasure (c • ν) := Measure.smul_finite ν hct
letI : IsFiniteMeasure μ := isFiniteMeasure_of_le (c • ν) hle
have hsmall : μ.rnDeriv (c • ν) ≤ᵐ[ν] 1 :=
(Measure.absolutelyContinuous_smul hc0).ae_le (Measure.rnDeriv_le_one_of_le hle)
filter_upwards [hsmall, Measure.rnDeriv_smul_right_of_ne_top μ ν hc0 hct] with z hz hs
rw [hs, Pi.smul_apply, smul_eq_mul] at hz
apply ENNReal.toReal_mono hct
calc
_ = c * (c⁻¹ * μ.rnDeriv ν z) := by
rw [← mul_assoc, ENNReal.mul_inv_cancel hc0 hct, one_mul]
_ ≤ c * 1 := mul_le_mul' le_rfl hz
_ = c := mul_one _
theorem cosineFamily_density_pair_integrable
Products of the alternative one-record densities are integrable, with bound sixty-four. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Integrable (fun z => ((Pobs (cosineFamily hL hhL lam)).rnDeriv (Pobs fairNull) z).toReal * ((Pobs (cosineFamily hL hhL lam')).rnDeriv (Pobs fairNull) z).toReal) (Pobs fairNull)
Proof (Lean source)
-- @node: cosineFamily_density_pair_integrable
lemma cosineFamily_density_pair_integrable (hL : ℝ)
(hhL : 0 < hL ∧ hL ≤ 1 / 4) (lam lam' : SignVector hL) :
Integrable (fun z => ((Pobs (cosineFamily hL hhL lam)).rnDeriv (Pobs fairNull) z).toReal *
((Pobs (cosineFamily hL hhL lam')).rnDeriv (Pobs fairNull) z).toReal)
(Pobs fairNull) := by
letI := observed_probability fairNull
have hb (l : SignVector hL) := rnDeriv_toReal_le_of_domination
(Pobs (cosineFamily hL hhL l)) (Pobs fairNull) 8 (by norm_num) (by norm_num)
(cosineFamily_observed_domination hL hhL l)
apply Integrable.of_mem_Icc 0 64 (by fun_prop)
filter_upwards [hb lam, hb lam'] with z hz hz'
norm_num at hz hz'
exact ⟨mul_nonneg ENNReal.toReal_nonneg ENNReal.toReal_nonneg,
(mul_le_mul hz hz' ENNReal.toReal_nonneg (by norm_num)).trans (by norm_num)⟩
The one-record density overlap is the nonnegative likelihood inner product.
Definition (Lean source)
parameters
def
signDensityOverlap hL hhL lam lam' :
ℝ
theorem signDensityOverlap_nonneg
Each one-record density overlap is nonnegative. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
0 ≤ signDensityOverlap hL hhL lam lam'
Proof (Lean source)
-- @node: signDensityOverlap_nonneg
lemma signDensityOverlap_nonneg (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(lam lam' : SignVector hL) : 0 ≤ signDensityOverlap hL hhL lam lam' := by
exact integral_nonneg (fun _ => mul_nonneg ENNReal.toReal_nonneg ENNReal.toReal_nonneg)
theorem signMixture_chiSq_overlap_identity
One plus the dataset mixture divergence is exactly the uniform average of nth powers of one-record overlaps; the iid product and finite-mixture identities come from Causalean. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1 / 4
n :
ℕ
conclusion
= (∑ lam : SignVector hL, ∑ lam' : SignVector hL, (signDensityOverlap hL hhL lam lam')^n) / (card (SignVector hL) : ℝ)^2
Proof (Lean source)
-- @node: signMixture_chiSq_overlap_identity
lemma signMixture_chiSq_overlap_identity (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) :
1 + chiSqDiv (signMixture hL hhL n) (dataLaw n fairNull) =
(∑ lam : SignVector hL, ∑ lam' : SignVector hL,
(signDensityOverlap hL hhL lam lam')^n) /
(card (SignVector hL) : ℝ)^2 := by
letI (P : CausalLaw) := observed_probability P
rw [signMixture_eq_uniformMixture]
exact one_add_chiSqDiv_uniformMixture_iid
(fun lam : SignVector hL => Pobs (cosineFamily hL hhL lam)) (Pobs fairNull)
(fun lam => Measure.absolutelyContinuous_of_le_smul
(cosineFamily_observed_domination hL hhL lam))
(cosineFamily_density_pair_integrable hL hhL) n
theorem signDensityOverlap_pow_le_exp
Nonnegative likelihood overlaps raised to n are dominated by the exponential of the centered overlap times n, as used in the fixed-sample roadmap. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
(signDensityOverlap hL hhL lam lam')^n
≤ exp ((n : ℝ)*(signDensityOverlap hL hhL lam lam'-1))
Proof (Lean source)
-- @node: signDensityOverlap_pow_le_exp
lemma signDensityOverlap_pow_le_exp (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) (lam lam' : SignVector hL) :
(signDensityOverlap hL hhL lam lam')^n ≤
exp ((n : ℝ)*(signDensityOverlap hL hhL lam lam'-1)) := by
have hb : signDensityOverlap hL hhL lam lam' ≤
exp (signDensityOverlap hL hhL lam lam'-1) := by
simpa using Real.add_one_le_exp (signDensityOverlap hL hhL lam lam'-1)
calc
_ ≤ (exp (signDensityOverlap hL hhL lam lam'-1))^n :=
pow_le_pow_left₀ (signDensityOverlap_nonneg hL hhL lam lam') hb n
_ = _ := by rw [← Real.exp_nat_mul]
theorem signMixture_chiSq_le_exp_average
The exact iid mixture identity reduces its divergence bound to a finite exponential average, without any additional integrability or absolute-continuity assumptions. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1 / 4
n :
ℕ
conclusion
≤ (∑ lam : SignVector hL, ∑ lam' : SignVector hL, exp ((n : ℝ)*(signDensityOverlap hL hhL lam lam'-1))) / (card (SignVector hL) : ℝ)^2
Proof (Lean source)
-- @node: signMixture_chiSq_le_exp_average
lemma signMixture_chiSq_le_exp_average (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) :
1 + chiSqDiv (signMixture hL hhL n) (dataLaw n fairNull) ≤
(∑ lam : SignVector hL, ∑ lam' : SignVector hL,
exp ((n : ℝ)*(signDensityOverlap hL hhL lam lam'-1))) /
(card (SignVector hL) : ℝ)^2 := by
rw [signMixture_chiSq_overlap_identity]
apply div_le_div_of_nonneg_right _ (sq_nonneg _)
exact Finset.sum_le_sum (fun lam _ => Finset.sum_le_sum
(fun lam' _ => signDensityOverlap_pow_le_exp hL hhL n lam lam'))
The centered conditional likelihood inner product in the paper's Fourier expansion.
Definition (Lean source)
parameters
def
signOverlapIntegrand hL lam lam' x :
ℝ
given by
separation hL * (1 + (separation hL*bump hL x-1)^2) * perturbation hL lam x * perturbation hL lam' x
+ (separation hL)^2 * (bump hL x-(perturbation hL lam x)^2) * (bump hL x-(perturbation hL lam' x)^2)
theorem conditionalLikelihood_pair_inner_product
Orthogonality of the four fair marks leaves exactly the two terms of the centered overlap integrand, pointwise in the covariate. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
(∑ av : Bool, ∑ yv : Bool, conditionalLikelihood hL lam x av yv * conditionalLikelihood hL lam' x av yv) / 4
= 1 + signOverlapIntegrand hL lam lam' x
Proof (Lean source)
-- @node: conditionalLikelihood_pair_inner_product
lemma conditionalLikelihood_pair_inner_product (hL : ℝ)
(hhL : 0 < hL ∧ hL ≤ 1 / 4) (lam lam' : SignVector hL) (x : Covariate) :
(∑ av : Bool, ∑ yv : Bool,
conditionalLikelihood hL lam x av yv * conditionalLikelihood hL lam' x av yv) / 4 =
1 + signOverlapIntegrand hL lam lam' x := by
have ht : 0 ≤ separation hL := (separation_le_sqrt hL hhL).1
simp_rw [conditionalLikelihood_fourier hL hhL]
simp only [Fintype.sum_bool, bit, if_true, Bool.false_eq_true, if_false]
dsimp [signOverlapIntegrand]
ring_nf
rw [Real.sq_sqrt ht]
ring
theorem measurable_signOverlapIntegrand
The centered overlap integrand is Borel measurable in the covariate. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
Proof (Lean source)
-- @node: measurable_signOverlapIntegrand
@[fun_prop] lemma measurable_signOverlapIntegrand (hL : ℝ)
(lam lam' : SignVector hL) : Measurable (signOverlapIntegrand hL lam lam') := by
unfold signOverlapIntegrand
fun_prop
The finite sign second moment is exactly the number of signs times the bump, including macro endpoints and lattice boundaries. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
(∑ lam : SignVector hL, (perturbation hL lam x)^2)
Proof (Lean source)
-- @node: perturbation_sign_square_sum_exact
lemma perturbation_sign_square_sum_exact (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(x : Covariate) :
(∑ lam : SignVector hL, (perturbation hL lam x)^2) =
(card (SignVector hL) : ℝ)*bump hL x := by
rw [perturbation_sign_square_sum, weighted_frame_square_partition hL hhL]
theorem signOverlapIntegrand_sign_sum
Averaging the centered integrand over the first sign vector gives zero for every fixed second sign vector and every covariate, exactly as required by the mgf step. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
(∑ lam : SignVector hL, signOverlapIntegrand hL lam lam' x) = 0
Proof (Lean source)
-- @node: signOverlapIntegrand_sign_sum
lemma signOverlapIntegrand_sign_sum (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(lam' : SignVector hL) (x : Covariate) :
(∑ lam : SignVector hL, signOverlapIntegrand hL lam lam' x) = 0 := by
classical
simp only [signOverlapIntegrand, Finset.sum_add_distrib, ← Finset.sum_mul,
← Finset.mul_sum, Finset.sum_sub_distrib, perturbation_sign_sum,
perturbation_sign_square_sum_exact hL hhL, Finset.sum_const, Finset.card_univ,
nsmul_eq_mul]
ring
theorem signOverlapIntegrand_sign_mean
The actual sign probability law gives the centered integrand mean zero pointwise. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
(∫ lam, signOverlapIntegrand hL lam lam' x ∂signLaw hL) = 0
Proof (Lean source)
-- @node: signOverlapIntegrand_sign_mean
lemma signOverlapIntegrand_sign_mean (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(lam' : SignVector hL) (x : Covariate) :
(∫ lam, signOverlapIntegrand hL lam lam' x ∂signLaw hL) = 0 := by
rw [integral_signLaw, signOverlapIntegrand_sign_sum hL hhL, mul_zero]
The fair observed null integrates four equally weighted mark cells at each covariate. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
f :
O → ℝ≥0∞
hf :
Proof (Lean source)
-- @node: fairNull_lintegral_marks
lemma fairNull_lintegral_marks (f : O → ℝ≥0∞) (hf : Measurable f) :
(∫⁻ z, f z ∂Pobs fairNull) =
∫⁻ x : Covariate, ∑ av : Bool, ∑ yv : Bool, (1 / 4 : ℝ≥0∞)*f (x,av,yv) := by
have hm : Measurable observe := by unfold observe X A Y; fun_prop
change (∫⁻ z, f z ∂(binaryLaw (fun _ => 1/2) (fun _ => 1/2)
(fun _ => 1/2)).map observe) = _
rw [lintegral_map hf hm, binaryLaw_lintegral (fun _ => 1/2) (fun _ => 1/2) (fun _ => 1/2) measurable_const
measurable_const measurable_const (fun z => f (observe z)) (hf.comp hm)]
apply lintegral_congr
intro x
simp only [Fintype.sum_prod_type, Fintype.sum_bool, Function.comp_apply,
observe, X, A, Y, bernoulliMass, Bool.false_eq_true, if_false, if_true]
norm_num [ENNReal.ofReal_div_of_pos]
have hc : (8 : ℝ≥0∞)⁻¹ * 2 = 4⁻¹ := by
rw [show (8 : ℝ≥0∞) = 4*2 by norm_num, ENNReal.mul_inv, mul_assoc,
ENNReal.inv_mul_cancel (by norm_num) (by norm_num), mul_one] <;> norm_num
ring_nf
simp only [mul_right_comm _ _ (2 : ℝ≥0∞), hc]
ring
theorem measurable_conditionalLikelihood_record Lemma measurable_conditionalLikelihood_record in the paper ↗
The actual conditional likelihood is measurable on the full observed record space. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
hL :
ℝ
lam :
SignVector hL
conclusion
Measurable (fun z : O => conditionalLikelihood hL lam z.1 z.2.1 z.2.2)
Proof (Lean source)
-- @node: measurable_conditionalLikelihood_record
@[fun_prop] lemma measurable_conditionalLikelihood_record (hL : ℝ) (lam : SignVector hL) :
Measurable (fun z : O => conditionalLikelihood hL lam z.1 z.2.1 z.2.2) := by
have ha : MeasurableSet {z : O | z.2.1 = true} :=
measurableSet_eq_fun (by fun_prop) measurable_const
have hy : MeasurableSet {z : O | z.2.2 = true} :=
measurableSet_eq_fun (by fun_prop) measurable_const
have he : Measurable (fun z : O => altE hL lam z.1) := by fun_prop
have h0 : Measurable (fun z : O => altMu0 hL lam z.1) := by fun_prop
have h1 : Measurable (fun z : O => altMu1 hL lam z.1) := by fun_prop
classical
have hm : Measurable (fun z : O => if z.2.1 then altMu1 hL lam z.1 else altMu0 hL lam z.1) := h1.ite ha h0
exact (measurable_const.mul (he.ite ha (measurable_const.sub he))).mul
(hm.ite hy (measurable_const.sub hm))
Every actual conditional mark likelihood is nonnegative for legal macro radii. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
0 ≤ conditionalLikelihood hL lam x av yv
Proof (Lean source)
-- @node: conditionalLikelihood_nonneg
lemma conditionalLikelihood_nonneg (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(lam : SignVector hL) (x : Covariate) (av yv : Bool) :
0 ≤ conditionalLikelihood hL lam x av yv := by
unfold conditionalLikelihood
apply mul_nonneg (mul_nonneg (by norm_num)
(bernoulliMass_nonneg (alt_parameters_range hL hhL lam x).1 av))
cases av
· exact bernoulliMass_nonneg (alt_parameters_range hL hhL lam x).2.1 yv
· exact bernoulliMass_nonneg (alt_parameters_range hL hhL lam x).2.2 yv
theorem cosineFamily_observed_withDensity
The conditional four-cell likelihood is the actual observed density relative to the fair null. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Pobs (cosineFamily hL hhL lam)
Proof (Lean source)
-- @node: cosineFamily_observed_withDensity
lemma cosineFamily_observed_withDensity (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(lam : SignVector hL) :
Pobs (cosineFamily hL hhL lam) = (Pobs fairNull).withDensity
(fun z => ofReal (conditionalLikelihood hL lam z.1 z.2.1 z.2.2)) := by
classical
ext s hs
rw [cosineFamily, binaryCausalLaw_observed_apply _ _ _
(measurable_altE hL lam) (measurable_altMu0 hL lam) (measurable_altMu1 hL lam)
(alt_parameters_range hL hhL lam) s hs, withDensity_apply _ hs,
← lintegral_indicator hs]
rw [fairNull_lintegral_marks _
(((measurable_conditionalLikelihood_record hL lam).ennreal_ofReal).indicator hs)]
apply lintegral_congr
intro x
rw [Fintype.sum_prod_type]
apply Finset.sum_congr rfl
intro av hav
apply Finset.sum_congr rfl
intro yv hyv
by_cases hx : (x,av,yv) ∈ s
· simp only [Set.indicator_of_mem hx, Pi.one_apply, mul_one]
rw [conditionalLikelihood, mul_assoc, ENNReal.ofReal_mul (by norm_num : (0 : ℝ) ≤ 4)]
rw [ENNReal.ofReal_ofNat, ← mul_assoc]
norm_num
rw [ENNReal.inv_mul_cancel (by norm_num) (by norm_num), one_mul]
· simp [Set.indicator_of_notMem hx]
theorem cosineFamily_rnDeriv_toReal_likelihood
The real observed density is the stated likelihood almost everywhere under the null. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: cosineFamily_rnDeriv_toReal_likelihood
lemma cosineFamily_rnDeriv_toReal_likelihood (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(lam : SignVector hL) :
(fun z => ((Pobs (cosineFamily hL hhL lam)).rnDeriv (Pobs fairNull) z).toReal)
=ᵐ[Pobs fairNull] (fun z => conditionalLikelihood hL lam z.1 z.2.1 z.2.2) := by
letI := observed_probability fairNull
rw [cosineFamily_observed_withDensity]
filter_upwards [Measure.rnDeriv_withDensity (Pobs fairNull)
(measurable_conditionalLikelihood_record hL lam).ennreal_ofReal] with z hz
rw [hz, ENNReal.toReal_ofReal (conditionalLikelihood_nonneg hL hhL lam z.1 z.2.1 z.2.2)]
Nonnegative real functions under the fair null integrate their four-cell average. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
Proof (Lean source)
-- @node: fairNull_integral_marks
lemma fairNull_integral_marks (f : O → ℝ) (hf : Measurable f) (hf0 : ∀ z, 0 ≤ f z) :
(∫ z, f z ∂Pobs fairNull) =
∫ x : Covariate, (∑ av : Bool, ∑ yv : Bool, f (x,av,yv)) / 4 := by
have hm : Measurable (fun x : Covariate => (∑ av : Bool, ∑ yv : Bool, f (x,av,yv)) / 4) := by
apply Measurable.div_const
apply Finset.measurable_fun_sum
intro av hav
apply Finset.measurable_fun_sum
intro yv hyv
exact hf.comp (by fun_prop)
rw [integral_eq_lintegral_of_nonneg_ae (ae_of_all _ hf0) hf.aestronglyMeasurable,
fairNull_lintegral_marks _ hf.ennreal_ofReal]
rw [integral_eq_lintegral_of_nonneg_ae
(ae_of_all _ (fun x => div_nonneg
(Finset.sum_nonneg (fun av _ => Finset.sum_nonneg (fun yv _ => hf0 (x,av,yv))))
(by norm_num))) hm.aestronglyMeasurable]
congr 1
apply lintegral_congr
intro x
rw [ENNReal.ofReal_div_of_pos (by norm_num : (0 : ℝ) < 4), ENNReal.ofReal_ofNat]
rw [ENNReal.ofReal_sum_of_nonneg
(fun av _ => Finset.sum_nonneg (fun yv _ => hf0 (x,av,yv)))]
simp_rw [ENNReal.ofReal_sum_of_nonneg (fun yv _ => hf0 (x,_,yv))]
simp only [div_eq_mul_inv, Finset.sum_mul, one_mul, mul_comm]
simp only [Finset.mul_sum]
Each actual conditional likelihood is at most four, since a Bernoulli cell mass is a product of two probabilities at most one. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
conditionalLikelihood hL lam x av yv ≤ 4
Proof (Lean source)
-- @node: conditionalLikelihood_le_four
lemma conditionalLikelihood_le_four (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(lam : SignVector hL) (x : Covariate) (av yv : Bool) :
conditionalLikelihood hL lam x av yv ≤ 4 := by
have hb (p : ℝ) (hp : p ∈ Icc 0 1) (b : Bool) : bernoulliMass p b ≤ 1 := by
cases b <;> simp only [bernoulliMass, Bool.false_eq_true, if_false, if_true] <;>
linarith [hp.1, hp.2]
have hm : (if av then altMu1 hL lam x else altMu0 hL lam x) ∈ Icc 0 1 := by
cases av
· exact (alt_parameters_range hL hhL lam x).2.1
· exact (alt_parameters_range hL hhL lam x).2.2
have hprod := mul_le_one₀ (hb _ (alt_parameters_range hL hhL lam x).1 av)
(bernoulliMass_nonneg hm yv) (hb _ hm yv)
unfold conditionalLikelihood
rw [mul_assoc]
linarith
theorem signOverlapIntegrand_range
The centered Fourier overlap integrand lies between minus one and fifteen. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
signOverlapIntegrand hL lam lam' x ∈ Icc (-1) 15
Proof (Lean source)
-- @node: signOverlapIntegrand_range
lemma signOverlapIntegrand_range (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(lam lam' : SignVector hL) (x : Covariate) :
signOverlapIntegrand hL lam lam' x ∈ Icc (-1) 15 := by
have hp (av yv : Bool) :
0 ≤ conditionalLikelihood hL lam x av yv * conditionalLikelihood hL lam' x av yv ∧
conditionalLikelihood hL lam x av yv * conditionalLikelihood hL lam' x av yv ≤ 16 := by
exact ⟨mul_nonneg (conditionalLikelihood_nonneg hL hhL lam x av yv)
(conditionalLikelihood_nonneg hL hhL lam' x av yv),
(mul_le_mul (conditionalLikelihood_le_four hL hhL lam x av yv)
(conditionalLikelihood_le_four hL hhL lam' x av yv)
(conditionalLikelihood_nonneg hL hhL lam' x av yv) (by norm_num)).trans
(by norm_num)⟩
have hl := Finset.sum_nonneg (s := univ)
(fun av _ => Finset.sum_nonneg (s := univ) (fun yv _ => (hp av yv).1))
have hu := Finset.sum_le_sum (s := univ)
(fun av _ => Finset.sum_le_sum (s := univ) (fun yv _ => (hp av yv).2))
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_bool, nsmul_eq_mul] at hu
have hu' : (∑ av : Bool, ∑ yv : Bool,
conditionalLikelihood hL lam x av yv * conditionalLikelihood hL lam' x av yv) ≤ 64 :=
hu.trans (by norm_num)
have he := conditionalLikelihood_pair_inner_product hL hhL lam lam' x
constructor <;> linarith [hu']
theorem integrable_signOverlapIntegrand
The covariate overlap integral is integrable by its measurable bounded construction. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
conclusion
Integrable (signOverlapIntegrand hL lam lam')
Proof (Lean source)
-- @node: integrable_signOverlapIntegrand
@[fun_prop] lemma integrable_signOverlapIntegrand (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(lam lam' : SignVector hL) : Integrable (signOverlapIntegrand hL lam lam') := by
exact Integrable.of_mem_Icc (-1) 15 (by fun_prop)
(ae_of_all _ (signOverlapIntegrand_range hL hhL lam lam'))
The paper's centered likelihood inner product integrates the Fourier overlap terms.
Definition (Lean source)
parameters
hL :
ℝ
lam lam' :
SignVector hL
def
signOverlap hL lam lam' :
ℝ
given by
∫ x : Covariate, signOverlapIntegrand hL lam lam' x
theorem signDensityOverlap_eq_one_add_signOverlap
The observed density overlap equals one plus the explicit Fourier covariate integral. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
signDensityOverlap hL hhL lam lam' = 1 + signOverlap hL lam lam'
Proof (Lean source)
-- @node: signDensityOverlap_eq_one_add_signOverlap
lemma signDensityOverlap_eq_one_add_signOverlap (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(lam lam' : SignVector hL) :
signDensityOverlap hL hhL lam lam' = 1 + signOverlap hL lam lam' := by
unfold signDensityOverlap
have he : (fun z =>
((Pobs (cosineFamily hL hhL lam)).rnDeriv (Pobs fairNull) z).toReal *
((Pobs (cosineFamily hL hhL lam')).rnDeriv (Pobs fairNull) z).toReal) =ᵐ[Pobs fairNull]
(fun z => conditionalLikelihood hL lam z.1 z.2.1 z.2.2 *
conditionalLikelihood hL lam' z.1 z.2.1 z.2.2) := by
filter_upwards [cosineFamily_rnDeriv_toReal_likelihood hL hhL lam,
cosineFamily_rnDeriv_toReal_likelihood hL hhL lam'] with z hz hz'
rw [hz, hz']
rw [integral_congr_ae he]
rw [fairNull_integral_marks _ (by fun_prop)
(fun z => mul_nonneg (conditionalLikelihood_nonneg hL hhL lam z.1 z.2.1 z.2.2)
(conditionalLikelihood_nonneg hL hhL lam' z.1 z.2.1 z.2.2))]
simp_rw [conditionalLikelihood_pair_inner_product hL hhL]
rw [integral_add (integrable_const 1) (integrable_signOverlapIntegrand hL hhL lam lam')]
simp [signOverlap]
theorem signOverlap_sign_sum
Finite averaging commutes with the bounded covariate integrals; the centered overlap has sum zero in its first sign vector for every fixed second vector. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
(∑ lam : SignVector hL, signOverlap hL lam lam') = 0
Proof (Lean source)
-- @node: signOverlap_sign_sum
lemma signOverlap_sign_sum (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(lam' : SignVector hL) : (∑ lam : SignVector hL, signOverlap hL lam lam') = 0 := by
classical
unfold signOverlap
rw [← integral_finsetSum _ (fun lam _ => integrable_signOverlapIntegrand hL hhL lam lam')]
simp_rw [signOverlapIntegrand_sign_sum hL hhL lam']
simp
theorem signOverlap_sign_mean
The centered likelihood overlap has expectation zero under the actual sign law. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
(∫ lam, signOverlap hL lam lam' ∂signLaw hL) = 0
Proof (Lean source)
-- @node: signOverlap_sign_mean
lemma signOverlap_sign_mean (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(lam' : SignVector hL) : (∫ lam, signOverlap hL lam lam' ∂signLaw hL) = 0 := by
rw [integral_signLaw, signOverlap_sign_sum hL hhL, mul_zero]
theorem signMixture_chiSq_fourier_identity
The exact fixed-sample chi-square identity uses the paper's centered Fourier overlap. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1 / 4
n :
ℕ
conclusion
= (∑ lam : SignVector hL, ∑ lam' : SignVector hL, (1 + signOverlap hL lam lam')^n) / (card (SignVector hL) : ℝ)^2
Proof (Lean source)
-- @node: signMixture_chiSq_fourier_identity
lemma signMixture_chiSq_fourier_identity (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(n : ℕ) :
1 + chiSqDiv (signMixture hL hhL n) (dataLaw n fairNull) =
(∑ lam : SignVector hL, ∑ lam' : SignVector hL,
(1 + signOverlap hL lam lam')^n) /
(card (SignVector hL) : ℝ)^2 := by
rw [signMixture_chiSq_overlap_identity]
simp_rw [signDensityOverlap_eq_one_add_signOverlap hL hhL]
theorem signMixture_chiSq_le_fourier_exp_average
The dataset divergence is bounded by the uniform exponential average of the explicit centered Fourier overlap; only the bounded-differences estimate remains to be assembled. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1 / 4
n :
ℕ
conclusion
≤ (∑ lam : SignVector hL, ∑ lam' : SignVector hL, exp ((n : ℝ)*signOverlap hL lam lam')) / (card (SignVector hL) : ℝ)^2
Proof (Lean source)
-- @node: signMixture_chiSq_le_fourier_exp_average
lemma signMixture_chiSq_le_fourier_exp_average (hL : ℝ)
(hhL : 0 < hL ∧ hL ≤ 1 / 4) (n : ℕ) :
1 + chiSqDiv (signMixture hL hhL n) (dataLaw n fairNull) ≤
(∑ lam : SignVector hL, ∑ lam' : SignVector hL,
exp ((n : ℝ)*signOverlap hL lam lam')) /
(card (SignVector hL) : ℝ)^2 := by
have hb := signMixture_chiSq_le_exp_average hL hhL n
simpa only [signDensityOverlap_eq_one_add_signOverlap hL hhL,
add_sub_cancel_left] using hb
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.Occupancy 27 declarations Occupancy cancellation and design-uniform population bias and denominator bounds.
Occupancy cancellation and design-uniform population bias and denominator bounds.
theorem design_mass_bounds
Density bounds compare every design-event mass with its Lebesgue mass. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: design_mass_bounds
lemma design_mass_bounds (P : CausalLaw) (hP : CompleteModel P) (s : Set Covariate) :
volume.real s / 2 ≤ (PX P).real s ∧
(PX P).real s ≤ 3 * volume.real s / 2 := by
let : IsProbabilityMeasure (PX P) := Measure.isProbabilityMeasure_map measurable_X.aemeasurable
have hlo : (ofReal (1 / 2 : ℝ)) • (volume : Measure Covariate) ≤ PX P := by
rw [PX, P.density_version hP.density.1, ← withDensity_const]
apply withDensity_mono
filter_upwards [hP.density.2] with x hx
exact ENNReal.ofReal_le_ofReal hx.1
have hhi : PX P ≤ (ofReal (3 / 2 : ℝ)) • (volume : Measure Covariate) := by
rw [PX, P.density_version hP.density.1, ← withDensity_const]
apply withDensity_mono
filter_upwards [hP.density.2] with x hx
exact ENNReal.ofReal_le_ofReal hx.2
have hl := ENNReal.toReal_mono (measure_ne_top (PX P) s) (hlo s)
have hu := ENNReal.toReal_mono
(show ((ofReal (3 / 2 : ℝ)) • (volume : Measure Covariate)) s ≠ ∞ by
simp only [Measure.smul_apply, smul_eq_mul]
exact ENNReal.mul_ne_top ENNReal.ofReal_ne_top (measure_ne_top volume s)) (hhi s)
simp only [Measure.smul_apply, smul_eq_mul, ENNReal.toReal_mul,
ENNReal.toReal_ofReal (by norm_num : (0 : ℝ) ≤ 1 / 2),
ENNReal.toReal_ofReal (by norm_num : (0 : ℝ) ≤ 3 / 2), ← measureReal_def] at hl hu
constructor <;> linarith
Each cell's endpoints lie in the unit interval and differ by the public width. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
conclusion 1
0 ≤ (x0 : ℝ) - h + j.val * cellWidth h k
conclusion 2
(x0 : ℝ) - h + (j.val + 1) * cellWidth h k ≤ 1
conclusion 3
(x0 : ℝ) - h + (j.val + 1) * cellWidth h k ≤ (x0 : ℝ) + h
conclusion 4
0 < cellWidth h k
Proof (Lean source)
-- @node: cell_endpoint_bounds
lemma cell_endpoint_bounds (k : ℕ) (h : ℝ) (hk : 0 < k)
(hh : 0 < h ∧ h ≤ 1 / 4) (j : Fin k) :
0 ≤ (x0 : ℝ) - h + j.val * cellWidth h k ∧
(x0 : ℝ) - h + (j.val + 1) * cellWidth h k ≤ 1 ∧
(x0 : ℝ) - h + (j.val + 1) * cellWidth h k ≤ (x0 : ℝ) + h ∧
0 < cellWidth h k := by
have hkR : (0 : ℝ) < k := by exact_mod_cast hk
have hh0 := hh.1
have hd : 0 < cellWidth h k := by unfold cellWidth; positivity
have hwidth : (k : ℝ) * cellWidth h k = 2 * h := by
unfold cellWidth
field_simp
have hj : (j.val : ℝ) + 1 ≤ k := by exact_mod_cast j.isLt
have hju := mul_le_mul_of_nonneg_right hj hd.le
have hjl := mul_nonneg (Nat.cast_nonneg j.val) hd.le
simp only [x0] at *
refine ⟨?_, ?_, ?_, hd⟩ <;> linarith
Including the final right endpoint does not change the Lebesgue mass of a cell. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
volume.real (cell h k j) = cellWidth h k
Proof (Lean source)
-- @node: cell_volume
lemma cell_volume (k : ℕ) (h : ℝ) (hk : 0 < k)
(hh : 0 < h ∧ h ≤ 1 / 4) (j : Fin k) :
volume.real (cell h k j) = cellWidth h k := by
obtain ⟨hl, hu, hwindow, hd⟩ := cell_endpoint_bounds k h hk hh j
let a : Covariate := ⟨(x0 : ℝ) - h + j.val * cellWidth h k,
hl, by nlinarith⟩
let b : Covariate := ⟨(x0 : ℝ) - h + (j.val + 1) * cellWidth h k,
by nlinarith, hu⟩
have hab : (b : ℝ) - (a : ℝ) = cellWidth h k := by dsimp [a, b]; ring
have hsub : Ico a b ⊆ cell h k j := by
intro x hx
exact ⟨hx.1, Or.inl hx.2⟩
have hsup : cell h k j ⊆ Icc a b := by
intro x hx
refine ⟨hx.1, ?_⟩
rcases hx.2 with hx | ⟨hj, hx⟩
· exact hx.le
· change (x : ℝ) ≤ (x0 : ℝ) - h + (j.val + 1) * cellWidth h k
have he : ((j.val + 1 : ℕ) : ℝ) = k := by exact_mod_cast hj
have hw : (k : ℝ) * cellWidth h k = 2 * h := by
unfold cellWidth
have : (k : ℝ) ≠ 0 := by exact_mod_cast (Nat.ne_of_gt hk)
field_simp
rw [Nat.cast_add, Nat.cast_one] at he
rw [hx, he, hw]
linarith
have hlo := measure_mono (μ := (volume : Measure Covariate)) hsub
have hhi := measure_mono (μ := (volume : Measure Covariate)) hsup
rw [unitInterval.volume_Ico, hab] at hlo
rw [unitInterval.volume_Icc, hab] at hhi
have heq := le_antisymm hhi hlo
rw [measureReal_def, heq, ENNReal.toReal_ofReal hd.le]
theorem cellMass_density_bounds
The complete model supplies both cellwise mass estimates without design smoothness. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
conclusion 1
cellWidth h k / 2 ≤ cellMass h k P j
conclusion 2
cellMass h k P j ≤ 3 * cellWidth h k / 2
Proof (Lean source)
-- @node: cellMass_density_bounds
lemma cellMass_density_bounds (k : ℕ) (h : ℝ) (hk : 0 < k)
(hh : 0 < h ∧ h ≤ 1 / 4) (P : CausalLaw) (hP : CompleteModel P) (j : Fin k) :
cellWidth h k / 2 ≤ cellMass h k P j ∧
cellMass h k P j ≤ 3 * cellWidth h k / 2 := by
have hb := design_mass_bounds P hP (cell h k j)
rw [cell_volume k h hk hh j] at hb
exact hb
theorem local_cellMass_sum_bounds
Summing equal-cell mass bounds gives the local-window mass bounds. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
Proof (Lean source)
-- @node: local_cellMass_sum_bounds
lemma local_cellMass_sum_bounds (k : ℕ) (h : ℝ) (hk : 0 < k)
(hh : 0 < h ∧ h ≤ 1 / 4) (P : CausalLaw) (hP : CompleteModel P) :
h ≤ ∑ j : Fin k, cellMass h k P j ∧
(∑ j : Fin k, cellMass h k P j) ≤ 3 * h := by
have hl := Finset.sum_le_sum (s := univ)
(fun (j : Fin k) _ => (cellMass_density_bounds k h hk hh P hP j).1)
have hu := Finset.sum_le_sum (s := univ)
(fun (j : Fin k) _ => (cellMass_density_bounds k h hk hh P hP j).2)
have hw : (k : ℝ) * cellWidth h k = 2 * h := by
unfold cellWidth
have : (k : ℝ) ≠ 0 := by exact_mod_cast (Nat.ne_of_gt hk)
field_simp
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul] at hl hu
constructor <;> nlinarith
theorem cellMass_bounds
Every population cell mass lies between zero and one. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: cellMass_bounds
lemma cellMass_bounds (h : ℝ) (k : ℕ) (P : CausalLaw) (j : Fin k) :
0 ≤ cellMass h k P j ∧ cellMass h k P j ≤ 1 := by
let : IsProbabilityMeasure (PX P) := Measure.isProbabilityMeasure_map measurable_X.aemeasurable
exact ⟨measureReal_nonneg, measureReal_le_one⟩
The probability of no partner has the usual exponential envelope. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: occupancy_no_partner_exp
lemma occupancy_no_partner_exp (p : ℝ) (m : ℕ) (hp : p ≤ 1) :
(1 - p) ^ m ≤ exp (-(m : ℝ) * p) := by
calc
_ ≤ (exp (-p)) ^ m :=
pow_le_pow_left₀ (sub_nonneg.mpr hp) (Real.one_sub_le_exp_neg p) m
_ = _ := by rw [← Real.exp_nat_mul]; congr 1; ring
The union bound controls the probability that a cell has a partner. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: occupancy_partner_union_bound
lemma occupancy_partner_union_bound (p : ℝ) (m : ℕ)
(hp : 0 ≤ p ∧ p ≤ 1) : 1 - (1 - p) ^ m ≤ min 1 ((m : ℝ) * p) := by
apply le_min
· have := pow_nonneg (sub_nonneg.mpr hp.2) m
linarith
· have hb := one_add_mul_le_pow (a := -p) (by linarith [hp.2]) m
have heq : 1 + -p = 1 - p := by ring
rw [heq] at hb
linarith
The exponential partner probability dominates a rational envelope. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: occupancy_exp_rational_lower
lemma occupancy_exp_rational_lower (x : ℝ) (hx : 0 ≤ x) :
x / (1 + x) ≤ 1 - exp (-x) := by
have hpos : 0 < 1 + x := by linarith
have hE : 1 + x ≤ exp x := by linarith [Real.add_one_le_exp x]
have hi : exp (-x) ≤ (1 + x)⁻¹ := by
rw [Real.exp_neg]
exact (inv_le_inv₀ (Real.exp_pos x) hpos).mpr hE
have hid : x / (1 + x) = 1 - (1 + x)⁻¹ := by
field_simp
<;> ring
rw [hid]
linarith
The rational envelope controls both sparse and dense occupancies uniformly. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: occupancy_rational_truncation
lemma occupancy_rational_truncation (x : ℝ) (hx : 0 ≤ x) :
min 1 x / 8 ≤ x / (4 + x) := by
have hd : 0 < 4 + x := by linarith
rw [le_div_iff₀ hd]
by_cases hsmall : x ≤ 1
· rw [min_eq_right hsmall]
nlinarith [mul_nonneg hx (sub_nonneg.mpr hsmall)]
· rw [min_eq_left (le_of_not_ge hsmall)]
linarith
The lower cell-mass bound gives a uniform partner-probability lower bound. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
n :
ℕ
delta p :
ℝ
hn :
2 ≤ n
hd :
0 ≤ delta
hp :
delta / 2 ≤ p
hp1 :
p ≤ 1
conclusion
min 1 ((n : ℝ) * delta) / 8 ≤ 1 - (1 - p) ^ (n - 1)
Proof (Lean source)
-- @node: occupancy_partner_lower
lemma occupancy_partner_lower (n : ℕ) (delta p : ℝ) (hn : 2 ≤ n)
(hd : 0 ≤ delta) (hp : delta / 2 ≤ p) (hp1 : p ≤ 1) :
min 1 ((n : ℝ) * delta) / 8 ≤ 1 - (1 - p) ^ (n - 1) := by
have hn0 : 0 ≤ (n : ℝ) := Nat.cast_nonneg n
have hn2 : (2 : ℝ) ≤ n := by exact_mod_cast hn
have hm : ((n - 1 : ℕ) : ℝ) = (n : ℝ) - 1 := by
rw [Nat.cast_sub (by omega), Nat.cast_one]
have hp0 : 0 ≤ p := by linarith
have hprod : (n : ℝ) * delta / 4 ≤ ((n - 1 : ℕ) : ℝ) * p := by
rw [hm]
nlinarith [mul_nonneg (show 0 ≤ (n : ℝ) / 2 by positivity)
(show 0 ≤ p - delta / 2 by linarith),
mul_nonneg (show 0 ≤ (n : ℝ) - 1 - (n : ℝ) / 2 by linarith) hp0]
have hpow : (1 - p) ^ (n - 1) ≤ exp (-((n : ℝ) * delta / 4)) :=
(occupancy_no_partner_exp p (n - 1) hp1).trans
(Real.exp_le_exp.mpr (by linarith))
calc
_ ≤ ((n : ℝ) * delta) / (4 + (n : ℝ) * delta) :=
occupancy_rational_truncation _ (mul_nonneg hn0 hd)
_ = ((n : ℝ) * delta / 4) / (1 + (n : ℝ) * delta / 4) := by
have : 4 + (n : ℝ) * delta ≠ 0 := by positivity
field_simp
<;> ring
_ ≤ 1 - exp (-((n : ℝ) * delta / 4)) :=
occupancy_exp_rational_lower _ (by positivity)
_ ≤ _ := by linarith
The upper cell-mass bound gives a uniform partner-probability upper bound. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
n :
ℕ
delta p :
ℝ
hd :
0 ≤ delta
hp :
0 ≤ p ∧ p ≤ 1
hpu :
p ≤ 3 * delta / 2
conclusion
1 - (1 - p) ^ (n - 1) ≤ (3 / 2) * min 1 ((n : ℝ) * delta)
Proof (Lean source)
-- @node: occupancy_partner_upper
lemma occupancy_partner_upper (n : ℕ) (delta p : ℝ)
(hd : 0 ≤ delta) (hp : 0 ≤ p ∧ p ≤ 1) (hpu : p ≤ 3 * delta / 2) :
1 - (1 - p) ^ (n - 1) ≤ (3 / 2) * min 1 ((n : ℝ) * delta) := by
have hb := occupancy_partner_union_bound p (n - 1) hp
have hm : ((n - 1 : ℕ) : ℝ) ≤ n := by exact_mod_cast sub_le n 1
have hn0 : 0 ≤ (n : ℝ) := Nat.cast_nonneg n
have hmp : ((n - 1 : ℕ) : ℝ) * p ≤ (3 / 2) * ((n : ℝ) * delta) := by
nlinarith [mul_nonneg (show 0 ≤ (n : ℝ) - ((n - 1 : ℕ) : ℝ) by linarith) hp.1,
mul_nonneg hn0 (show 0 ≤ 3 * delta / 2 - p by linarith)]
by_cases hs : (n : ℝ) * delta ≤ 1
· rw [min_eq_right hs]
exact (hb.trans (min_le_right _ _)).trans hmp
· rw [min_eq_left (le_of_not_ge hs)]
linarith [hb.trans (min_le_left _ _)]
theorem occupancy_comparisons_of_cell_masses
Summing the partner envelopes gives both design-uniform occupancy comparisons. The premises are the cell-mass estimates supplied by the model's density bounds. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
n k :
ℕ
h :
ℝ
P :
hn :
2 ≤ n
hd :
0 ≤ cellWidth h k
hp :
∀ j : Fin k,
cellWidth h k / 2 ≤ cellMass h k P j ∧ cellMass h k P j ≤ 3 * cellWidth h k / 2
Proof (Lean source)
-- @node: occupancy_comparisons_of_cell_masses
lemma occupancy_comparisons_of_cell_masses (n k : ℕ) (h : ℝ) (P : CausalLaw)
(hn : 2 ≤ n) (hd : 0 ≤ cellWidth h k)
(hp : ∀ j : Fin k, cellWidth h k / 2 ≤ cellMass h k P j ∧
cellMass h k P j ≤ 3 * cellWidth h k / 2)
(hs : h ≤ ∑ j : Fin k, cellMass h k P j ∧
(∑ j : Fin k, cellMass h k P j) ≤ 3 * h) :
info n h k / 8 ≤ Wocc n h k P ∧ Wocc n h k P ≤ 9 * info n h k / 2 := by
have hn0 : 0 ≤ (n : ℝ) := Nat.cast_nonneg n
have ht : 0 ≤ min 1 ((n : ℝ) * cellWidth h k) :=
le_min (by norm_num) (mul_nonneg hn0 hd)
have hsumlo :
(∑ j : Fin k, (n : ℝ) * cellMass h k P j *
(min 1 ((n : ℝ) * cellWidth h k) / 8)) ≤ Wocc n h k P := by
apply Finset.sum_le_sum
intro j hj
exact mul_le_mul_of_nonneg_left
(occupancy_partner_lower n _ _ hn hd (hp j).1 (cellMass_bounds h k P j).2)
(mul_nonneg hn0 (cellMass_bounds h k P j).1)
have hsumhi : Wocc n h k P ≤
∑ j : Fin k, (n : ℝ) * cellMass h k P j *
((3 / 2) * min 1 ((n : ℝ) * cellWidth h k)) := by
apply Finset.sum_le_sum
intro j hj
exact mul_le_mul_of_nonneg_left
(occupancy_partner_upper n _ _ hd (cellMass_bounds h k P j) (hp j).2)
(mul_nonneg hn0 (cellMass_bounds h k P j).1)
simp only [← Finset.sum_mul, ← Finset.mul_sum] at hsumlo hsumhi
constructor
· have hmass := mul_le_mul_of_nonneg_right
(mul_le_mul_of_nonneg_left hs.1 hn0) (show 0 ≤ min 1 ((n : ℝ) * cellWidth h k) / 8 by positivity)
unfold info
nlinarith
· have hmass := mul_le_mul_of_nonneg_right
(mul_le_mul_of_nonneg_left hs.2 hn0)
(show 0 ≤ (3 / 2 : ℝ) * min 1 ((n : ℝ) * cellWidth h k) by positivity)
unfold info
nlinarith
theorem occupancy_comparisons
The density bounds imply both occupancy comparisons for every law in the complete model. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: occupancy_comparisons
lemma occupancy_comparisons (n k : ℕ) (h : ℝ) (hn : 2 ≤ n) (hk : 0 < k)
(hh : 0 < h ∧ h ≤ 1 / 4) (P : CausalLaw) (hP : CompleteModel P) :
info n h k / 8 ≤ Wocc n h k P ∧ Wocc n h k P ≤ 9 * info n h k / 2 := by
exact occupancy_comparisons_of_cell_masses n k h P hn
(cell_endpoint_bounds k h hk hh ⟨0, hk⟩).2.2.2.le
(cellMass_density_bounds k h hk hh P hP)
(local_cellMass_sum_bounds k h hk hh P hP)
theorem occupancyWeight_nonneg
The common occupancy weights are nonnegative, even without model assumptions. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: occupancyWeight_nonneg
lemma occupancyWeight_nonneg (n k : ℕ) (h : ℝ) (P : CausalLaw) (j : Fin k) :
0 ≤ occupancyWeight n h k P j := by
obtain ⟨hp0, hp1⟩ := cellMass_bounds h k P j
have hpow : (1 - cellMass h k P j) ^ (n - 1) ≤ 1 :=
pow_le_one₀ (sub_nonneg.mpr hp1) (by linarith)
exact mul_nonneg (mul_nonneg (Nat.cast_nonneg n) hp0) (sub_nonneg.mpr hpow)
The public information and denominator floor are strictly positive on the public domain. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: info_d0_pos
lemma info_d0_pos (n k : ℕ) (h : ℝ) (hn : 0 < n) (hk : 0 < k) (hh : 0 < h) :
0 < info n h k ∧ 0 < d0 n h k := by
have hn' : 0 < (n : ℝ) := Nat.cast_pos.mpr hn
have hk' : 0 < (k : ℝ) := Nat.cast_pos.mpr hk
have hw : 0 < cellWidth h k := by unfold cellWidth; positivity
have hi : 0 < info n h k := by unfold info; positivity
exact ⟨hi, by unfold d0; positivity⟩
theorem occupancy_denominator_lower
A common nonnegative weight preserves the cellwise denominator lower bound. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: occupancy_denominator_lower
lemma occupancy_denominator_lower (n k : ℕ) (h : ℝ) (P : CausalLaw)
(hd : ∀ j : Fin k, 3 / 8 ≤ dj h k P j) :
(3 / 8) * Wocc n h k P ≤
∑ j : Fin k, occupancyWeight n h k P j * dj h k P j := by
unfold Wocc
rw [Finset.mul_sum]
apply Finset.sum_le_sum
intro j hj
calc
(3 / 8) * occupancyWeight n h k P j = occupancyWeight n h k P j * (3 / 8) := mul_comm _ _
_ ≤ _ := mul_le_mul_of_nonneg_left (hd j) (occupancyWeight_nonneg n k h P j)
theorem occupancy_weighted_bias
Common nonnegative weights preserve the conditional cell bias certificate. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
n k :
ℕ
h :
ℝ
P :
conclusion
Proof (Lean source)
-- @node: occupancy_weighted_bias
lemma occupancy_weighted_bias (n k : ℕ) (h : ℝ) (P : CausalLaw)
(hb : ∀ j : Fin k,
|nu h k P j - theta P * dj h k P j| ≤ bias h k * dj h k P j) :
|(∑ j : Fin k, occupancyWeight n h k P j * nu h k P j) -
theta P * (∑ j : Fin k, occupancyWeight n h k P j * dj h k P j)| ≤
bias h k * (∑ j : Fin k, occupancyWeight n h k P j * dj h k P j) := by
classical
calc
_ = |∑ j : Fin k, occupancyWeight n h k P j *
(nu h k P j - theta P * dj h k P j)| := by
congr 1
simp only [mul_sub, Finset.sum_sub_distrib, Finset.mul_sum]
congr 1
apply Finset.sum_congr rfl
intro j hj
ring
_ ≤ ∑ j : Fin k, |occupancyWeight n h k P j *
(nu h k P j - theta P * dj h k P j)| := Finset.abs_sum_le_sum_abs _ _
_ ≤ ∑ j : Fin k, occupancyWeight n h k P j * (bias h k * dj h k P j) := by
apply Finset.sum_le_sum
intro j hj
rw [abs_mul, abs_of_nonneg (occupancyWeight_nonneg n k h P j)]
exact mul_le_mul_of_nonneg_left (hb j) (occupancyWeight_nonneg n k h P j)
_ = _ := by
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro j hj
ring
Dividing a weighted bias certificate by a positive denominator gives the ratio bound. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
N D target b :
ℝ
hD :
0 < D
hb :
|N - target * D| ≤ b * D
conclusion
|N / D - target| ≤ b
Proof (Lean source)
-- @node: ratio_bias_of_positive_denominator
lemma ratio_bias_of_positive_denominator (N D target b : ℝ) (hD : 0 < D)
(hb : |N - target * D| ≤ b * D) : |N / D - target| ≤ b := by
have heq : N / D - target = (N - target * D) / D := by
field_simp
rw [heq, abs_div, abs_of_pos hD]
exact (div_le_iff₀ hD).mpr hb
The independent cross-treatment probability is at least three eighths under overlap. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
e e' :
ℝ
he :
1 / 4 ≤ e ∧ e ≤ 3 / 4
he' :
1 / 4 ≤ e' ∧ e' ≤ 3 / 4
conclusion
conclusion 1
3 / 8 ≤ e * (1 - e') + e' * (1 - e)
conclusion 2
e * (1 - e') + e' * (1 - e) ≤ 1
Proof (Lean source)
-- @node: cross_treatment_denominator_bounds
lemma cross_treatment_denominator_bounds (e e' : ℝ)
(he : 1 / 4 ≤ e ∧ e ≤ 3 / 4) (he' : 1 / 4 ≤ e' ∧ e' ≤ 3 / 4) :
3 / 8 ≤ e * (1 - e') + e' * (1 - e) ∧
e * (1 - e') + e' * (1 - e) ≤ 1 := by
have h1 := mul_nonneg (sub_nonneg.mpr he.1) (sub_nonneg.mpr he'.2)
have h2 := mul_nonneg (sub_nonneg.mpr he.2) (sub_nonneg.mpr he'.1)
have h3 := mul_nonneg (show 0 ≤ e by linarith) (show 0 ≤ e' by linarith)
have h4 := mul_nonneg (show 0 ≤ 1 - e by linarith) (show 0 ≤ 1 - e' by linarith)
constructor <;> nlinarith
The pointwise conditional numerator identity gives the paper's cellwise bias envelope. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
e e' residual tau0 tau1 target h u :
ℝ
he :
1 / 4 ≤ e ∧ e ≤ 3 / 4
he' :
1 / 4 ≤ e' ∧ e' ≤ 3 / 4
hr :
|residual| ≤ 9 * u
ht0 :
|tau0 - target| ≤ 3 * h
ht1 :
|tau1 - target| ≤ 3 * h
conclusion
|residual
+ e * (1 - e') * tau0
+ e' * (1 - e) * tau1
- target * (e * (1 - e') + e' * (1 - e))|
≤ (3 * h + 24 * u) * (e * (1 - e') + e' * (1 - e))
Proof (Lean source)
-- @node: cross_treatment_bias_bound
lemma cross_treatment_bias_bound (e e' residual tau0 tau1 target h u : ℝ)
(he : 1 / 4 ≤ e ∧ e ≤ 3 / 4) (he' : 1 / 4 ≤ e' ∧ e' ≤ 3 / 4)
(hr : |residual| ≤ 9 * u)
(ht0 : |tau0 - target| ≤ 3 * h) (ht1 : |tau1 - target| ≤ 3 * h) :
|residual + e * (1 - e') * tau0 + e' * (1 - e) * tau1 -
target * (e * (1 - e') + e' * (1 - e))| ≤
(3 * h + 24 * u) * (e * (1 - e') + e' * (1 - e)) := by
have hw0 : 0 ≤ e * (1 - e') :=
mul_nonneg (by linarith [he.1]) (by linarith [he'.2])
have hw1 : 0 ≤ e' * (1 - e) :=
mul_nonneg (by linarith [he'.1]) (by linarith [he.2])
have hu : 0 ≤ u := by nlinarith [abs_nonneg residual]
have hd := (cross_treatment_denominator_bounds e e' he he').1
calc
_ = |residual + (e * (1 - e') * (tau0 - target) +
e' * (1 - e) * (tau1 - target))| := by congr 1; ring
_ ≤ |residual| + |e * (1 - e') * (tau0 - target) +
e' * (1 - e) * (tau1 - target)| := abs_add_le _ _
_ ≤ |residual| + (|e * (1 - e') * (tau0 - target)| +
|e' * (1 - e) * (tau1 - target)|) := by
exact add_le_add (le_refl |residual|)
(abs_add_le (e * (1 - e') * (tau0 - target)) (e' * (1 - e) * (tau1 - target)))
_ ≤ 9 * u + (e * (1 - e') * (3 * h) + e' * (1 - e) * (3 * h)) := by
rw [abs_mul (e * (1 - e')), abs_mul (e' * (1 - e)),
abs_of_nonneg hw0, abs_of_nonneg hw1]
exact add_le_add hr (add_le_add
(mul_le_mul_of_nonneg_left ht0 hw0) (mul_le_mul_of_nonneg_left ht1 hw1))
_ ≤ _ := by nlinarith [mul_nonneg hu (sub_nonneg.mpr hd)]
Every point of a public cell lies within the public radius of the target covariate. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: cell_distance_to_target
lemma cell_distance_to_target (k : ℕ) (h : ℝ) (hk : 0 < k)
(hh : 0 < h ∧ h ≤ 1 / 4) (j : Fin k) (x : Covariate)
(hx : x ∈ cell h k j) : |(x : ℝ) - x0| ≤ h := by
obtain ⟨hl, hu, hw, hd⟩ := cell_endpoint_bounds k h hk hh j
have hj : 0 ≤ (j.val : ℝ) * cellWidth h k :=
mul_nonneg (Nat.cast_nonneg _) hd.le
have hxlo : (x0 : ℝ) - h ≤ (x : ℝ) := by linarith [hx.1]
have hxhi : (x : ℝ) ≤ (x0 : ℝ) + h := by
rcases hx.2 with hx | ⟨_, hx⟩
· linarith
· exact hx.le
exact abs_le.mpr ⟨by linarith, by linarith⟩
Two covariates in one cell are separated by at most the public cell width, including the right endpoint of the final cell. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: cell_pair_distance
lemma cell_pair_distance (k : ℕ) (h : ℝ) (hk : 0 < k)
(hh : 0 < h ∧ h ≤ 1 / 4) (j : Fin k) (x x' : Covariate)
(hx : x ∈ cell h k j) (hx' : x' ∈ cell h k j) :
|(x : ℝ) - x'| ≤ cellWidth h k := by
have hwidth : (k : ℝ) * cellWidth h k = 2 * h := by
unfold cellWidth
have hkR : (k : ℝ) ≠ 0 := by exact_mod_cast (Nat.ne_of_gt hk)
field_simp
have hupper : ∀ z : Covariate, z ∈ cell h k j →
(z : ℝ) ≤ (x0 : ℝ) - h + (j.val + 1) * cellWidth h k := by
intro z hz
rcases hz.2 with hz | ⟨hj, hz⟩
· exact hz.le
· have hjR : (j.val : ℝ) + 1 = k := by exact_mod_cast hj
rw [hz, hjR, hwidth]
linarith
have hu := hupper x hx
have hu' := hupper x' hx'
exact abs_le.mpr ⟨by linarith [hx.1], by linarith [hx'.1]⟩
theorem cell_nuisance_product_bound
The two one-tenth Holder remainders multiply to the one-fifth cell-width term. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
k :
ℕ
h :
ℝ
hk :
0 < k
hh :
0 < h ∧ h ≤ 1 / 4
P :
hP :
j :
Fin k
x x' :
hx :
x ∈ cell h k j
hx' :
x' ∈ cell h k j
conclusion
|(P.e x - P.e x') * (P.mu0 x - P.mu0 x')| ≤ 9 * (cellWidth h k) ^ (1 / 5 : ℝ)
Proof (Lean source)
-- @node: cell_nuisance_product_bound
lemma cell_nuisance_product_bound (k : ℕ) (h : ℝ) (hk : 0 < k)
(hh : 0 < h ∧ h ≤ 1 / 4) (P : CausalLaw) (hP : CompleteModel P)
(j : Fin k) (x x' : Covariate) (hx : x ∈ cell h k j)
(hx' : x' ∈ cell h k j) :
|(P.e x - P.e x') * (P.mu0 x - P.mu0 x')| ≤
9 * (cellWidth h k) ^ (1 / 5 : ℝ) := by
have hdelta := (cell_endpoint_bounds k h hk hh j).2.2.2
have hd := cell_pair_distance k h hk hh j x x' hx hx'
have he := hP.propensity_holder x x'
have hm := hP.control_holder x x'
change |P.e x - P.e x'| ≤ 3 * |(x : ℝ) - x'| ^ (1 / 10 : ℝ) at he
change |P.mu0 x - P.mu0 x'| ≤ 3 * |(x : ℝ) - x'| ^ (1 / 10 : ℝ) at hm
have hr : |(x : ℝ) - x'| ^ (1 / 10 : ℝ) ≤
cellWidth h k ^ (1 / 10 : ℝ) :=
Real.rpow_le_rpow (abs_nonneg _) hd (by norm_num)
have hpow : cellWidth h k ^ (1 / 10 : ℝ) *
cellWidth h k ^ (1 / 10 : ℝ) = cellWidth h k ^ (1 / 5 : ℝ) := by
rw [← Real.rpow_add (cell_endpoint_bounds k h hk hh j).2.2.2]
norm_num
calc
_ = |P.e x - P.e x'| * |P.mu0 x - P.mu0 x'| := abs_mul _ _
_ ≤ (3 * cellWidth h k ^ (1 / 10 : ℝ)) *
(3 * cellWidth h k ^ (1 / 10 : ℝ)) :=
mul_le_mul (he.trans (by linarith)) (hm.trans (by linarith))
(abs_nonneg _) (by positivity)
_ = 9 * cellWidth h k ^ (1 / 5 : ℝ) := by nlinarith [hpow]
theorem independent_pair_numerator_expansion Lemma independent_pair_numerator_expansion in the paper ↗
Expanding the two cross-treatment cases separates nuisance variation from the contrast. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
e e' m m' t t' :
ℝ
conclusion
e * (1 - e') * (m + t - m') + e' * (1 - e) * (m' + t' - m)
= (e - e') * (m - m') + e * (1 - e') * t + e' * (1 - e) * t'
Proof (Lean source)
-- @node: independent_pair_numerator_expansion
lemma independent_pair_numerator_expansion (e e' m m' t t' : ℝ) :
e * (1 - e') * (m + t - m') + e' * (1 - e) * (m' + t' - m) =
(e - e') * (m - m') + e * (1 - e') * t + e' * (1 - e) * t' := by
ring
theorem cell_cross_treatment_certificate
At any two covariates in one cell, the independent binary cross-treatment moments satisfy both the denominator lower bound and the paper's complete bias envelope. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
k :
ℕ
h :
ℝ
hk :
0 < k
hh :
0 < h ∧ h ≤ 1 / 4
P :
hP :
j :
Fin k
x x' :
hx :
x ∈ cell h k j
hx' :
x' ∈ cell h k j
conclusion
conclusion 1
3 / 8 ≤ P.e x * (1 - P.e x') + P.e x' * (1 - P.e x)
conclusion 2
|P.e x * (1 - P.e x') * (P.mu1 x - P.mu0 x')
+ P.e x' * (1 - P.e x) * (P.mu1 x' - P.mu0 x)
- theta P * (P.e x * (1 - P.e x') + P.e x' * (1 - P.e x))|
≤ bias h k * (P.e x * (1 - P.e x') + P.e x' * (1 - P.e x))
Proof (Lean source)
-- @node: cell_cross_treatment_certificate
lemma cell_cross_treatment_certificate (k : ℕ) (h : ℝ) (hk : 0 < k)
(hh : 0 < h ∧ h ≤ 1 / 4) (P : CausalLaw) (hP : CompleteModel P)
(j : Fin k) (x x' : Covariate) (hx : x ∈ cell h k j)
(hx' : x' ∈ cell h k j) :
3 / 8 ≤ P.e x * (1 - P.e x') + P.e x' * (1 - P.e x) ∧
|P.e x * (1 - P.e x') * (P.mu1 x - P.mu0 x') +
P.e x' * (1 - P.e x) * (P.mu1 x' - P.mu0 x) -
theta P * (P.e x * (1 - P.e x') + P.e x' * (1 - P.e x))| ≤
bias h k * (P.e x * (1 - P.e x') + P.e x' * (1 - P.e x)) := by
have ht : ∀ z : Covariate, z ∈ cell h k j → |tau P z - theta P| ≤ 3 * h := by
intro z hz
have hc : |tau P z - theta P| ≤ 3 * |(z : ℝ) - x0| :=
hP.contrast_lipschitz z x0
exact hc.trans
(mul_le_mul_of_nonneg_left (cell_distance_to_target k h hk hh j z hz) (by norm_num))
refine ⟨(cross_treatment_denominator_bounds _ _ (hP.overlap x) (hP.overlap x')).1, ?_⟩
have hn := independent_pair_numerator_expansion (P.e x) (P.e x')
(P.mu0 x) (P.mu0 x') (tau P x) (tau P x')
simp only [tau] at hn
have hb := cross_treatment_bias_bound (P.e x) (P.e x')
((P.e x - P.e x') * (P.mu0 x - P.mu0 x')) (tau P x) (tau P x')
(theta P) h (cellWidth h k ^ (1 / 5 : ℝ)) (hP.overlap x) (hP.overlap x')
(cell_nuisance_product_bound k h hk hh P hP j x x' hx hx') (ht x hx) (ht x' hx')
rw [show P.mu0 x + (P.mu1 x - P.mu0 x) = P.mu1 x by ring,
show P.mu0 x' + (P.mu1 x' - P.mu0 x') = P.mu1 x' by ring] at hn
rw [hn]
exact hb
theorem occupancy_ratio_assembly
The population identities and cellwise certificates imply the floor and ratio bias bounds. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
n k :
ℕ
h :
ℝ
P :
hn :
0 < n
hk :
0 < k
hh :
0 < h
hN :
hD :
Proof (Lean source)
-- @node: occupancy_ratio_assembly
lemma occupancy_ratio_assembly (n k : ℕ) (h : ℝ) (P : CausalLaw)
(Q : Measure (Dataset n)) (hn : 0 < n) (hk : 0 < k) (hh : 0 < h)
(hN : Nbar n h k Q = ∑ j : Fin k, occupancyWeight n h k P j * nu h k P j)
(hD : Dbar n h k Q = ∑ j : Fin k, occupancyWeight n h k P j * dj h k P j)
(hc : ∀ j : Fin k, 3 / 8 ≤ dj h k P j ∧
|nu h k P j - theta P * dj h k P j| ≤ bias h k * dj h k P j)
(hW : info n h k / 8 ≤ Wocc n h k P) :
d0 n h k ≤ Dbar n h k Q ∧
|Nbar n h k Q / Dbar n h k Q - theta P| ≤ bias h k := by
have hden : (3 / 8) * Wocc n h k P ≤ Dbar n h k Q := by
rw [hD]
exact occupancy_denominator_lower n k h P (fun j => (hc j).1)
have hfloor : d0 n h k ≤ Dbar n h k Q := by
unfold d0
linarith
have hpos : 0 < Dbar n h k Q :=
lt_of_lt_of_le (info_d0_pos n k h hn hk hh).2 hfloor
have hb : |Nbar n h k Q - theta P * Dbar n h k Q| ≤ bias h k * Dbar n h k Q := by
rw [hN, hD]
exact occupancy_weighted_bias n k h P (fun j => (hc j).2)
exact ⟨hfloor, ratio_bias_of_positive_denominator _ _ _ _ hpos hb⟩
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.PartnerProbability 13 declarations Product-law probabilities of a record having another record in a specified cell.
Product-law probabilities of a record having another record in a specified cell.
theorem observed_cell_mass
The observed probability of a covariate cell equals its design probability. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: observed_cell_mass
lemma observed_cell_mass (h : ℝ) (k : ℕ) (P : CausalLaw) (j : Fin k) :
(Pobs P).real {z : O | z.1 ∈ cell h k j} = cellMass h k P j := by
have hs : MeasurableSet {z : O | z.1 ∈ cell h k j} :=
(measurableSet_cell h k j).preimage measurable_fst
simp only [cellMass, PX, Pobs, measureReal_def]
rw [Measure.map_apply measurable_observe hs,
Measure.map_apply measurable_X (measurableSet_cell h k j)]
rfl
theorem iid_rectangle_real
A product rectangle has real mass equal to the product of its coordinate real masses. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: iid_rectangle_real
lemma iid_rectangle_real (n : ℕ) (P : CausalLaw) (s : Fin n → Set O) :
(dataLaw n P).real (Set.univ.pi s) = ∏ i, (Pobs P).real (s i) := by
let : IsProbabilityMeasure (Pobs P) :=
Measure.isProbabilityMeasure_map measurable_observe.aemeasurable
rw [dataLaw, measureReal_def, Measure.pi_pi, ENNReal.toReal_prod]
rfl
theorem iid_no_partner_probability
Under iid sampling, none of the other records belongs to a fixed measurable event with probability p with probability (1-p)^(n-1). the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: iid_no_partner_probability
lemma iid_no_partner_probability (n : ℕ) (P : CausalLaw) (r : Fin n)
(s : Set O) (hs : MeasurableSet s) :
(dataLaw n P).real {D | ∀ i, i ≠ r → D i ∉ s} =
(1 - (Pobs P).real s) ^ (n - 1) := by
classical
let : IsProbabilityMeasure (Pobs P) :=
Measure.isProbabilityMeasure_map measurable_observe.aemeasurable
have he : {D : Dataset n | ∀ i, i ≠ r → D i ∉ s} =
Set.univ.pi (fun i => if i = r then univ else sᶜ) := by
ext D
simp only [Set.mem_setOf_eq, Set.mem_pi, Set.mem_univ, forall_const]
constructor
· intro h i
by_cases hi : i = r <;> simp [hi, h i]
· intro h i hi
simpa [hi] using h i
rw [he, iid_rectangle_real, ← Finset.mul_prod_erase _ _ (Finset.mem_univ r)]
simp only [if_pos rfl, probReal_univ, one_mul]
have hp : (∏ i ∈ Finset.univ.erase r, (Pobs P).real
(if i = r then univ else sᶜ)) =
∏ _i ∈ Finset.univ.erase r, (1 - (Pobs P).real s) := by
apply Finset.prod_congr rfl
intro i hi
rw [if_neg (Finset.mem_erase.mp hi).1, probReal_compl_eq_one_sub hs]
rw [hp]
simp
theorem iid_isolated_record_probability
The probability that the distinguished record belongs to an event and all other records avoid it is p(1-p)^(n-1). the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: iid_isolated_record_probability
lemma iid_isolated_record_probability (n : ℕ) (P : CausalLaw) (r : Fin n)
(s : Set O) (hs : MeasurableSet s) :
(dataLaw n P).real {D | D r ∈ s ∧ ∀ i, i ≠ r → D i ∉ s} =
(Pobs P).real s * (1 - (Pobs P).real s) ^ (n - 1) := by
classical
let : IsProbabilityMeasure (Pobs P) :=
Measure.isProbabilityMeasure_map measurable_observe.aemeasurable
have he : {D : Dataset n | D r ∈ s ∧ ∀ i, i ≠ r → D i ∉ s} =
Set.univ.pi (fun i => if i = r then s else sᶜ) := by
ext D
simp only [Set.mem_setOf_eq, Set.mem_pi, Set.mem_univ, forall_const]
constructor
· rintro ⟨hr, h⟩ i
by_cases hi : i = r <;> simp [hi, hr, h i]
· intro h
exact ⟨by simpa using h r, fun i hi => by simpa [hi] using h i⟩
rw [he, iid_rectangle_real, ← Finset.mul_prod_erase _ _ (Finset.mem_univ r)]
simp only [if_pos rfl]
congr 1
calc
_ = ∏ _i ∈ Finset.univ.erase r, (1 - (Pobs P).real s) := by
apply Finset.prod_congr rfl
intro i hi
rw [if_neg (Finset.mem_erase.mp hi).1, probReal_compl_eq_one_sub hs]
_ = _ := by simp
Avoidance by all records except the distinguished one is a measurable event. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
MeasurableSet {D : Dataset n | ∀ i, i ≠ r → D i ∉ s}
Proof (Lean source)
-- @node: measurableSet_no_partner
lemma measurableSet_no_partner (n : ℕ) (r : Fin n) (s : Set O)
(hs : MeasurableSet s) :
MeasurableSet {D : Dataset n | ∀ i, i ≠ r → D i ∉ s} := by
simp only [Set.setOf_forall]
apply MeasurableSet.iInter
intro i
by_cases hi : i = r
· simp [hi]
· have hm : MeasurableSet {D : Dataset n | D i ∉ s} :=
hs.compl.preimage (measurable_pi_apply i)
simpa [hi] using hm
theorem iid_record_probability
The real mass of a single iid coordinate event is the observed marginal mass. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: iid_record_probability
lemma iid_record_probability (n : ℕ) (P : CausalLaw) (r : Fin n) (s : Set O) :
(dataLaw n P).real {D | D r ∈ s} = (Pobs P).real s := by
classical
let : IsProbabilityMeasure (Pobs P) :=
Measure.isProbabilityMeasure_map measurable_observe.aemeasurable
have he : {D : Dataset n | D r ∈ s} =
Set.univ.pi (fun i => if i = r then s else univ) := by
ext D
simp only [Set.mem_setOf_eq, Set.mem_pi, Set.mem_univ, forall_const]
constructor
· intro hr i; by_cases hi : i = r <;> simp [hi, hr]
· intro h; simpa using h r
rw [he, iid_rectangle_real, ← Finset.mul_prod_erase _ _ (Finset.mem_univ r)]
simp only [if_pos rfl]
have hp : (∏ i ∈ Finset.univ.erase r, (Pobs P).real
(if i = r then s else univ)) = 1 := by
apply Finset.prod_eq_one
intro i hi
rw [if_neg (Finset.mem_erase.mp hi).1, probReal_univ]
simp [hp]
theorem iid_old_partner_probability
The old record has a partner in a measurable event with probability p[1-(1-p)^(n-1)]. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: iid_old_partner_probability
lemma iid_old_partner_probability (n : ℕ) (P : CausalLaw) (r : Fin n)
(s : Set O) (hs : MeasurableSet s) :
(dataLaw n P).real {D | D r ∈ s ∧ ∃ i, i ≠ r ∧ D i ∈ s} =
(Pobs P).real s * (1 - (1 - (Pobs P).real s) ^ (n - 1)) := by
let : IsProbabilityMeasure (Pobs P) :=
Measure.isProbabilityMeasure_map measurable_observe.aemeasurable
let : IsProbabilityMeasure (dataLaw n P) := by unfold dataLaw; infer_instance
have his : MeasurableSet {D : Dataset n | D r ∈ s ∧ ∀ i, i ≠ r → D i ∉ s} :=
(hs.preimage (measurable_pi_apply r)).inter (measurableSet_no_partner n r s hs)
have he : {D : Dataset n | D r ∈ s ∧ ∃ i, i ≠ r ∧ D i ∈ s} =
{D | D r ∈ s} \ {D | D r ∈ s ∧ ∀ i, i ≠ r → D i ∉ s} := by
ext D; simp only [Set.mem_setOf_eq, Set.mem_sdiff]
simp only [not_and, not_forall, _root_.not_imp, not_not, exists_prop]
tauto
rw [he, measureReal_sdiff (s₁ := {D : Dataset n | D r ∈ s}) (fun D hD => hD.1) his,
iid_record_probability, iid_isolated_record_probability n P r s hs]
ring
theorem iid_new_partner_probability
An independent replacement has a partner among the unreplaced records with the same probability p[1-(1-p)^(n-1)]. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: iid_new_partner_probability
lemma iid_new_partner_probability (n : ℕ) (P : CausalLaw) (r : Fin n)
(s : Set O) (hs : MeasurableSet s) :
((dataLaw n P).prod (Pobs P)).real
{zz : Dataset n × O | zz.2 ∈ s ∧ ∃ i, i ≠ r ∧ zz.1 i ∈ s} =
(Pobs P).real s * (1 - (1 - (Pobs P).real s) ^ (n - 1)) := by
let : IsProbabilityMeasure (Pobs P) :=
Measure.isProbabilityMeasure_map measurable_observe.aemeasurable
let : IsProbabilityMeasure (dataLaw n P) := by unfold dataLaw; infer_instance
have he : {zz : Dataset n × O | zz.2 ∈ s ∧ ∃ i, i ≠ r ∧ zz.1 i ∈ s} =
{D : Dataset n | ∀ i, i ≠ r → D i ∉ s}ᶜ ×ˢ s := by
ext zz; simp only [Set.mem_setOf_eq, Set.mem_prod, Set.mem_compl_iff]
simp only [not_forall, _root_.not_imp, not_not, exists_prop]
exact and_comm
rw [he, measureReal_prod_prod,
probReal_compl_eq_one_sub (measurableSet_no_partner n r s hs),
iid_no_partner_probability n P r s hs]
ring
Having a partner in a measurable event is itself a measurable dataset event. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
MeasurableSet {D : Dataset n | D r ∈ s ∧ ∃ i, i ≠ r ∧ D i ∈ s}
Proof (Lean source)
-- @node: measurableSet_old_partner
lemma measurableSet_old_partner (n : ℕ) (r : Fin n) (s : Set O)
(hs : MeasurableSet s) :
MeasurableSet {D : Dataset n | D r ∈ s ∧ ∃ i, i ≠ r ∧ D i ∈ s} := by
have he : {D : Dataset n | D r ∈ s ∧ ∃ i, i ≠ r ∧ D i ∈ s} =
{D | D r ∈ s} ∩ {D | ∀ i, i ≠ r → D i ∉ s}ᶜ := by
ext D
simp only [Set.mem_setOf_eq, Set.mem_inter_iff, Set.mem_compl_iff,
not_forall, _root_.not_imp, not_not, exists_prop]
rw [he]
exact (hs.preimage (measurable_pi_apply r)).inter
(measurableSet_no_partner n r s hs).compl
Counting members with a distinct partner discards exactly the empty and singleton counts. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: finite_partner_count
lemma finite_partner_count {α : Type*} [Fintype α] (s : Finset α) :
(∑ r : α, if r ∈ s ∧ ∃ i ∈ s, i ≠ r then (1 : ℝ) else 0) =
if s.card < 2 then 0 else (s.card : ℝ) := by
classical
by_cases hs : s.card < 2
· rw [if_pos hs]
apply Finset.sum_eq_zero
intro r hr
have hn : ¬ (r ∈ s ∧ ∃ i ∈ s, i ≠ r) := by
rintro ⟨hr, i, hi, hir⟩
have hc := Finset.one_lt_card.mpr ⟨i, hi, r, hr, hir⟩
omega
simp only [if_neg hn]
· rw [if_neg hs]
have he (r : α) : (r ∈ s ∧ ∃ i ∈ s, i ≠ r) ↔ r ∈ s := by
constructor
· exact left
· intro hr
exact ⟨hr, Finset.exists_mem_ne (by omega) r⟩
simp_rw [he]
simp
The eligible cell occupancy is the sum of record-level partner indicators. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: cell_partner_count
lemma cell_partner_count (n k : ℕ) (h : ℝ) (D : Dataset n) (j : Fin k) :
(if cellCount n h k D j < 2 then 0 else (cellCount n h k D j : ℝ)) =
∑ r : Fin n, if (D r).1 ∈ cell h k j ∧
∃ i, i ≠ r ∧ (D i).1 ∈ cell h k j then (1 : ℝ) else 0 := by
classical
have he (r : Fin n) : (r ∈ cellRecords n h k D j ∧
∃ i ∈ cellRecords n h k D j, i ≠ r) ↔
((D r).1 ∈ cell h k j ∧ ∃ i, i ≠ r ∧ (D i).1 ∈ cell h k j) := by
simp only [cellRecords, mem_filter, Finset.mem_univ, true_and]
constructor
· rintro ⟨hr, i, hi, hir⟩
exact ⟨hr, i, hir, hi⟩
· rintro ⟨hr, i, hir, hi⟩
exact ⟨hr, i, hi, hir⟩
rw [cellCount, ← finite_partner_count]
apply Finset.sum_congr rfl
intro r hr
by_cases hp : (D r).1 ∈ cell h k j ∧ ∃ i, i ≠ r ∧ (D i).1 ∈ cell h k j
· rw [if_pos hp, if_pos ((he r).mpr hp)]
· rw [if_neg hp, if_neg (fun hc => hp ((he r).mp hc))]
Eligible occupancy is integrable under every finite dataset measure. The displayed conclusion follows.
Formal statement
hypotheses
conclusion
Integrable (fun D => if cellCount n h k D j < 2 then 0 else (cellCount n h k D j : ℝ)) Q
Proof (Lean source)
-- @node: integrable_cell_partner_count
@[fun_prop] lemma integrable_cell_partner_count (n k : ℕ) (h : ℝ) (j : Fin k)
(Q : Measure (Dataset n)) [IsFiniteMeasure Q] :
Integrable (fun D => if cellCount n h k D j < 2 then 0 else
(cellCount n h k D j : ℝ)) Q := by
classical
simp_rw [cell_partner_count]
apply integrable_finset_sum
intro r hr
have hs := measurableSet_old_partner n r {z : O | z.1 ∈ cell h k j}
((measurableSet_cell h k j).preimage measurable_fst)
have hi := (integrable_const (μ := Q) (1 : ℝ)).indicator hs
apply hi.congr
filter_upwards with D
by_cases hp : (D r).1 ∈ cell h k j ∧ ∃ i, i ≠ r ∧ (D i).1 ∈ cell h k j
· simp [Set.indicator_apply, hp]
· simp [Set.indicator_apply, hp]
theorem iid_cell_partner_count_expectation
The binomial occupancy identity follows by summing iid record-partner probabilities, without requiring independence between different cell counts. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
(∫ D, (if cellCount n h k D j < 2 then 0 else (cellCount n h k D j : ℝ)) ∂dataLaw n P)
= occupancyWeight n h k P j
Proof (Lean source)
-- @node: iid_cell_partner_count_expectation
lemma iid_cell_partner_count_expectation (n k : ℕ) (h : ℝ) (P : CausalLaw)
(j : Fin k) :
(∫ D, (if cellCount n h k D j < 2 then 0 else
(cellCount n h k D j : ℝ)) ∂dataLaw n P) = occupancyWeight n h k P j := by
classical
let : IsProbabilityMeasure (Pobs P) :=
Measure.isProbabilityMeasure_map measurable_observe.aemeasurable
let : IsProbabilityMeasure (dataLaw n P) := by unfold dataLaw; infer_instance
let s : Set O := {z | z.1 ∈ cell h k j}
have hs : MeasurableSet s := (measurableSet_cell h k j).preimage measurable_fst
have hi (r : Fin n) : Integrable (fun D : Dataset n =>
if D r ∈ s ∧ ∃ i, i ≠ r ∧ D i ∈ s then (1 : ℝ) else 0) (dataLaw n P) := by
have hi := (integrable_const (μ := dataLaw n P) (1 : ℝ)).indicator (measurableSet_old_partner n r s hs)
apply hi.congr
filter_upwards with D
by_cases hp : D r ∈ s ∧ ∃ i, i ≠ r ∧ D i ∈ s
· simp [Set.indicator_apply, hp]
· simp [Set.indicator_apply, hp]
simp_rw [cell_partner_count]
change (∫ D : Dataset n, (∑ r : Fin n,
if D r ∈ s ∧ ∃ i, i ≠ r ∧ D i ∈ s then (1 : ℝ) else 0) ∂dataLaw n P) = _
rw [integral_finset_sum _ (fun r _ => hi r)]
have hm (r : Fin n) : (∫ D : Dataset n,
(if D r ∈ s ∧ ∃ i, i ≠ r ∧ D i ∈ s then (1 : ℝ) else 0) ∂dataLaw n P) =
cellMass h k P j * (1 - (1 - cellMass h k P j) ^ (n - 1)) := by
have hv := integral_indicator_one (μ := dataLaw n P)
(measurableSet_old_partner n r s hs)
simp only [indicator, Set.mem_setOf_eq, Pi.one_apply] at hv
rw [hv, iid_old_partner_probability n P r s hs]
rw [show (Pobs P).real s = cellMass h k P j from observed_cell_mass h k P j]
change (∑ r : Fin n, (∫ D : Dataset n,
(if D r ∈ s ∧ ∃ i, i ≠ r ∧ D i ∈ s then (1 : ℝ) else 0) ∂dataLaw n P)) = _
simp_rw [hm]
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul,
occupancyWeight]
ring
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.PartnerVariance 23 declarations Partner-sensitive replacement increments and the occupancy-weighted Efron--Stein assembly.
Partner-sensitive replacement increments and the occupancy-weighted Efron--Stein assembly.
theorem selected_scalar_insertion_bound
Inserting one record into an arbitrary selected set changes each scalar causal contribution by at most three, at every count. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
conclusion 1
conclusion 2
Proof (Lean source)
-- @node: selected_scalar_insertion_bound
lemma selected_scalar_insertion_bound {n : ℕ} (I : Finset (Fin n)) (D : Dataset n)
(r : Fin n) (hr : r ∉ I) :
|selectedNumerator (insert r I) D - selectedNumerator I D| ≤ 3 ∧
|selectedDenominator (insert r I) D - selectedDenominator I D| ≤ 3 := by
classical
let e : Fin I.card ≃ {i // i ∈ I} :=
(Fintype.equivFinOfCardEq (by simp)).symm
let z : Fin (I.card + 1) → O := Fin.lastCases (D r) (fun i => D (e i))
have hz (i : Fin I.card) : z i.castSucc = D (e i) := by simp [z]
have hlast : z (last I.card) = D r := by simp [z]
have hsum (v : O → ℝ) :
(∑ i, v (z i)) = ∑ i ∈ insert r I, v (D i) := by
rw [Fin.sum_univ_castSucc]
simp_rw [hz]
rw [selected_sum n _ I e (fun i => v (D i)), Finset.sum_insert hr, hlast]
exact add_comm _ _
obtain ⟨hN, hD⟩ := pair_moment_formulas _ z
rw [hsum (fun z => bit z.2.1 * bit z.2.2),
hsum (fun z => bit z.2.1), hsum (fun z => bit z.2.2)] at hN
rw [hsum (fun z => bit z.2.1)] at hD
have hcard := Finset.card_insert_of_notMem hr
have hNi : pairContribution KN z = selectedNumerator (insert r I) D := by
simpa only [selectedNumerator, hcard, Nat.cast_add, Nat.cast_one] using hN
have hDi : pairContribution KD z = selectedDenominator (insert r I) D := by
simpa only [selectedDenominator, hcard, Nat.cast_add, Nat.cast_one] using hD
obtain ⟨hNo, hDo⟩ := selected_pair_formulas I D e
have hbN := pairContribution_insertion_bound I.card KN z
(fun x y => (causal_pair_kernel_unit_bounds x y).1)
have hbD := pairContribution_insertion_bound I.card KD z
(fun x y => (causal_pair_kernel_unit_bounds x y).2)
simp_rw [hz] at hbN hbD
rw [hNi, hNo] at hbN
rw [hDi, hDo] at hbD
exact ⟨hbN, hbD⟩
theorem selected_partner_erasure_bound
Without a remaining selected record, both scalar contributions before and after insertion are zero. With a partner, the scalar insertion bound is three. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
conclusion 1
|selectedNumerator I D - selectedNumerator (I.erase r) D|
≤ (if r ∈ I ∧ (I.erase r).Nonempty then 3 else 0)
conclusion 2
|selectedDenominator I D - selectedDenominator (I.erase r) D|
≤ (if r ∈ I ∧ (I.erase r).Nonempty then 3 else 0)
Proof (Lean source)
-- @node: selected_partner_erasure_bound
lemma selected_partner_erasure_bound {n : ℕ} (I : Finset (Fin n)) (D : Dataset n)
(r : Fin n) :
|selectedNumerator I D - selectedNumerator (I.erase r) D| ≤
(if r ∈ I ∧ (I.erase r).Nonempty then 3 else 0) ∧
|selectedDenominator I D - selectedDenominator (I.erase r) D| ≤
(if r ∈ I ∧ (I.erase r).Nonempty then 3 else 0) := by
classical
by_cases hr : r ∈ I
· by_cases hp : (I.erase r).Nonempty
· simp only [hr, hp, and_self, if_true]
have hb := selected_scalar_insertion_bound (I.erase r) D r (Finset.notMem_erase r I)
rwa [Finset.insert_erase hr] at hb
· have he : I.erase r = ∅ := Finset.not_nonempty_iff_eq_empty.mp hp
have hI : I = {r} := by
have hi := Finset.insert_erase hr
rw [he] at hi
simpa using hi.symm
simp [hI, selectedNumerator, selectedDenominator]
· simp [hr]
The partner indicator records whether a proposed record shares a public cell with any record other than the replaced index.
Definition (Lean source)
parameters
def
partnerIndicator n k h D r z :
ℝ
given by
theorem partnerIndicator_values
Partner indicators take only the values zero and one. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
partnerIndicator n k h D r z = 0 ∨ partnerIndicator n k h D r z = 1
Proof (Lean source)
-- @node: partnerIndicator_values
lemma partnerIndicator_values (n k : ℕ) (h : ℝ) (D : Dataset n) (r : Fin n) (z : O) :
partnerIndicator n k h D r z = 0 ∨ partnerIndicator n k h D r z = 1 := by
classical
unfold partnerIndicator
split_ifs <;> simp
The two datasets in a replacement have identical cell selections after erasing its index. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: erased_cellRecords_congr
lemma erased_cellRecords_congr (n k : ℕ) (h : ℝ) (D D' : Dataset n) (r : Fin n)
(heq : ∀ i, i ≠ r → D i = D' i) (j : Fin k) :
(cellRecords n h k D j).erase r = (cellRecords n h k D' j).erase r := by
classical
ext i
by_cases hi : i = r
· subst i; simp
· simp [cellRecords, hi, heq i hi]
A scalar cell replacement is bounded by the old and new partner indicators in that cell. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
B :
(cellRecords n h k D j).erase r
old :
if (D r).1 ∈ cell h k j ∧ B.Nonempty then (3 : ℝ) else 0
new :
if (D' r).1 ∈ cell h k j ∧ B.Nonempty then (3 : ℝ) else 0
|numeratorCell n h k D j - numeratorCell n h k D' j| ≤ old + new
|denominatorCell n h k D j - denominatorCell n h k D' j| ≤ old + new
Proof (Lean source)
-- @node: cell_partner_replacement_bound
lemma cell_partner_replacement_bound (n k : ℕ) (h : ℝ) (D D' : Dataset n) (r : Fin n)
(heq : ∀ i, i ≠ r → D i = D' i) (j : Fin k) :
let B := (cellRecords n h k D j).erase r
let old := if (D r).1 ∈ cell h k j ∧ B.Nonempty then (3 : ℝ) else 0
let new := if (D' r).1 ∈ cell h k j ∧ B.Nonempty then (3 : ℝ) else 0
|numeratorCell n h k D j - numeratorCell n h k D' j| ≤ old + new ∧
|denominatorCell n h k D j - denominatorCell n h k D' j| ≤ old + new := by
classical
dsimp only
let I := cellRecords n h k D j
let I' := cellRecords n h k D' j
have hbase : I.erase r = I'.erase r := erased_cellRecords_congr n k h D D' r heq j
have hcommon : ∀ i ∈ I.erase r, D i = D' i := by
intro i hi
exact heq i (Finset.mem_erase.mp hi).1
obtain ⟨hcN, hcD⟩ := selected_congr (I.erase r) D D' hcommon
rw [hbase] at hcN hcD
obtain ⟨hbN, hbD⟩ := selected_partner_erasure_bound I D r
obtain ⟨hbN', hbD'⟩ := selected_partner_erasure_bound I' D' r
rw [← hcN, ← hbase] at hbN'
rw [← hcD, ← hbase] at hbD'
have hN := abs_sub_le (selectedNumerator I D)
(selectedNumerator (I.erase r) D) (selectedNumerator I' D')
have hD := abs_sub_le (selectedDenominator I D)
(selectedDenominator (I.erase r) D) (selectedDenominator I' D')
rw [abs_sub_comm (selectedNumerator (I.erase r) D)] at hN
rw [abs_sub_comm (selectedDenominator (I.erase r) D)] at hD
have hmem : (r ∈ I) ↔ (D r).1 ∈ cell h k j := by simp [I, cellRecords]
have hmem' : (r ∈ I') ↔ (D' r).1 ∈ cell h k j := by simp [I', cellRecords]
simp only [hmem] at hbN hbD
simp only [hmem'] at hbN' hbD'
change |selectedNumerator I D - selectedNumerator I' D'| ≤ _ ∧
|selectedDenominator I D - selectedDenominator I' D'| ≤ _
constructor <;> linarith
theorem cell_partner_sum
Disjointness reduces the sum of cellwise partner envelopes to the single partner indicator. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
= 3 * partnerIndicator n k h D r z
Proof (Lean source)
-- @node: cell_partner_sum
lemma cell_partner_sum (n k : ℕ) (h : ℝ) (hh : 0 < h) (hk : 0 < k)
(D : Dataset n) (r : Fin n) (z : O) :
(∑ j : Fin k, if z.1 ∈ cell h k j ∧
((cellRecords n h k D j).erase r).Nonempty then (3 : ℝ) else 0) =
3 * partnerIndicator n k h D r z := by
classical
unfold partnerIndicator
by_cases hx : ∃ j : Fin k, z.1 ∈ cell h k j ∧
((cellRecords n h k D j).erase r).Nonempty
· rw [if_pos hx]
obtain ⟨j, hj⟩ := hx
rw [Finset.sum_eq_single j]
· simp [hj]
· intro i hi hij
have hnot : z.1 ∉ cell h k i := by
intro hz
exact Set.disjoint_left.mp (cell_disjoint k h hh hk i j hij) hz hj.1
simp [hnot]
· simp
· rw [if_neg hx]
have hn : ∀ j : Fin k, ¬(z.1 ∈ cell h k j ∧
((cellRecords n h k D j).erase r).Nonempty) := by simpa using hx
simp [hn]
theorem statistic_partner_replacement_bound
Each scalar replacement increment vanishes without partners and is at most three times the sum of its old and new partner indicators. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
J :
partnerIndicator n k h D r (D r)
J' :
partnerIndicator n k h D r (D' r)
Proof (Lean source)
-- @node: statistic_partner_replacement_bound
lemma statistic_partner_replacement_bound (n k : ℕ) (h : ℝ) (hh : 0 < h) (hk : 0 < k)
(D D' : Dataset n) (r : Fin n) (heq : ∀ i, i ≠ r → D i = D' i) :
let J := partnerIndicator n k h D r (D r)
let J' := partnerIndicator n k h D r (D' r)
|SN n h k D - SN n h k D'| ≤ 3 * (J + J') ∧
|SD n h k D - SD n h k D'| ≤ 3 * (J + J') := by
classical
dsimp only
have hbN := Finset.sum_le_sum (s := univ)
(fun j _ => (cell_partner_replacement_bound n k h D D' r heq j).1)
have hbD := Finset.sum_le_sum (s := univ)
(fun j _ => (cell_partner_replacement_bound n k h D D' r heq j).2)
simp only [Finset.sum_add_distrib, cell_partner_sum n k h hh hk D r] at hbN hbD
have hN : |SN n h k D - SN n h k D'| ≤
∑ j : Fin k, |numeratorCell n h k D j - numeratorCell n h k D' j| := by
rw [SN, SN, ← Finset.sum_sub_distrib]
exact Finset.abs_sum_le_sum_abs _ _
have hD : |SD n h k D - SD n h k D'| ≤
∑ j : Fin k, |denominatorCell n h k D j - denominatorCell n h k D' j| := by
rw [SD, SD, ← Finset.sum_sub_distrib]
exact Finset.abs_sum_le_sum_abs _ _
constructor <;> linarith
theorem statistic_partner_square_bound
Squared scalar replacement increments are bounded by eighteen times the two partner indicators, exactly the deterministic envelope in the variance roadmap. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
J :
partnerIndicator n k h D r (D r)
J' :
partnerIndicator n k h D r z
Proof (Lean source)
-- @node: statistic_partner_square_bound
lemma statistic_partner_square_bound (n k : ℕ) (h : ℝ) (hh : 0 < h) (hk : 0 < k)
(D : Dataset n) (r : Fin n) (z : O) :
let J := partnerIndicator n k h D r (D r)
let J' := partnerIndicator n k h D r z
(SN n h k D - SN n h k (update D r z))^2 ≤ 18 * (J + J') ∧
(SD n h k D - SD n h k (update D r z))^2 ≤ 18 * (J + J') := by
have hb := statistic_partner_replacement_bound n k h hh hk D
(update D r z) r (fun i hi => (Function.update_of_ne hi z D).symm)
simp only [Function.update_self] at hb
exact ⟨replacement_square_indicator_bound _ _ _
(partnerIndicator_values n k h D r (D r)) (partnerIndicator_values n k h D r z) hb.1,
replacement_square_indicator_bound _ _ _
(partnerIndicator_values n k h D r (D r)) (partnerIndicator_values n k h D r z) hb.2⟩
theorem measurable_partnerIndicator
Partner indicators are Borel functions of the dataset and proposed record. The displayed conclusion follows.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: measurable_partnerIndicator
@[fun_prop] lemma measurable_partnerIndicator (n k : ℕ) (h : ℝ) (r : Fin n) :
Measurable (fun zz : Dataset n × O => partnerIndicator n k h zz.1 r zz.2) := by
classical
unfold partnerIndicator
simp only [Nonempty, Finset.mem_erase, cellRecords, mem_filter,
Finset.mem_univ, true_and]
refine measurable_const.ite ?_ measurable_const
simp only [Set.ofPred_exists]
apply MeasurableSet.iUnion
intro j
apply MeasurableSet.inter
· exact (measurableSet_cell h k j).preimage (by fun_prop)
· change MeasurableSet {zz : Dataset n × O | ∃ i : Fin n,
i ≠ r ∧ (zz.1 i).1 ∈ cell h k j}
rw [Set.ofPred_exists]
apply MeasurableSet.iUnion
intro i
by_cases hi : i = r
· simp [hi]
· simp only [ne_eq, hi, not_false_eq_true, true_and]
exact (measurableSet_cell h k j).preimage (by fun_prop)
theorem partnerIndicator_integrable
Partner indicators are bounded by one and integrable under every finite measure. The displayed conclusion follows.
Formal statement
hypotheses
conclusion
Integrable (fun zz => partnerIndicator n k h zz.1 r zz.2) Q
Proof (Lean source)
-- @node: partnerIndicator_integrable
@[fun_prop] lemma partnerIndicator_integrable (n k : ℕ) (h : ℝ) (r : Fin n)
(Q : Measure (Dataset n × O)) [IsFiniteMeasure Q] :
Integrable (fun zz => partnerIndicator n k h zz.1 r zz.2) Q := by
apply Integrable.of_bound (by fun_prop) 1
apply ae_of_all
intro zz
rcases partnerIndicator_values n k h zz.1 r zz.2 with hz | hz <;> simp [hz]
theorem statistic_replacement_square_integrable Lemma statistic_replacement_square_integrable in the paper ↗
The squared replacement differences of the bounded statistics are integrable under every finite measure on dataset and replacement pairs. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
conclusion 1
conclusion 2
Integrable (fun zz => (SD n h k zz.1 - SD n h k (update zz.1 r zz.2))^2) Q
Proof (Lean source)
-- @node: statistic_replacement_square_integrable
lemma statistic_replacement_square_integrable (n k : ℕ) (h : ℝ)
(hh : 0 < h) (hk : 0 < k) (r : Fin n) (Q : Measure (Dataset n × O))
[IsFiniteMeasure Q] :
Integrable (fun zz => (SN n h k zz.1 - SN n h k (update zz.1 r zz.2))^2) Q ∧
Integrable (fun zz => (SD n h k zz.1 - SD n h k (update zz.1 r zz.2))^2) Q := by
have hb (zz : Dataset n × O) :
(SN n h k zz.1 - SN n h k (update zz.1 r zz.2))^2 ≤ 36 ∧
(SD n h k zz.1 - SD n h k (update zz.1 r zz.2))^2 ≤ 36 := by
obtain ⟨hN, hD⟩ := statistic_partner_square_bound n k h hh hk zz.1 r zz.2
have hv := partnerIndicator_values n k h zz.1 r (zz.1 r)
have hv' := partnerIndicator_values n k h zz.1 r zz.2
rcases hv with hv | hv <;> rcases hv' with hv' | hv' <;>
simp only [hv, hv'] at hN hD <;> constructor <;> linarith
constructor
· apply Integrable.of_bound (by fun_prop) 36
exact ae_of_all _ (fun zz => by simpa only [Real.norm_eq_abs, abs_sq]
using (hb zz).1)
· apply Integrable.of_bound (by fun_prop) 36
exact ae_of_all _ (fun zz => by simpa only [Real.norm_eq_abs, abs_sq]
using (hb zz).2)
theorem partnerIndicator_eq_sum
Disjoint public cells express the partner indicator as a sum of cell event indicators. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
partnerIndicator n k h D r z
= ∑ j : Fin k, (if z.1 ∈ cell h k j ∧ ∃ i, i ≠ r ∧ (D i).1 ∈ cell h k j then 1 else 0)
Proof (Lean source)
-- @node: partnerIndicator_eq_sum
lemma partnerIndicator_eq_sum (n k : ℕ) (h : ℝ) (hh : 0 < h) (hk : 0 < k)
(D : Dataset n) (r : Fin n) (z : O) :
partnerIndicator n k h D r z = ∑ j : Fin k,
(if z.1 ∈ cell h k j ∧ ∃ i, i ≠ r ∧ (D i).1 ∈ cell h k j then 1 else 0) := by
classical
have he (j : Fin k) : ((cellRecords n h k D j).erase r).Nonempty ↔
∃ i, i ≠ r ∧ (D i).1 ∈ cell h k j := by
simp [Nonempty, cellRecords]
have hs := cell_partner_sum n k h hh hk D r z
simp only [he] at hs
have ht : (∑ j : Fin k, if z.1 ∈ cell h k j ∧
∃ i, i ≠ r ∧ (D i).1 ∈ cell h k j then (3 : ℝ) else 0) =
3 * ∑ j : Fin k, if z.1 ∈ cell h k j ∧
∃ i, i ≠ r ∧ (D i).1 ∈ cell h k j then (1 : ℝ) else 0 := by
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro j hj
split_ifs <;> norm_num
rw [ht] at hs
linarith
Existence of an unreplaced record in an event is measurable. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
MeasurableSet {D : Dataset n | ∃ i, i ≠ r ∧ D i ∈ s}
Proof (Lean source)
-- @node: measurableSet_has_partner
lemma measurableSet_has_partner (n : ℕ) (r : Fin n) (s : Set O)
(hs : MeasurableSet s) :
MeasurableSet {D : Dataset n | ∃ i, i ≠ r ∧ D i ∈ s} := by
have he : {D : Dataset n | ∃ i, i ≠ r ∧ D i ∈ s} =
{D : Dataset n | ∀ i, i ≠ r → D i ∉ s}ᶜ := by
ext D
simp [not_forall, _root_.not_imp]
rw [he]
exact (measurableSet_no_partner n r s hs).compl
theorem integral_cell_partner_indicators
Old and independent-copy cell partner indicators both have the Bernoulli occupancy mean. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: integral_cell_partner_indicators
lemma integral_cell_partner_indicators (n k : ℕ) (h : ℝ) (P : CausalLaw)
(r : Fin n) (j : Fin k) :
let Q := (dataLaw n P).prod (Pobs P)
let p := cellMass h k P j
(∫ zz : Dataset n × O, if (zz.1 r).1 ∈ cell h k j ∧
∃ i, i ≠ r ∧ (zz.1 i).1 ∈ cell h k j then (1 : ℝ) else 0 ∂Q) =
p * (1 - (1 - p)^(n - 1)) ∧
(∫ zz : Dataset n × O, if zz.2.1 ∈ cell h k j ∧
∃ i, i ≠ r ∧ (zz.1 i).1 ∈ cell h k j then (1 : ℝ) else 0 ∂Q) =
p * (1 - (1 - p)^(n - 1)) := by
classical
let : IsProbabilityMeasure (Pobs P) :=
Measure.isProbabilityMeasure_map measurable_observe.aemeasurable
let : IsProbabilityMeasure (dataLaw n P) := by unfold dataLaw; infer_instance
dsimp only
let s : Set O := {z | z.1 ∈ cell h k j}
have hs : MeasurableSet s := (measurableSet_cell h k j).preimage measurable_fst
have hp := measurableSet_has_partner n r s hs
have hor : MeasurableSet {D : Dataset n | D r ∈ s} :=
hs.preimage (measurable_pi_apply r)
have ho : MeasurableSet {zz : Dataset n × O | (zz.1 r) ∈ s ∧
∃ i, i ≠ r ∧ zz.1 i ∈ s} :=
(hor.inter hp).preimage measurable_fst
have hn : MeasurableSet {zz : Dataset n × O | zz.2 ∈ s ∧
∃ i, i ≠ r ∧ zz.1 i ∈ s} :=
(hs.preimage measurable_snd).inter (hp.preimage measurable_fst)
have he : {zz : Dataset n × O | (zz.1 r) ∈ s ∧ ∃ i, i ≠ r ∧ zz.1 i ∈ s} =
{D : Dataset n | D r ∈ s ∧ ∃ i, i ≠ r ∧ D i ∈ s} ×ˢ univ := by
ext zz; simp
constructor
· have ht := integral_indicator_one (μ := (dataLaw n P).prod (Pobs P)) ho
simp only [indicator, Set.mem_setOf_eq, Pi.one_apply, s] at ht
simp only [s, Set.mem_setOf_eq] at he
rw [ht, he, measureReal_prod_prod, probReal_univ, mul_one]
simpa only [s, Set.mem_setOf_eq, observed_cell_mass] using iid_old_partner_probability n P r s hs
· have ht := integral_indicator_one (μ := (dataLaw n P).prod (Pobs P)) hn
simp only [indicator, Set.mem_setOf_eq, Pi.one_apply, s] at ht
rw [ht]
simpa only [s, Set.mem_setOf_eq, observed_cell_mass] using iid_new_partner_probability n P r s hs
theorem measurable_oldPartnerIndicator
The old-record partner indicator is Borel on the dataset and independent-copy space. The displayed conclusion follows.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: measurable_oldPartnerIndicator
@[fun_prop] lemma measurable_oldPartnerIndicator (n k : ℕ) (h : ℝ) (r : Fin n) :
Measurable (fun zz : Dataset n × O => partnerIndicator n k h zz.1 r (zz.1 r)) := by
fun_prop
theorem oldPartnerIndicator_integrable
The old-record partner indicator is integrable under any finite pair measure. The displayed conclusion follows.
Formal statement
hypotheses
conclusion
Integrable (fun zz => partnerIndicator n k h zz.1 r (zz.1 r)) Q
Proof (Lean source)
-- @node: oldPartnerIndicator_integrable
@[fun_prop] lemma oldPartnerIndicator_integrable (n k : ℕ) (h : ℝ) (r : Fin n)
(Q : Measure (Dataset n × O)) [IsFiniteMeasure Q] :
Integrable (fun zz => partnerIndicator n k h zz.1 r (zz.1 r)) Q := by
apply Integrable.of_bound (measurable_oldPartnerIndicator n k h r |>.aestronglyMeasurable) 1
exact ae_of_all _ (fun zz => by
rcases partnerIndicator_values n k h zz.1 r (zz.1 r) with hz | hz <;> simp [hz])
Cellwise partner indicators are integrable for any finite pair measure. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
conclusion 1
Integrable (fun zz : Dataset n × O => if (zz.1 r).1 ∈ cell h k j ∧ ∃ i, i ≠ r ∧ (zz.1 i).1 ∈ cell h k j then (1 : ℝ) else 0) Q
conclusion 2
Integrable (fun zz : Dataset n × O => if zz.2.1 ∈ cell h k j ∧ ∃ i, i ≠ r ∧ (zz.1 i).1 ∈ cell h k j then (1 : ℝ) else 0) Q
Proof (Lean source)
-- @node: integrable_cell_partner_indicators
lemma integrable_cell_partner_indicators (n k : ℕ) (h : ℝ) (r : Fin n) (j : Fin k)
(Q : Measure (Dataset n × O)) [IsFiniteMeasure Q] :
Integrable (fun zz : Dataset n × O => if (zz.1 r).1 ∈ cell h k j ∧
∃ i, i ≠ r ∧ (zz.1 i).1 ∈ cell h k j then (1 : ℝ) else 0) Q ∧
Integrable (fun zz : Dataset n × O => if zz.2.1 ∈ cell h k j ∧
∃ i, i ≠ r ∧ (zz.1 i).1 ∈ cell h k j then (1 : ℝ) else 0) Q := by
let s : Set O := {z | z.1 ∈ cell h k j}
have hs : MeasurableSet s := (measurableSet_cell h k j).preimage measurable_fst
have hp := measurableSet_has_partner n r s hs
have hor : MeasurableSet {D : Dataset n | D r ∈ s} :=
hs.preimage (measurable_pi_apply r)
have ho : MeasurableSet {zz : Dataset n × O | (zz.1 r) ∈ s ∧
∃ i, i ≠ r ∧ zz.1 i ∈ s} := (hor.inter hp).preimage measurable_fst
have hn : MeasurableSet {zz : Dataset n × O | zz.2 ∈ s ∧
∃ i, i ≠ r ∧ zz.1 i ∈ s} :=
(hs.preimage measurable_snd).inter (hp.preimage measurable_fst)
constructor
· apply Integrable.of_bound (measurable_const.ite ho measurable_const).aestronglyMeasurable 1
exact ae_of_all _ (fun zz => by split_ifs <;> norm_num)
· apply Integrable.of_bound (measurable_const.ite hn measurable_const).aestronglyMeasurable 1
exact ae_of_all _ (fun zz => by split_ifs <;> norm_num)
theorem iid_partner_expectations
Each of the two partner indicators has mean W/n under the iid law and its independent copy. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
Proof (Lean source)
-- @node: iid_partner_expectations
lemma iid_partner_expectations (n k : ℕ) (h : ℝ) (hn : 0 < n)
(hh : 0 < h) (hk : 0 < k) (P : CausalLaw) (r : Fin n) :
(∫ zz : Dataset n × O, partnerIndicator n k h zz.1 r (zz.1 r)
∂((dataLaw n P).prod (Pobs P))) = Wocc n h k P / n ∧
(∫ zz : Dataset n × O, partnerIndicator n k h zz.1 r zz.2
∂((dataLaw n P).prod (Pobs P))) = Wocc n h k P / n := by
classical
let : IsProbabilityMeasure (Pobs P) :=
Measure.isProbabilityMeasure_map measurable_observe.aemeasurable
let : IsProbabilityMeasure (dataLaw n P) := by unfold dataLaw; infer_instance
have hw : (∑ j : Fin k, cellMass h k P j *
(1 - (1 - cellMass h k P j)^(n - 1))) = Wocc n h k P / n := by
have hnR : (n : ℝ) ≠ 0 := by exact_mod_cast hn.ne'
unfold Wocc occupancyWeight
simp only [mul_assoc, ← Finset.mul_sum]
field_simp
constructor
· simp_rw [partnerIndicator_eq_sum n k h hh hk]
rw [integral_finset_sum _ (fun j _ =>
(integrable_cell_partner_indicators n k h r j _).1)]
calc
_ = ∑ j : Fin k, cellMass h k P j *
(1 - (1 - cellMass h k P j)^(n - 1)) := by
apply Finset.sum_congr rfl
intro j hj
exact (integral_cell_partner_indicators n k h P r j).1
_ = _ := hw
· simp_rw [partnerIndicator_eq_sum n k h hh hk]
rw [integral_finset_sum _ (fun j _ =>
(integrable_cell_partner_indicators n k h r j _).2)]
calc
_ = ∑ j : Fin k, cellMass h k P j *
(1 - (1 - cellMass h k P j)^(n - 1)) := by
apply Finset.sum_congr rfl
intro j hj
exact (integral_cell_partner_indicators n k h P r j).2
_ = _ := hw
theorem iid_partner_sum_expectation
The old-plus-new partner expectation is exactly twice W/n. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
(∫ zz : Dataset n × O, (partnerIndicator n k h zz.1 r (zz.1 r) + partnerIndicator n k h zz.1 r zz.2) ∂((dataLaw n P).prod (Pobs P)))
= 2 * Wocc n h k P / n
Proof (Lean source)
-- @node: iid_partner_sum_expectation
lemma iid_partner_sum_expectation (n k : ℕ) (h : ℝ) (hn : 0 < n)
(hh : 0 < h) (hk : 0 < k) (P : CausalLaw) (r : Fin n) :
(∫ zz : Dataset n × O,
(partnerIndicator n k h zz.1 r (zz.1 r) + partnerIndicator n k h zz.1 r zz.2)
∂((dataLaw n P).prod (Pobs P))) = 2 * Wocc n h k P / n := by
let : IsProbabilityMeasure (Pobs P) :=
Measure.isProbabilityMeasure_map measurable_observe.aemeasurable
let : IsProbabilityMeasure (dataLaw n P) := by unfold dataLaw; infer_instance
obtain ⟨ho, hn'⟩ := iid_partner_expectations n k h hn hh hk P r
rw [integral_add (oldPartnerIndicator_integrable n k h r _)
(partnerIndicator_integrable n k h r _), ho, hn']
ring
theorem statistic_replacement_moments_of_partners
Integrating the deterministic squared replacement bound reduces both second moments to the common old-plus-new partner expectation. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
n k :
ℕ
h :
ℝ
hh :
0 < h
hk :
0 < k
P :
r :
Fin n
hpartners :
(∫ zz : Dataset n × O, (partnerIndicator n k h zz.1 r (zz.1 r) + partnerIndicator n k h zz.1 r zz.2) ∂((dataLaw n P).prod (Pobs P)))
= 2 * Wocc n h k P / n
Proof (Lean source)
-- @node: statistic_replacement_moments_of_partners
lemma statistic_replacement_moments_of_partners (n k : ℕ) (h : ℝ)
(hh : 0 < h) (hk : 0 < k) (P : CausalLaw) (r : Fin n)
(hpartners : (∫ zz : Dataset n × O,
(partnerIndicator n k h zz.1 r (zz.1 r) + partnerIndicator n k h zz.1 r zz.2)
∂((dataLaw n P).prod (Pobs P))) = 2 * Wocc n h k P / n) :
(∫ zz, (SN n h k zz.1 - SN n h k (update zz.1 r zz.2))^2
∂((dataLaw n P).prod (Pobs P))) ≤ 36 * Wocc n h k P / n ∧
(∫ zz, (SD n h k zz.1 - SD n h k (update zz.1 r zz.2))^2
∂((dataLaw n P).prod (Pobs P))) ≤ 36 * Wocc n h k P / n := by
let : IsProbabilityMeasure (Pobs P) :=
Measure.isProbabilityMeasure_map measurable_observe.aemeasurable
let : IsProbabilityMeasure (dataLaw n P) := by unfold dataLaw; infer_instance
obtain ⟨hiN, hiD⟩ := statistic_replacement_square_integrable n k h hh hk r
((dataLaw n P).prod (Pobs P))
have hi : Integrable (fun zz : Dataset n × O =>
18 * (partnerIndicator n k h zz.1 r (zz.1 r) + partnerIndicator n k h zz.1 r zz.2))
((dataLaw n P).prod (Pobs P)) := by fun_prop
have he : (∫ zz : Dataset n × O,
18 * (partnerIndicator n k h zz.1 r (zz.1 r) + partnerIndicator n k h zz.1 r zz.2)
∂((dataLaw n P).prod (Pobs P))) = 36 * Wocc n h k P / n := by
rw [integral_const_mul, hpartners]
ring
constructor
· calc
_ ≤ _ := integral_mono hiN hi
(fun zz => (statistic_partner_square_bound n k h hh hk zz.1 r zz.2).1)
_ = _ := he
· calc
_ ≤ _ := integral_mono hiD hi
(fun zz => (statistic_partner_square_bound n k h hh hk zz.1 r zz.2).2)
_ = _ := he
theorem statistic_efronStein_assembly
Efron--Stein converts the two occupancy-weighted replacement second-moment estimates into the asserted variance bound. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
efronStein_of_gate :
n k :
ℕ
h :
ℝ
hn :
0 < n
hk :
0 < k
hh :
0 < h
P :
Proof (Lean source)
-- @node: statistic_efronStein_assembly
lemma statistic_efronStein_assembly (efronStein_of_gate : EfronSteinReplacement)
(n k : ℕ) (h : ℝ) (hn : 0 < n) (hk : 0 < k) (hh : 0 < h) (P : CausalLaw)
(hstep : ∀ r : Fin n,
(∫ zz, (SN n h k zz.1 - SN n h k (update zz.1 r zz.2))^2
∂((dataLaw n P).prod (Pobs P))) ≤ 36 * Wocc n h k P / n ∧
(∫ zz, (SD n h k zz.1 - SD n h k (update zz.1 r zz.2))^2
∂((dataLaw n P).prod (Pobs P))) ≤ 36 * Wocc n h k P / n) :
max (variance (SN n h k) (dataLaw n P))
(variance (SD n h k) (dataLaw n P)) ≤ 18 * Wocc n h k P := by
let : IsProbabilityMeasure (Pobs P) :=
Measure.isProbabilityMeasure_map measurable_observe.aemeasurable
let : IsProbabilityMeasure (dataLaw n P) := by unfold dataLaw; infer_instance
have hLp := statistic_memLp n k h hh hk (dataLaw n P) 2
have hN := efronStein_of_gate n (fun _ => O) (fun _ => inferInstance)
(fun _ => Pobs P) (fun _ => inferInstance) (SN n h k) (by fun_prop) hLp.1
have hD := efronStein_of_gate n (fun _ => O) (fun _ => inferInstance)
(fun _ => Pobs P) (fun _ => inferInstance) (SD n h k) (by fun_prop) hLp.2
have hsN := Finset.sum_le_sum (s := univ) (fun r _ => (hstep r).1)
have hsD := Finset.sum_le_sum (s := univ) (fun r _ => (hstep r).2)
have hnR : (n : ℝ) ≠ 0 := by exact_mod_cast hn.ne'
have heq : (∑ _ : Fin n, 36 * Wocc n h k P / (n : ℝ)) = 36 * Wocc n h k P := by
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul]
field_simp
rw [heq] at hsN hsD
dsimp only [dataLaw] at hsN hsD
apply max_le
· exact hN.trans (by linarith)
· exact hD.trans (by linarith)
theorem all_count_stability
Sensitivity includes every input and every occupancy; iid variance on the complete model is controlled by the population occupancy weight. The replacement-copy inequality is an explicit assumed gate. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
efronStein_of_gate :
n k :
ℕ
h :
ℝ
hn :
2 ≤ n
hk :
2 ≤ k
hh :
0 < h ∧ h ≤ 1 / 4
P :
hiid :
IIDSampling n P Q
hP :
Proof (Lean source)
-- @node: lem:all-count-stability
lemma all_count_stability (efronStein_of_gate : EfronSteinReplacement)
(n k : ℕ) (h : ℝ) (hn : 2 ≤ n) (hk : 2 ≤ k) (hh : 0 < h ∧ h ≤ 1 / 4)
(P : CausalLaw) (Q : Measure (Dataset n)) (hiid : IIDSampling n P Q)
(hP : CompleteModel P) :
(∀ D Dprime : Dataset n, dHam n D Dprime = 1 →
|SN n h k D - SN n h k Dprime| + |SD n h k D - SD n h k Dprime| ≤ 12) ∧
max (variance (SN n h k) Q) (variance (SD n h k) Q) ≤ 18*Wocc n h k P := by
constructor
· intro D Dprime hadj
exact statistic_hamming_sensitivity n k h hh.1 (by omega) D Dprime hadj
· -- The iid product calculation gives both old and new partner expectations.
have hpartners : ∀ r : Fin n, (∫ zz : Dataset n × O,
(partnerIndicator n k h zz.1 r (zz.1 r) + partnerIndicator n k h zz.1 r zz.2)
∂((dataLaw n P).prod (Pobs P))) = 2 * Wocc n h k P / n := by
intro r
exact iid_partner_sum_expectation n k h (by omega) hh.1 (by omega) P r
have hvariance := statistic_efronStein_assembly efronStein_of_gate n k h
(by omega) (by omega) hh.1 P
(fun r => statistic_replacement_moments_of_partners n k h hh.1 (by omega) P r
(hpartners r))
rw [show Q = dataLaw n P from hiid]
exact hvariance
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.PointVersion 26 declarations Continuity selects the uniquely identified point-CATE representative from the observed law.
Continuity selects the uniquely identified point-CATE representative from the observed law.
theorem contrast_lipschitzWith
The model's contrast modulus is an ordinary Lipschitz bound. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: contrast_lipschitzWith
lemma contrast_lipschitzWith (P : CausalLaw) (hP : CompleteModel P) :
LipschitzWith 3 (tau P) := by
apply LipschitzWith.of_dist_le_mul
intro x y
simpa only [Real.dist_eq, Subtype.dist_eq, NNReal.coe_ofNat, L] using
hP.contrast_lipschitz x y
theorem continuous_tau
The selected contrast is continuous under the complete model. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
Proof (Lean source)
-- @node: continuous_tau
@[fun_prop] lemma continuous_tau (P : CausalLaw) (hP : CompleteModel P) :
Continuous (tau P) := by
exact (contrast_lipschitzWith P hP).continuous
theorem volume_absolutelyContinuous_PX
The positive design density transfers design-null sets to Lebesgue-null sets. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
Proof (Lean source)
-- @node: volume_absolutelyContinuous_PX
lemma volume_absolutelyContinuous_PX (P : CausalLaw) (hP : CompleteModel P) :
(volume : Measure Covariate) ≪ PX P := by
rw [PX, P.density_version hP.density.1]
apply withDensity_absolutelyContinuous'
· exact (ENNReal.measurable_ofReal.comp P.f_measurable).aemeasurable
· filter_upwards [hP.density.2] with x hx
exact ENNReal.ofReal_ne_zero_iff.mpr (lt_of_lt_of_le (by norm_num) hx.1)
Continuous functions on the unit interval that agree Lebesgue-almost everywhere agree at every point, including the two boundary points. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
v = w
Proof (Lean source)
-- @node: continuous_eq_of_volume_ae_eq
lemma continuous_eq_of_volume_ae_eq {v w : Covariate → ℝ}
(hv : Continuous v) (hw : Continuous w) (heq : v =ᵐ[volume] w) : v = w := by
let c : ℝ → Covariate := projIcc 0 1 (by norm_num)
have hc : Continuous c := by fun_prop
have hae : (v ∘ c) =ᵐ[(volume : Measure ℝ).restrict (Icc 0 1)] (w ∘ c) := by
rw [← unitInterval.measurePreserving_coe.map_eq]
apply (ae_map_iff (by fun_prop)
(isClosed_eq (hv.comp hc) (hw.comp hc)).measurableSet).mpr
filter_upwards [heq] with x hx
simpa [c, projIcc_of_mem (by norm_num : (0 : ℝ) ≤ 1) x.property] using hx
have hevery := Measure.eqOn_Icc_of_ae_eq (volume : Measure ℝ)
(by norm_num : (0 : ℝ) ≠ 1) hae (hv.comp hc).continuousOn (hw.comp hc).continuousOn
funext x
simpa [c, projIcc_of_mem (by norm_num : (0 : ℝ) ≤ 1) x.property] using hevery x.property
theorem continuous_tau_unique
Positive design density makes the continuous representative of the contrast unique. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
v = tau P
Proof (Lean source)
-- @node: continuous_tau_unique
lemma continuous_tau_unique (P : CausalLaw) (hP : CompleteModel P)
(v : Covariate → ℝ) (hv : Continuous v) (heq : v =ᵐ[PX P] tau P) : v = tau P := by
exact continuous_eq_of_volume_ae_eq hv (continuous_tau P hP)
((volume_absolutelyContinuous_PX P hP).ae_eq heq)
The treatment coordinate is measurable. The displayed conclusion follows.
Formal statement
conclusion
Proof (Lean source)
-- @node: measurable_A
@[fun_prop] lemma measurable_A : Measurable A := by unfold A; fun_prop
The observed outcome coordinate is measurable. The displayed conclusion follows.
Formal statement
conclusion
Proof (Lean source)
-- @node: measurable_Y
@[fun_prop] lemma measurable_Y : Measurable Y := by unfold Y; fun_prop
Each treatment-arm event is measurable. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: arm_event_measurable
lemma arm_event_measurable (d : Bool) : MeasurableSet {z : Record | A z = d} := by
exact measurableSet_eq_fun measurable_A measurable_const
theorem arm_design_apply
The arm-specific design measure evaluates a covariate set as its joint arm event. the theorem's stated inputs and assumptions, and the asserted conclusion follows.
Formal statement
Proof (Lean source)
-- @node: arm_design_apply
lemma arm_design_apply (P : CausalLaw) (d : Bool) (s : Set Covariate)
(hs : MeasurableSet s) :
((P.law.restrict {z | A z = d}).map X) s = P.law {z | X z ∈ s ∧ A z = d} := by
rw [Measure.map_apply measurable_X hs, Measure.restrict_apply (measurable_X hs)]
rfl
theorem arm_prob_setIntegral
The selected propensity and its complement integrate to the two arm probabilities. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
∫ x in s, (if d then P.e x else 1 - P.e x) ∂PX P = P.law.real {z | X z ∈ s ∧ A z = d}
Proof (Lean source)
-- @node: arm_prob_setIntegral
lemma arm_prob_setIntegral (P : CausalLaw) (d : Bool) (s : Set Covariate)
(hs : MeasurableSet s) :
∫ x in s, (if d then P.e x else 1 - P.e x) ∂PX P =
P.law.real {z | X z ∈ s ∧ A z = d} := by
let : IsProbabilityMeasure (PX P) := Measure.isProbabilityMeasure_map measurable_X.aemeasurable
have he : Integrable P.e (PX P) := P.e_integrable
cases d
· have hpartition : (X ⁻¹' s) = {z | X z ∈ s ∧ A z = false} ∪
{z | X z ∈ s ∧ A z = true} := by
ext z
cases h : A z <;> simp [h]
have hd : Disjoint {z | X z ∈ s ∧ A z = false}
{z | X z ∈ s ∧ A z = true} := by
rw [Set.disjoint_left]
intro z hz hz'
exact Bool.false_ne_true (hz.2.symm.trans hz'.2)
have ht : MeasurableSet {z | X z ∈ s ∧ A z = true} :=
(measurable_X hs).inter (arm_event_measurable true)
have hm : (PX P).real s = P.law.real {z | X z ∈ s ∧ A z = false} +
P.law.real {z | X z ∈ s ∧ A z = true} := by
rw [PX, measureReal_def, Measure.map_apply measurable_X hs, hpartition,
← measureReal_def, measureReal_union hd ht]
simp only [Bool.false_eq_true, if_false]
rw [integral_sub (integrable_const _).integrableOn he.integrableOn,
setIntegral_const, smul_eq_mul, mul_one]
have hv : ∫ x in s, P.e x ∂PX P = P.law.real {z | X z ∈ s ∧ A z = true} :=
P.e_version s hs
linarith
· exact P.e_version s hs
theorem PX_absolutelyContinuous_arm
Overlap transfers arm-specific null sets to design null sets. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
Proof (Lean source)
-- @node: PX_absolutelyContinuous_arm
lemma PX_absolutelyContinuous_arm (P : CausalLaw) (hP : CompleteModel P) (d : Bool) :
PX P ≪ (P.law.restrict {z | A z = d}).map X := by
let : IsProbabilityMeasure (PX P) := Measure.isProbabilityMeasure_map measurable_X.aemeasurable
apply Measure.AbsolutelyContinuous.mk
intro s hs hz
have hzero : P.law.real {z | X z ∈ s ∧ A z = d} = 0 := by
rw [measureReal_def, ← arm_design_apply P d s hs, hz, ENNReal.toReal_zero]
have hi : Integrable (fun x => if d then P.e x else 1 - P.e x) (PX P) := by
cases d
· exact (integrable_const _).sub P.e_integrable
· exact P.e_integrable
have hl : ∫ x in s, (1/4 : ℝ) ∂PX P ≤
∫ x in s, (if d then P.e x else 1 - P.e x) ∂PX P := by
apply integral_mono_ae (integrable_const _) hi.integrableOn
filter_upwards with x
cases d <;> simp only [Bool.false_eq_true, if_false, if_true]
· linarith [(hP.overlap x).2]
· exact (hP.overlap x).1
rw [arm_prob_setIntegral P d s hs, hzero, setIntegral_const, smul_eq_mul] at hl
have hr : (PX P).real s = 0 := by
have := measureReal_nonneg (μ := PX P) (s := s)
nlinarith
exact (ENNReal.toReal_eq_zero_iff (PX P s)).mp hr |>.resolve_right (measure_ne_top _ _)
theorem continuous_mu0
The control Holder modulus gives continuity of the selected control mean. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
Proof (Lean source)
-- @node: continuous_mu0
@[fun_prop] lemma continuous_mu0 (P : CausalLaw) (hP : CompleteModel P) :
Continuous P.mu0 := by
have hh : HolderWith 3 (1/10 : ℝ≥0) P.mu0 := by
intro x y
rw [edist_dist, edist_dist]
have hr := hP.control_holder x y
have hn : (0 : ℝ) ≤ 1/10 := by norm_num
have he := ENNReal.ofReal_le_ofReal hr
simpa only [L, ← Real.dist_eq, ← Subtype.dist_eq,
ENNReal.ofReal_mul (by norm_num : (0 : ℝ) ≤ 3),
ENNReal.ofReal_rpow_of_nonneg (dist_nonneg : 0 ≤ dist x y) hn,
ENNReal.ofReal_ofNat, NNReal.coe_div, NNReal.coe_one, NNReal.coe_ofNat,
ENNReal.coe_ofNat] using he
exact hh.continuous (by norm_num)
theorem continuous_mu1
Continuity of the control mean and contrast gives continuity of the treated mean. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
Proof (Lean source)
-- @node: continuous_mu1
@[fun_prop] lemma continuous_mu1 (P : CausalLaw) (hP : CompleteModel P) :
Continuous P.mu1 := by
have hid : P.mu1 = fun x => P.mu0 x + tau P x := by
funext x
simp [tau]
rw [hid]
exact (continuous_mu0 P hP).add (continuous_tau P hP)
theorem arm_mean_ae_mem_Icc
The observed binary response bounds each selected arm mean almost everywhere. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: arm_mean_ae_mem_Icc
lemma arm_mean_ae_mem_Icc (P : CausalLaw) (d : Bool) :
∀ᵐ x ∂((P.law.restrict {z | A z = d}).map X),
(if d then P.mu1 x else P.mu0 x) ∈ Icc (0 : ℝ) 1 := by
have hi : Integrable (fun x => if d then P.mu1 x else P.mu0 x)
((P.law.restrict {z | A z = d}).map X) := by
cases d
· exact P.mu0_integrable
· exact P.mu1_integrable
have hv (s : Set Covariate) (hs : MeasurableSet s) :
∫ x in s, (if d then P.mu1 x else P.mu0 x)
∂((P.law.restrict {z | A z = d}).map X) =
∫ z in {z | X z ∈ s ∧ A z = d}, bit (Y z) ∂P.law := by
cases d
· exact P.mu0_version s hs
· exact P.mu1_version s hs
have hlo : 0 ≤ᵐ[((P.law.restrict {z | A z = d}).map X)]
(fun x => if d then P.mu1 x else P.mu0 x) := by
apply ae_nonneg_of_forall_setIntegral_nonneg hi
intro s hs _
rw [hv s hs]
apply integral_nonneg
intro z
simp only [bit]
split_ifs <;> norm_num
have hup : (fun x => if d then P.mu1 x else P.mu0 x) ≤ᵐ[
((P.law.restrict {z | A z = d}).map X)] (fun _ => (1 : ℝ)) := by
apply ae_le_of_forall_setIntegral_le hi (integrable_const _)
intro s hs _
rw [hv s hs, setIntegral_const, smul_eq_mul, mul_one, measureReal_def,
arm_design_apply P d s hs, ← measureReal_def, ← setIntegral_one_eq_measureReal]
apply integral_mono_ae
· apply Integrable.of_mem_Icc 0 1 (by fun_prop)
filter_upwards with z
simp only [bit]
split_ifs <;> norm_num
· exact integrable_const _
· filter_upwards with z
simp only [bit]
split_ifs <;> norm_num
filter_upwards [hlo, hup] with x hx hy
exact ⟨hx, hy⟩
theorem arm_mean_mem_Icc
Continuity and positive arm support extend the binary mean bounds to every covariate. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
conclusion
(if d then P.mu1 x else P.mu0 x) ∈ Icc (0 : ℝ) 1
Proof (Lean source)
-- @node: arm_mean_mem_Icc
lemma arm_mean_mem_Icc (P : CausalLaw) (hP : CompleteModel P) (d : Bool) (x : Covariate) :
(if d then P.mu1 x else P.mu0 x) ∈ Icc (0 : ℝ) 1 := by
let m : Covariate → ℝ := fun x => if d then P.mu1 x else P.mu0 x
have hc : Continuous m := by
cases d
· exact continuous_mu0 P hP
· exact continuous_mu1 P hP
have hae : ∀ᵐ x ∂(volume : Measure Covariate), m x ∈ Icc (0 : ℝ) 1 :=
((volume_absolutelyContinuous_PX P hP).trans (PX_absolutelyContinuous_arm P hP d)).ae_le
(arm_mean_ae_mem_Icc P d)
have heq : m = fun x => max 0 (min 1 (m x)) := by
apply continuous_eq_of_volume_ae_eq hc (by fun_prop)
filter_upwards [hae] with x hx
rw [min_eq_right hx.2, max_eq_right hx.1]
change m x ∈ Icc (0 : ℝ) 1
rw [congrFun heq x]
exact ⟨le_max_left _ _, max_le (by norm_num) (min_le_left _ _)⟩
theorem theta_mem_Icc
The difference of the two binary arm means bounds the point target between minus one and one. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: theta_mem_Icc
lemma theta_mem_Icc (P : CausalLaw) (hP : CompleteModel P) : theta P ∈ Icc (-1 : ℝ) 1 := by
have h0 := arm_mean_mem_Icc P hP false x0
have h1 := arm_mean_mem_Icc P hP true x0
simp only [Bool.false_eq_true, if_false, if_true] at h0 h1
change P.mu1 x0 - P.mu0 x0 ∈ Icc (-1 : ℝ) 1
constructor <;> linarith [h0.1, h0.2, h1.1, h1.2]
The observation map is measurable. The displayed conclusion follows.
Formal statement
conclusion
Measurable observe
Proof (Lean source)
-- @node: measurable_observe
@[fun_prop] lemma measurable_observe : Measurable observe := by
unfold observe
fun_prop
theorem arm_design_eq_observed
The observed marginal determines each arm-specific covariate measure. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: arm_design_eq_observed
lemma arm_design_eq_observed (P : CausalLaw) (d : Bool) :
(P.law.restrict {z | A z = d}).map X =
((Pobs P).restrict {z : O | z.2.1 = d}).map fst := by
rw [Pobs, Measure.restrict_map measurable_observe
(measurableSet_eq_fun (by fun_prop) measurable_const),
Measure.map_map measurable_fst measurable_observe]
rfl
theorem observed_arm_outcome_integral
The observed marginal determines the response integral on every joint covariate-arm event. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: observed_arm_outcome_integral
lemma observed_arm_outcome_integral (P : CausalLaw) (d : Bool)
(s : Set Covariate) (hs : MeasurableSet s) :
∫ z in {z | X z ∈ s ∧ A z = d}, bit (Y z) ∂P.law =
∫ z in {z : O | z.1 ∈ s ∧ z.2.1 = d}, bit z.2.2 ∂Pobs P := by
have ht : MeasurableSet {z : O | z.1 ∈ s ∧ z.2.1 = d} :=
(hs.preimage measurable_fst).inter
(measurableSet_eq_fun (by fun_prop) measurable_const)
rw [Pobs, setIntegral_map ht (by fun_prop) measurable_observe.aemeasurable]
rfl
theorem arm_mean_eq_of_observed_eq
Equal observed laws have identical continuous arm-mean versions at every covariate. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
conclusion
(fun x => if d then P'.mu1 x else P'.mu0 x) = (fun x => if d then P.mu1 x else P.mu0 x)
Proof (Lean source)
-- @node: arm_mean_eq_of_observed_eq
lemma arm_mean_eq_of_observed_eq (P P' : CausalLaw) (hP : CompleteModel P)
(hP' : CompleteModel P') (hobs : Pobs P' = Pobs P) (d : Bool) :
(fun x => if d then P'.mu1 x else P'.mu0 x) =
(fun x => if d then P.mu1 x else P.mu0 x) := by
have hmeasure : (P'.law.restrict {z | A z = d}).map X =
(P.law.restrict {z | A z = d}).map X := by
rw [arm_design_eq_observed P' d, arm_design_eq_observed P d, hobs]
have hi (R : CausalLaw) : Integrable (fun x => if d then R.mu1 x else R.mu0 x)
((R.law.restrict {z | A z = d}).map X) := by
cases d
· exact R.mu0_integrable
· exact R.mu1_integrable
have hv (R : CausalLaw) (s : Set Covariate) (hs : MeasurableSet s) :
∫ x in s, (if d then R.mu1 x else R.mu0 x)
∂((R.law.restrict {z | A z = d}).map X) =
∫ z in {z : O | z.1 ∈ s ∧ z.2.1 = d}, bit z.2.2 ∂Pobs R := by
rw [← observed_arm_outcome_integral R d s hs]
cases d
· exact R.mu0_version s hs
· exact R.mu1_version s hs
have hae : (fun x => if d then P'.mu1 x else P'.mu0 x) =ᵐ[
((P.law.restrict {z | A z = d}).map X)] (fun x => if d then P.mu1 x else P.mu0 x) := by
apply Integrable.ae_eq_of_forall_setIntegral_eq
· rw [← hmeasure]
exact hi P'
· exact hi P
· intro s hs _
rw [← hmeasure, hv P' s hs, hmeasure, hv P s hs, hobs]
apply continuous_eq_of_volume_ae_eq
· cases d
· exact continuous_mu0 P' hP'
· exact continuous_mu1 P' hP'
· cases d
· exact continuous_mu0 P hP
· exact continuous_mu1 P hP
· exact ((volume_absolutelyContinuous_PX P hP).trans
(PX_absolutelyContinuous_arm P hP d)).ae_eq hae
theorem theta_eq_of_observed_eq
The observed law identifies the point target through uniqueness of the continuous arm means. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
conclusion
theta P' = theta P
Proof (Lean source)
-- @node: theta_eq_of_observed_eq
lemma theta_eq_of_observed_eq (P P' : CausalLaw) (hP : CompleteModel P)
(hP' : CompleteModel P') (hobs : Pobs P' = Pobs P) : theta P' = theta P := by
have h0 := congrFun (arm_mean_eq_of_observed_eq P P' hP hP' hobs false) x0
have h1 := congrFun (arm_mean_eq_of_observed_eq P P' hP hP' hobs true) x0
simp only [Bool.false_eq_true, if_false, if_true] at h0 h1
change P'.mu1 x0 - P'.mu0 x0 = P.mu1 x0 - P.mu0 x0
rw [h0, h1]
theorem arm_design_withDensity
Each arm-specific design measure has the selected arm probability as its density relative to the design. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: arm_design_withDensity
lemma arm_design_withDensity (P : CausalLaw) (hP : CompleteModel P) (d : Bool) :
(P.law.restrict {z | A z = d}).map X =
(PX P).withDensity (fun x => ofReal (if d then P.e x else 1-P.e x)) := by
have hq : Measurable (fun x => if d then P.e x else 1-P.e x) := by
have he := P.e_measurable
cases d <;> simp only [Bool.false_eq_true, if_false, if_true] <;> fun_prop
have hi : Integrable (fun x => if d then P.e x else 1-P.e x) (PX P) := by
let : IsProbabilityMeasure (PX P) := Measure.isProbabilityMeasure_map measurable_X.aemeasurable
cases d
· exact (integrable_const _).sub P.e_integrable
· exact P.e_integrable
have hn (x : Covariate) : 0 ≤ (if d then P.e x else 1-P.e x) := by
cases d <;> simp only [Bool.false_eq_true, if_false, if_true] <;>
linarith [(hP.overlap x).1, (hP.overlap x).2]
ext s hs
rw [withDensity_apply _ hs,
← ofReal_integral_eq_lintegral_ofReal hi.integrableOn (ae_of_all _ hn),
arm_prob_setIntegral P d s hs, measureReal_def,
ENNReal.ofReal_toReal (measure_ne_top _ _), arm_design_apply P d s hs]
A measurable binary response is integrable under any finite measure. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Integrable (fun z => bit (b z)) μ
Proof (Lean source)
-- @node: integrable_bit_comp
lemma integrable_bit_comp {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)
[IsFiniteMeasure μ] (b : Ω → Bool) (hb : Measurable b) :
Integrable (fun z => bit (b z)) μ := by
apply Integrable.of_mem_Icc 0 1 (by fun_prop)
filter_upwards with z
simp only [bit]
split_ifs <;> norm_num
theorem arm_conditional_mean
Exchangeability factors the arm-potential product; consistency and overlap identify its conditional mean. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
=ᵐ[P.law] (fun z => if d then P.mu1 (X z) else P.mu0 (X z))
Proof (Lean source)
-- @node: arm_conditional_mean
lemma arm_conditional_mean (P : CausalLaw) (hP : CompleteModel P) (d : Bool) :
P.law[(fun z => bit (if d then (Ypot z).2 else (Ypot z).1)) |
comap X inferInstance] =ᵐ[P.law]
(fun z => if d then P.mu1 (X z) else P.mu0 (X z)) := by
have hσ : comap X inferInstance ≤
(inferInstance : MeasurableSpace Record) := measurable_X.comap_le
let q : Covariate → ℝ := fun x => if d then P.e x else 1-P.e x
let m : Covariate → ℝ := fun x => if d then P.mu1 x else P.mu0 x
let r : Bool → ℝ := fun a => if a = d then 1 else 0
let b : Record → Bool := fun z => if d then (Ypot z).2 else (Ypot z).1
have hq : Measurable q := by
have he := P.e_measurable
cases d <;> dsimp [q] <;> fun_prop
have hm : Measurable m := by cases d <;> dsimp [m] <;> fun_prop
have hr : Measurable r := by dsimp [r]; fun_prop
have hb : Measurable b := by cases d <;> dsimp [b, Ypot] <;> fun_prop
have hqr (x : Covariate) : q x ∈ Icc (0 : ℝ) 1 := by
dsimp [q]
cases d <;> simp only [Bool.false_eq_true, if_false, if_true] <;>
constructor <;> linarith [(hP.overlap x).1, (hP.overlap x).2]
have hmr (x : Covariate) : m x ∈ Icc (0 : ℝ) 1 := arm_mean_mem_Icc P hP d x
have hqi : Integrable (fun z => q (X z)) P.law :=
Integrable.of_mem_Icc 0 1 (by fun_prop) (ae_of_all _ (fun z => hqr (X z)))
have hqmi : Integrable (fun z => q (X z) * m (X z)) P.law := by
apply Integrable.of_mem_Icc 0 1 (by fun_prop)
filter_upwards with z
exact ⟨mul_nonneg (hqr _).1 (hmr _).1,
mul_le_one₀ (hqr _).2 (hmr _).1 (hmr _).2⟩
have hri : Integrable (fun z => r (A z)) P.law := by
apply Integrable.of_mem_Icc 0 1 (by fun_prop)
filter_upwards with z
dsimp [r]
split_ifs <;> norm_num
have hyi := integrable_bit_comp P.law Y measurable_Y
have hbi := integrable_bit_comp P.law b hb
have hryi : Integrable (fun z => r (A z) * bit (Y z)) P.law := by
have heq : (fun z => r (A z) * bit (Y z)) =
{z | A z = d}.indicator (fun z => bit (Y z)) := by
funext z
simp [r, indicator]
rw [heq]
exact hyi.indicator (arm_event_measurable d)
have harm : P.law[(fun z => r (A z)) | comap X inferInstance]
=ᵐ[P.law] fun z => q (X z) := by
symm
apply ae_eq_condExp_of_forall_setIntegral_eq hσ hri
(fun _ _ _ => hqi.integrableOn)
· intro s hs _
obtain ⟨t, ht, rfl⟩ := MeasurableSpace.measurableSet_comap.mp hs
have hi := setIntegral_map (μ := P.law) ht hq.aestronglyMeasurable measurable_X.aemeasurable
have hv := arm_prob_setIntegral P d t ht
have heq : (fun z => r (A z)) = {z | A z = d}.indicator (fun _ => (1 : ℝ)) := by
funext z
simp [r, indicator]
rw [heq, setIntegral_indicator (arm_event_measurable d), setIntegral_one_eq_measureReal]
exact hi.symm.trans hv
· exact (hq.comp (comap_measurable X)).aestronglyMeasurable
have hprod : P.law[(fun z => r (A z) * bit (Y z)) |
comap X inferInstance] =ᵐ[P.law]
fun z => q (X z) * m (X z) := by
symm
apply ae_eq_condExp_of_forall_setIntegral_eq hσ hryi
(fun _ _ _ => hqmi.integrableOn)
· intro s hs _
obtain ⟨t, ht, rfl⟩ := MeasurableSpace.measurableSet_comap.mp hs
have hi := setIntegral_map (μ := P.law) ht
(hq.mul hm).aestronglyMeasurable measurable_X.aemeasurable
have hw : ∫ x in t, m x ∂((P.law.restrict {z | A z = d}).map X) =
∫ x in t, q x * m x ∂PX P := by
rw [arm_design_withDensity P hP d]
change (∫ x in t, m x ∂(PX P).withDensity (ofReal ∘ q)) = _
rw [setIntegral_withDensity_eq_setIntegral_toReal_smul₀ (μ := PX P)
(ENNReal.measurable_ofReal.comp hq).aemeasurable
(ae_of_all _ (fun x => ENNReal.ofReal_lt_top)) m ht]
apply integral_congr_ae
filter_upwards with x
simp [q, ENNReal.toReal_ofReal (hqr x).1, smul_eq_mul]
have hv : ∫ x in t, m x ∂((P.law.restrict {z | A z = d}).map X) =
∫ z in {z | X z ∈ t ∧ A z = d}, bit (Y z) ∂P.law := by
cases d
· exact P.mu0_version t ht
· exact P.mu1_version t ht
have heq : (fun z => r (A z) * bit (Y z)) =
{z | A z = d}.indicator (fun z => bit (Y z)) := by
funext z
simp [r, indicator]
rw [heq, setIntegral_indicator (arm_event_measurable d)]
exact hi.symm.trans (hw.symm.trans hv)
· exact ((hq.mul hm).comp (comap_measurable X)).aestronglyMeasurable
have hci : CondIndepFun (comap X inferInstance) hσ A b P.law := by
cases d
· exact hP.exchangeability.symm.comp measurable_id measurable_fst
· exact hP.exchangeability.symm.comp measurable_id measurable_snd
have hcons : (fun z => r (A z) * bit (Y z)) =ᵐ[P.law]
fun z => r (A z) * bit (b z) := by
filter_upwards [hP.consistency] with z hz
dsimp [r, b]
by_cases ha : A z = d
· cases d <;> simp_all
· simp [ha]
have hf := condExp_mul_of_condIndep
hσ measurable_A hb hci hr measurable_bit hri hbi (hryi.congr hcons)
have hfact := (condExp_congr_ae hcons).trans hf
filter_upwards [harm, hprod, hfact] with z ha hp hf
simp only [Pi.mul_apply] at hf
rw [ha] at hf
have hn : q (X z) ≠ 0 := by
have hpos : 1/4 ≤ q (X z) := by
dsimp [q]
cases d <;> simp only [Bool.false_eq_true, if_false, if_true] <;>
linarith [(hP.overlap (X z)).1, (hP.overlap (X z)).2]
linarith
exact mul_left_cancel₀ hn (hf.symm.trans hp)
theorem tau_conditional_causal_contrast
Conditional-expectation linearity identifies the selected contrast with the conditional causal effect. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
P :
hP :
Proof (Lean source)
-- @node: tau_conditional_causal_contrast
lemma tau_conditional_causal_contrast (P : CausalLaw) (hP : CompleteModel P) :
(fun z => tau P (X z)) =ᵐ[P.law]
P.law[(fun z => bit (Ypot z).2 - bit (Ypot z).1) |
comap X inferInstance] := by
have h1 := arm_conditional_mean P hP true
have h0 := arm_conditional_mean P hP false
have hi1 := integrable_bit_comp P.law (fun z => (Ypot z).2) (by unfold Ypot; fun_prop)
have hi0 := integrable_bit_comp P.law (fun z => (Ypot z).1) (by unfold Ypot; fun_prop)
exact ((condExp_sub hi1 hi0 _).trans (h1.sub h0)).symm
theorem point_version
Model regularity selects a unique continuous contrast, bounds the target, and identifies both the conditional causal contrast and its point value from the observed marginal. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
P :
hP :
Proof (Lean source)
-- @node: lem:point-version
lemma point_version (P : CausalLaw) (hP : CompleteModel P) :
Continuous (tau P) ∧
(∀ v : Covariate → ℝ, Continuous v → v =ᵐ[PX P] tau P → v = tau P) ∧
theta P ∈ Icc (-1) 1 ∧
(fun z => tau P (X z)) =ᵐ[P.law]
P.law[(fun z => bit (Ypot z).2 - bit (Ypot z).1) |
comap X inferInstance] ∧
(∀ P' : CausalLaw, CompleteModel P' → Pobs P' = Pobs P → theta P' = theta P) := by
refine ⟨continuous_tau P hP, ?_, ?_⟩
· intro v hv heq
exact continuous_tau_unique P hP v hv heq
· refine ⟨theta_mem_Icc P hP, tau_conditional_causal_contrast P hP, ?_⟩
intro P' hP' hobs
exact theta_eq_of_observed_eq P P' hP hP' hobs
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.PrivateComparison 9 declarations Private Hamming comparison, unrestricted Markov contraction, and finite-chi-square TV bounds.
Private Hamming comparison, unrestricted Markov contraction, and finite-chi-square TV bounds.
A composed probability kernel evaluates an event by averaging its row probabilities. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Proof (Lean source)
-- @node: kernel_comp_real
lemma kernel_comp_real {α β : Type*} [MeasurableSpace α] [MeasurableSpace β]
(K : Kernel α β) [IsMarkovKernel K] (Q : Measure α) [IsProbabilityMeasure Q]
(E : Set β) (hE : MeasurableSet E) :
(K ∘ₘ Q).real E = ∫ x, (K x).real E ∂Q := by
rw [measureReal_def, Measure.bind_apply hE K.aemeasurable]
exact (integral_toReal (K.measurable_coe hE).aemeasurable
(Filter.Eventually.of_forall fun x => measure_lt_top (K x) E)).symm
Averaging a Markov kernel contracts total variation, by the bounded-function dual bound. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
Proof (Lean source)
-- @node: kernel_TV_contraction
lemma kernel_TV_contraction {α β : Type*} [MeasurableSpace α] [MeasurableSpace β]
(K : Kernel α β) [IsMarkovKernel K] (Q Q' : Measure α)
[IsProbabilityMeasure Q] [IsProbabilityMeasure Q'] :
TV (K ∘ₘ Q) (K ∘ₘ Q') ≤ TV Q Q' := by
refine ciSup_le fun E => ?_
rw [kernel_comp_real K Q E.1 E.2, kernel_comp_real K Q' E.1 E.2]
simpa using tvDist_integral_range Q Q'
(fun x => (K x).real E.1) (K.measurable_coe E.2).ennreal_toReal
0 1 (by norm_num) (fun x => ⟨measureReal_nonneg, by simp [measureReal_le_one]⟩)
Finite squared density deviation supplies the integrability needed for the chi-square TV bound. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
α :
Type*
Q Q' :
hac :
Q ≪ Q'
Proof (Lean source)
-- @node: TV_le_sqrt_chiSq_of_finite
lemma TV_le_sqrt_chiSq_of_finite {α : Type*} [MeasurableSpace α]
(Q Q' : Measure α) [IsProbabilityMeasure Q] [IsProbabilityMeasure Q']
(hac : Q ≪ Q')
(hf : (∫⁻ z, ofReal (((Q.rnDeriv Q' z).toReal - 1)^2) ∂Q') < ⊤) :
TV Q Q' ≤ 1/2 * sqrt (chiSqDiv Q Q') := by
apply tvDist_le_half_sqrt_chiSqDiv Q Q' hac
apply (lintegral_ofReal_ne_top_iff_integrable
(((Q.measurable_rnDeriv Q').ennreal_toReal.sub_const 1).pow_const 2).aestronglyMeasurable
(Filter.Eventually.of_forall fun z => sq_nonneg _)).mp
exact hf.ne
Convexity on the unit interval bounds the privacy exponential increment. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: exp_privacy_increment_le
lemma exp_privacy_increment_le (epsilon : ℝ) (he : 0 ≤ epsilon ∧ epsilon ≤ 1) :
exp epsilon - 1 ≤ 2 * epsilon := by
have hc := convexOn_exp.2 (Set.mem_univ (0 : ℝ)) (Set.mem_univ (1 : ℝ))
(show 0 ≤ 1 - epsilon by linarith) he.1 (show 1 - epsilon + epsilon = 1 by ring)
simp only [smul_eq_mul, mul_zero, zero_add, mul_one, Real.exp_zero] at hc
nlinarith [Real.exp_one_lt_three]
theorem private_adjacent_event_bound
Privacy in both directions bounds the event-probability change on an adjacent pair. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
n :
ℕ
epsilon :
ℝ
he :
0 < epsilon ∧ epsilon ≤ 1
B :
Type*
M :
hM :
PrivateKernel n epsilon M
D D' :
Dataset n
hadj :
dHam n D D' = 1
E :
Set B
hE :
Proof (Lean source)
-- @node: private_adjacent_event_bound
lemma private_adjacent_event_bound (n : ℕ) (epsilon : ℝ)
(he : 0 < epsilon ∧ epsilon ≤ 1) {B : Type*} [MeasurableSpace B]
(M : Kernel (Dataset n) B) [IsMarkovKernel M]
(hM : PrivateKernel n epsilon M) (D D' : Dataset n)
(hadj : dHam n D D' = 1) (E : Set B) (hE : MeasurableSet E) :
|(M D).real E - (M D').real E| ≤ 2 * epsilon := by
have hf := hM.pure_privacy D D' hadj E hE
have hr := hM.pure_privacy D' D (by simpa [dHam, hammingDist_comm] using hadj) E hE
have hc : 0 ≤ exp epsilon - 1 := sub_nonneg.mpr (Real.one_le_exp he.1.le)
have hD := mul_le_mul_of_nonneg_left
(measureReal_le_one (μ := M D) (s := E)) hc
have hD' := mul_le_mul_of_nonneg_left
(measureReal_le_one (μ := M D') (s := E)) hc
have hexp := exp_privacy_increment_le epsilon ⟨he.1.le, he.2⟩
rw [abs_le]
constructor <;> nlinarith
A one-coordinate oscillation bound sums along the finite Hamming replacement path. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: event_bound_of_hamming_path
lemma event_bound_of_hamming_path (n : ℕ) (f : Dataset n → ℝ) (c : ℝ) (hc : 0 ≤ c)
(hstep : ∀ D D', dHam n D D' = 1 → |f D - f D'| ≤ c)
(D D' : Dataset n) : |f D - f D'| ≤ c * dHam n D D' := by
classical
have hfinite (s : Finset (Fin n)) : ∀ D D' : Dataset n,
(∀ i, i ∉ s → D i = D' i) → |f D - f D'| ≤ c * s.card := by
induction s using Finset.induction_on with
| empty =>
intro D D' h
have heq : D = D' := funext fun i => h i (by simp)
simp [heq]
| @insert i s hi ih =>
intro D D' h
let Dmid := update D i (D' i)
have hagree : ∀ j, j ∉ s → Dmid j = D' j := by
intro j hj
by_cases hji : j = i
· subst j; simp [Dmid]
· simp only [Dmid, Function.update_of_ne hji]
exact h j (by simp [hj, hji])
have hrest := ih Dmid D' hagree
have hfirst : |f D - f Dmid| ≤ c := by
by_cases heq : D i = D' i
· have hm : Dmid = D := by simp [Dmid, ← heq]
simpa [hm] using hc
· apply hstep D Dmid
have hfilter : Finset.univ.filter (fun j => D j ≠ Dmid j) = {i} := by
ext j
by_cases hji : j = i
· subst j; simp [Dmid, heq]
· simp [Dmid, Function.update_of_ne hji, hji]
simp [dHam, hammingDist, hfilter]
calc
|f D - f D'| ≤ |f D - f Dmid| + |f Dmid - f D'| := abs_sub_le _ _ _
_ ≤ c + c * s.card := add_le_add hfirst hrest
_ = c * (insert i s).card := by rw [Finset.card_insert_of_notMem hi]; push_cast; ring
exact hfinite (Finset.univ.filter (fun i => D i ≠ D' i)) D D'
(by intro i hi; simpa using hi)
theorem private_hamming_event_bound
Pure privacy controls every event along the Hamming path, including out-of-model datasets. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
n :
ℕ
epsilon :
ℝ
he :
0 < epsilon ∧ epsilon ≤ 1
B :
Type*
M :
hM :
PrivateKernel n epsilon M
D D' :
Dataset n
E :
Set B
hE :
Proof (Lean source)
-- @node: private_hamming_event_bound
lemma private_hamming_event_bound (n : ℕ) (epsilon : ℝ)
(he : 0 < epsilon ∧ epsilon ≤ 1) {B : Type*} [MeasurableSpace B]
(M : Kernel (Dataset n) B) [IsMarkovKernel M]
(hM : PrivateKernel n epsilon M) (D D' : Dataset n)
(E : Set B) (hE : MeasurableSet E) :
|(M D).real E - (M D').real E| ≤ 2 * epsilon * dHam n D D' := by
exact event_bound_of_hamming_path n (fun D => (M D).real E) (2 * epsilon)
(mul_nonneg (by norm_num) he.1.le) (fun D D' h => private_adjacent_event_bound n epsilon he M hM D D' h E hE)
D D'
theorem private_TV_le_coupling_cost
Integrating the Hamming event bound against any coupling controls output total variation. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
n :
ℕ
epsilon :
ℝ
he :
0 < epsilon ∧ epsilon ≤ 1
B :
Type*
M :
hM :
PrivateKernel n epsilon M
Q Q' :
hγ :
IsCoupling γ Q Q'
Proof (Lean source)
-- @node: private_TV_le_coupling_cost
lemma private_TV_le_coupling_cost (n : ℕ) (epsilon : ℝ)
(he : 0 < epsilon ∧ epsilon ≤ 1) {B : Type*} [MeasurableSpace B]
(M : Kernel (Dataset n) B) [IsMarkovKernel M]
(hM : PrivateKernel n epsilon M) (Q Q' : Measure (Dataset n))
[IsProbabilityMeasure Q] [IsProbabilityMeasure Q']
(γ : Measure (Dataset n × Dataset n)) (hγ : IsCoupling γ Q Q') :
ofReal (TV (M ∘ₘ Q) (M ∘ₘ Q')) ≤
ofReal (2 * epsilon) * ∫⁻ z, (dHam n z.1 z.2 : ℝ≥0∞) ∂γ := by
letI := hγ.isProbabilityMeasure
have hcost : (∫⁻ z, (dHam n z.1 z.2 : ℝ≥0∞) ∂γ) ≤ (n : ℝ≥0∞) := by
calc
_ ≤ ∫⁻ _z, (n : ℝ≥0∞) ∂γ := lintegral_mono fun z => by
exact Nat.cast_le.mpr (show dHam n z.1 z.2 ≤ n from by
simpa [dHam] using (hammingDist_le_card_fintype (x := z.1) (y := z.2)))
_ = n := by simp
have hfin : ofReal (2 * epsilon) *
(∫⁻ z, (dHam n z.1 z.2 : ℝ≥0∞) ∂γ) ≠ ⊤ :=
ENNReal.mul_ne_top ENNReal.ofReal_ne_top (ne_top_of_le_ne_top (by simp) hcost)
apply (ENNReal.ofReal_le_iff_le_toReal hfin).mpr
refine ciSup_le fun E => ?_
let f : Dataset n → ℝ := fun D => (M D).real E.1
have hf : Measurable f := (M.measurable_coe E.2).ennreal_toReal
have hbound : ∀ D, ‖f D‖ ≤ 1 := fun D => by
rw [Real.norm_eq_abs, abs_of_nonneg measureReal_nonneg]
exact measureReal_le_one
have hi1 : Integrable (fun z => f z.1) γ :=
Integrable.of_bound (hf.comp measurable_fst).aestronglyMeasurable 1
(Filter.Eventually.of_forall fun z => hbound z.1)
have hi2 : Integrable (fun z => f z.2) γ :=
Integrable.of_bound (hf.comp measurable_snd).aestronglyMeasurable 1
(Filter.Eventually.of_forall fun z => hbound z.2)
rw [kernel_comp_real M Q E.1 E.2, kernel_comp_real M Q' E.1 E.2]
change |(∫ D, f D ∂Q) - ∫ D, f D ∂Q'| ≤ _
rw [← hγ.map_fst, ← hγ.map_snd,
integral_map measurable_fst.aemeasurable hf.aestronglyMeasurable,
integral_map measurable_snd.aemeasurable hf.aestronglyMeasurable]
apply (ENNReal.ofReal_le_iff_le_toReal hfin).mp
calc
ofReal |(∫ z, f z.1 ∂γ) - ∫ z, f z.2 ∂γ| ≤
∫⁻ z, ofReal |f z.1 - f z.2| ∂γ := by
simpa only [edist_dist, Real.dist_eq] using edist_integral_le_lintegral_edist hi1 hi2
_ ≤ ∫⁻ z, ofReal (2 * epsilon) * (dHam n z.1 z.2 : ℝ≥0∞) ∂γ := by
apply lintegral_mono
intro z
simpa only [f, ENNReal.ofReal_mul (show 0 ≤ 2 * epsilon from
mul_nonneg (by norm_num) he.1.le),
ENNReal.ofReal_natCast] using ENNReal.ofReal_le_ofReal
(private_hamming_event_bound n epsilon he M hM z.1 z.2 E.1 E.2)
_ = _ := lintegral_const_mul' _ _ ENNReal.ofReal_ne_top
theorem private_kernel_comparison
Replacement privacy contracts expected Hamming transport. Ordinary Markov contraction needs no privacy, and finite chi-square divergence gives the square-root TV bound. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
n :
ℕ
epsilon :
ℝ
he :
0 < epsilon ∧ epsilon ≤ 1
B :
@realizes B(standard Borel output space)
M :
@realizes M(everywhere probability-valued kernel)
hM :
PrivateKernel n epsilon M
Q Q' :
Proof (Lean source)
-- @node: lem:private-kernel-comparison
lemma private_kernel_comparison (n : ℕ) (epsilon : ℝ) (he : 0 < epsilon ∧ epsilon ≤ 1)
{B : Type*} [MeasurableSpace B]
[StandardBorelSpace B] -- @realizes B(standard Borel output space)
(M : Kernel (Dataset n) B)
[IsMarkovKernel M] -- @realizes M(everywhere probability-valued kernel)
(hM : PrivateKernel n epsilon M)
(Q Q' : Measure (Dataset n)) [IsProbabilityMeasure Q] [IsProbabilityMeasure Q'] :
ofReal (TV (M ∘ₘ Q) (M ∘ₘ Q')) ≤ ofReal (2*epsilon) * WH n Q Q' ∧
(∀ K : Kernel (Dataset n) B, IsMarkovKernel K →
TV (K ∘ₘ Q) (K ∘ₘ Q') ≤ TV Q Q') ∧
(Q ≪ Q' → (∫⁻ z, ofReal (((Q.rnDeriv Q' z).toReal - 1)^2) ∂Q') < ⊤ →
TV Q Q' ≤ 1/2 * sqrt (chiSqDiv Q Q')) := by
refine ⟨?_, ?_, ?_⟩
· rw [WH, ENNReal.mul_iInf_of_ne
(ENNReal.ofReal_ne_zero_iff.mpr (mul_pos (by norm_num) he.1)) ENNReal.ofReal_ne_top]
apply le_iInf
intro γ
rw [ENNReal.mul_iInf_of_ne
(ENNReal.ofReal_ne_zero_iff.mpr (mul_pos (by norm_num) he.1)) ENNReal.ofReal_ne_top]
apply le_iInf
intro hγ
exact private_TV_le_coupling_cost n epsilon he M hM Q Q' γ hγ
· intro K hK
letI := hK
exact kernel_TV_contraction K Q Q'
· exact TV_le_sqrt_chiSq_of_finite Q Q'
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.RateRegimes 13 declarations Exact comparisons of the benchmark terms and the four public budget regimes.
Exact comparisons of the benchmark terms and the four public budget regimes.
The sparse-privacy term meets the sampling term at the fourth-root budget. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
x epsilon :
ℝ
hx :
0 < x
he :
0 < epsilon
conclusion
(x ^ 2 * epsilon) ^ (-1 / 7 : ℝ) ≤ x ^ (-1 / 4 : ℝ) ↔ x ^ (-1 / 4 : ℝ) ≤ epsilon
Proof (Lean source)
-- @node: sparse_le_sampling_iff
lemma sparse_le_sampling_iff (x epsilon : ℝ) (hx : 0 < x) (he : 0 < epsilon) :
(x ^ 2 * epsilon) ^ (-1 / 7 : ℝ) ≤ x ^ (-1 / 4 : ℝ) ↔ x ^ (-1 / 4 : ℝ) ≤ epsilon := by
have hxe : 0 < x ^ 2 * epsilon := by positivity
rw [← Real.log_le_log_iff (Real.rpow_pos_of_pos hxe _) (Real.rpow_pos_of_pos hx _),
← Real.log_le_log_iff (Real.rpow_pos_of_pos hx _) he]
rw [Real.log_rpow hxe, Real.log_rpow hx, Real.log_mul (by positivity) he.ne',
Real.log_pow]
norm_num
constructor <;> intro h <;> linarith
The dense-privacy term meets the sampling term at the square-root budget. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
x epsilon :
ℝ
hx :
0 < x
he :
0 < epsilon
conclusion
(x * epsilon) ^ (-1 / 2 : ℝ) ≤ x ^ (-1 / 4 : ℝ) ↔ x ^ (-1 / 2 : ℝ) ≤ epsilon
Proof (Lean source)
-- @node: dense_le_sampling_iff
lemma dense_le_sampling_iff (x epsilon : ℝ) (hx : 0 < x) (he : 0 < epsilon) :
(x * epsilon) ^ (-1 / 2 : ℝ) ≤ x ^ (-1 / 4 : ℝ) ↔ x ^ (-1 / 2 : ℝ) ≤ epsilon := by
have hxe : 0 < x * epsilon := by positivity
rw [← Real.log_le_log_iff (Real.rpow_pos_of_pos hxe _) (Real.rpow_pos_of_pos hx _),
← Real.log_le_log_iff (Real.rpow_pos_of_pos hx _) he]
rw [Real.log_rpow hxe, Real.log_rpow hx, Real.log_rpow hx,
Real.log_mul hx.ne' he.ne']
constructor <;> intro h <;> linarith
The two privacy terms meet at the three-fifths budget. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
x epsilon :
ℝ
hx :
0 < x
he :
0 < epsilon
conclusion
(x * epsilon) ^ (-1 / 2 : ℝ)
≤ (x ^ 2 * epsilon) ^ (-1 / 7 : ℝ) ↔ x ^ (-3 / 5 : ℝ) ≤ epsilon
Proof (Lean source)
-- @node: dense_le_sparse_iff
lemma dense_le_sparse_iff (x epsilon : ℝ) (hx : 0 < x) (he : 0 < epsilon) :
(x * epsilon) ^ (-1 / 2 : ℝ) ≤ (x ^ 2 * epsilon) ^ (-1 / 7 : ℝ) ↔
x ^ (-3 / 5 : ℝ) ≤ epsilon := by
have hxe : 0 < x * epsilon := by positivity
have hx2e : 0 < x ^ 2 * epsilon := by positivity
rw [← Real.log_le_log_iff (Real.rpow_pos_of_pos hxe _) (Real.rpow_pos_of_pos hx2e _),
← Real.log_le_log_iff (Real.rpow_pos_of_pos hx _) he]
rw [Real.log_rpow hxe, Real.log_rpow hx2e, Real.log_rpow hx,
Real.log_mul hx.ne' he.ne', Real.log_mul (by positivity) he.ne',
Real.log_pow]
norm_num
constructor <;> intro h <;> linarith
Reversing the budget comparison reverses the order of the two privacy terms. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
x epsilon :
ℝ
hx :
0 < x
he :
0 < epsilon
conclusion
(x ^ 2 * epsilon) ^ (-1 / 7 : ℝ)
≤ (x * epsilon) ^ (-1 / 2 : ℝ) ↔ epsilon ≤ x ^ (-3 / 5 : ℝ)
Proof (Lean source)
-- @node: sparse_le_dense_iff
lemma sparse_le_dense_iff (x epsilon : ℝ) (hx : 0 < x) (he : 0 < epsilon) :
(x ^ 2 * epsilon) ^ (-1 / 7 : ℝ) ≤ (x * epsilon) ^ (-1 / 2 : ℝ) ↔
epsilon ≤ x ^ (-3 / 5 : ℝ) := by
have hxe : 0 < x * epsilon := by positivity
have hx2e : 0 < x ^ 2 * epsilon := by positivity
rw [← Real.log_le_log_iff (Real.rpow_pos_of_pos hx2e _) (Real.rpow_pos_of_pos hxe _),
← Real.log_le_log_iff he (Real.rpow_pos_of_pos hx _)]
rw [Real.log_rpow hxe, Real.log_rpow hx2e, Real.log_rpow hx,
Real.log_mul hx.ne' he.ne', Real.log_mul (by positivity) he.ne',
Real.log_pow]
norm_num
constructor <;> intro h <;> linarith
theorem rate_sampling_regime
In the large-budget regime both privacy terms are below the sampling term. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
n :
ℕ
epsilon :
ℝ
hn :
2 ≤ n
he :
0 < epsilon
hb :
(n : ℝ) ^ (-1 / 4 : ℝ) ≤ epsilon
conclusion
rate n epsilon = (n : ℝ) ^ (-1 / 4 : ℝ)
Proof (Lean source)
-- @node: rate_sampling_regime
lemma rate_sampling_regime (n : ℕ) (epsilon : ℝ) (hn : 2 ≤ n) (he : 0 < epsilon)
(hb : (n : ℝ) ^ (-1 / 4 : ℝ) ≤ epsilon) :
rate n epsilon = (n : ℝ) ^ (-1 / 4 : ℝ) := by
have hn1 : (1 : ℝ) ≤ n := by exact_mod_cast (show 1 ≤ n by omega)
have hn0 : (0 : ℝ) < n := by linarith
have hba := (sparse_le_sampling_iff _ _ hn0 he).mpr hb
have hca := (dense_le_sampling_iff _ _ hn0 he).mpr
((Real.rpow_le_rpow_of_exponent_le hn1 (by norm_num : (-1 / 2 : ℝ) ≤ -1 / 4)).trans hb)
have ha1 := Real.rpow_le_one_of_one_le_of_nonpos hn1 (by norm_num : (-1 / 4 : ℝ) ≤ 0)
simp only [rate, max_eq_left (max_le hba hca), min_eq_right ha1]
theorem rate_sparse_regime
In the intermediate sparse regime the sparse-privacy term is the maximum and below one. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
n :
ℕ
epsilon :
ℝ
hn :
2 ≤ n
he :
0 < epsilon
hb :
(n : ℝ) ^ (-3 / 5 : ℝ) ≤ epsilon ∧ epsilon ≤ (n : ℝ) ^ (-1 / 4 : ℝ)
conclusion
rate n epsilon = ((n : ℝ) ^ 2 * epsilon) ^ (-1 / 7 : ℝ)
Proof (Lean source)
-- @node: rate_sparse_regime
lemma rate_sparse_regime (n : ℕ) (epsilon : ℝ) (hn : 2 ≤ n) (he : 0 < epsilon)
(hb : (n : ℝ) ^ (-3 / 5 : ℝ) ≤ epsilon ∧ epsilon ≤ (n : ℝ) ^ (-1 / 4 : ℝ)) :
rate n epsilon = ((n : ℝ) ^ 2 * epsilon) ^ (-1 / 7 : ℝ) := by
have hn1 : (1 : ℝ) ≤ n := by exact_mod_cast (show 1 ≤ n by omega)
have hn0 : (0 : ℝ) < n := by linarith
have hab : (n : ℝ) ^ (-1 / 4 : ℝ) ≤ ((n : ℝ) ^ 2 * epsilon) ^ (-1 / 7 : ℝ) := by
have hx : 0 < (n : ℝ) ^ 2 * epsilon := by positivity
rw [← Real.log_le_log_iff (Real.rpow_pos_of_pos hn0 _) (Real.rpow_pos_of_pos hx _)]
have hl := (Real.log_le_log_iff he (Real.rpow_pos_of_pos hn0 _)).mpr hb.2
rw [Real.log_rpow hn0] at hl
rw [Real.log_rpow hn0, Real.log_rpow hx, Real.log_mul (by positivity) he.ne',
Real.log_pow]
norm_num
linarith
have hcb := (dense_le_sparse_iff _ _ hn0 he).mpr hb.1
have hi : (n : ℝ) ^ (-1 : ℝ) ≤ epsilon :=
(Real.rpow_le_rpow_of_exponent_le hn1 (by norm_num : (-1 : ℝ) ≤ -3 / 5)).trans hb.1
have hprod : 1 ≤ (n : ℝ) * epsilon := by
rw [Real.rpow_neg_one] at hi
have := mul_le_mul_of_nonneg_left hi hn0.le
simpa [mul_inv_cancel₀ hn0.ne'] using this
have hprod2 : 1 ≤ (n : ℝ) ^ 2 * epsilon := by nlinarith
have hb1 := Real.rpow_le_one_of_one_le_of_nonpos hprod2 (by norm_num : (-1 / 7 : ℝ) ≤ 0)
simp only [rate, max_eq_left hcb, max_eq_right hab, min_eq_right hb1]
theorem rate_dense_regime
In the dense regime the dense-privacy term is the maximum and below one. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
n :
ℕ
epsilon :
ℝ
hn :
2 ≤ n
he :
0 < epsilon
hb :
(n : ℝ) ^ (-1 : ℤ) ≤ epsilon ∧ epsilon ≤ (n : ℝ) ^ (-3 / 5 : ℝ)
conclusion
rate n epsilon = ((n : ℝ) * epsilon) ^ (-1 / 2 : ℝ)
Proof (Lean source)
-- @node: rate_dense_regime
lemma rate_dense_regime (n : ℕ) (epsilon : ℝ) (hn : 2 ≤ n) (he : 0 < epsilon)
(hb : (n : ℝ) ^ (-1 : ℤ) ≤ epsilon ∧ epsilon ≤ (n : ℝ) ^ (-3 / 5 : ℝ)) :
rate n epsilon = ((n : ℝ) * epsilon) ^ (-1 / 2 : ℝ) := by
have hn1 : (1 : ℝ) ≤ n := by exact_mod_cast (show 1 ≤ n by omega)
have hn0 : (0 : ℝ) < n := by linarith
have hbc := (sparse_le_dense_iff _ _ hn0 he).mpr hb.2
have hac : (n : ℝ) ^ (-1 / 4 : ℝ) ≤ ((n : ℝ) * epsilon) ^ (-1 / 2 : ℝ) := by
have hx : 0 < (n : ℝ) * epsilon := by positivity
have hi : epsilon ≤ (n : ℝ) ^ (-1 / 2 : ℝ) := hb.2.trans
(Real.rpow_le_rpow_of_exponent_le hn1 (by norm_num : (-3 / 5 : ℝ) ≤ -1 / 2))
have hl := (Real.log_le_log_iff he (Real.rpow_pos_of_pos hn0 _)).mpr hi
rw [Real.log_rpow hn0] at hl
rw [← Real.log_le_log_iff (Real.rpow_pos_of_pos hn0 _) (Real.rpow_pos_of_pos hx _),
Real.log_rpow hn0, Real.log_rpow hx, Real.log_mul hn0.ne' he.ne']
linarith
have hprod : 1 ≤ (n : ℝ) * epsilon := by
have := mul_le_mul_of_nonneg_left hb.1 hn0.le
simpa [zpow_neg_one, mul_inv_cancel₀ hn0.ne'] using this
have hc1 := Real.rpow_le_one_of_one_le_of_nonpos hprod (by norm_num : (-1 / 2 : ℝ) ≤ 0)
simp only [rate, max_eq_right hbc, max_eq_right hac, min_eq_right hc1]
theorem rate_capped_regime
Below the inverse-sample-size budget the dense term reaches the cap. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
n :
ℕ
epsilon :
ℝ
hn :
2 ≤ n
he :
0 < epsilon
hb :
epsilon ≤ (n : ℝ) ^ (-1 : ℤ)
conclusion
rate n epsilon = 1
Proof (Lean source)
-- @node: rate_capped_regime
lemma rate_capped_regime (n : ℕ) (epsilon : ℝ) (hn : 2 ≤ n) (he : 0 < epsilon)
(hb : epsilon ≤ (n : ℝ) ^ (-1 : ℤ)) : rate n epsilon = 1 := by
have hn0 : (0 : ℝ) < n := by exact_mod_cast (show 0 < n by omega)
have hx : 0 < (n : ℝ) * epsilon := by positivity
have hprod : (n : ℝ) * epsilon ≤ 1 := by
have := mul_le_mul_of_nonneg_left hb hn0.le
simpa [zpow_neg_one, mul_inv_cancel₀ hn0.ne'] using this
have hc : 1 ≤ ((n : ℝ) * epsilon) ^ (-1 / 2 : ℝ) := by
simpa using Real.rpow_le_rpow_of_nonpos hx hprod (by norm_num : (-1 / 2 : ℝ) ≤ 0)
exact min_eq_left (hc.trans ((le_max_right _ _).trans (le_max_right _ _)))
theorem rate_unit_budget
Unit budget gives exactly the nonprivate sampling benchmark. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: rate_unit_budget
lemma rate_unit_budget (n : ℕ) (hn : 2 ≤ n) : rate n 1 = (n : ℝ) ^ (-1 / 4 : ℝ) := by
apply rate_sampling_regime n 1 hn zero_lt_one
exact Real.rpow_le_one_of_one_le_of_nonpos
(by exact_mod_cast (show 1 ≤ n by omega)) (by norm_num)
theorem capped_dense_le_rate
The capped dense-privacy term is always a lower bound for the benchmark. the theorem's stated inputs and assumptions, and the asserted conclusion follows.
Formal statement
Proof (Lean source)
-- @node: capped_dense_le_rate
lemma capped_dense_le_rate (n : ℕ) (epsilon : ℝ) :
min 1 (((n : ℝ) * epsilon) ^ (-1 / 2 : ℝ)) ≤ rate n epsilon := by
exact min_le_min_left 1 ((le_max_right _ _).trans (le_max_right _ _))
theorem rate_tendsto_zero_of_budget_diverges
Divergence of sample size times budget makes all three benchmark terms vanish. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
Proof (Lean source)
-- @node: rate_tendsto_zero_of_budget_diverges
lemma rate_tendsto_zero_of_budget_diverges (budgets : ℕ → ℝ)
(he : ∀ j, 2 ≤ j → 0 < budgets j)
(hdiv : Tendsto (fun j : ℕ => (j : ℝ) * budgets j) atTop atTop) :
Tendsto (fun j => rate j (budgets j)) atTop (𝓝 0) := by
have ha : Tendsto (fun j : ℕ => (j : ℝ) ^ (-1 / 4 : ℝ)) atTop (𝓝 0) := by
simpa [Function.comp_def, neg_div] using
(tendsto_rpow_neg_atTop (by norm_num : (0 : ℝ) < 1 / 4)).comp
tendsto_natCast_atTop_atTop
have hdiv2 : Tendsto (fun j : ℕ => (j : ℝ) ^ 2 * budgets j) atTop atTop := by
apply tendsto_atTop_mono' atTop _ hdiv
filter_upwards [eventually_ge_atTop 2] with j hj
have hj1 : (1 : ℝ) ≤ j := by exact_mod_cast (show 1 ≤ j by omega)
have heb := (he j hj).le
have hs : (j : ℝ) ≤ (j : ℝ) ^ 2 := by nlinarith
exact mul_le_mul_of_nonneg_right hs heb
have hb : Tendsto (fun j : ℕ => ((j : ℝ) ^ 2 * budgets j) ^ (-1 / 7 : ℝ)) atTop (𝓝 0) := by
simpa [Function.comp_def, neg_div] using
(tendsto_rpow_neg_atTop (by norm_num : (0 : ℝ) < 1 / 7)).comp hdiv2
have hc : Tendsto (fun j : ℕ => ((j : ℝ) * budgets j) ^ (-1 / 2 : ℝ)) atTop (𝓝 0) := by
simpa [Function.comp_def, neg_div] using
(tendsto_rpow_neg_atTop (by norm_num : (0 : ℝ) < 1 / 2)).comp hdiv
simpa [rate] using
(tendsto_const_nhds (x := (1 : ℝ))).min (ha.max (hb.max hc))
theorem budget_diverges_of_rate_tendsto_zero
A vanishing benchmark forces sample size times budget past every finite bound. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: budget_diverges_of_rate_tendsto_zero
lemma budget_diverges_of_rate_tendsto_zero (budgets : ℕ → ℝ)
(he : ∀ j, 2 ≤ j → 0 < budgets j)
(hr : Tendsto (fun j => rate j (budgets j)) atTop (𝓝 0)) :
Tendsto (fun j : ℕ => (j : ℝ) * budgets j) atTop atTop := by
apply tendsto_atTop.2
intro K
let B : ℝ := max 1 K
have hB : 0 < B := lt_of_lt_of_le zero_lt_one (le_max_left _ _)
have hsmall : 0 < min 1 (B ^ (-1 / 2 : ℝ)) :=
lt_min zero_lt_one (Real.rpow_pos_of_pos hB _)
filter_upwards [(tendsto_order.1 hr).2 _ hsmall, eventually_ge_atTop 2] with j hj hj2
have hp : 0 < (j : ℝ) * budgets j := by
have : (0 : ℝ) < j := by exact_mod_cast (show 0 < j by omega)
exact mul_pos this (he j hj2)
have hm := lt_of_le_of_lt (capped_dense_le_rate j (budgets j)) hj
have hc : ((j : ℝ) * budgets j) ^ (-1 / 2 : ℝ) < B ^ (-1 / 2 : ℝ) := by
by_contra hh
have : min 1 (B ^ (-1 / 2 : ℝ)) ≤ min 1 (((j : ℝ) * budgets j) ^ (-1 / 2 : ℝ)) :=
min_le_min_left 1 (le_of_not_gt hh)
exact (not_lt_of_ge this) hm
have hlt := (Real.rpow_lt_rpow_iff_of_neg hp hB (by norm_num : (-1 / 2 : ℝ) < 0)).mp hc
exact (le_max_right 1 K).trans hlt.le
theorem rate_consistency_iff
The numerical benchmark vanishes exactly for diverging sample size times budget. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: rate_consistency_iff
lemma rate_consistency_iff (budgets : ℕ → ℝ)
(he : ∀ j, 2 ≤ j → 0 < budgets j) :
Tendsto (fun j => rate j (budgets j)) atTop (𝓝 0) ↔
Tendsto (fun j : ℕ => (j : ℝ) * budgets j) atTop atTop := by
exact ⟨budget_diverges_of_rate_tendsto_zero budgets he,
rate_tendsto_zero_of_budget_diverges budgets he⟩
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.ReleaseMoments 3 declarations Exact centered Laplace release moments and the uniform density-free squared-error guarantee.
Exact centered Laplace release moments and the uniform density-free squared-error guarantee.
theorem privateRatioRelease_eq_laplaceKernel
The constructed pair release is the measurable Laplace kernel. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: privateRatioRelease_eq_laplaceKernel
lemma privateRatioRelease_eq_laplaceKernel (n k : ℕ) (epsilon h : ℝ)
(he : 0 < epsilon) :
privateRatioRelease n epsilon h k = laplaceMechPiKernel (12/epsilon)
(by positivity) (pairQuery n h k) (measurable_pairQuery n h k) := by
ext D E : 2
rw [laplaceMechPiKernel_apply]
rfl
theorem privateRatioRelease_centered_moments
The released coordinate squares add exactly two independent Laplace variances to the input variances, with no inflation of the noise coefficient. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
Proof (Lean source)
-- @node: privateRatioRelease_centered_moments
lemma privateRatioRelease_centered_moments (n k : ℕ) (epsilon h : ℝ)
(he : 0 < epsilon) (hh : 0 < h) (hk : 0 < k) (P : CausalLaw) :
(∫ v, (v 0-Nbar n h k (dataLaw n P))^2 +
(v 1-Dbar n h k (dataLaw n P))^2
∂(privateRatioRelease n epsilon h k ∘ₘ dataLaw n P)) =
variance (SN n h k) (dataLaw n P) + variance (SD n h k) (dataLaw n P) +
576/epsilon^2 := by
letI : IsProbabilityMeasure (Pobs P) :=
Measure.isProbabilityMeasure_map measurable_observe.aemeasurable
letI : IsProbabilityMeasure (dataLaw n P) := by unfold dataLaw; infer_instance
have hLp := statistic_memLp n k h hh hk (dataLaw n P) 2
have hN : Integrable (fun D => (pairQuery n h k D 0)^2) (dataLaw n P) := by
simpa [pairQuery] using hLp.1.integrable_sq
have hD : Integrable (fun D => (pairQuery n h k D 1)^2) (dataLaw n P) := by
simpa [pairQuery] using hLp.2.integrable_sq
rw [privateRatioRelease_eq_laplaceKernel n k epsilon h he]
have hiN := laplaceMechPiKernel_integrable_coord_sub_sq (12/epsilon) (by positivity)
(pairQuery n h k) (measurable_pairQuery n h k) (dataLaw n P) 0 hN
(Nbar n h k (dataLaw n P))
have hiD := laplaceMechPiKernel_integrable_coord_sub_sq (12/epsilon) (by positivity)
(pairQuery n h k) (measurable_pairQuery n h k) (dataLaw n P) 1 hD
(Dbar n h k (dataLaw n P))
rw [integral_add hiN hiD,
laplaceMechPiKernel_integral_coord_sub_sq _ _ _ _ _ 0 hN,
laplaceMechPiKernel_integral_coord_sub_sq _ _ _ _ _ 1 hD,
variance_eq_integral (by fun_prop : AEMeasurable (SN n h k) (dataLaw n P)),
variance_eq_integral (by fun_prop : AEMeasurable (SD n h k) (dataLaw n P))]
simp only [pairQuery, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.head_cons,
Nbar, Dbar]
ring
theorem Thk_model_squared_error_le
Integrating the pointwise floored-ratio bound and exact release moments proves the public second-moment envelope throughout the complete model. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
efronStein_of_gate :
n k :
ℕ
epsilon h :
ℝ
hn :
2 ≤ n
hk :
2 ≤ k
he :
0 < epsilon
hh :
0 < h ∧ h ≤ 1/4
P :
hP :
Proof (Lean source)
-- @node: Thk_model_squared_error_le
lemma Thk_model_squared_error_le (efronStein_of_gate : EfronSteinReplacement)
(n k : ℕ) (epsilon h : ℝ) (hn : 2 ≤ n) (hk : 2 ≤ k)
(he : 0 < epsilon) (hh : 0 < h ∧ h ≤ 1/4) (P : CausalLaw)
(hP : CompleteModel P) :
(∫⁻ u, ofReal ((u-theta P)^2) ∂(Thk n epsilon h k ∘ₘ dataLaw n P)) ≤
ofReal (Vbound n epsilon h k) := by
letI : IsProbabilityMeasure (Pobs P) :=
Measure.isProbabilityMeasure_map measurable_observe.aemeasurable
letI : IsProbabilityMeasure (dataLaw n P) := by unfold dataLaw; infer_instance
letI := privateRatioRelease_markov n k epsilon h he
let Q := privateRatioRelease n epsilon h k ∘ₘ dataLaw n P
let f := fun v : Fin 2 → ℝ => (v 0-Nbar n h k (dataLaw n P))^2 +
(v 1-Dbar n h k (dataLaw n P))^2
have hLp := statistic_memLp n k h hh.1 (by omega) (dataLaw n P) 2
have hi (i : Fin 2) (c : ℝ) : Integrable (fun v => (v i-c)^2) Q := by
dsimp [Q]
rw [privateRatioRelease_eq_laplaceKernel n k epsilon h he]
apply laplaceMechPiKernel_integrable_coord_sub_sq
fin_cases i
· simpa [pairQuery] using hLp.1.integrable_sq
· simpa [pairQuery] using hLp.2.integrable_sq
have hif : Integrable f Q := (hi 0 _).add (hi 1 _)
have hrhs : Integrable (fun v => 2*(bias h k)^2 + 4/(d0 n h k)^2 * f v) Q :=
(integrable_const _).add (hif.const_mul _)
have hbound := ratioMap_model_squared_error_le n k h hn hk hh P hP
have hierr : Integrable (fun v => (ratioMap n h k v-theta P)^2) Q := by
apply hrhs.mono' (by fun_prop)
apply ae_of_all
intro v
rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg (ratioMap n h k v-theta P))]
exact hbound v
have hvar := (all_count_stability efronStein_of_gate n k h hn hk hh P
(dataLaw n P) rfl hP).2
have hmom : (∫ v, f v ∂Q) ≤ 36*Wocc n h k P + 576/epsilon^2 := by
rw [show (∫ v, f v ∂Q) = _ from
privateRatioRelease_centered_moments n k epsilon h he hh.1 (by omega) P]
linarith [le_max_left (variance (SN n h k) (dataLaw n P))
(variance (SD n h k) (dataLaw n P)),
le_max_right (variance (SN n h k) (dataLaw n P))
(variance (SD n h k) (dataLaw n P))]
have herr : (∫ v, (ratioMap n h k v-theta P)^2 ∂Q) ≤ Vbound n epsilon h k := by
calc
_ ≤ ∫ v, (2*(bias h k)^2 + 4/(d0 n h k)^2 * f v) ∂Q :=
integral_mono hierr hrhs hbound
_ = 2*(bias h k)^2 + 4/(d0 n h k)^2 * (∫ v, f v ∂Q) := by
rw [integral_add (integrable_const _) (hif.const_mul _), integral_const_mul]
simp [integral_const_mul]
_ ≤ 2*(bias h k)^2 + 4/(d0 n h k)^2 *
(36*Wocc n h k P + 576/epsilon^2) := by
gcongr
_ = 2*(bias h k)^2 + 144*Wocc n h k P/(d0 n h k)^2 +
2304/(epsilon^2*(d0 n h k)^2) := by ring
_ ≤ _ := model_variance_envelope_le n k epsilon h hn hk he hh P hP
rw [Thk_squared_error_eq, ← ofReal_integral_eq_lintegral_ofReal hierr
(ae_of_all _ (fun v => sq_nonneg (ratioMap n h k v-theta P)))]
exact ENNReal.ofReal_le_ofReal herr
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.SelectionLaw 11 declarations Membership-label rectangles and their normalized iid product laws.
Membership-label rectangles and their normalized iid product laws.
Specifying every cell membership label is a rectangle in the iid sample space. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
{D : Dataset n | cellRecords n h k D j = I}
= Set.univ.pi (fun i => if i ∈ I then {z : O | z.1 ∈ cell h k j} else {z : O | z.1 ∈ cell h k j}ᶜ)
Proof (Lean source)
-- @node: cell_membership_rectangle
lemma cell_membership_rectangle (n k : ℕ) (h : ℝ) (j : Fin k)
(I : Finset (Fin n)) :
{D : Dataset n | cellRecords n h k D j = I} =
Set.univ.pi (fun i => if i ∈ I then {z : O | z.1 ∈ cell h k j}
else {z : O | z.1 ∈ cell h k j}ᶜ) := by
classical
ext D
simp only [Set.mem_ofPred_eq, Set.mem_pi, Set.mem_univ, forall_const]
constructor
· intro hD i
have hi : i ∈ cellRecords n h k D j ↔ i ∈ I := by rw [hD]
simp only [cellRecords, mem_filter, Finset.mem_univ, true_and] at hi
by_cases hI : i ∈ I <;> simp [hI, hi]
· intro hD
apply Finset.ext
intro i
simp only [cellRecords, mem_filter, Finset.mem_univ, true_and]
by_cases hI : i ∈ I <;> simpa [hI] using hD i
Every exact membership-label event is measurable. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: measurableSet_cell_membership
lemma measurableSet_cell_membership (n k : ℕ) (h : ℝ) (j : Fin k)
(I : Finset (Fin n)) :
MeasurableSet {D : Dataset n | cellRecords n h k D j = I} := by
classical
rw [cell_membership_rectangle]
apply MeasurableSet.univ_pi
intro i
split_ifs
· exact (measurableSet_cell h k j).preimage measurable_fst
· exact ((measurableSet_cell h k j).preimage measurable_fst).compl
theorem iid_membership_restriction
Restricting the iid sample law to specified labels factors coordinate by coordinate. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: iid_membership_restriction
lemma iid_membership_restriction (n k : ℕ) (h : ℝ) (P : CausalLaw)
(j : Fin k) (I : Finset (Fin n)) :
(dataLaw n P).restrict {D | cellRecords n h k D j = I} =
Measure.pi (fun i => (Pobs P).restrict
(if i ∈ I then {z : O | z.1 ∈ cell h k j}
else {z : O | z.1 ∈ cell h k j}ᶜ)) := by
classical
let : IsProbabilityMeasure (Pobs P) :=
Measure.isProbabilityMeasure_map measurable_observe.aemeasurable
rw [cell_membership_rectangle, dataLaw, Measure.restrict_pi_pi]
theorem normalized_record_restriction_probability Lemma normalized_record_restriction_probability in the paper ↗
Normalizing any positive-mass restriction gives a probability law. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
IsProbabilityMeasure ((μ s)⁻¹ • μ.restrict s)
Proof (Lean source)
-- @node: normalized_record_restriction_probability
lemma normalized_record_restriction_probability (μ : Measure O) [IsFiniteMeasure μ]
(s : Set O) (hs : μ s ≠ 0) :
IsProbabilityMeasure ((μ s)⁻¹ • μ.restrict s) := by
refine ⟨?_⟩
rw [Measure.smul_apply, Measure.restrict_apply MeasurableSet.univ,
Set.univ_inter, smul_eq_mul]
exact ENNReal.inv_mul_cancel hs (measure_ne_top μ s)
Dividing a positive rectangle restriction by its mass preserves the product structure. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: iid_normalized_rectangle
lemma iid_normalized_rectangle (n : ℕ) (μ : Measure O) [IsFiniteMeasure μ]
(s : Fin n → Set O) (hs : ∀ i, μ (s i) ≠ 0) :
((Measure.pi (fun _ : Fin n => μ)) (Set.univ.pi s))⁻¹ •
(Measure.pi (fun _ : Fin n => μ)).restrict (Set.univ.pi s) =
Measure.pi (fun i => (μ (s i))⁻¹ • μ.restrict (s i)) := by
let ν := fun i => (μ (s i))⁻¹ • μ.restrict (s i)
let : ∀ i, IsProbabilityMeasure (ν i) :=
fun i => normalized_record_restriction_probability μ (s i) (hs i)
apply (Measure.pi_eq (μ := ν) _).symm
intro t ht
rw [Measure.smul_apply, smul_eq_mul, Measure.pi_pi,
Measure.restrict_pi_pi, Measure.pi_pi]
simp only [ν, Measure.smul_apply, smul_eq_mul]
rw [Finset.prod_mul_distrib, ENNReal.prod_inv_distrib]
intro i hi l hl hil
exact inl (hs i)
theorem iid_conditional_membership_law
Given a specified positive-probability label pattern, all records are independent with coordinate laws conditioned on their own labels. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
conclusion
((dataLaw n P) {D | cellRecords n h k D j = I})⁻¹ • (dataLaw n P).restrict {D | cellRecords n h k D j = I}
Proof (Lean source)
-- @node: iid_conditional_membership_law
lemma iid_conditional_membership_law (n k : ℕ) (h : ℝ) (P : CausalLaw)
(j : Fin k) (I : Finset (Fin n))
(hp : ∀ i : Fin n, (Pobs P)
(if i ∈ I then {z : O | z.1 ∈ cell h k j}
else {z : O | z.1 ∈ cell h k j}ᶜ) ≠ 0) :
((dataLaw n P) {D | cellRecords n h k D j = I})⁻¹ •
(dataLaw n P).restrict {D | cellRecords n h k D j = I} =
Measure.pi (fun i =>
((Pobs P) (if i ∈ I then {z : O | z.1 ∈ cell h k j}
else {z : O | z.1 ∈ cell h k j}ᶜ))⁻¹ •
(Pobs P).restrict (if i ∈ I then {z : O | z.1 ∈ cell h k j}
else {z : O | z.1 ∈ cell h k j}ᶜ)) := by
classical
let : IsProbabilityMeasure (Pobs P) :=
Measure.isProbabilityMeasure_map measurable_observe.aemeasurable
rw [cell_membership_rectangle, dataLaw]
exact iid_normalized_rectangle n (Pobs P) _ hp
theorem iid_conditional_membership_pair
On an exact label event, two selected coordinates have the independent conditional cell law, irrespective of the membership of the other records. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
conclusion
MeasurePreserving (fun D : Dataset n => (D i, D l)) (((dataLaw n P) {D | cellRecords n h k D j = I})⁻¹ • (dataLaw n P).restrict {D | cellRecords n h k D j = I}) ((conditionalCellLaw h k P j).prod (conditionalCellLaw h k P j))
Proof (Lean source)
-- @node: iid_conditional_membership_pair
lemma iid_conditional_membership_pair (n k : ℕ) (h : ℝ) (P : CausalLaw)
(j : Fin k) (I : Finset (Fin n))
(hp : ∀ r : Fin n, (Pobs P)
(if r ∈ I then {z : O | z.1 ∈ cell h k j}
else {z : O | z.1 ∈ cell h k j}ᶜ) ≠ 0)
(i l : Fin n) (hi : i ∈ I) (hl : l ∈ I) (hil : i ≠ l) :
MeasurePreserving (fun D : Dataset n => (D i, D l))
(((dataLaw n P) {D | cellRecords n h k D j = I})⁻¹ •
(dataLaw n P).restrict {D | cellRecords n h k D j = I})
((conditionalCellLaw h k P j).prod (conditionalCellLaw h k P j)) := by
classical
let : IsProbabilityMeasure (Pobs P) :=
Measure.isProbabilityMeasure_map measurable_observe.aemeasurable
let s := fun r : Fin n => if r ∈ I then {z : O | z.1 ∈ cell h k j}
else {z : O | z.1 ∈ cell h k j}ᶜ
let ν := fun r => ((Pobs P) (s r))⁻¹ • (Pobs P).restrict (s r)
let : ∀ r, IsProbabilityMeasure (ν r) :=
fun r => normalized_record_restriction_probability (Pobs P) (s r) (hp r)
have hind := ProbabilityTheory.iIndepFun_pi
(μ := ν) (X := fun _ (z : O) => z) (fun _ => measurable_id.aemeasurable)
have hm := (hind.indepFun hil).map_prod_eq_prod_map_map
(measurable_pi_apply i).aemeasurable (measurable_pi_apply l).aemeasurable
rw [(measurePreserving_eval ν i).map_eq, (measurePreserving_eval ν l).map_eq] at hm
have hvi : ν i = conditionalCellLaw h k P j := by
simp [ν, s, hi, conditionalCellLaw]
have hvl : ν l = conditionalCellLaw h k P j := by
simp [ν, s, hl, conditionalCellLaw]
rw [hvi, hvl] at hm
rw [iid_conditional_membership_law n k h P j I hp]
exact ⟨by fun_prop, hm⟩
theorem iid_conditional_membership_pair_integral
Every selected pair on a fixed label event has the same kernel expectation. This is the conditional pair identity needed before summing the unordered pairs and then the label events. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
n k :
ℕ
h :
ℝ
P :
j :
Fin k
hp :
i l :
Fin n
hi :
i ∈ I
hl :
l ∈ I
hil :
i ≠ l
K :
O → O → ℝ
hK :
Integrable (fun z : O × O => K z.1 z.2) ((conditionalCellLaw h k P j).prod (conditionalCellLaw h k P j))
conclusion
conclusion 1
Integrable (fun D : Dataset n => K (D i) (D l)) (((dataLaw n P) {D | cellRecords n h k D j = I})⁻¹ • (dataLaw n P).restrict {D | cellRecords n h k D j = I})
conclusion 2
(∫ D : Dataset n, K (D i) (D l) ∂(((dataLaw n P) {D | cellRecords n h k D j = I})⁻¹ • (dataLaw n P).restrict {D | cellRecords n h k D j = I}))
Proof (Lean source)
-- @node: iid_conditional_membership_pair_integral
lemma iid_conditional_membership_pair_integral (n k : ℕ) (h : ℝ) (P : CausalLaw)
(j : Fin k) (I : Finset (Fin n))
(hp : ∀ r : Fin n, (Pobs P)
(if r ∈ I then {z : O | z.1 ∈ cell h k j}
else {z : O | z.1 ∈ cell h k j}ᶜ) ≠ 0)
(i l : Fin n) (hi : i ∈ I) (hl : l ∈ I) (hil : i ≠ l)
(K : O → O → ℝ)
(hK : Integrable (fun z : O × O => K z.1 z.2)
((conditionalCellLaw h k P j).prod (conditionalCellLaw h k P j))) :
Integrable (fun D : Dataset n => K (D i) (D l))
(((dataLaw n P) {D | cellRecords n h k D j = I})⁻¹ •
(dataLaw n P).restrict {D | cellRecords n h k D j = I}) ∧
(∫ D : Dataset n, K (D i) (D l)
∂(((dataLaw n P) {D | cellRecords n h k D j = I})⁻¹ •
(dataLaw n P).restrict {D | cellRecords n h k D j = I})) =
∫ z, K z.1 z.2
∂(conditionalCellLaw h k P j).prod (conditionalCellLaw h k P j) := by
have hf := iid_conditional_membership_pair n k h P j I hp i l hi hl hil
constructor
· exact hf.integrable_comp_of_integrable hK
· rw [← hf.map_eq] at hK ⊢
exact (integral_map hf.measurable.aemeasurable hK.aestronglyMeasurable).symm
theorem cell_membership_label_positive
The complete density model gives positive probability to either membership label. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: cell_membership_label_positive
lemma cell_membership_label_positive (n k : ℕ) (h : ℝ) (hk : 2 ≤ k)
(hh : 0 < h ∧ h ≤ 1 / 4) (P : CausalLaw) (hP : CompleteModel P)
(j : Fin k) (I : Finset (Fin n)) (i : Fin n) :
(Pobs P) (if i ∈ I then {z : O | z.1 ∈ cell h k j}
else {z : O | z.1 ∈ cell h k j}ᶜ) ≠ 0 := by
classical
let : IsProbabilityMeasure (Pobs P) :=
Measure.isProbabilityMeasure_map measurable_observe.aemeasurable
have hd := (cell_endpoint_bounds k h (by omega) hh j).2.2.2
obtain ⟨hlo, hhi⟩ := cellMass_density_bounds k h (by omega) hh P hP j
have hw : cellWidth h k ≤ 1 / 4 := by
unfold cellWidth
apply (div_le_iff₀ (by positivity : (0 : ℝ) < k)).mpr
have hkR : (2 : ℝ) ≤ k := by exact_mod_cast hk
nlinarith [hh.2]
have hp : 0 < (Pobs P).real {z : O | z.1 ∈ cell h k j} := by
rw [observed_cell_mass]
linarith
have hc : 0 < (Pobs P).real {z : O | z.1 ∈ cell h k j}ᶜ := by
have hs : MeasurableSet {z : O | z.1 ∈ cell h k j} :=
(measurableSet_cell h k j).preimage measurable_fst
rw [probReal_compl_eq_one_sub (μ := Pobs P) hs,
observed_cell_mass]
linarith
split_ifs
· intro hz
simp [measureReal_def, hz] at hp
· intro hz
simp [measureReal_def, hz] at hc
theorem conditional_membership_probability
A normalized exact label event is a probability measure. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
conclusion
IsProbabilityMeasure (((dataLaw n P) {D | cellRecords n h k D j = I})⁻¹ • (dataLaw n P).restrict {D | cellRecords n h k D j = I})
Proof (Lean source)
-- @node: conditional_membership_probability
lemma conditional_membership_probability (n k : ℕ) (h : ℝ) (P : CausalLaw)
(j : Fin k) (I : Finset (Fin n))
(hp : ∀ r : Fin n, (Pobs P)
(if r ∈ I then {z : O | z.1 ∈ cell h k j}
else {z : O | z.1 ∈ cell h k j}ᶜ) ≠ 0) :
IsProbabilityMeasure (((dataLaw n P) {D | cellRecords n h k D j = I})⁻¹ •
(dataLaw n P).restrict {D | cellRecords n h k D j = I}) := by
classical
let : IsProbabilityMeasure (Pobs P) :=
Measure.isProbabilityMeasure_map measurable_observe.aemeasurable
rw [iid_conditional_membership_law n k h P j I hp]
let : ∀ r : Fin n, IsProbabilityMeasure
(((Pobs P) (if r ∈ I then {z : O | z.1 ∈ cell h k j}
else {z : O | z.1 ∈ cell h k j}ᶜ))⁻¹ •
(Pobs P).restrict (if r ∈ I then {z : O | z.1 ∈ cell h k j}
else {z : O | z.1 ∈ cell h k j}ᶜ)) :=
fun r => normalized_record_restriction_probability (Pobs P) _ (hp r)
infer_instance
theorem conditional_selected_pairContribution_expectation
Summing the selected unordered pairs at fixed membership labels gives occupancy times the common conditional pair moment. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
n k :
ℕ
h :
ℝ
P :
j :
Fin k
hp :
e :
Fin I.card ≃ {i // i ∈ I}
K :
O → O → ℝ
hK :
Integrable (fun z : O × O => K z.1 z.2) ((conditionalCellLaw h k P j).prod (conditionalCellLaw h k P j))
conclusion
(∫ D : Dataset n, pairContribution K (fun i => D (e i)) ∂(((dataLaw n P) {D | cellRecords n h k D j = I})⁻¹ • (dataLaw n P).restrict {D | cellRecords n h k D j = I}))
= (if I.card < 2 then 0 else (I.card : ℝ)) * ∫ z, K z.1 z.2 ∂(conditionalCellLaw h k P j).prod (conditionalCellLaw h k P j)
Proof (Lean source)
-- @node: conditional_selected_pairContribution_expectation
lemma conditional_selected_pairContribution_expectation (n k : ℕ) (h : ℝ)
(P : CausalLaw) (j : Fin k) (I : Finset (Fin n))
(hp : ∀ r : Fin n, (Pobs P)
(if r ∈ I then {z : O | z.1 ∈ cell h k j}
else {z : O | z.1 ∈ cell h k j}ᶜ) ≠ 0)
(e : Fin I.card ≃ {i // i ∈ I}) (K : O → O → ℝ)
(hK : Integrable (fun z : O × O => K z.1 z.2)
((conditionalCellLaw h k P j).prod (conditionalCellLaw h k P j))) :
(∫ D : Dataset n, pairContribution K (fun i => D (e i))
∂(((dataLaw n P) {D | cellRecords n h k D j = I})⁻¹ •
(dataLaw n P).restrict {D | cellRecords n h k D j = I})) =
(if I.card < 2 then 0 else (I.card : ℝ)) *
∫ z, K z.1 z.2
∂(conditionalCellLaw h k P j).prod (conditionalCellLaw h k P j) := by
classical
let Q := ((dataLaw n P) {D | cellRecords n h k D j = I})⁻¹ •
(dataLaw n P).restrict {D | cellRecords n h k D j = I}
by_cases hs : I.card < 2
· simp [pairContribution, hs]
have hc : (choose I.card 2 : ℝ) ≠ 0 := by
exact_mod_cast (Nat.choose_pos (show 2 ≤ I.card by omega)).ne'
have hpairs (i l : Fin I.card) (hil : i ≠ l) :=
iid_conditional_membership_pair_integral n k h P j I hp (e i) (e l)
(e i).property (e l).property
(fun he => hil (e.injective (Subtype.ext he))) K hK
have hi (i l : Fin I.card) : Integrable (fun D : Dataset n =>
if i < l then K (D (e i)) (D (e l)) else 0) Q := by
by_cases hil : i < l
· simpa only [if_pos hil] using (hpairs i l hil.ne).1
· simp only [if_neg hil]
exact integrable_zero _ _ _
have hm (i l : Fin I.card) :
(∫ D : Dataset n, (if i < l then K (D (e i)) (D (e l)) else 0) ∂Q) =
(if i < l then (1 : ℝ) else 0) *
∫ z, K z.1 z.2
∂(conditionalCellLaw h k P j).prod (conditionalCellLaw h k P j) := by
by_cases hil : i < l
· simp only [if_pos hil, one_mul]
exact (hpairs i l hil.ne).2
· simp [hil]
change (∫ D : Dataset n, pairContribution K (fun i => D (e i)) ∂Q) = _
simp only [pairContribution, if_neg hs]
rw [integral_const_mul, integral_finset_sum _ (fun i _ =>
integrable_finset_sum _ (fun l _ => hi i l))]
simp_rw [integral_finset_sum _ (fun l _ => hi _ l), hm, ← Finset.sum_mul]
rw [ordered_pair_count]
field_simp [hc]
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.SignBlockFactorization 4 declarations Finite fair-sign averages factor over disjoint coordinate blocks, by iterating Causalean's two-block expectation identity.
Finite fair-sign averages factor over disjoint coordinate blocks, by iterating Causalean's two-block expectation identity.
A single latent sign is a fair bit.
Definition (Lean source)
def
given by
clause 1
p _ := 1 / 2
clause 2
p_nonneg _ := by norm_num
clause 3
p_sum := by simp [Fintype.sum_bool]
theorem fairSignDesign_E
The independent fair-bit design expectation is the uniform finite sign average. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
ι :
F :
(ι → Bool) → ℝ
conclusion
(prodDesign (fun _ : ι => fairBitDesign)).E F
= (2 : ℝ)^(-((card ι) : ℤ)) * ∑ lam, F lam
Proof (Lean source)
-- @node: fairSignDesign_E
lemma fairSignDesign_E {ι : Type*} [Fintype ι] [DecidableEq ι]
(F : (ι → Bool) → ℝ) :
(prodDesign (fun _ : ι => fairBitDesign)).E F =
(2 : ℝ)^(-((card ι) : ℤ)) * ∑ lam, F lam := by
simp only [FiniteDesign.E, prodDesign_p, fairBitDesign, Finset.prod_const,
Finset.card_univ, ← Finset.mul_sum]
simp [zpow_neg, zpow_natCast, one_div]
An arbitrary finite family of functions using pairwise disjoint coordinate blocks has factorized product expectation under an independent coordinate design. the theorem's stated inputs and assumptions, and the asserted conclusion follows.
Formal statement
hypotheses
ι κ :
∀ i, Fintype (α i)
D :
∀ i, FiniteDesign (α i)
s :
Finset κ
A :
κ → Finset ι
F :
κ → (∀ i, α i) → ℝ
hdisj :
PairwiseDisjoint (s : Set κ) A
hdep :
∀ k ∈ s, ∀ w w', (∀ i ∈ A k, w i = w' i) → F k w = F k w'
conclusion
(prodDesign D).E (fun w => ∏ k ∈ s, F k w) = ∏ k ∈ s, (prodDesign D).E (F k)
Proof (Lean source)
-- @node: finiteDesign_disjointBlocks_prod
lemma finiteDesign_disjointBlocks_prod
{ι κ : Type*} [Fintype ι] [DecidableEq ι] [DecidableEq κ]
{α : ι → Type*} [∀ i, Fintype (α i)]
(D : ∀ i, FiniteDesign (α i)) (s : Finset κ) (A : κ → Finset ι)
(F : κ → (∀ i, α i) → ℝ)
(hdisj : PairwiseDisjoint (s : Set κ) A)
(hdep : ∀ k ∈ s, ∀ w w', (∀ i ∈ A k, w i = w' i) → F k w = F k w') :
(prodDesign D).E (fun w => ∏ k ∈ s, F k w) =
∏ k ∈ s, (prodDesign D).E (F k) := by
classical
induction s using Finset.induction_on with
| empty => simp [FiniteDesign.E_const]
| @insert k s hk ih =>
simp only [Finset.prod_insert hk]
rw [FiniteDesign.E_prod_block_mul D (A k) (F k) (fun w => ∏ l ∈ s, F l w)]
· rw [ih]
· intro i hi j hj hij
exact hdisj (Finset.mem_insert_of_mem hi) (Finset.mem_insert_of_mem hj) hij
· intro l hl
exact hdep l (Finset.mem_insert_of_mem hl)
· exact hdep k (Finset.mem_insert_self k s)
· intro w w' hagree
apply Finset.prod_congr rfl
intro l hl
apply hdep l (Finset.mem_insert_of_mem hl) w w'
intro i hi
apply hagree i
intro hik
have hkl : k ≠ l := by intro heq; subst l; exact hk hl
exact Finset.disjoint_left.mp
(hdisj (Finset.mem_insert_self k s) (Finset.mem_insert_of_mem hl) hkl) hik hi
theorem fairSignAverage_disjointBlocks_prod Lemma fairSignAverage_disjointBlocks_prod in the paper ↗
Uniform fair-sign averaging commutes with a product of functions of disjoint sign blocks. the theorem's stated inputs and assumptions, and the asserted conclusion follows.
Formal statement
hypotheses
ι κ :
s :
Finset κ
A :
κ → Finset ι
F :
κ → (ι → Bool) → ℝ
hdisj :
PairwiseDisjoint (s : Set κ) A
hdep :
∀ k ∈ s, ∀ w w', (∀ i ∈ A k, w i = w' i) → F k w = F k w'
Proof (Lean source)
-- @node: fairSignAverage_disjointBlocks_prod
lemma fairSignAverage_disjointBlocks_prod
{ι κ : Type*} [Fintype ι] [DecidableEq ι] [DecidableEq κ]
(s : Finset κ) (A : κ → Finset ι) (F : κ → (ι → Bool) → ℝ)
(hdisj : PairwiseDisjoint (s : Set κ) A)
(hdep : ∀ k ∈ s, ∀ w w', (∀ i ∈ A k, w i = w' i) → F k w = F k w') :
(2 : ℝ)^(-((card ι) : ℤ)) * ∑ lam, ∏ k ∈ s, F k lam =
∏ k ∈ s, ((2 : ℝ)^(-((card ι) : ℤ)) * ∑ lam, F k lam) := by
simpa only [fairSignDesign_E] using
finiteDesign_disjointBlocks_prod (fun _ => fairBitDesign) s A F hdisj hdep
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.SignProduct 3 declarations Uniform finite sign averages as independent fair-bit product integrals, and the published bounded-differences gate transported to arbitrary finite sign indices.
Uniform finite sign averages as independent fair-bit product integrals, and the published bounded-differences gate transported to arbitrary finite sign indices.
Every atom of the independent fair-bit product has the same reciprocal-cardinality mass. The displayed conclusion follows.
Formal statement
hypotheses
ι :
z :
ι → Bool
conclusion
Measure.pi (fun _ : ι => (uniformOfFintype Bool).toMeasure) {z}
Proof (Lean source)
-- @node: fairSignProduct_singleton
lemma fairSignProduct_singleton {ι : Type*} [Fintype ι] [DecidableEq ι] (z : ι → Bool) :
Measure.pi (fun _ : ι => (uniformOfFintype Bool).toMeasure) {z} =
(card (ι → Bool) : ENNReal)⁻¹ := by
classical
rw [Measure.pi_singleton]
simp only [(uniformOfFintype Bool).toMeasure_apply_singleton _ (measurableSet_singleton _),
PMF.uniformOfFintype_apply, Fintype.card_bool, Finset.prod_const,
Finset.card_univ, Fintype.card_fun, Nat.cast_pow, ENNReal.inv_pow]
Integrating a function of independent fair bits is exactly its uniform finite average. The displayed conclusion follows.
Formal statement
hypotheses
ι :
F :
(ι → Bool) → ℝ
conclusion
(∫ z, F z ∂Measure.pi (fun _ : ι => (uniformOfFintype Bool).toMeasure))
Proof (Lean source)
-- @node: integral_fairSignProduct
lemma integral_fairSignProduct {ι : Type*} [Fintype ι] [DecidableEq ι] (F : (ι → Bool) → ℝ) :
(∫ z, F z ∂Measure.pi (fun _ : ι => (uniformOfFintype Bool).toMeasure)) =
(∑ z, F z) / (card (ι → Bool) : ℝ) := by
classical
rw [integral_fintype Integrable.of_finite]
simp only [Measure.real, fairSignProduct_singleton, ENNReal.toReal_inv,
ENNReal.toReal_natCast, smul_eq_mul]
rw [← Finset.mul_sum, div_eq_mul_inv, mul_comm]
theorem finite_sign_mgf_average
The published independent-coordinate mgf gate bounds a centered uniform finite-sign average on any finite index type, after reindexing its coordinates by Fin. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: finite_sign_mgf_average
lemma finite_sign_mgf_average (gate : BoundedDifferencesMGF)
{ι : Type*} [Fintype ι] [DecidableEq ι] (F : (ι → Bool) → ℝ)
(hzero : (∑ z, F z) = 0) (hbound : ∃ C : ℝ, ∀ z, |F z| ≤ C)
(c : ι → ℝ) (hc : ∀ i, 0 ≤ c i)
(hchange : ∀ i z b, |F z - F (update z i b)| ≤ c i)
(u : ℝ) (hu : 0 ≤ u) :
(∑ z, exp (u * F z)) / (card (ι → Bool) : ℝ) ≤
exp (u^2 / 8 * ∑ i, (c i)^2) := by
classical
let e : Fin (card ι) ≃ ι := (Fintype.equivFin ι).symm
let E : (Fin (card ι) → Bool) ≃ (ι → Bool) :=
e.arrowCongr (Equiv.refl Bool)
have hupdate (z : Fin (card ι) → Bool) (i : Fin (card ι)) (b : Bool) :
E (update z i b) = update (E z) (e i) b := by
funext j
change update z i b (e.symm j) = update (E z) (e i) b j
by_cases hj : j = e i
· subst j; simp
· have hi : e.symm j ≠ i := by intro h; apply hj; exact (e.symm_apply_eq).mp h
simp [Function.update_of_ne hj, Function.update_of_ne hi, E, Equiv.arrowCongr]
have hmean :
(∫ z, F (E z) ∂Measure.pi
(fun _ : Fin (card ι) => (uniformOfFintype Bool).toMeasure)) = 0 := by
rw [integral_fairSignProduct, E.sum_comp F, hzero, zero_div]
have hm := gate (card ι) (fun _ => Bool) (fun _ => inferInstance)
(fun _ => (uniformOfFintype Bool).toMeasure) (fun _ => inferInstance)
(fun z => F (E z)) (by fun_prop) (by
obtain ⟨C, hC⟩ := hbound
exact ⟨C, fun z => hC (E z)⟩)
(fun i => c (e i)) (fun i => hc (e i)) (by
intro i z b
rw [hupdate]
exact hchange (e i) (E z) b) u hu
rw [hmean] at hm
simp only [sub_zero] at hm
rw [integral_fairSignProduct, E.sum_comp (fun z => exp (u * F z))] at hm
have hcard : card (Fin (card ι) → Bool) = card (ι → Bool) :=
Fintype.card_congr E
rw [hcard, e.sum_comp (fun i => (c i)^2)] at hm
exact hm
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.Stability 31 declarations All-count sensitivity and Efron--Stein-conditional iid variance stability.
All-count sensitivity and Efron--Stein-conditional iid variance stability.
Appending a last index adds exactly its pairs with the existing indices. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: ordered_pair_sum_append
lemma ordered_pair_sum_append (s : ℕ) (f : Fin (s + 1) → Fin (s + 1) → ℝ) :
(∑ i, ∑ l, if i < l then f i l else 0) =
(∑ i : Fin s, ∑ l : Fin s,
if i < l then f i.castSucc l.castSucc else 0) +
∑ i : Fin s, f i.castSucc (last s) := by
classical
rw [Fin.sum_univ_castSucc]
have hlast : ∀ i : Fin s, ¬ last s < i.castSucc :=
fun i => not_lt.mpr (Fin.castSucc_lt_last i).le
simp [Fin.sum_univ_castSucc, hlast, Finset.sum_add_distrib]
The number of ordered index pairs with the first strictly smaller is the binomial count. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: ordered_pair_count
lemma ordered_pair_count (s : ℕ) :
(∑ i : Fin s, ∑ l : Fin s, if i < l then (1 : ℝ) else 0) =
(choose s 2 : ℝ) := by
induction s with
| zero => simp
| succ s ih =>
rw [ordered_pair_sum_append, ih]
simp [Nat.choose_succ_succ, Nat.choose_one_right, add_comm]
A unit-bounded kernel has a pair sum bounded by the number of pairs. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: ordered_pair_sum_bound
lemma ordered_pair_sum_bound (s : ℕ) (K : O → O → ℝ) (z : Fin s → O)
(hK : ∀ x y, |K x y| ≤ 1) :
|∑ i : Fin s, ∑ l : Fin s, if i < l then K (z i) (z l) else 0| ≤
(choose s 2 : ℝ) := by
calc
_ ≤ ∑ i : Fin s, |∑ l : Fin s, if i < l then K (z i) (z l) else 0| :=
Finset.abs_sum_le_sum_abs _ _
_ ≤ ∑ i : Fin s, ∑ l : Fin s, |if i < l then K (z i) (z l) else 0| := by
apply Finset.sum_le_sum
intro i hi
exact Finset.abs_sum_le_sum_abs _ _
_ ≤ ∑ i : Fin s, ∑ l : Fin s, if i < l then (1 : ℝ) else 0 := by
apply Finset.sum_le_sum
intro i hi
apply Finset.sum_le_sum
intro l hl
by_cases hil : i < l
· simpa only [if_pos hil] using hK (z i) (z l)
· simp only [if_neg hil, abs_zero, le_refl]
_ = _ := ordered_pair_count s
Count-weighted pair contributions are bounded by the occupancy, at every count. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
|pairContribution K z| ≤ s
Proof (Lean source)
-- @node: pairContribution_abs_le
lemma pairContribution_abs_le (s : ℕ) (K : O → O → ℝ) (z : Fin s → O)
(hK : ∀ x y, |K x y| ≤ 1) : |pairContribution K z| ≤ s := by
classical
by_cases hs : s < 2
· simp [pairContribution, hs]
have hc : 0 < (choose s 2 : ℝ) := by
exact_mod_cast Nat.choose_pos (show 2 ≤ s by omega)
rw [pairContribution, if_neg hs, abs_mul,
abs_of_nonneg (div_nonneg (Nat.cast_nonneg s) hc.le)]
calc
_ ≤ ((s : ℝ) / choose s 2) * (choose s 2 : ℝ) :=
mul_le_mul_of_nonneg_left (ordered_pair_sum_bound s K z hK) (by positivity)
_ = _ := div_mul_cancel₀ _ hc.ne'
theorem pairContribution_insertion_identity Lemma pairContribution_insertion_identity in the paper ↗
At occupancy at least two, insertion equals twice the new-pair average minus the old pair average, as in the all-count stability roadmap. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: pairContribution_insertion_identity
lemma pairContribution_insertion_identity (s : ℕ) (hs : 2 ≤ s)
(K : O → O → ℝ) (z : Fin (s + 1) → O) :
pairContribution K z - pairContribution K (fun i : Fin s => z i.castSucc) =
2 / (s : ℝ) * (∑ i : Fin s, K (z i.castSucc) (z (last s))) -
pairContribution K (fun i : Fin s => z i.castSucc) / (s : ℝ) := by
classical
have hs0 : (s : ℝ) ≠ 0 := by exact_mod_cast (show s ≠ 0 by omega)
have hs1 : (s : ℝ) - 1 ≠ 0 := by
have : (2 : ℝ) ≤ s := by exact_mod_cast hs
linarith
simp only [pairContribution, show ¬s < 2 by omega, show ¬s + 1 < 2 by omega,
if_false]
rw [ordered_pair_sum_append]
simp only [Nat.cast_choose_two ℝ, Nat.cast_add, Nat.cast_one]
have hsp : (s : ℝ) + 1 ≠ 0 := by positivity
field_simp [hs0, hs1, hsp]
ring
The new pairs have total absolute value at most the old occupancy. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: insertion_pair_sum_bound
lemma insertion_pair_sum_bound (s : ℕ) (K : O → O → ℝ)
(z : Fin (s + 1) → O) (hK : ∀ x y, |K x y| ≤ 1) :
|∑ i : Fin s, K (z i.castSucc) (z (last s))| ≤ s := by
calc
_ ≤ ∑ i : Fin s, |K (z i.castSucc) (z (last s))| :=
Finset.abs_sum_le_sum_abs _ _
_ ≤ ∑ i : Fin s, (1 : ℝ) := Finset.sum_le_sum (fun i hi => hK _ _)
_ = _ := by simp
theorem pairContribution_insertion_bound_large Lemma pairContribution_insertion_bound_large in the paper ↗
Inserting into a cell with at least two records changes a scalar contribution by at most three, using the exact pair identity rather than a count-specific approximation. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: pairContribution_insertion_bound_large
lemma pairContribution_insertion_bound_large (s : ℕ) (hs : 2 ≤ s)
(K : O → O → ℝ) (z : Fin (s + 1) → O) (hK : ∀ x y, |K x y| ≤ 1) :
|pairContribution K z - pairContribution K (fun i : Fin s => z i.castSucc)| ≤ 3 := by
have hsR : 0 < (s : ℝ) := by exact_mod_cast (show 0 < s by omega)
have hb := insertion_pair_sum_bound s K z hK
have hf := pairContribution_abs_le s K (fun i => z i.castSucc) hK
rw [pairContribution_insertion_identity s hs K z]
calc
_ ≤ |2 / (s : ℝ) * ∑ i : Fin s, K (z i.castSucc) (z (last s))| +
|pairContribution K (fun i : Fin s => z i.castSucc) / (s : ℝ)| :=
abs_sub _ _
_ = (2 / (s : ℝ)) * |∑ i : Fin s, K (z i.castSucc) (z (last s))| +
|pairContribution K (fun i : Fin s => z i.castSucc)| / (s : ℝ) := by
simp only [abs_mul, abs_div, abs_of_pos hsR, abs_of_nonneg (by norm_num : (0 : ℝ) ≤ 2)]
_ ≤ (2 / (s : ℝ)) * (s : ℝ) + (s : ℝ) / (s : ℝ) :=
add_le_add (mul_le_mul_of_nonneg_left hb (by positivity))
(div_le_div_of_nonneg_right hf hsR.le)
_ = 3 := by field_simp; ring
A singleton contributes zero, and the first pair contributes twice its kernel value. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: pairContribution_small_counts
lemma pairContribution_small_counts (K : O → O → ℝ) (z : Fin 2 → O) :
pairContribution K (fun i : Fin 1 => z i.castSucc) = 0 ∧
pairContribution K z = 2 * K (z 0) (z 1) := by
classical
simp [pairContribution, Fin.sum_univ_two]
The insertion bound includes empty and singleton cells as well as all larger occupancies. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: pairContribution_insertion_bound
lemma pairContribution_insertion_bound (s : ℕ) (K : O → O → ℝ)
(z : Fin (s + 1) → O) (hK : ∀ x y, |K x y| ≤ 1) :
|pairContribution K z - pairContribution K (fun i : Fin s => z i.castSucc)| ≤ 3 := by
by_cases hs : 2 ≤ s
· exact pairContribution_insertion_bound_large s hs K z hK
have hsmall : s = 0 ∨ s = 1 := by omega
rcases hsmall with rfl | rfl
· simp [pairContribution]
· obtain ⟨hzero, hpair⟩ := pairContribution_small_counts K z
rw [hzero, hpair, sub_zero, abs_mul]
have hb := hK (z 0) (z 1)
rw [abs_of_pos (by norm_num : (0 : ℝ) < 2)]
linarith
Both causal pair kernels are symmetric, and their joint absolute mass is at most two. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
The causal numerator and denominator satisfy the scalar unit-kernel bound. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Proof (Lean source)
-- @node: causal_pair_kernel_unit_bounds
lemma causal_pair_kernel_unit_bounds (z z' : O) : |KN z z'| ≤ 1 ∧ |KD z z'| ≤ 1 := by
obtain ⟨hsN, hsD, hND, hD0, hD1⟩ := causal_pair_kernel_bounds z z'
exact ⟨hND.trans hD1, by rwa [abs_of_nonneg hD0]⟩
The two causal contributions have joint insertion sensitivity at most six at every count. the theorem's stated inputs and assumptions, and the asserted conclusion follows.
Formal statement
Proof (Lean source)
-- @node: causal_pair_insertion_bound
lemma causal_pair_insertion_bound (s : ℕ) (z : Fin (s + 1) → O) :
|pairContribution KN z - pairContribution KN (fun i : Fin s => z i.castSucc)| +
|pairContribution KD z - pairContribution KD (fun i : Fin s => z i.castSucc)| ≤ 6 := by
have hN := pairContribution_insertion_bound s KN z
(fun x y => (causal_pair_kernel_unit_bounds x y).1)
have hD := pairContribution_insertion_bound s KD z
(fun x y => (causal_pair_kernel_unit_bounds x y).2)
linarith
A replacement through a common deleted dataset has joint sensitivity at most twelve. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: causal_pair_replacement_bound
lemma causal_pair_replacement_bound (s : ℕ) (z z' : Fin (s + 1) → O)
(hcommon : ∀ i : Fin s, z i.castSucc = z' i.castSucc) :
|pairContribution KN z - pairContribution KN z'| +
|pairContribution KD z - pairContribution KD z'| ≤ 12 := by
have hN : pairContribution KN (fun i : Fin s => z i.castSucc) =
pairContribution KN (fun i : Fin s => z' i.castSucc) := by
congr 1
funext i
exact hcommon i
have hD : pairContribution KD (fun i : Fin s => z i.castSucc) =
pairContribution KD (fun i : Fin s => z' i.castSucc) := by
congr 1
funext i
exact hcommon i
have hz := causal_pair_insertion_bound s z
have hz' := causal_pair_insertion_bound s z'
rw [← hN, ← hD] at hz'
have htN := abs_sub_le (pairContribution KN z)
(pairContribution KN (fun i : Fin s => z i.castSucc)) (pairContribution KN z')
have htD := abs_sub_le (pairContribution KD z)
(pairContribution KD (fun i : Fin s => z i.castSucc)) (pairContribution KD z')
rw [abs_sub_comm (pairContribution KN (fun i : Fin s => z i.castSucc))] at htN
rw [abs_sub_comm (pairContribution KD (fun i : Fin s => z i.castSucc))] at htD
linarith
The insertion indicator bound yields the squared replacement bound used by Efron--Stein. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
change J J' :
ℝ
hJ :
J = 0 ∨ J = 1
hJ' :
J' = 0 ∨ J' = 1
hchange :
|change| ≤ 3 * (J + J')
conclusion
change^2 ≤ 18 * (J + J')
Proof (Lean source)
-- @node: replacement_square_indicator_bound
lemma replacement_square_indicator_bound (change J J' : ℝ)
(hJ : J = 0 ∨ J = 1) (hJ' : J' = 0 ∨ J' = 1)
(hchange : |change| ≤ 3 * (J + J')) : change^2 ≤ 18 * (J + J') := by
have hsq : change^2 ≤ (3 * (J + J'))^2 := by
have hnonneg := abs_nonneg change
nlinarith [sq_abs change]
rcases hJ with rfl | rfl <;> rcases hJ' with rfl | rfl <;> nlinarith
Reindexing the selected record set by any finite enumeration preserves its moment sums. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: selected_sum
lemma selected_sum (n s : ℕ) (I : Finset (Fin n))
(e : Fin s ≃ {i // i ∈ I}) (v : Fin n → ℝ) :
(∑ i : Fin s, v (e i)) = ∑ i ∈ I, v i := by
classical
rw [e.sum_comp (fun i => v i.val)]
exact Finset.sum_attach I v
Every enumeration of the records in a cell realizes its closed numerator and denominator as count-weighted pair contributions. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
n k :
ℕ
h :
ℝ
D :
Dataset n
j :
Fin k
e :
Fin (cellCount n h k D j) ≃ {i // i ∈ cellRecords n h k D j}
conclusion
conclusion 1
pairContribution KN (fun i => D (e i)) = numeratorCell n h k D j
conclusion 2
pairContribution KD (fun i => D (e i)) = denominatorCell n h k D j
Proof (Lean source)
-- @node: cell_pair_formulas
lemma cell_pair_formulas (n k : ℕ) (h : ℝ) (D : Dataset n) (j : Fin k)
(e : Fin (cellCount n h k D j) ≃ {i // i ∈ cellRecords n h k D j}) :
pairContribution KN (fun i => D (e i)) = numeratorCell n h k D j ∧
pairContribution KD (fun i => D (e i)) = denominatorCell n h k D j := by
classical
obtain ⟨hN, hD⟩ := pair_moment_formulas _ (fun i => D (e i))
have hsum (v : O → ℝ) :
(∑ i, v (D (e i))) = cellSum n h k D j v :=
selected_sum n _ _ e (fun i => v (D i))
rw [hsum (fun z => bit z.2.1 * bit z.2.2),
hsum (fun z => bit z.2.1), hsum (fun z => bit z.2.2)] at hN
rw [hsum (fun z => bit z.2.1)] at hD
exact ⟨hN, hD⟩
Each scalar cell contribution is bounded by its occupancy, including empty and singleton cells. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
conclusion 1
|numeratorCell n h k D j| ≤ cellCount n h k D j
conclusion 2
|denominatorCell n h k D j| ≤ cellCount n h k D j
Proof (Lean source)
-- @node: cell_contribution_bounds
lemma cell_contribution_bounds (n k : ℕ) (h : ℝ) (D : Dataset n) (j : Fin k) :
|numeratorCell n h k D j| ≤ cellCount n h k D j ∧
|denominatorCell n h k D j| ≤ cellCount n h k D j := by
classical
let e : Fin (cellCount n h k D j) ≃ {i // i ∈ cellRecords n h k D j} :=
(Fintype.equivFinOfCardEq (by simp [cellCount])).symm
obtain ⟨hN, hD⟩ := cell_pair_formulas n k h D j e
rw [← hN, ← hD]
exact ⟨pairContribution_abs_le _ KN _ (fun x y => (causal_pair_kernel_unit_bounds x y).1),
pairContribution_abs_le _ KD _ (fun x y => (causal_pair_kernel_unit_bounds x y).2)⟩
Distinct public cells are disjoint, including the final right endpoint. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: cell_disjoint
lemma cell_disjoint (k : ℕ) (h : ℝ) (hh : 0 < h) (hk : 0 < k)
(i j : Fin k) (hij : i ≠ j) : Disjoint (cell h k i) (cell h k j) := by
have hkR : (0 : ℝ) < k := by exact_mod_cast hk
have hd : 0 < cellWidth h k := by unfold cellWidth; positivity
have hw : (k : ℝ) * cellWidth h k = 2 * h := by
unfold cellWidth
field_simp
have hlt (i j : Fin k) (hij : i < j) : Disjoint (cell h k i) (cell h k j) := by
apply Set.disjoint_left.mpr
intro x hxi hxj
have horder : (i.val : ℝ) + 1 ≤ j.val := by exact_mod_cast hij
have hupper := mul_le_mul_of_nonneg_right horder hd.le
rcases hxi.2 with hi | ⟨hi, hx⟩
· linarith [hxj.1]
· have : i.val < j.val := hij
omega
rcases lt_or_gt_of_ne hij with hij | hij
· exact hlt i j hij
· exact (hlt j i hij).symm
Disjoint cell selection counts each dataset record at most once. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: total_cell_counts
lemma total_cell_counts (n k : ℕ) (h : ℝ) (hh : 0 < h) (hk : 0 < k)
(D : Dataset n) : (∑ j : Fin k, cellCount n h k D j) ≤ n := by
classical
have hd : ∀ i j : Fin k, i ≠ j →
Disjoint (cellRecords n h k D i) (cellRecords n h k D j) := by
intro i j hij
apply Finset.disjoint_left.mpr
intro r hi hj
have hx := Set.disjoint_left.mp (cell_disjoint k h hh hk i j hij)
exact hx (by simpa [cellRecords] using hi) (by simpa [cellRecords] using hj)
have hcard := Finset.card_biUnion (s := (Finset.univ : Finset (Fin k)))
(t := fun j => cellRecords n h k D j) (fun i _ j _ hij => hd i j hij)
have hle := Finset.card_le_card
(show Finset.univ.biUnion (fun j : Fin k => cellRecords n h k D j) ⊆
(Finset.univ : Finset (Fin n)) from Finset.subset_univ _)
simpa only [hcard, Finset.card_univ, Fintype.card_fin, cellCount] using hle
Both scalar statistics are bounded by the sample size on every input dataset. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: statistic_bounds
lemma statistic_bounds (n k : ℕ) (h : ℝ) (hh : 0 < h) (hk : 0 < k)
(D : Dataset n) : |SN n h k D| ≤ n ∧ |SD n h k D| ≤ n := by
have hc : (∑ j : Fin k, (cellCount n h k D j : ℝ)) ≤ n := by
exact_mod_cast total_cell_counts n k h hh hk D
constructor
· exact (Finset.abs_sum_le_sum_abs _ _).trans
((Finset.sum_le_sum (fun j _ => (cell_contribution_bounds n k h D j).1)).trans hc)
· exact (Finset.abs_sum_le_sum_abs _ _).trans
((Finset.sum_le_sum (fun j _ => (cell_contribution_bounds n k h D j).2)).trans hc)
The bounded measurable statistics belong to every finite-measure Lp space, in particular the square-integrability class needed for Efron--Stein. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: statistic_memLp
lemma statistic_memLp (n k : ℕ) (h : ℝ) (hh : 0 < h) (hk : 0 < k)
(Q : Measure (Dataset n)) [IsFiniteMeasure Q] (p : ENNReal) :
MemLp (SN n h k) p Q ∧ MemLp (SD n h k) p Q := by
constructor
· apply MemLp.of_bound (by fun_prop) (n : ℝ)
exact ae_of_all _ (fun D => by simpa only [Real.norm_eq_abs] using
(statistic_bounds n k h hh hk D).1)
· apply MemLp.of_bound (by fun_prop) (n : ℝ)
exact ae_of_all _ (fun D => by simpa only [Real.norm_eq_abs] using
(statistic_bounds n k h hh hk D).2)
The closed numerator moment formula on an arbitrary finite selected record set.
The closed denominator moment formula on an arbitrary finite selected record set.
theorem selected_pair_formulas
An arbitrary finite record selection has the same closed moments as its enumerated pair contributions. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
conclusion 1
pairContribution KN (fun i => D (e i)) = selectedNumerator I D
conclusion 2
pairContribution KD (fun i => D (e i)) = selectedDenominator I D
Proof (Lean source)
-- @node: selected_pair_formulas
lemma selected_pair_formulas {n : ℕ} (I : Finset (Fin n)) (D : Dataset n)
(e : Fin I.card ≃ {i // i ∈ I}) :
pairContribution KN (fun i => D (e i)) = selectedNumerator I D ∧
pairContribution KD (fun i => D (e i)) = selectedDenominator I D := by
obtain ⟨hN, hD⟩ := pair_moment_formulas _ (fun i => D (e i))
have hsum (v : O → ℝ) :
(∑ i, v (D (e i))) = ∑ i ∈ I, v (D i) :=
selected_sum n _ _ e (fun i => v (D i))
rw [hsum (fun z => bit z.2.1 * bit z.2.2),
hsum (fun z => bit z.2.1), hsum (fun z => bit z.2.2)] at hN
rw [hsum (fun z => bit z.2.1)] at hD
exact ⟨hN, hD⟩
theorem selected_insertion_bound
Inserting one record into an arbitrary selected set changes the joint causal contribution by at most six, at every count. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
|selectedNumerator (insert r I) D
- selectedNumerator I D|
+ |selectedDenominator (insert r I) D
- selectedDenominator I D|
≤ 6
Proof (Lean source)
-- @node: selected_insertion_bound
lemma selected_insertion_bound {n : ℕ} (I : Finset (Fin n)) (D : Dataset n)
(r : Fin n) (hr : r ∉ I) :
|selectedNumerator (insert r I) D - selectedNumerator I D| +
|selectedDenominator (insert r I) D - selectedDenominator I D| ≤ 6 := by
classical
let e : Fin I.card ≃ {i // i ∈ I} :=
(Fintype.equivFinOfCardEq (by simp)).symm
let z : Fin (I.card + 1) → O := Fin.lastCases (D r) (fun i => D (e i))
have hz (i : Fin I.card) : z i.castSucc = D (e i) := by simp [z]
have hlast : z (last I.card) = D r := by simp [z]
have hsum (v : O → ℝ) :
(∑ i, v (z i)) = ∑ i ∈ insert r I, v (D i) := by
rw [Fin.sum_univ_castSucc]
simp_rw [hz]
rw [selected_sum n _ I e (fun i => v (D i)), Finset.sum_insert hr, hlast]
exact add_comm _ _
obtain ⟨hN, hD⟩ := pair_moment_formulas _ z
rw [hsum (fun z => bit z.2.1 * bit z.2.2),
hsum (fun z => bit z.2.1), hsum (fun z => bit z.2.2)] at hN
rw [hsum (fun z => bit z.2.1)] at hD
have hcard := Finset.card_insert_of_notMem hr
have hNi : pairContribution KN z = selectedNumerator (insert r I) D := by
simpa only [selectedNumerator, hcard, Nat.cast_add, Nat.cast_one] using hN
have hDi : pairContribution KD z = selectedDenominator (insert r I) D := by
simpa only [selectedDenominator, hcard, Nat.cast_add, Nat.cast_one] using hD
obtain ⟨hNo, hDo⟩ := selected_pair_formulas I D e
have hb := causal_pair_insertion_bound I.card z
simp_rw [hz] at hb
rwa [hNi, hDi, hNo, hDo] at hb
theorem selected_congr
Selected contributions depend only on records at the selected indices. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
conclusion 1
selectedNumerator I D = selectedNumerator I D'
conclusion 2
selectedDenominator I D = selectedDenominator I D'
Proof (Lean source)
-- @node: selected_congr
lemma selected_congr {n : ℕ} (I : Finset (Fin n)) (D D' : Dataset n)
(heq : ∀ i ∈ I, D i = D' i) :
selectedNumerator I D = selectedNumerator I D' ∧
selectedDenominator I D = selectedDenominator I D' := by
have hs (v : O → ℝ) : (∑ i ∈ I, v (D i)) = ∑ i ∈ I, v (D' i) :=
Finset.sum_congr rfl (fun i hi => congrArg v (heq i hi))
simp only [selectedNumerator, selectedDenominator]
rw [hs (fun z => bit z.2.1 * bit z.2.2), hs (fun z => bit z.2.1),
hs (fun z => bit z.2.2)]
exact ⟨rfl, rfl⟩
theorem selected_erasure_bound
Erasing one selected record has joint sensitivity at most six; erasing an absent record has zero effect. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
|selectedNumerator I D
- selectedNumerator (I.erase r) D|
+ |selectedDenominator I D
- selectedDenominator (I.erase r) D|
≤ if r ∈ I then 6 else 0
Proof (Lean source)
-- @node: selected_erasure_bound
lemma selected_erasure_bound {n : ℕ} (I : Finset (Fin n)) (D : Dataset n)
(r : Fin n) :
|selectedNumerator I D - selectedNumerator (I.erase r) D| +
|selectedDenominator I D - selectedDenominator (I.erase r) D| ≤
if r ∈ I then 6 else 0 := by
classical
by_cases hr : r ∈ I
· rw [if_pos hr]
have hb := selected_insertion_bound (I.erase r) D r (Finset.notMem_erase r I)
rwa [Finset.insert_erase hr] at hb
· simp [hr]
A replacement changes only its old and new cells, using their common erased selection. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
|numeratorCell n h k D j
- numeratorCell n h k D' j|
+ |denominatorCell n h k D j
- denominatorCell n h k D' j|
≤ (if (D r).1 ∈ cell h k j then 6 else 0) + (if (D' r).1 ∈ cell h k j then 6 else 0)
Proof (Lean source)
-- @node: cell_replacement_bound
lemma cell_replacement_bound (n k : ℕ) (h : ℝ) (D D' : Dataset n) (r : Fin n)
(heq : ∀ i, i ≠ r → D i = D' i) (j : Fin k) :
|numeratorCell n h k D j - numeratorCell n h k D' j| +
|denominatorCell n h k D j - denominatorCell n h k D' j| ≤
(if (D r).1 ∈ cell h k j then 6 else 0) +
(if (D' r).1 ∈ cell h k j then 6 else 0) := by
classical
let I := cellRecords n h k D j
let I' := cellRecords n h k D' j
have hbase : I.erase r = I'.erase r := by
ext i
by_cases hi : i = r
· subst i; simp
· simp [I, I', cellRecords, hi, heq i hi]
have hcommon : ∀ i ∈ I.erase r, D i = D' i := by
intro i hi
exact heq i (Finset.mem_erase.mp hi).1
obtain ⟨hcN, hcD⟩ := selected_congr (I.erase r) D D' hcommon
rw [hbase] at hcN hcD
have hb := selected_erasure_bound I D r
have hb' := selected_erasure_bound I' D' r
rw [← hcN, ← hcD, ← hbase] at hb'
have hN := abs_sub_le (selectedNumerator I D)
(selectedNumerator (I.erase r) D) (selectedNumerator I' D')
have hD := abs_sub_le (selectedDenominator I D)
(selectedDenominator (I.erase r) D) (selectedDenominator I' D')
rw [abs_sub_comm (selectedNumerator (I.erase r) D)] at hN
rw [abs_sub_comm (selectedDenominator (I.erase r) D)] at hD
change |selectedNumerator I D - selectedNumerator I' D'| +
|selectedDenominator I D - selectedDenominator I' D'| ≤ _
have hmem : (r ∈ I) ↔ (D r).1 ∈ cell h k j := by simp [I, cellRecords]
have hmem' : (r ∈ I') ↔ (D' r).1 ∈ cell h k j := by simp [I', cellRecords]
simp only [hmem] at hb
simp only [hmem'] at hb'
linarith
A covariate belongs to at most one public cell, so its summed insertion envelope is at most six. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: cell_membership_sum
lemma cell_membership_sum (k : ℕ) (h : ℝ) (hh : 0 < h) (hk : 0 < k)
(x : Covariate) :
(∑ j : Fin k, if x ∈ cell h k j then (6 : ℝ) else 0) ≤ 6 := by
classical
by_cases hx : ∃ j : Fin k, x ∈ cell h k j
· obtain ⟨j, hj⟩ := hx
rw [Finset.sum_eq_single j]
· simp [hj]
· intro i hi hij
have hnot : x ∉ cell h k i := by
intro hx
exact Set.disjoint_left.mp (cell_disjoint k h hh hk i j hij) hx hj
simp [hnot]
· simp
· have hn : ∀ j : Fin k, x ∉ cell h k j := by simpa using hx
simp [hn]
Deletion followed by insertion bounds the joint statistic change by twelve, including replacements between different cells. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: statistic_replacement_bound
lemma statistic_replacement_bound (n k : ℕ) (h : ℝ) (hh : 0 < h) (hk : 0 < k)
(D D' : Dataset n) (r : Fin n) (heq : ∀ i, i ≠ r → D i = D' i) :
|SN n h k D - SN n h k D'| + |SD n h k D - SD n h k D'| ≤ 12 := by
have hb := Finset.sum_le_sum (s := univ)
(fun j _ => cell_replacement_bound n k h D D' r heq j)
rw [Finset.sum_add_distrib, Finset.sum_add_distrib] at hb
have hN : |SN n h k D - SN n h k D'| ≤
∑ j : Fin k, |numeratorCell n h k D j - numeratorCell n h k D' j| := by
rw [SN, SN, ← Finset.sum_sub_distrib]
exact Finset.abs_sum_le_sum_abs _ _
have hD : |SD n h k D - SD n h k D'| ≤
∑ j : Fin k, |denominatorCell n h k D j - denominatorCell n h k D' j| := by
rw [SD, SD, ← Finset.sum_sub_distrib]
exact Finset.abs_sum_le_sum_abs _ _
have ho := cell_membership_sum k h hh hk (D r).1
have hn := cell_membership_sum k h hh hk (D' r).1
linarith
Hamming-distance-one datasets satisfy the asserted joint sensitivity bound on the entire input space. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: statistic_hamming_sensitivity
lemma statistic_hamming_sensitivity (n k : ℕ) (h : ℝ) (hh : 0 < h) (hk : 0 < k)
(D D' : Dataset n) (hadj : dHam n D D' = 1) :
|SN n h k D - SN n h k D'| + |SD n h k D - SD n h k D'| ≤ 12 := by
classical
unfold dHam hammingDist at hadj
obtain ⟨r, hr⟩ := Finset.card_eq_one.mp hadj
apply statistic_replacement_bound n k h hh hk D D' r
intro i hi
by_contra hne
have hmem : i ∈ (Finset.univ.filter (fun i => D i ≠ D' i)) := by simp [hne]
rw [hr] at hmem
exact hi (Finset.mem_singleton.mp hmem)
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.TestingPriors 12 declarations Two finite, class-supported priors with fixed target separation and uniformly close outputs under every private Markov kernel.
Two finite, class-supported priors with fixed target separation and uniformly close outputs under every private Markov kernel.
def finiteProductMixture
A finite prior uniformly averages the observed product laws of its constituent causal laws.
theorem directAlternative_contrast
The direct alternative has the localized contrast prescribed by the construction. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: directAlternative_contrast
lemma directAlternative_contrast (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(x : Covariate) : tau (directAlternative hL hhL) x = separation hL * bump hL x := by
change (1/2 + separation hL * bump hL x) - 1/2 = _
ring
theorem directAlternative_target
The direct alternative has exactly the common target required by the testing prior. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1 / 4
conclusion
theta (directAlternative hL hhL) = separation hL
Proof (Lean source)
-- @node: directAlternative_target
lemma directAlternative_target (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4) :
theta (directAlternative hL hhL) = separation hL := by
rw [theta, directAlternative_contrast, bump_at_target hL hhL.1.le, mul_one]
theorem directAlternative_contrast_lipschitz
Scaling the bump cancels its inverse-radius modulus, giving radius-three regularity. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: directAlternative_contrast_lipschitz
lemma directAlternative_contrast_lipschitz (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4) :
ContrastLipschitz (directAlternative hL hhL) := by
intro x y
rw [directAlternative_contrast, directAlternative_contrast, ← mul_sub, abs_mul]
have ht : 0 ≤ separation hL := (separation_le_sqrt hL hhL).1
rw [abs_of_nonneg ht]
calc
_ ≤ separation hL * ((2*pi/hL)*|(x : ℝ)-y|) :=
mul_le_mul_of_nonneg_left (bump_difference_bound hL hhL.1 x y) ht
_ = (2*pi*kappa)*|(x : ℝ)-y| := by
unfold separation
field_simp [hhL.1.ne']
_ ≤ L*|(x : ℝ)-y| := by
apply mul_le_mul_of_nonneg_right _ (abs_nonneg _)
norm_num [kappa, L]
linarith [Real.pi_lt_four]
theorem directAlternative_completeModel
The explicit direct law satisfies every condition of the complete unknown-density model. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: directAlternative_completeModel
lemma directAlternative_completeModel (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4) :
CompleteModel (directAlternative hL hhL) := by
apply binaryCausalLaw_completeModel (fun _ => 1/2) (fun _ => 1/2) (directMu1 hL)
measurable_const measurable_const (measurable_directMu1 hL)
(direct_parameters_range hL hhL)
· exact binaryCausalLaw_exchangeability _ _ _ measurable_const measurable_const
(measurable_directMu1 hL) (direct_parameters_range hL hhL)
· intro x
norm_num
· intro x y
simp only [sub_self, abs_zero]
exact mul_nonneg (by norm_num [L]) (Real.rpow_nonneg (abs_nonneg _) _)
· intro x y
simp only [sub_self, abs_zero]
exact mul_nonneg (by norm_num [L]) (Real.rpow_nonneg (abs_nonneg _) _)
· exact directAlternative_contrast_lipschitz hL hhL
theorem finiteProductMixture_singleton
A singleton uniform prior is the observed product law itself. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: finiteProductMixture_singleton
lemma finiteProductMixture_singleton (n : ℕ) (P : CausalLaw) :
finiteProductMixture n 1 (fun _ => P) = dataLaw n P := by
simp [finiteProductMixture]
theorem finiteProductMixture_signs
Enumerating all sign vectors gives exactly the uniform sign mixture. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
= signMixture hL hhL n
Proof (Lean source)
-- @node: finiteProductMixture_signs
lemma finiteProductMixture_signs (n : ℕ) (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(e : Fin (card (SignVector hL)) ≃ SignVector hL) :
finiteProductMixture n (card (SignVector hL))
(fun i => cosineFamily hL hhL (e i)) = signMixture hL hhL n := by
classical
have hc0 : (card (SignVector hL) : ℝ≥0∞) ≠ 0 := by positivity
have hcT : (card (SignVector hL) : ℝ≥0∞) ≠ ⊤ := by finiteness
have hw : ofReal ((2 : ℝ)^(-((activeSigns hL).card : ℤ))) =
(card (SignVector hL) : ℝ≥0∞)⁻¹ := by
calc
_ = _ * ((card (SignVector hL) : ℝ≥0∞) *
(card (SignVector hL) : ℝ≥0∞)⁻¹) := by
rw [ENNReal.mul_inv_cancel hc0 hcT, mul_one]
_ = _ := by rw [← mul_assoc, sign_weight_normalization, one_mul]
have hs : (∑ i : Fin (card (SignVector hL)),
dataLaw n (cosineFamily hL hhL (e i))) =
∑ lam : SignVector hL, dataLaw n (cosineFamily hL hhL lam) := by
exact Finset.sum_equiv e (by simp) (by simp)
simp only [finiteProductMixture, signMixture, hw, hs]
theorem sampling_sign_output_TV_le
The sampling-radius sign experiment remains close after every Markov release. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
gate :
n :
ℕ
hL :
ℝ
hhL :
0 < hL ∧ hL ≤ 1 / 4
hscale :
(n : ℝ) ^ 2 * hL ^ 8 ≤ 1 / 4 ^ 8
B :
Type*
M :
conclusion
Proof (Lean source)
-- @node: sampling_sign_output_TV_le
lemma sampling_sign_output_TV_le (gate : BoundedDifferencesMGF)
(n : ℕ) (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(hscale : (n : ℝ) ^ 2 * hL ^ 8 ≤ 1 / 4 ^ 8)
{B : Type*} [MeasurableSpace B] (M : Kernel (Dataset n) B) [IsMarkovKernel M] :
TV (M ∘ₘ dataLaw n fairNull) (M ∘ₘ signMixture hL hhL n) ≤ 1 / 4 := by
let := signMixture_probability hL hhL n
have hm : Measurable observe := by unfold observe X A Y; fun_prop
let : IsProbabilityMeasure (Pobs fairNull) := Measure.isProbabilityMeasure_map hm.aemeasurable
let : IsProbabilityMeasure (dataLaw n fairNull) := by
unfold dataLaw
infer_instance
have hcert := positive_family_certificate hL hhL gate n
let v := 160*(n : ℝ)^2*(separation hL)^2*hL*deltaL hL
have hv : 0 ≤ v ∧ v ≤ 1 / 16 := by
have hid : v = 160*kappa^2*((n : ℝ)^2*hL^8) := by
dsimp [v, separation, deltaL]
ring
rw [hid]
constructor
· positivity
· have hb := mul_le_mul_of_nonneg_left hscale (by positivity : 0 ≤ 160*kappa^2)
norm_num [kappa] at hb ⊢
linarith
have hchi : chiSqDiv (signMixture hL hhL n) (dataLaw n fairNull) ≤
1 / 4 := by
have hx := exp_privacy_increment_le v ⟨hv.1, hv.2.trans (by norm_num)⟩
have hc := hcert.2.2.2.2.2.1
change chiSqDiv _ _ ≤ exp v - 1 at hc
linarith
have htv := tvDist_le_half_sqrt_chiSqDiv
(signMixture hL hhL n) (dataLaw n fairNull) hcert.2.2.2.2.2.2.1 hcert.2.2.2.2.2.2.2
have hroot : sqrt (chiSqDiv (signMixture hL hhL n)
(dataLaw n fairNull)) ≤ 1 / 2 := by
apply (Real.sqrt_le_iff).mpr
exact ⟨by norm_num, by nlinarith⟩
change tvDist _ _ ≤ _
rw [tvDist_symm]
exact (kernel_TV_contraction M _ _).trans (by change TV _ _ ≤ _ at htv; linarith)
def testingUsesSigns
The proof selects signs when the sampling scale is a maximum, or when the sparse scale strictly dominates both other scales. At sample size times budget at most one it selects the direct family; otherwise ties favor sampling, then direct, then sparse.
Definition (Lean source)
parameters
n :
ℕ
epsilon :
ℝ
def
testingUsesSigns n epsilon :
Prop
given by
a :
(n : ℝ)^(-1/4 : ℝ)
b :
((n : ℝ)^2*epsilon)^(-1/7 : ℝ)
c :
((n : ℝ)*epsilon)^(-1/2 : ℝ)
1 < (n : ℝ)*epsilon
(b ≤ a ∧ c ≤ a) ∨ (a < b ∧ c < b)
theorem testing_direct_rate
In every branch selecting the direct experiment, the benchmark is the capped square-root privacy scale, including ties with the sparse scale. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
Proof (Lean source)
-- @node: testing_direct_rate
lemma testing_direct_rate (n : ℕ) (epsilon : ℝ) (hn : 2 ≤ n) (he : 0 < epsilon)
(hnot : ¬ testingUsesSigns n epsilon) :
rate n epsilon = min 1 (((n : ℝ)*epsilon)^(-1/2 : ℝ)) := by
have hn0 : (0 : ℝ) < n := by exact_mod_cast (show 0 < n by omega)
have hbase : 0 < (n : ℝ)*epsilon := mul_pos hn0 he
by_cases hbudget : 1 < (n : ℝ)*epsilon
· let a := (n : ℝ)^(-1/4 : ℝ)
let b := ((n : ℝ)^2*epsilon)^(-1/7 : ℝ)
let c := ((n : ℝ)*epsilon)^(-1/2 : ℝ)
have hac : a ≤ c := by
by_contra h
have hca : c < a := lt_of_not_ge h
by_cases hba : b ≤ a
· exact hnot ⟨hbudget, inl ⟨hba, hca.le⟩⟩
· have hab : a < b := lt_of_not_ge hba
exact hnot ⟨hbudget, inr ⟨hab, hca.trans hab⟩⟩
have hbc : b ≤ c := by
by_contra h
have hcb : c < b := lt_of_not_ge h
exact hnot ⟨hbudget, inr ⟨hac.trans_lt hcb, hcb⟩⟩
dsimp [a, b, c] at hac hbc
simp only [rate, max_eq_right hbc, max_eq_right hac]
· have hc : 1 ≤ ((n : ℝ)*epsilon)^(-1/2 : ℝ) := by
simpa using Real.rpow_le_rpow_of_nonpos hbase (le_of_not_gt hbudget)
(by norm_num : (-1/2 : ℝ) ≤ 0)
rw [rate, min_eq_left (hc.trans ((le_max_right _ _).trans (le_max_right _ _))),
min_eq_left hc]
theorem testing_direct_privacy_scale_le
The selected direct radius has the uniformly small privacy-transport scale computed in the testing-prior roadmap. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: testing_direct_privacy_scale_le
lemma testing_direct_privacy_scale_le (n : ℕ) (epsilon : ℝ) (hn : 2 ≤ n)
(he : 0 < epsilon) (hnot : ¬ testingUsesSigns n epsilon) :
2*epsilon*(n : ℝ)*separation (rate n epsilon / 4)*(rate n epsilon / 4) ≤ kappa/8 := by
have hn0 : (0 : ℝ) < n := by exact_mod_cast (show 0 < n by omega)
have hbase : 0 < (n : ℝ)*epsilon := mul_pos hn0 he
have hr0 : 0 ≤ rate n epsilon := (rate_pos n epsilon (by omega)).le
have hr := testing_direct_rate n epsilon hn he hnot
have hrc : rate n epsilon ≤ ((n : ℝ)*epsilon)^(-1/2 : ℝ) := by
rw [hr]
exact min_le_right _ _
have hc0 : 0 ≤ ((n : ℝ)*epsilon)^(-1/2 : ℝ) := Real.rpow_nonneg hbase.le _
have hp : (((n : ℝ)*epsilon)^(-1/2 : ℝ))^2 = ((n : ℝ)*epsilon)⁻¹ := by
rw [← Real.rpow_mul_natCast hbase.le]
norm_num [Real.rpow_neg_one]
have hsquare : (rate n epsilon)^2 ≤ ((n : ℝ)*epsilon)⁻¹ := by
rw [← hp]
nlinarith
have hscale : ((n : ℝ)*epsilon)*(rate n epsilon)^2 ≤ 1 := by
have hm := mul_le_mul_of_nonneg_left hsquare hbase.le
simpa only [mul_inv_cancel₀ hbase.ne'] using hm
calc
_ = (kappa/8)*(((n : ℝ)*epsilon)*(rate n epsilon)^2) := by
unfold separation
ring
_ ≤ (kappa/8)*1 := mul_le_mul_of_nonneg_left hscale (by norm_num [kappa])
_ = _ := mul_one _
theorem frontier_testing_priors
The proof-selected uniform sign mixture or direct product law at one quarter of the benchmark has the stated separation and TV bound after every private standard-Borel release. the theorem's stated inputs and assumptions, and the asserted conclusion follows.
Formal statement
hypotheses
boundedDifferences_of_gate :
cayley_of_gate :
n :
ℕ
epsilon :
ℝ
hn :
2 ≤ n
he :
0 < epsilon ∧ epsilon ≤ 1
conclusion
conclusion 3
∃ hL : ℝ, ∃ hhL : 0 < hL ∧ hL ≤ 1/4,
hL = rate n epsilon / 4
i :
F0 i = fairNull
testingUsesSigns n epsilon
i :
∃ lam : SignVector hL, F1 i = cosineFamily hL hhL lam
finiteProductMixture n s1 F1 = signMixture hL hhL n
¬ testingUsesSigns n epsilon
i :
F1 i = directAlternative hL hhL
Proof (Lean source)
-- @node: lem:frontier-testing-priors
lemma frontier_testing_priors
(boundedDifferences_of_gate : BoundedDifferencesMGF)
(cayley_of_gate : CayleyLabeledTreeCount)
(n : ℕ) (epsilon : ℝ) (hn : 2 ≤ n) (he : 0 < epsilon ∧ epsilon ≤ 1) :
∃ (s0 s1 : ℕ) (F0 : Fin s0 → CausalLaw) (F1 : Fin s1 → CausalLaw) (Delta : ℝ),
0 < s0 ∧ 0 < s1 ∧ (2 : ℝ)^(-14 : ℤ)*rate n epsilon ≤ Delta ∧
(∀ i, CompleteModel (F0 i) ∧ theta (F0 i) = 0) ∧
(∀ i, CompleteModel (F1 i) ∧ theta (F1 i) = Delta) ∧
(∃ hL : ℝ, ∃ hhL : 0 < hL ∧ hL ≤ 1/4,
hL = rate n epsilon / 4 ∧
(∀ i, F0 i = fairNull) ∧
(testingUsesSigns n epsilon →
(∀ i, ∃ lam : SignVector hL, F1 i = cosineFamily hL hhL lam) ∧
finiteProductMixture n s1 F1 = signMixture hL hhL n) ∧
(¬ testingUsesSigns n epsilon →
(∀ i, F1 i = directAlternative hL hhL) ∧
finiteProductMixture n s1 F1 = dataLaw n (directAlternative hL hhL))) ∧
(∀ (B : Type*) [MeasurableSpace B] [StandardBorelSpace B]
(M : Kernel (Dataset n) B) [IsMarkovKernel M], PrivateKernel n epsilon M →
TV (M ∘ₘ finiteProductMixture n s0 F0)
(M ∘ₘ finiteProductMixture n s1 F1) ≤ 1 / 4) := by
classical
let hL := rate n epsilon / 4
have hhL : 0 < hL ∧ hL ≤ 1 / 4 := by
constructor
· exact div_pos (rate_pos n epsilon (by omega)) (by norm_num)
· have hr : rate n epsilon ≤ 1 := min_le_left _ _
dsimp [hL]
linarith
have hsep : separation hL = (2 : ℝ)^(-14 : ℤ) * rate n epsilon := by
dsimp [separation, kappa, hL]
norm_num
ring
by_cases hsign : testingUsesSigns n epsilon
· let s := card (SignVector hL)
let e : Fin s ≃ SignVector hL := (Fintype.equivFin (SignVector hL)).symm
refine ⟨1, s, (fun _ => fairNull), (fun i => cosineFamily hL hhL (e i)),
separation hL, by omega, Fintype.card_pos, hsep.ge, ?_, ?_, ?_, ?_⟩
· intro i
exact ⟨fairNull_completeModel, fairNull_target⟩
· intro i
exact ⟨cosineFamily_completeModel hL hhL (e i), cosineFamily_target hL hhL (e i)⟩
· refine ⟨hL, hhL, rfl, (fun _ => rfl), ?_, ?_⟩
· intro _
exact ⟨fun i => ⟨e i, rfl⟩, finiteProductMixture_signs n hL hhL e⟩
· intro hnot
exact False.elim (hnot hsign)
· intro B _ _ M _ hM
rw [finiteProductMixture_singleton, finiteProductMixture_signs n hL hhL e]
rcases hsign with ⟨hbudget, hsampling | hsparse⟩
· have hn0 : (0 : ℝ) < n := by exact_mod_cast (show 0 < n by omega)
have hn1 : (1 : ℝ) ≤ n := by exact_mod_cast (show 1 ≤ n by omega)
have ha1 := Real.rpow_le_one_of_one_le_of_nonpos hn1
(by norm_num : (-1 / 4 : ℝ) ≤ 0)
have hr : rate n epsilon = (n : ℝ)^(-1 / 4 : ℝ) := by
simp only [rate, max_eq_left (max_le hsampling.1 hsampling.2), min_eq_right ha1]
refine sampling_sign_output_TV_le boundedDifferences_of_gate n hL hhL ?_ M
have hp : ((n : ℝ)^(-1 / 4 : ℝ))^8 = ((n : ℝ)^2)⁻¹ := by
rw [← Real.rpow_mul_natCast hn0.le]
norm_num
dsimp [hL]
rw [hr, div_pow, hp]
have hn2 : (n : ℝ)^2 ≠ 0 := pow_ne_zero _ hn0.ne'
field_simp [hn2]
norm_num
· have hn0 : (0 : ℝ) < n := by exact_mod_cast (show 0 < n by omega)
have hn1 : (1 : ℝ) ≤ n := by exact_mod_cast (show 1 ≤ n by omega)
let b := ((n : ℝ)^2*epsilon)^(-1 / 7 : ℝ)
have he0 := he.1
have hbase : 0 < (n : ℝ)^2*epsilon := by positivity
have hb : 0 < b := Real.rpow_pos_of_pos hbase _
have hbase1 : 1 ≤ (n : ℝ)^2*epsilon := by nlinarith
have hb1 : b ≤ 1 := Real.rpow_le_one_of_one_le_of_nonpos hbase1 (by norm_num)
have hr : rate n epsilon = b := by
simp only [rate, max_eq_left hsparse.2.le, max_eq_right hsparse.1.le]
exact min_eq_right hb1
have hocc : (n : ℝ)*b^5 ≤ 1 := by
have hc0 : 0 < ((n : ℝ)*epsilon)^(-1 / 2 : ℝ) := by positivity
have hl := (Real.log_le_log_iff hc0 hb).mpr hsparse.2.le
dsimp [b] at hl
rw [Real.log_rpow (by positivity), Real.log_rpow hbase,
Real.log_mul hn0.ne' he.1.ne', Real.log_mul (by positivity) he.1.ne',
Real.log_pow] at hl
rw [← Real.log_le_log_iff (by positivity : 0 < (n : ℝ)*b^5) zero_lt_one,
Real.log_mul hn0.ne' (pow_pos hb 5).ne', Real.log_pow, Real.log_one]
dsimp [b]
rw [Real.log_rpow hbase, Real.log_mul (by positivity) he.1.ne', Real.log_pow]
linarith
have hcap : (n : ℝ)*deltaL hL ≤ 1 / 128 := by
dsimp [deltaL, hL]
rw [hr, div_pow]
nlinarith
have hout := ((measurable_component_contraction hL hhL n cayley_of_gate
boundedDifferences_of_gate epsilon he M hM).2.2.2.2 hcap).2
have hp : b^7 = ((n : ℝ)^2*epsilon)⁻¹ := by
dsimp [b]
rw [← Real.rpow_mul_natCast hbase.le]
norm_num [Real.rpow_neg_one]
have hid : 2048*epsilon*separation hL*(n : ℝ)^2*hL*deltaL hL =
2048*kappa / 4^7 := by
dsimp [separation, deltaL, hL]
rw [hr]
calc
_ = (2048*kappa / 4^7) * (((n : ℝ)^2*epsilon)*b^7) := by ring
_ = _ := by rw [hp, mul_inv_cancel₀ hbase.ne', mul_one]
change tvDist _ _ ≤ _
rw [tvDist_symm]
rw [hid] at hout
exact hout.trans (by norm_num [kappa])
· refine ⟨1, 1, (fun _ => fairNull), (fun _ => directAlternative hL hhL),
separation hL, by omega, by omega, hsep.ge, ?_, ?_, ?_, ?_⟩
· intro i
exact ⟨fairNull_completeModel, fairNull_target⟩
· intro i
exact ⟨directAlternative_completeModel hL hhL, directAlternative_target hL hhL⟩
· refine ⟨hL, hhL, rfl, (fun _ => rfl), ?_, ?_⟩
· intro hy
exact False.elim (hsign hy)
· intro _
exact ⟨fun _ => rfl, finiteProductMixture_singleton n _⟩
· intro B _ _ M _ hM
rw [finiteProductMixture_singleton, finiteProductMixture_singleton]
have hm : Measurable observe := by unfold observe X A Y; fun_prop
let : IsProbabilityMeasure (Pobs fairNull) :=
Measure.isProbabilityMeasure_map hm.aemeasurable
let : IsProbabilityMeasure (Pobs (directAlternative hL hhL)) :=
Measure.isProbabilityMeasure_map hm.aemeasurable
let : IsProbabilityMeasure (dataLaw n fairNull) := by unfold dataLaw; infer_instance
let : IsProbabilityMeasure (dataLaw n (directAlternative hL hhL)) := by
unfold dataLaw; infer_instance
have hcoupling : ∃ γ : Measure (Dataset n × Dataset n),
IsCoupling γ (dataLaw n fairNull)
(dataLaw n (directAlternative hL hhL)) ∧
(∫⁻ z, (dHam n z.1 z.2 : ℝ≥0∞) ∂γ) ≤
ofReal ((n : ℝ)*separation hL*hL) := by
exact ⟨directDatasetCoupling hL n, directDatasetCoupling_isCoupling hL hhL n,
directDatasetCoupling_cost_le hL hhL n⟩
obtain ⟨γ, hγ, hcost⟩ := hcoupling
have hout := private_TV_le_coupling_cost n epsilon he M hM _ _ γ hγ
have hscale : 2*epsilon*((n : ℝ)*separation hL*hL) ≤ kappa/8 := by
simpa only [mul_assoc] using testing_direct_privacy_scale_le n epsilon hn he.1 hsign
have htv : ofReal (TV (M ∘ₘ dataLaw n fairNull)
(M ∘ₘ dataLaw n (directAlternative hL hhL))) ≤ ofReal (kappa/8) := by
calc
_ ≤ ofReal (2*epsilon)*ofReal ((n : ℝ)*separation hL*hL) :=
hout.trans (mul_le_mul' le_rfl hcost)
_ = ofReal (2*epsilon*((n : ℝ)*separation hL*hL)) := by
rw [ENNReal.ofReal_mul (mul_nonneg (by norm_num) he.1.le : 0 ≤ 2*epsilon)]
_ ≤ _ := ENNReal.ofReal_le_ofReal hscale
have hreal := (ENNReal.ofReal_le_ofReal_iff (by norm_num [kappa] : 0 ≤ kappa/8)).mp htv
exact hreal.trans (by norm_num [kappa])
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.TreeGeometry 10 declarations Geometric support restrictions and rooted integration bounds for shared-sign trees.
Geometric support restrictions and rooted integration bounds for shared-sign trees. Each child is integrated in its parent's micro neighborhood; the root is in the macro window.
A nonzero localized frame is strictly inside both the macro and micro supports. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: weighted_frame_nonzero_radii
lemma weighted_frame_nonzero_radii (hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4)
(j : ℤ) (x : Covariate) (hx : envelope hL x * frame hL j x ≠ 0) :
|(x : ℝ) - x0| < hL ∧ |(x : ℝ) - j * deltaL hL| < deltaL hL := by
constructor
· by_contra h
exact hx (by rw [envelope_zero_of_radius_le hL hhL x (le_of_not_gt h), zero_mul])
· by_contra h
exact hx (by rw [frame_zero_of_radius_le hL hhL j x (le_of_not_gt h), mul_zero])
Any subset of a real-radius neighborhood in the unit interval has volume at most its length. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: volume_covariate_neighborhood_le
lemma volume_covariate_neighborhood_le (c r : ℝ) :
(volume : Measure Covariate) {x | |(x : ℝ) - c| < r} ≤ ofReal (2 * r) := by
rw [unitInterval.volume_apply]
calc
_ ≤ (volume : Measure ℝ) (Ioo (c-r) (c+r)) := by
apply measure_mono
rintro z ⟨x, hx, rfl⟩
have hx' := abs_lt.mp (show |(x : ℝ) - c| < r from hx)
constructor <;> linarith
_ = _ := by rw [Real.volume_Ioo]; congr 1; ring
The potential child region of a frozen parent has length at most four micro radii. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
Proof (Lean source)
-- @node: volume_tree_child_region_le
lemma volume_tree_child_region_le (hL : ℝ) (x : Covariate) :
(volume : Measure Covariate) {y | |(y : ℝ) - x| < 2 * deltaL hL} ≤
ofReal (4 * deltaL hL) := by
simpa only [← mul_assoc, show (2 : ℝ) * 2 = 4 by norm_num] using
volume_covariate_neighborhood_le (x : ℝ) (2 * deltaL hL)
The root region has length at most two macro radii. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
Proof (Lean source)
-- @node: volume_tree_root_region_le
lemma volume_tree_root_region_le (hL : ℝ) :
(volume : Measure Covariate) {x | |(x : ℝ) - x0| < hL} ≤ ofReal (2 * hL) :=
volume_covariate_neighborhood_le x0 hL
The exact rooted-order integral for successive children constrained to their parent's neighborhood. The parent selector reads only the already sampled covariates.
Definition (Lean source)
theorem treeExtensionVolume_le
Each newly integrated child costs at most four micro radii, regardless of the previous covariates or the choice of its parent among them. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: treeExtensionVolume_le
lemma treeExtensionVolume_le (d : ℝ) (parent : List Covariate → Covariate)
(m : ℕ) (xs : List Covariate) :
treeExtensionVolume d parent m xs ≤ (ofReal (4 * d)) ^ m := by
classical
induction m generalizing xs with
| zero => simp [treeExtensionVolume]
| succ m ih =>
rw [treeExtensionVolume]
calc
_ ≤ ∫⁻ y : Covariate,
{y : Covariate | |(y : ℝ) - parent xs| < 2 * d}.indicator
(fun _ => (ofReal (4 * d)) ^ m) y := by
apply lintegral_mono
intro y
exact Set.indicator_le_indicator (ih _)
_ = (ofReal (4 * d)) ^ m *
(volume : Measure Covariate) {y | |(y : ℝ) - parent xs| < 2 * d} :=
lintegral_indicator_const (measurableSet_lt (by fun_prop) measurable_const) _
_ ≤ (ofReal (4 * d)) ^ m * ofReal (4 * d) := by
apply mul_le_mul' le_rfl
simpa only [← mul_assoc, show (2 : ℝ) * 2 = 4 by norm_num] using
volume_covariate_neighborhood_le (parent xs : ℝ) (2 * d)
_ = _ := (pow_succ _ m).symm
theorem rooted_tree_iterated_volume_le
Integrating the root and then the children gives the roadmap's specified-tree envelope. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: rooted_tree_iterated_volume_le
lemma rooted_tree_iterated_volume_le (h d : ℝ) (parent : List Covariate → Covariate)
(m : ℕ) :
(∫⁻ x : Covariate, {x : Covariate | |(x : ℝ) - x0| < h}.indicator
(fun x => treeExtensionVolume d parent m [x]) x) ≤
ofReal (2 * h) * (ofReal (4 * d)) ^ m := by
classical
calc
_ ≤ ∫⁻ x : Covariate, {x : Covariate | |(x : ℝ) - x0| < h}.indicator
(fun _ => (ofReal (4 * d)) ^ m) x := by
apply lintegral_mono
intro x
exact Set.indicator_le_indicator (treeExtensionVolume_le d parent m _)
_ = (ofReal (4 * d)) ^ m *
(volume : Measure Covariate) {x | |(x : ℝ) - x0| < h} :=
lintegral_indicator_const (measurableSet_lt (by fun_prop) measurable_const) _
_ ≤ (ofReal (4 * d)) ^ m * ofReal (2 * h) :=
mul_le_mul' le_rfl (volume_tree_root_region_le h)
_ = _ := mul_comm _ _
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.TreeOrdering 6 declarations Parent-before-child enumerations of finite connected graphs and the resulting specified labeled-tree probability bound.
Parent-before-child enumerations of finite connected graphs and the resulting specified labeled-tree probability bound. This closes the geometric input to the general Cayley component-count estimate and sparse common-mass Hamming-cost bound.
Every finite connected graph has an ordering in which each non-root vertex has an earlier neighbor. Removing a non-cut vertex constructs the ordering inductively. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
Proof (Lean source)
-- @node: connected_parent_order
lemma connected_parent_order (m : ℕ) (G : SimpleGraph (Fin (m + 1)))
(hG : G.Connected) :
∃ e : Fin (m + 1) ≃ Fin (m + 1),
∀ i, i ≠ 0 → ∃ j, j < i ∧ G.Adj (e i) (e j) := by
classical
induction m with
| zero =>
refine ⟨Equiv.refl _, ?_⟩
intro i hi
apply (hi _).elim
apply Fin.ext
have := i.isLt
simp only [Fin.val_zero]
omega
| succ m ih =>
obtain ⟨v, hv⟩ := hG.exists_connected_induce_compl_singleton_of_finite_nontrivial
let a : Fin (m + 1) ≃ {x : Fin (m + 2) // x ≠ v} := finSuccAboveEquiv v
let H := (G.induce {v}ᶜ).comap a
have hH : H.Connected := (SimpleGraph.Iso.comap a _).connected_iff.mpr hv
obtain ⟨e, he⟩ := ih H hH
let f : Fin (m + 2) ≃ Fin (m + 2) :=
(finSuccEquivLast.trans (Equiv.optionCongr e)).trans (finSuccEquiv' v).symm
have hf (i : Fin (m + 1)) : f i.castSucc = v.succAbove (e i) := by
simp [f, Equiv.trans_apply, finSuccEquivLast_castSucc]
have hfl : f (last (m + 1)) = v := by
simp [f, Equiv.trans_apply]
refine ⟨f, ?_⟩
intro i
refine Fin.lastCases ?_ (fun k => ?_) i
· intro hi
obtain ⟨w, hw⟩ := hG.preconnected.exists_adj_of_nontrivial v
let b : {x : Fin (m + 2) // x ≠ v} := ⟨w, hw.ne.symm⟩
let j := e.symm (a.symm b)
refine ⟨j.castSucc, Fin.castSucc_lt_last j, ?_⟩
rw [hfl, hf]
have hb : v.succAbove (e j) = w := by
change (a (e j)).val = w
simp [j, b]
rw [hb]
exact hw
· intro hi
have hk : k ≠ 0 := by
intro hk
apply hi
simp [hk]
obtain ⟨j, hj, hadj⟩ := he k hk
refine ⟨j.castSucc, by simpa using hj, ?_⟩
rw [hf, hf]
exact hadj
theorem uniform_labeled_tree_probability_bound
Every specified labeled tree has the roadmap probability envelope, without an assumed enumeration or probability bound. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
Proof (Lean source)
-- @node: uniform_labeled_tree_probability_bound
lemma uniform_labeled_tree_probability_bound (hL : ℝ)
(hhL : 0 < hL ∧ hL ≤ 1 / 4) (s : ℕ) (hs : 2 ≤ s)
(T : {T : SimpleGraph (Fin s) // T.IsTree}) :
(Measure.pi (fun _ : Fin s => (volume : Measure Covariate)))
(labeledSharedTreeEvent hL s s id T.val) ≤
ofReal (2 * hL * (4 * deltaL hL) ^ (s - 1)) := by
obtain ⟨m, rfl⟩ : ∃ m, s = m + 1 := ⟨s - 1, by omega⟩
simpa using labeledSharedTreeEvent_volume_le hL hhL m (by omega)
T.val T.property.connected
theorem lintegral_sizeComponentCount_le_cayley
The actual size-s component count has the binomial Cayley expectation bound under iid uniform design. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: lintegral_sizeComponentCount_le_cayley
lemma lintegral_sizeComponentCount_le_cayley (cayley : CayleyLabeledTreeCount)
(hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4) (n s : ℕ) (hs : 2 ≤ s) :
(∫⁻ x, (sizeComponentCount hL n x s : ℝ≥0∞)
∂(Measure.pi (fun _ : Fin n => (volume : Measure Covariate)))) ≤
ofReal ((n.choose s : ℝ) * 2 * hL * (s : ℝ) ^ (s - 2) *
(4 * deltaL hL) ^ (s - 1)) := by
exact lintegral_sizeComponentCount_le_of_uniform_labeled_tree_bound cayley hL n s hs
(uniform_labeled_tree_probability_bound hL hhL s hs)
theorem commonMassCoupling_hamming_cost_le_sparse
The constructed common-mass coupling has expected Hamming cost at most 1024 times separation, squared sample size, macro radius, and micro radius in the sparse regime. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
(∫⁻ zw, (dHam n zw.1 zw.2 : ℝ≥0∞) ∂commonMassCoupling hL n)
Proof (Lean source)
-- @node: commonMassCoupling_hamming_cost_le_sparse
lemma commonMassCoupling_hamming_cost_le_sparse (cayley : CayleyLabeledTreeCount)
(hL : ℝ) (hhL : 0 < hL ∧ hL ≤ 1 / 4) (n : ℕ) (hn : 1 ≤ n)
(hcap : (n : ℝ) * deltaL hL ≤ 1 / 128) :
(∫⁻ zw, (dHam n zw.1 zw.2 : ℝ≥0∞) ∂commonMassCoupling hL n) ≤
ofReal (1024 * separation hL * (n : ℝ) ^ 2 * hL * deltaL hL) := by
exact commonMassCoupling_hamming_cost_le_of_labeled_tree_bound cayley hL hhL n hn hcap
(fun s hs _ => uniform_labeled_tree_probability_bound hL hhL s hs)
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.TreeProbability 8 declarations Product-measure bounds for rooted trees enumerated in parent-before-child order.
Product-measure bounds for rooted trees enumerated in parent-before-child order.
Appending an independent uniform child multiplies any prefix event's probability by at most four micro radii. The selector can be any measurable function of the prefix. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
Proof (Lean source)
-- @node: uniform_child_extension_volume_le
lemma uniform_child_extension_volume_le (m : ℕ) (d : ℝ)
(A : Set (Fin m → Covariate)) (hA : MeasurableSet A)
(parent : (Fin m → Covariate) → Covariate) (hp : Measurable parent) :
((volume : Measure Covariate).prod
(Measure.pi (fun _ : Fin m => (volume : Measure Covariate))))
{yx | yx.2 ∈ A ∧ |(yx.1 : ℝ) - parent yx.2| < 2 * d} ≤
ofReal (4 * d) *
(Measure.pi (fun _ : Fin m => (volume : Measure Covariate))) A := by
classical
have hS : MeasurableSet {yx : Covariate × (Fin m → Covariate) |
yx.2 ∈ A ∧ |(yx.1 : ℝ) - parent yx.2| < 2 * d} := by
exact (hA.preimage measurable_snd).inter
(measurableSet_lt (show Measurable (fun yx : Covariate × (Fin m → Covariate) =>
|(yx.1 : ℝ) - parent yx.2|) by fun_prop) measurable_const)
rw [Measure.prod_apply_symm hS]
calc
_ ≤ ∫⁻ x : Fin m → Covariate, A.indicator (fun _ => ofReal (4 * d)) x
∂(Measure.pi (fun _ : Fin m => (volume : Measure Covariate))) := by
apply lintegral_mono
intro x
by_cases hx : x ∈ A
· simp only [Set.indicator_of_mem hx]
have he : (fun y : Covariate => (y, x)) ⁻¹'
{yx | yx.2 ∈ A ∧ |(yx.1 : ℝ) - parent yx.2| < 2 * d} =
{y : Covariate | |(y : ℝ) - parent x| < 2 * d} := by ext y; simp [hx]
rw [he]
simpa only [← mul_assoc, show (2 : ℝ) * 2 = 4 by norm_num] using
volume_covariate_neighborhood_le (parent x : ℝ) (2 * d)
· simp [hx]
_ = _ := lintegral_indicator_const hA _
theorem pi_uniform_child_extension_volume_le Lemma pi_uniform_child_extension_volume_le in the paper ↗
An independent last coordinate obeys the child-extension bound in the original finite product space, without introducing a conditional probability version. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
Proof (Lean source)
-- @node: pi_uniform_child_extension_volume_le
lemma pi_uniform_child_extension_volume_le (m : ℕ) (d : ℝ)
(A : Set (Fin m → Covariate)) (hA : MeasurableSet A)
(parent : (Fin m → Covariate) → Covariate) (hp : Measurable parent) :
(Measure.pi (fun _ : Fin (m + 1) => (volume : Measure Covariate)))
{x | (fun i : Fin m => x i.castSucc) ∈ A ∧
|(x (last m) : ℝ) - parent (fun i : Fin m => x i.castSucc)| < 2 * d} ≤
ofReal (4 * d) *
(Measure.pi (fun _ : Fin m => (volume : Measure Covariate))) A := by
let e := piFinSuccAbove (fun _ : Fin (m + 1) => Covariate) (last m)
have hS : MeasurableSet {yx : Covariate × (Fin m → Covariate) |
yx.2 ∈ A ∧ |(yx.1 : ℝ) - parent yx.2| < 2 * d} := by
exact (hA.preimage measurable_snd).inter
(measurableSet_lt (show Measurable (fun yx : Covariate × (Fin m → Covariate) =>
|(yx.1 : ℝ) - parent yx.2|) by fun_prop) measurable_const)
have hmap := (measurePreserving_piFinSuccAbove
(fun _ : Fin (m + 1) => (volume : Measure Covariate)) (last m)).map_eq
have he : e ⁻¹' {yx : Covariate × (Fin m → Covariate) |
yx.2 ∈ A ∧ |(yx.1 : ℝ) - parent yx.2| < 2 * d} =
{x | (fun i : Fin m => x i.castSucc) ∈ A ∧
|(x (last m) : ℝ) - parent (fun i : Fin m => x i.castSucc)| < 2 * d} := by
ext x
simp [e, piFinSuccAbove, Fin.init_def]
have hm : (Measure.pi (fun _ : Fin (m + 1) => (volume : Measure Covariate)))
(e ⁻¹' {yx : Covariate × (Fin m → Covariate) |
yx.2 ∈ A ∧ |(yx.1 : ℝ) - parent yx.2| < 2 * d}) =
((volume : Measure Covariate).prod
(Measure.pi (fun _ : Fin m => (volume : Measure Covariate))))
{yx | yx.2 ∈ A ∧ |(yx.1 : ℝ) - parent yx.2| < 2 * d} := by
rw [← hmap, Measure.map_apply e.measurable hS]
rw [he] at hm
rw [hm]
exact uniform_child_extension_volume_le m d A hA parent hp
The rooted geometric event obtained by successively adding a child whose parent is a function of the earlier covariates. Coordinate projections give ordinary ordered labeled trees.
Definition (Lean source)
parameters
given by
clause 1
| 0 => {x | |(x 0 : ℝ) - x0| < h}
clause 2
| m+1 => {x | (fun i : Fin (m + 1) => x i.castSucc) ∈ orderedTreeRegion h d parent m ∧
theorem measurableSet_orderedTreeRegion
The ordered rooted-tree event is Borel whenever each parent selector is Borel. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
MeasurableSet (orderedTreeRegion h d parent m)
Proof (Lean source)
-- @node: measurableSet_orderedTreeRegion
lemma measurableSet_orderedTreeRegion (h d : ℝ)
(parent : (m : ℕ) → (Fin (m + 1) → Covariate) → Covariate)
(hp : ∀ m, Measurable (parent m)) (m : ℕ) :
MeasurableSet (orderedTreeRegion h d parent m) := by
induction m with
| zero => exact measurableSet_lt (by fun_prop) measurable_const
| succ m ih =>
change MeasurableSet {x : Fin (m + 2) → Covariate |
(fun i : Fin (m + 1) => x i.castSucc) ∈ orderedTreeRegion h d parent m ∧
|(x (last (m + 1)) : ℝ) - parent m (fun i : Fin (m + 1) => x i.castSucc)| < 2 * d}
exact (ih.preimage (by fun_prop)).inter
(measurableSet_lt (show Measurable (fun x : Fin (m + 2) → Covariate =>
|(x (last (m + 1)) : ℝ) - parent m (fun i : Fin (m + 1) => x i.castSucc)|)
by fun_prop) measurable_const)
theorem orderedTreeRegion_volume_le
Under iid uniform design, any parent-before-child rooted tree has probability bounded by its root length times one child length per edge. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
Proof (Lean source)
-- @node: orderedTreeRegion_volume_le
lemma orderedTreeRegion_volume_le (h d : ℝ)
(parent : (m : ℕ) → (Fin (m + 1) → Covariate) → Covariate)
(hp : ∀ m, Measurable (parent m)) (m : ℕ) :
(Measure.pi (fun _ : Fin (m + 1) => (volume : Measure Covariate)))
(orderedTreeRegion h d parent m) ≤
ofReal (2 * h) * (ofReal (4 * d)) ^ m := by
induction m with
| zero =>
have he : orderedTreeRegion h d parent 0 =
pi univ (fun _ : Fin 1 => {x : Covariate | |(x : ℝ) - x0| < h}) := by
ext x
simp [orderedTreeRegion, Set.mem_pi, Fin.forall_fin_one]
rw [he, Measure.pi_pi]
simpa using volume_tree_root_region_le h
| succ m ih =>
calc
_ ≤ ofReal (4 * d) *
(Measure.pi (fun _ : Fin (m + 1) => (volume : Measure Covariate)))
(orderedTreeRegion h d parent m) :=
pi_uniform_child_extension_volume_le (m + 1) d _
(measurableSet_orderedTreeRegion h d parent hp m) (parent m) (hp m)
_ ≤ ofReal (4 * d) *
(ofReal (2 * h) * (ofReal (4 * d)) ^ m) :=
mul_le_mul' le_rfl ih
_ = _ := by rw [pow_succ]; ac_rfl
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.TreeSeries 17 declarations Numerical summation of the labeled-tree envelope for sparse shared-sign components.
Numerical summation of the labeled-tree envelope for sparse shared-sign components.
In the sparse regime, the differentiated geometric envelope is at most three. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
z :
ℝ
hz :
0 ≤ z
hzcap :
z ≤ 1 / 8
N :
ℕ
conclusion
(∑ i ∈ range N, ((i : ℝ) + 2) * z ^ i) ≤ 3
Proof (Lean source)
-- @node: shifted_geometric_sum_le_three
lemma shifted_geometric_sum_le_three (z : ℝ) (hz : 0 ≤ z) (hzcap : z ≤ 1 / 8)
(N : ℕ) : (∑ i ∈ range N, ((i : ℝ) + 2) * z ^ i) ≤ 3 := by
apply (shiftedArithmeticGeometric_sum_le z hz (by linarith) N).trans
apply (div_le_iff₀ (sq_pos_of_pos (by linarith : 0 < 1 - z))).mpr
have hs : (7/8 : ℝ) ^ 2 ≤ (1 - z) ^ 2 := by nlinarith
nlinarith
Each tree-series coefficient is bounded by the differentiated geometric majorant. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
Proof (Lean source)
-- @node: tree_series_term_le
lemma tree_series_term_le (a : ℝ) (ha : 0 ≤ a) (i : ℕ) :
((i + 2 : ℕ) : ℝ) ^ ((i + 2) + 1) / ((i + 2).factorial : ℝ) * a ^ i ≤
(exp 1) ^ 2 * ((i : ℝ) + 2) * ((exp 1) * a) ^ i := by
have h :
(((i + 2 : ℕ) : ℝ) ^ (i + 2)) / (((i + 2).factorial : ℕ) : ℝ) ≤
(exp 1) ^ (i + 2) := by
have hExp := Real.pow_div_factorial_le_exp
(x := ((i + 2 : ℕ) : ℝ)) (by positivity) (i + 2)
simpa only [← Real.exp_nat_mul, mul_one] using hExp
have hc : 0 ≤ ((i + 2 : ℕ) : ℝ) * a ^ i := by positivity
have hm := mul_le_mul_of_nonneg_right h hc
calc
_ = ((i + 2 : ℕ) : ℝ) ^ (i + 2) / ((i + 2).factorial : ℝ) * (((i + 2 : ℕ) : ℝ) * a ^ i) := by rw [pow_succ]; ring
_ ≤ (exp 1) ^ (i + 2) * (((i + 2 : ℕ) : ℝ) * a ^ i) := hm
_ = _ := by rw [pow_add, mul_pow]; push_cast; ring
Summing the tree coefficients under the sparse cap gives the roadmap's constant thirty-two. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: tree_series_sum_le_thirty_two
lemma tree_series_sum_le_thirty_two (a : ℝ) (ha : 0 ≤ a)
(hcap : (exp 1) * a ≤ 1 / 8) (N : ℕ) :
(∑ i ∈ range N,
((i + 2 : ℕ) : ℝ) ^ ((i + 2) + 1) / ((i + 2).factorial : ℝ) * a ^ i) ≤ 32 := by
calc
_ ≤ ∑ i ∈ range N,
(exp 1) ^ 2 * ((i : ℝ) + 2) * ((exp 1) * a) ^ i :=
Finset.sum_le_sum (fun i _ => tree_series_term_le a ha i)
_ = (exp 1) ^ 2 * ∑ i ∈ range N,
((i : ℝ) + 2) * ((exp 1) * a) ^ i := by
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro i _
ring
_ ≤ (exp 1) ^ 2 * 3 := mul_le_mul_of_nonneg_left
(shifted_geometric_sum_le_three _ (mul_nonneg (Real.exp_pos 1).le ha) hcap N)
(sq_nonneg _)
_ ≤ 32 := by
have he := Real.exp_one_lt_d9
have hp := Real.exp_pos 1
nlinarith
The sample sparsity cap places the exponential tree-series argument below one eighth. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
n :
ℕ
d :
ℝ
hd :
0 ≤ d
hcap :
(n : ℝ) * d ≤ 1 / 128
conclusion
(exp 1) * (4 * (n : ℝ) * d) ≤ 1 / 8
Proof (Lean source)
-- @node: sparse_tree_series_argument_le
lemma sparse_tree_series_argument_le (n : ℕ) (d : ℝ) (hd : 0 ≤ d)
(hcap : (n : ℝ) * d ≤ 1 / 128) :
(exp 1) * (4 * (n : ℝ) * d) ≤ 1 / 8 := by
have he := Real.exp_one_lt_three
have hn : 0 ≤ (n : ℝ) * d := mul_nonneg (Nat.cast_nonneg n) hd
have hm := mul_le_mul_of_nonneg_right he.le hn
nlinarith
The finite numerical tree series is uniformly bounded in the sparse sample regime. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
n N :
ℕ
d :
ℝ
hd :
0 ≤ d
hcap :
(n : ℝ) * d ≤ 1 / 128
conclusion
(∑ i ∈ range N, ((i + 2 : ℕ) : ℝ) ^ ((i + 2) + 1) / ((i + 2).factorial : ℝ) * (4 * (n : ℝ) * d) ^ i)
≤ 32
Proof (Lean source)
-- @node: sparse_tree_series_sum_le
lemma sparse_tree_series_sum_le (n N : ℕ) (d : ℝ) (hd : 0 ≤ d)
(hcap : (n : ℝ) * d ≤ 1 / 128) :
(∑ i ∈ range N,
((i + 2 : ℕ) : ℝ) ^ ((i + 2) + 1) / ((i + 2).factorial : ℝ) * (4 * (n : ℝ) * d) ^ i) ≤ 32 := by
exact tree_series_sum_le_thirty_two _ (by positivity)
(sparse_tree_series_argument_le n d hd hcap) N
Each cubic component-count envelope factors into the numerical tree coefficient. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
n i :
ℕ
t h d :
ℝ
ht :
0 ≤ t
hh :
0 ≤ h
hd :
0 ≤ d
conclusion
4 * t * ((i + 2 : ℕ) : ℝ) ^ 3 * (n.choose (i + 2) : ℝ) * (2 * h * ((i + 2 : ℕ) : ℝ) ^ i * (4 * d) ^ (i + 1))
≤ 32 * t * (n : ℝ) ^ 2 * h * d * (((i + 2 : ℕ) : ℝ) ^ ((i + 2) + 1) / ((i + 2).factorial : ℝ) * (4 * (n : ℝ) * d) ^ i)
Proof (Lean source)
-- @node: cubic_tree_envelope_term_le
lemma cubic_tree_envelope_term_le (n i : ℕ) (t h d : ℝ)
(ht : 0 ≤ t) (hh : 0 ≤ h) (hd : 0 ≤ d) :
4 * t * ((i + 2 : ℕ) : ℝ) ^ 3 * (n.choose (i + 2) : ℝ) * (2 * h * ((i + 2 : ℕ) : ℝ) ^ i * (4 * d) ^ (i + 1)) ≤
32 * t * (n : ℝ) ^ 2 * h * d * (((i + 2 : ℕ) : ℝ) ^ ((i + 2) + 1) / ((i + 2).factorial : ℝ) * (4 * (n : ℝ) * d) ^ i) := by
have hb : (n.choose (i + 2) : ℝ) ≤
(n : ℝ) ^ (i + 2) / (((i + 2).factorial : ℕ) : ℝ) :=
Nat.choose_le_pow_div (i + 2) n
have hm := mul_le_mul_of_nonneg_left hb
(show 0 ≤ 4 * t * ((i + 2 : ℕ) : ℝ) ^ 3 * (2 * h * ((i + 2 : ℕ) : ℝ) ^ i * (4 * d) ^ (i + 1)) by positivity)
calc
_ ≤ 4 * t * ((i + 2 : ℕ) : ℝ) ^ 3 * ((n : ℝ) ^ (i + 2) / ((i + 2).factorial : ℝ)) * (2 * h * ((i + 2 : ℕ) : ℝ) ^ i * (4 * d) ^ (i + 1)) := by
simpa only [mul_assoc, mul_left_comm, mul_comm] using hm
_ = _ := by
rw [show (i + 2) + 1 = i + 3 by omega]
simp only [pow_add, mul_pow]
ring
The full labeled-tree cubic envelope is at most the transport constant in the roadmap. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
n N :
ℕ
t h d :
ℝ
ht :
0 ≤ t
hh :
0 ≤ h
hd :
0 ≤ d
hcap :
(n : ℝ) * d ≤ 1 / 128
conclusion
(∑ i ∈ range N, 4 * t * ((i + 2 : ℕ) : ℝ) ^ 3 * (n.choose (i + 2) : ℝ) * (2 * h * ((i + 2 : ℕ) : ℝ) ^ i * (4 * d) ^ (i + 1)))
≤ 1024 * t * (n : ℝ) ^ 2 * h * d
Proof (Lean source)
-- @node: cubic_tree_envelope_sum_le
lemma cubic_tree_envelope_sum_le (n N : ℕ) (t h d : ℝ)
(ht : 0 ≤ t) (hh : 0 ≤ h) (hd : 0 ≤ d)
(hcap : (n : ℝ) * d ≤ 1 / 128) :
(∑ i ∈ range N, 4 * t * ((i + 2 : ℕ) : ℝ) ^ 3 * (n.choose (i + 2) : ℝ) * (2 * h * ((i + 2 : ℕ) : ℝ) ^ i * (4 * d) ^ (i + 1))) ≤
1024 * t * (n : ℝ) ^ 2 * h * d := by
calc
_ ≤ ∑ i ∈ range N, 32 * t * (n : ℝ) ^ 2 * h * d * (((i + 2 : ℕ) : ℝ) ^ ((i + 2) + 1) / ((i + 2).factorial : ℝ) * (4 * (n : ℝ) * d) ^ i) :=
Finset.sum_le_sum (fun i _ => cubic_tree_envelope_term_le n i t h d ht hh hd)
_ = 32 * t * (n : ℝ) ^ 2 * h * d * ∑ i ∈ range N,
(((i + 2 : ℕ) : ℝ) ^ ((i + 2) + 1) / ((i + 2).factorial : ℝ) * (4 * (n : ℝ) * d) ^ i) := by rw [Finset.mul_sum]
_ ≤ 32 * t * (n : ℝ) ^ 2 * h * d * 32 :=
mul_le_mul_of_nonneg_left (sparse_tree_series_sum_le n N d hd hcap) (by positivity)
_ = _ := by ring
The number of shared-sign components with a specified size, counted without repeated roots.
Definition (Lean source)
parameters
def
sizeComponentCount hL n x s :
ℕ
given by
((orderedComponents hL n x).toFinset.filter (fun C => C.card = s)).card
Every ordered component has between one and the sample size many vertices. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
conclusion 1
1 ≤ C.card
conclusion 2
C.card ≤ n
Proof (Lean source)
-- @node: orderedComponents_card_range
lemma orderedComponents_card_range (hL : ℝ) (n : ℕ) (x : Fin n → Covariate)
(C : Finset (Fin n)) (hC : C ∈ orderedComponents hL n x) :
1 ≤ C.card ∧ C.card ≤ n := by
exact ⟨(orderedComponents_nonempty hL n x C hC).card_pos,
(Finset.card_le_univ C).trans_eq (Fintype.card_fin n)⟩
theorem sizeComponentCount_zero
Empty components never contribute to the component-size count. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: sizeComponentCount_zero
lemma sizeComponentCount_zero (hL : ℝ) (n : ℕ) (x : Fin n → Covariate) :
sizeComponentCount hL n x 0 = 0 := by
classical
unfold sizeComponentCount
apply Finset.card_eq_zero.mpr
apply Finset.eq_empty_iff_forall_notMem.mpr
intro C hC
obtain ⟨hC, hc⟩ := Finset.mem_filter.mp hC
have hp := (orderedComponents_card_range hL n x C (List.mem_toFinset.mp hC)).1
omega
theorem orderedComponents_sum_by_size
Grouping a component observable by size is an exact finite counting identity. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
(∑ C ∈ (orderedComponents hL n x).toFinset, f C.card)
= ∑ s ∈ range (n + 1), (sizeComponentCount hL n x s : R) * f s
Proof (Lean source)
-- @node: orderedComponents_sum_by_size
lemma orderedComponents_sum_by_size {R : Type*} [CommSemiring R]
(hL : ℝ) (n : ℕ) (x : Fin n → Covariate) (f : ℕ → R) :
(∑ C ∈ (orderedComponents hL n x).toFinset, f C.card) =
∑ s ∈ range (n + 1), (sizeComponentCount hL n x s : R) * f s := by
classical
have hsingle (C : Finset (Fin n)) (hC : C ∈ (orderedComponents hL n x).toFinset) :
f C.card = ∑ s ∈ range (n + 1), if C.card = s then f s else 0 := by
have hc := (orderedComponents_card_range hL n x C (List.mem_toFinset.mp hC)).2
simp [Finset.mem_range, show C.card < n + 1 by omega]
rw [Finset.sum_congr rfl (fun C hC => hsingle C hC), Finset.sum_comm]
apply Finset.sum_congr rfl
intro s _
rw [← Finset.sum_filter]
have he : (∑ C ∈ (orderedComponents hL n x).toFinset.filter (fun C => C.card = s),
f s) = (sizeComponentCount hL n x s : R) * f s := by
simp [sizeComponentCount, nsmul_eq_mul]
exact he
theorem measurable_sizeComponentCount
Component counts are Borel because they are constant on each finite labeled-graph stratum. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
Proof (Lean source)
-- @node: measurable_sizeComponentCount
@[fun_prop] lemma measurable_sizeComponentCount (hL : ℝ) (n s : ℕ) :
Measurable (fun x => sizeComponentCount hL n x s) := by
classical
have hm : Measurable (fun x : Fin n → Covariate => ∑ G : SimpleGraph (Fin n),
{x | sharedGraph hL n x = G}.indicator (fun _ =>
((componentsOfGraph n G).toFinset.filter (fun C => C.card = s)).card) x) := by
apply Finset.measurable_sum
intro G _
exact measurable_const.indicator (measurableSet_sharedGraph_eq hL n G)
convert hm using 1
funext x
simp only [indicator, Set.mem_ofPred_eq]
rw [Finset.sum_ite_eq, if_pos (Finset.mem_univ _)]
rfl
theorem lintegral_component_cubic_sum_by_size
The expected nonsingleton cubic sum is exactly the sum of the expected component counts. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
(∫⁻ x, ∑ C ∈ (orderedComponents hL n x).toFinset, if C.card = 1 then 0 else ofReal (4 * t * (C.card : ℝ) ^ 3) ∂μ)
Proof (Lean source)
-- @node: lintegral_component_cubic_sum_by_size
lemma lintegral_component_cubic_sum_by_size (hL : ℝ) (n : ℕ) (t : ℝ)
(μ : Measure (Fin n → Covariate)) :
(∫⁻ x, ∑ C ∈ (orderedComponents hL n x).toFinset,
if C.card = 1 then 0 else ofReal (4 * t * (C.card : ℝ) ^ 3) ∂μ) =
∑ s ∈ range (n + 1), (if s = 1 then 0 else ofReal (4 * t * (s : ℝ) ^ 3)) * ∫⁻ x, (sizeComponentCount hL n x s : ℝ≥0∞) ∂μ := by
classical
simp_rw [orderedComponents_sum_by_size hL n _
(fun s => if s = 1 then 0 else ofReal (4 * t * (s : ℝ) ^ 3))]
rw [lintegral_finsetSum _ (fun s _ =>
by fun_prop)]
apply Finset.sum_congr rfl
intro s _
simp_rw [mul_comm (sizeComponentCount hL n _ s : ℝ≥0∞)]
exact lintegral_const_mul _ (by fun_prop)
Removing sizes zero and one reindexes any vanishing observable by nonsingleton sizes. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
Proof (Lean source)
-- @node: sum_sizes_eq_shifted
lemma sum_sizes_eq_shifted {R : Type*} [AddCommMonoid R] (n : ℕ) (hn : 1 ≤ n)
(f : ℕ → R) (h0 : f 0 = 0) (h1 : f 1 = 0) :
(∑ s ∈ range (n + 1), f s) = ∑ i ∈ range (n - 1), f (i + 2) := by
rw [show n + 1 = 2 + (n - 1) by omega, Finset.sum_range_add]
simp only [Finset.sum_range_succ, Finset.sum_range_zero, h0, h1, add_zero, zero_add]
apply Finset.sum_congr rfl
intro i _
rw [Nat.add_comm]
theorem lintegral_component_cubic_sum_shifted
The expected nonsingleton cubic component cost is indexed by sizes two through n. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
(∫⁻ x, ∑ C ∈ (orderedComponents hL n x).toFinset, if C.card = 1 then 0 else ofReal (4 * t * (C.card : ℝ) ^ 3) ∂μ)
= ∑ i ∈ range (n - 1), ofReal (4 * t * ((i + 2 : ℕ) : ℝ) ^ 3) * ∫⁻ x, (sizeComponentCount hL n x (i + 2) : ℝ≥0∞) ∂μ
Proof (Lean source)
-- @node: lintegral_component_cubic_sum_shifted
lemma lintegral_component_cubic_sum_shifted (hL : ℝ) (n : ℕ) (hn : 1 ≤ n) (t : ℝ)
(μ : Measure (Fin n → Covariate)) :
(∫⁻ x, ∑ C ∈ (orderedComponents hL n x).toFinset,
if C.card = 1 then 0 else ofReal (4 * t * (C.card : ℝ) ^ 3) ∂μ) =
∑ i ∈ range (n - 1), ofReal (4 * t * ((i + 2 : ℕ) : ℝ) ^ 3) * ∫⁻ x, (sizeComponentCount hL n x (i + 2) : ℝ≥0∞) ∂μ := by
rw [lintegral_component_cubic_sum_by_size]
rw [sum_sizes_eq_shifted n hn _ (by simp) (by simp)]
apply Finset.sum_congr rfl
intro i _
rw [if_neg (by omega : i + 2 ≠ 1)]
theorem lintegral_component_cubic_sum_le_tree_envelope
Component-count expectations bounded by the labeled-tree envelope give the claimed cubic cost. This implication leaves the geometric spanning - tree probability estimate as an explicit input, without inserting it into a frozen paper theorem. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
hL :
ℝ
n :
ℕ
hn :
1 ≤ n
t h d :
ℝ
ht :
0 ≤ t
hh :
0 ≤ h
hd :
0 ≤ d
hcap :
(n : ℝ) * d ≤ 1 / 128
hcounts :
∀ i ∈ range (n - 1),
(∫⁻ x, (sizeComponentCount hL n x (i + 2) : ℝ≥0∞) ∂μ)
≤ ofReal ((n.choose (i + 2) : ℝ) * (2 * h * ((i + 2 : ℕ) : ℝ) ^ i * (4 * d) ^ (i + 1)))
conclusion
(∫⁻ x, ∑ C ∈ (orderedComponents hL n x).toFinset, if C.card = 1 then 0 else ofReal (4 * t * (C.card : ℝ) ^ 3) ∂μ)
≤ ofReal (1024 * t * (n : ℝ) ^ 2 * h * d)
Proof (Lean source)
-- @node: lintegral_component_cubic_sum_le_tree_envelope
lemma lintegral_component_cubic_sum_le_tree_envelope (hL : ℝ) (n : ℕ) (hn : 1 ≤ n)
(t h d : ℝ) (ht : 0 ≤ t) (hh : 0 ≤ h) (hd : 0 ≤ d)
(hcap : (n : ℝ) * d ≤ 1 / 128) (μ : Measure (Fin n → Covariate))
(hcounts : ∀ i ∈ range (n - 1),
(∫⁻ x, (sizeComponentCount hL n x (i + 2) : ℝ≥0∞) ∂μ) ≤
ofReal ((n.choose (i + 2) : ℝ) * (2 * h * ((i + 2 : ℕ) : ℝ) ^ i * (4 * d) ^ (i + 1)))) :
(∫⁻ x, ∑ C ∈ (orderedComponents hL n x).toFinset,
if C.card = 1 then 0 else ofReal (4 * t * (C.card : ℝ) ^ 3) ∂μ) ≤
ofReal (1024 * t * (n : ℝ) ^ 2 * h * d) := by
rw [lintegral_component_cubic_sum_shifted hL n hn]
calc
_ ≤ ∑ i ∈ range (n - 1), ofReal (4 * t * ((i + 2 : ℕ) : ℝ) ^ 3) * ofReal ((n.choose (i + 2) : ℝ) * (2 * h * ((i + 2 : ℕ) : ℝ) ^ i * (4 * d) ^ (i + 1))) :=
Finset.sum_le_sum (fun i hi => mul_le_mul le_rfl (hcounts i hi) zero_le zero_le)
_ = ofReal (∑ i ∈ range (n - 1),
4 * t * ((i + 2 : ℕ) : ℝ) ^ 3 * (n.choose (i + 2) : ℝ) * (2 * h * ((i + 2 : ℕ) : ℝ) ^ i * (4 * d) ^ (i + 1))) := by
rw [ENNReal.ofReal_sum_of_nonneg (fun i _ => by positivity)]
apply Finset.sum_congr rfl
intro i _
rw [← ENNReal.ofReal_mul (by positivity)]
congr 1
ring
_ ≤ _ := ENNReal.ofReal_le_ofReal
(cubic_tree_envelope_sum_le n (n - 1) t h d ht hh hd hcap)
theorem commonMassCoupling_hamming_cost_le_of_tree_counts
For the actual dataset coupling, the remaining tree-count probability estimate implies its full sparse Hamming cost certificate. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
conclusion
(∫⁻ zw, (dHam n zw.1 zw.2 : ℝ≥0∞) ∂commonMassCoupling hL n)
Proof (Lean source)
-- @node: commonMassCoupling_hamming_cost_le_of_tree_counts
lemma commonMassCoupling_hamming_cost_le_of_tree_counts (hL : ℝ)
(hhL : 0 < hL ∧ hL ≤ 1 / 4) (n : ℕ) (hn : 1 ≤ n)
(hcap : (n : ℝ) * deltaL hL ≤ 1 / 128)
(hcounts : ∀ i ∈ range (n - 1),
(∫⁻ x, (sizeComponentCount hL n x (i + 2) : ℝ≥0∞)
∂(Measure.pi (fun _ : Fin n => (volume : Measure Covariate)))) ≤
ofReal ((n.choose (i + 2) : ℝ) * (2 * hL * ((i + 2 : ℕ) : ℝ) ^ i * (4 * deltaL hL) ^ (i + 1)))) :
(∫⁻ zw, (dHam n zw.1 zw.2 : ℝ≥0∞) ∂commonMassCoupling hL n) ≤
ofReal (1024 * separation hL * (n : ℝ) ^ 2 * hL * deltaL hL) := by
have hp : 0 < hL := hhL.1
apply (commonMassCoupling_hamming_cost_le_cubic_sum hL hhL n).trans
apply lintegral_component_cubic_sum_le_tree_envelope hL n hn
(separation hL) hL (deltaL hL) _ hhL.1.le (by unfold deltaL; positivity) hcap _ hcounts
unfold separation kappa
positivity
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.TunedLength 4 declarations Uniform expected interval length under the public tuning, including its fallback branch.
Uniform expected interval length under the public tuning, including its fallback branch.
theorem tuned_interval_radius_le
The tuned variance envelope bounds twice the conservative interval radius. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: tuned_interval_radius_le
lemma tuned_interval_radius_le (n : ℕ) (epsilon : ℝ) (hn : 2 ≤ n)
(he : 0 < epsilon ∧ epsilon ≤ 1) (hr : rate n epsilon < 1 / 8) :
2 * sqrt (10 * Vbound n epsilon (tunedH n epsilon) (tunedK n epsilon)) ≤
2^22 * rate n epsilon := by
have hv := tuned_Vbound_le n epsilon hn he hr
have hr0 := (rate_pos n epsilon (by omega)).le
have hs : sqrt (10 * Vbound n epsilon (tunedH n epsilon) (tunedK n epsilon)) ≤
2^21 * rate n epsilon := by
apply (Real.sqrt_le_iff).mpr
constructor
· positivity
· nlinarith [sq_nonneg (rate n epsilon)]
nlinarith
theorem full_interval_expectedLength
The public full-range fallback has length two under every model law. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
n :
ℕ
P :
conclusion
Proof (Lean source)
-- @node: full_interval_expectedLength
lemma full_interval_expectedLength (n : ℕ) (P : CausalLaw) :
expectedLength n (Kernel.const (Dataset n) (Measure.dirac (closedInterval (-1) 1))) P = 2 := by
let : IsProbabilityMeasure (Pobs P) := by
unfold Pobs
exact Measure.isProbabilityMeasure_map measurable_observe.aemeasurable
let : IsProbabilityMeasure (dataLaw n P) := by unfold dataLaw; infer_instance
unfold expectedLength
change (∫⁻ c, c.leb ∂(dataLaw n P).bind
(fun _ => Measure.dirac (closedInterval (-1) 1))) = 2
rw [Measure.bind_const, measure_univ, one_smul, lintegral_dirac]
rw [closedInterval_leb]
norm_num
theorem publicTunedInterval_expectedLength_le
The tuned interval obeys the stated expected-length rate at every law, from the deterministic radius bound on the active branch and the full-range length on the fallback branch. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
expectedLength n (publicTunedInterval n epsilon) P
Proof (Lean source)
-- @node: publicTunedInterval_expectedLength_le
lemma publicTunedInterval_expectedLength_le (n : ℕ) (epsilon : ℝ) (hn : 2 ≤ n)
(he : 0 < epsilon ∧ epsilon ≤ 1) (P : CausalLaw) :
expectedLength n (publicTunedInterval n epsilon) P ≤
ofReal (2^22 * rate n epsilon) := by
classical
unfold publicTunedInterval
split_ifs with hr
· rw [full_interval_expectedLength]
have hb : (2 : ℝ) ≤ 2^22 * rate n epsilon := by nlinarith
calc
(2 : ℝ≥0∞) = ofReal (2 : ℝ) := by norm_num
_ ≤ _ := ENNReal.ofReal_le_ofReal hb
· exact (Ihk_expectedLength_le n _ epsilon _ he.1 P).trans
(ENNReal.ofReal_le_ofReal ((min_le_right _ _).trans
(tuned_interval_radius_le n epsilon hn he (lt_of_not_ge hr))))
theorem publicTunedInterval_worstLength_le
Taking the supremum over the full model preserves the public interval-length bound. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
n :
ℕ
epsilon :
ℝ
hn :
2 ≤ n
he :
0 < epsilon ∧ epsilon ≤ 1
conclusion
Proof (Lean source)
-- @node: publicTunedInterval_worstLength_le
lemma publicTunedInterval_worstLength_le (n : ℕ) (epsilon : ℝ) (hn : 2 ≤ n)
(he : 0 < epsilon ∧ epsilon ≤ 1) :
worstLength n (publicTunedInterval n epsilon) ≤ ofReal (2^22 * rate n epsilon) := by
unfold worstLength
exact iSup_le fun P => iSup_le fun _ => publicTunedInterval_expectedLength_le n epsilon hn he P
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.TunedRisk 4 declarations Absolute risk from the squared-error envelope under public tuning.
Absolute risk from the squared-error envelope under public tuning.
theorem absolute_moment_le_of_squared_moment Lemma absolute_moment_le_of_squared_moment in the paper ↗
Young's quadratic inequality converts a second moment to an absolute first moment. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: absolute_moment_le_of_squared_moment
lemma absolute_moment_le_of_squared_moment (Q : Measure ℝ) [IsProbabilityMeasure Q]
(t B : ℝ) (hB : 0 < B)
(hmse : (∫⁻ u, ofReal ((u-t)^2) ∂Q) ≤ ofReal (B^2)) :
(∫⁻ u, ofReal |u-t| ∂Q) ≤ ofReal B := by
have hy (u : ℝ) : |u-t| ≤ (2*B)⁻¹ * (u-t)^2 + B/2 := by
rw [inv_mul_eq_div]
have hs := sq_nonneg (|u-t|-B)
have heq : (2*B) * ((u-t)^2 / (2*B) + B/2) = (u-t)^2 + B^2 := by
field_simp
apply (mul_le_mul_iff_right₀ (show 0 < 2*B by positivity)).mp
rw [heq]
nlinarith [sq_abs (u-t)]
calc
(∫⁻ u, ofReal |u-t| ∂Q) ≤
∫⁻ u, ofReal ((2*B)⁻¹) * ofReal ((u-t)^2) +
ofReal (B/2) ∂Q := by
apply lintegral_mono
intro u
dsimp only
rw [← ENNReal.ofReal_mul (by positivity), ← ENNReal.ofReal_add (by positivity) (by positivity)]
exact ENNReal.ofReal_le_ofReal (hy u)
_ = ofReal ((2*B)⁻¹) * (∫⁻ u, ofReal ((u-t)^2) ∂Q) +
ofReal (B/2) := by
rw [lintegral_add_right _ measurable_const, lintegral_const_mul]
· simp
· fun_prop
_ ≤ ofReal ((2*B)⁻¹) * ofReal (B^2) + ofReal (B/2) :=
add_le_add (mul_le_mul le_rfl hmse zero_le zero_le) le_rfl
_ = ofReal B := by
rw [← ENNReal.ofReal_mul (by positivity), ← ENNReal.ofReal_add (by positivity) (by positivity)]
congr 1
field_simp
ring
theorem zero_release_scalarRisk_le
The data-independent zero release has absolute risk at most one throughout the model. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
scalarRisk n (Kernel.const (Dataset n) (Measure.dirac 0)) P ≤ 1
Proof (Lean source)
-- @node: zero_release_scalarRisk_le
lemma zero_release_scalarRisk_le (n : ℕ) (P : CausalLaw) (hP : CompleteModel P) :
scalarRisk n (Kernel.const (Dataset n) (Measure.dirac 0)) P ≤ 1 := by
letI : IsProbabilityMeasure (Pobs P) := by
unfold Pobs
exact Measure.isProbabilityMeasure_map measurable_observe.aemeasurable
letI : IsProbabilityMeasure (dataLaw n P) := by unfold dataLaw; infer_instance
unfold scalarRisk
change (∫⁻ u, ofReal |u-theta P| ∂(dataLaw n P).bind
(fun _ => Measure.dirac 0)) ≤ 1
rw [Measure.bind_const, measure_univ, one_smul, lintegral_dirac]
have ht := theta_mem_Icc P hP
have ha : |0-theta P| ≤ 1 := by rw [zero_sub, abs_neg]; exact abs_le.mpr ht
simpa using ENNReal.ofReal_le_ofReal ha
theorem tuned_scalarRisk_le_of_squared_error_le
The active tuned scalar release attains the absolute-risk rate from its second moment. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
Proof (Lean source)
-- @node: tuned_scalarRisk_le_of_squared_error_le
lemma tuned_scalarRisk_le_of_squared_error_le (n : ℕ) (epsilon : ℝ) (hn : 2 ≤ n)
(he : 0 < epsilon ∧ epsilon ≤ 1) (hr : rate n epsilon < 1/8)
(P : CausalLaw)
(hmse : (∫⁻ u, ofReal ((u-theta P)^2)
∂(Thk n epsilon (tunedH n epsilon) (tunedK n epsilon) ∘ₘ dataLaw n P)) ≤
ofReal (Vbound n epsilon (tunedH n epsilon) (tunedK n epsilon))) :
scalarRisk n (Thk n epsilon (tunedH n epsilon) (tunedK n epsilon)) P ≤
ofReal (2^18 * rate n epsilon) := by
letI : IsProbabilityMeasure (Pobs P) := by
unfold Pobs
exact Measure.isProbabilityMeasure_map measurable_observe.aemeasurable
letI : IsProbabilityMeasure (dataLaw n P) := by unfold dataLaw; infer_instance
letI := (Thk_private n (tunedK n epsilon) epsilon (tunedH n epsilon) he.1
(public_tuning_parameters n epsilon hn he hr).1
(by have hp := public_tuning_parameters n epsilon hn he hr; omega)).markov
have hv := tuned_Vbound_le n epsilon hn he hr
have hb : Vbound n epsilon (tunedH n epsilon) (tunedK n epsilon) ≤
(2^18 * rate n epsilon)^2 := by nlinarith [sq_nonneg (rate n epsilon)]
exact absolute_moment_le_of_squared_moment _ _ _
(by have hp := rate_pos n epsilon (by omega); positivity)
(hmse.trans (ENNReal.ofReal_le_ofReal hb))
theorem publicTunedRelease_worstRisk_of_squared_error_le
The fallback and active tuning branches give the uniform scalar-risk rate once the uniform statistical second moment has been proved. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
n :
ℕ
epsilon :
ℝ
hn :
2 ≤ n
he :
0 < epsilon ∧ epsilon ≤ 1
conclusion
Proof (Lean source)
-- @node: publicTunedRelease_worstRisk_of_squared_error_le
lemma publicTunedRelease_worstRisk_of_squared_error_le (n : ℕ) (epsilon : ℝ)
(hn : 2 ≤ n) (he : 0 < epsilon ∧ epsilon ≤ 1)
(hmse : ∀ (k : ℕ) (h : ℝ), 2 ≤ k → (0 < h ∧ h ≤ 1/4) →
∀ P : CausalLaw, CompleteModel P →
(∫⁻ u, ofReal ((u-theta P)^2) ∂(Thk n epsilon h k ∘ₘ dataLaw n P)) ≤
ofReal (Vbound n epsilon h k)) :
worstRisk n (publicTunedRelease n epsilon) ≤ ofReal (2^18 * rate n epsilon) := by
classical
unfold worstRisk
refine iSup_le fun P => iSup_le fun hP => ?_
unfold publicTunedRelease
split_ifs with hr
· have hb : (1 : ℝ) ≤ 2^18 * rate n epsilon := by nlinarith
exact (zero_release_scalarRisk_le n P hP).trans (by
simpa using ENNReal.ofReal_le_ofReal hb)
· have hr' := lt_of_not_ge hr
have hp := public_tuning_parameters n epsilon hn he hr'
exact tuned_scalarRisk_le_of_squared_error_le n epsilon hn he hr' P
(hmse _ _ hp.2.2.1 ⟨hp.1, hp.2.1⟩ P hP)
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.TuningEnvelope 7 declarations The benchmark constraints and the dyadic public tuning variance envelope.
The benchmark constraints and the dyadic public tuning variance envelope.
A reciprocal-root lower bound is equivalent to the corresponding polynomial constraint. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
x r :
ℝ
m :
ℕ
hx :
0 < x
hm :
0 < m
_hr :
0 ≤ r
h :
x ^ (-1 / (m : ℝ)) ≤ r
conclusion
1 ≤ x * r^m
Proof (Lean source)
-- @node: reciprocal_root_constraint
lemma reciprocal_root_constraint (x r : ℝ) (m : ℕ) (hx : 0 < x) (hm : 0 < m)
(_hr : 0 ≤ r) (h : x ^ (-1 / (m : ℝ)) ≤ r) : 1 ≤ x * r^m := by
have hp := pow_le_pow_left₀ (Real.rpow_nonneg hx.le _) h m
have heq : (x ^ (-1 / (m : ℝ))) ^ m = x⁻¹ := by
rw [← Real.rpow_mul_natCast hx.le]
have hm' : (m : ℝ) ≠ 0 := by exact_mod_cast hm.ne'
rw [div_mul_cancel₀ _ hm', Real.rpow_neg_one]
rw [heq] at hp
have := mul_le_mul_of_nonneg_left hp hx.le
simpa [hx.ne'] using this
theorem active_rate_constraints
Below the public fallback threshold the cap is inactive, so each candidate scale bounds its polynomial information constraint. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: active_rate_constraints
lemma active_rate_constraints (n : ℕ) (epsilon : ℝ) (hn : 2 ≤ n)
(he : 0 < epsilon) (hr : rate n epsilon < 1 / 8) :
1 ≤ (n : ℝ) * (rate n epsilon)^4 ∧
1 ≤ (n : ℝ)^2 * epsilon * (rate n epsilon)^7 ∧
1 ≤ (n : ℝ) * epsilon * (rate n epsilon)^2 := by
have hn0 : (0 : ℝ) < n := Nat.cast_pos.mpr (by omega)
have hr0 := (rate_pos n epsilon (by omega)).le
have hcap : rate n epsilon = max ((n : ℝ)^(-1/4 : ℝ))
(max (((n : ℝ)^2*epsilon)^(-1/7 : ℝ)) (((n : ℝ)*epsilon)^(-1/2 : ℝ))) := by
unfold rate at hr ⊢
exact min_eq_right (by rcases min_lt_iff.mp hr with h | h <;> linarith)
have ha : (n : ℝ)^(-1/4 : ℝ) ≤ rate n epsilon := by
rw [hcap]; exact le_max_left _ _
have hb : ((n : ℝ)^2*epsilon)^(-1/7 : ℝ) ≤ rate n epsilon := by
rw [hcap]; exact (le_max_left _ _).trans (le_max_right _ _)
have hc : ((n : ℝ)*epsilon)^(-1/2 : ℝ) ≤ rate n epsilon := by
rw [hcap]; exact (le_max_right _ _).trans (le_max_right _ _)
exact ⟨reciprocal_root_constraint _ _ 4 hn0 (by norm_num) hr0 ha,
reciprocal_root_constraint _ _ 7 (by positivity) (by norm_num) hr0 hb,
reciprocal_root_constraint _ _ 2 (by positivity) (by norm_num) hr0 hc⟩
theorem tuned_info_lower
Dyadic rounding loses at most a factor of sixty-four in the occupancy scale. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: tuned_info_lower
lemma tuned_info_lower (n : ℕ) (epsilon : ℝ) (hn : 2 ≤ n)
(he : 0 < epsilon ∧ epsilon ≤ 1) (hr : rate n epsilon < 1 / 8) :
min ((n : ℝ) * rate n epsilon) ((n : ℝ)^2 * (rate n epsilon)^6) / 64 ≤
info n (tunedH n epsilon) (tunedK n epsilon) := by
have hp := public_tuning_parameters n epsilon hn he hr
have hb := tunedH_bounds n epsilon (by omega)
have hr0 := (rate_pos n epsilon (by omega)).le
have hn0 : (0 : ℝ) ≤ n := Nat.cast_nonneg n
have h6 := pow_le_pow_left₀ (by positivity : 0 ≤ rate n epsilon / 2) hb.2.1 6
have h1 : rate n epsilon / 64 ≤ tunedH n epsilon := by linarith
have h6' : (rate n epsilon)^6 / 64 ≤ (tunedH n epsilon)^6 := by
nlinarith [h6]
have hi : info n (tunedH n epsilon) (tunedK n epsilon) =
min ((n : ℝ) * tunedH n epsilon) ((n : ℝ)^2 * (tunedH n epsilon)^6) := by
unfold info
rw [hp.2.2.2.2, mul_min_of_nonneg]
· congr 1 <;> ring
· exact mul_nonneg hn0 hb.1.le
rw [hi]
apply le_min
· calc
_ ≤ ((n : ℝ) * rate n epsilon) / 64 :=
div_le_div_of_nonneg_right (min_le_left _ _) (by norm_num)
_ ≤ _ := by nlinarith [mul_le_mul_of_nonneg_left h1 hn0]
· calc
_ ≤ ((n : ℝ)^2 * (rate n epsilon)^6) / 64 :=
div_le_div_of_nonneg_right (min_le_right _ _) (by norm_num)
_ ≤ _ := by nlinarith [mul_le_mul_of_nonneg_left h6' (sq_nonneg (n : ℝ))]
theorem tuned_scaled_info_lower
The three candidate rate constraints imply the two scaled information lower bounds used by the public variance envelope. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: tuned_scaled_info_lower
lemma tuned_scaled_info_lower (n : ℕ) (epsilon : ℝ) (hn : 2 ≤ n)
(he : 0 < epsilon ∧ epsilon ≤ 1) (hr : rate n epsilon < 1 / 8) :
1 / 64 ≤ (rate n epsilon)^2 * info n (tunedH n epsilon) (tunedK n epsilon) ∧
1 / 64 ≤ rate n epsilon * epsilon * info n (tunedH n epsilon) (tunedK n epsilon) := by
have hc := active_rate_constraints n epsilon hn he.1 hr
have hi := tuned_info_lower n epsilon hn he hr
have hr0 := (rate_pos n epsilon (by omega)).le
have hn0 : (0 : ℝ) ≤ n := Nat.cast_nonneg n
have h3 : 1 ≤ (n : ℝ) * (rate n epsilon)^3 := by
have hpow : (rate n epsilon)^4 ≤ (rate n epsilon)^3 := by
have := mul_le_mul_of_nonneg_left (show rate n epsilon ≤ 1 by linarith) (pow_nonneg hr0 3)
nlinarith
exact hc.1.trans (mul_le_mul_of_nonneg_left hpow hn0)
have h8 : 1 ≤ (n : ℝ)^2 * (rate n epsilon)^8 := by
nlinarith [sq_nonneg ((n : ℝ) * (rate n epsilon)^4 - 1)]
constructor
· have hm : 1 ≤ (rate n epsilon)^2 *
min ((n : ℝ) * rate n epsilon) ((n : ℝ)^2 * (rate n epsilon)^6) := by
rw [mul_min_of_nonneg _ _ (sq_nonneg _)]
apply le_min <;> nlinarith
nlinarith [mul_le_mul_of_nonneg_left hi (sq_nonneg (rate n epsilon))]
· have hm : 1 ≤ (rate n epsilon) * epsilon *
min ((n : ℝ) * rate n epsilon) ((n : ℝ)^2 * (rate n epsilon)^6) := by
rw [mul_min_of_nonneg _ _ (mul_nonneg hr0 he.1.le)]
apply le_min <;> nlinarith [hc.2.1, hc.2.2]
nlinarith [mul_le_mul_of_nonneg_left hi (mul_nonneg hr0 he.1.le)]
theorem tuned_inverse_info_upper
The scaled information bounds control the inverse occupancy and inverse privacy scales. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
Proof (Lean source)
-- @node: tuned_inverse_info_upper
lemma tuned_inverse_info_upper (n : ℕ) (epsilon : ℝ) (hn : 2 ≤ n)
(he : 0 < epsilon ∧ epsilon ≤ 1) (hr : rate n epsilon < 1 / 8) :
(info n (tunedH n epsilon) (tunedK n epsilon))⁻¹ ≤ 64 * (rate n epsilon)^2 ∧
(epsilon * info n (tunedH n epsilon) (tunedK n epsilon))^(-2 : ℤ) ≤
4096 * (rate n epsilon)^2 := by
have hp := public_tuning_parameters n epsilon hn he hr
have hn0 : (0 : ℝ) < n := Nat.cast_pos.mpr (by omega)
have hh0 := hp.1
have ha : 0 < info n (tunedH n epsilon) (tunedK n epsilon) := by
unfold info
rw [hp.2.2.2.2]
positivity
have hs := tuned_scaled_info_lower n epsilon hn he hr
have hi : (info n (tunedH n epsilon) (tunedK n epsilon))⁻¹ ≤
64 * (rate n epsilon)^2 := by
rw [inv_eq_one_div]
apply (div_le_iff₀ ha).mpr
nlinarith [hs.1]
have hea : 0 < epsilon * info n (tunedH n epsilon) (tunedK n epsilon) :=
mul_pos he.1 ha
have hie : (epsilon * info n (tunedH n epsilon) (tunedK n epsilon))⁻¹ ≤
64 * rate n epsilon := by
rw [inv_eq_one_div]
apply (div_le_iff₀ hea).mpr
nlinarith [hs.2]
refine ⟨hi, ?_⟩
have hsq := pow_le_pow_left₀ (inv_nonneg.mpr hea.le) hie 2
simpa only [zpow_neg, zpow_ofNat, inv_pow, mul_pow,
show (64 : ℝ)^2 = 4096 by norm_num] using hsq
theorem tuned_cell_roughness
At the tuned cell width, the roughness term equals the squared macro radius. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: tuned_cell_roughness
lemma tuned_cell_roughness (n : ℕ) (epsilon : ℝ) (hn : 2 ≤ n)
(he : 0 < epsilon ∧ epsilon ≤ 1) (hr : rate n epsilon < 1 / 8) :
(cellWidth (tunedH n epsilon) (tunedK n epsilon))^(2/5 : ℝ) =
(tunedH n epsilon)^2 := by
have hp := public_tuning_parameters n epsilon hn he hr
rw [hp.2.2.2.2, ← Real.rpow_natCast, ← Real.rpow_mul hp.1.le]
norm_num
theorem tuned_Vbound_le
The public tuning puts the entire second-moment envelope below the roadmap's squared-rate constant, independently of the law or the statistical second-moment proof. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: tuned_Vbound_le
lemma tuned_Vbound_le (n : ℕ) (epsilon : ℝ) (hn : 2 ≤ n)
(he : 0 < epsilon ∧ epsilon ≤ 1) (hr : rate n epsilon < 1 / 8) :
Vbound n epsilon (tunedH n epsilon) (tunedK n epsilon) ≤
2^33 * (rate n epsilon)^2 := by
have hi := tuned_inverse_info_upper n epsilon hn he hr
have hb := tunedH_bounds n epsilon (by omega)
have hsq := pow_le_pow_left₀ hb.1.le hb.2.2 2
unfold Vbound
rw [tuned_cell_roughness n epsilon hn he hr]
nlinarith [hi.1, hi.2, sq_nonneg (rate n epsilon)]
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.Helpers.UpperGuarantees 35 declarations Uniform private scalar-risk, coverage and length guarantees for the total single-release construction and its public tuning, over the full unknown measurable-density model.
Uniform private scalar-risk, coverage and length guarantees for the total single-release construction and its public tuning, over the full unknown measurable-density model.
theorem privateRatioRelease_markov
The positive public budget normalizes the everywhere-defined noisy pair release. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
n k :
ℕ
epsilon h :
ℝ
he :
0 < epsilon
conclusion
IsMarkovKernel (privateRatioRelease n epsilon h k)
Proof (Lean source)
-- @node: privateRatioRelease_markov
lemma privateRatioRelease_markov (n k : ℕ) (epsilon h : ℝ) (he : 0 < epsilon) :
IsMarkovKernel (privateRatioRelease n epsilon h k) := by
constructor
intro D
exact laplaceMechPi_isProbabilityMeasure
(12 / epsilon) (by positivity) (pairQuery n h k) D
theorem privateRatioRelease_private
The all-count sensitivity bound gives pure privacy for the single joint release. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
n k :
ℕ
epsilon h :
ℝ
he :
0 < epsilon
hh :
0 < h
hk :
0 < k
conclusion
PrivateKernel n epsilon (privateRatioRelease n epsilon h k)
Proof (Lean source)
-- @node: privateRatioRelease_private
lemma privateRatioRelease_private (n k : ℕ) (epsilon h : ℝ)
(he : 0 < epsilon) (hh : 0 < h) (hk : 0 < k) :
PrivateKernel n epsilon (privateRatioRelease n epsilon h k) := by
refine ⟨privateRatioRelease_markov n k epsilon h he, ?_⟩
apply laplaceMechPi_pure_dp
(fun D D' => dHam n D D' = 1) (pairQuery n h k) (by norm_num) he
intro D D' hadj
simpa [pairQuery, Fin.sum_univ_two] using
statistic_hamming_sensitivity n k h hh hk D D' hadj
theorem privateKernel_mapOfMeasurable
Measurable postprocessing preserves normalization and replacement privacy for any output carrier. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
PrivateKernel n epsilon (M.mapOfMeasurable f hf)
Proof (Lean source)
-- @node: privateKernel_mapOfMeasurable
lemma privateKernel_mapOfMeasurable {n : ℕ} {epsilon : ℝ}
{B C : Type*} [MeasurableSpace B] [MeasurableSpace C]
(M : Kernel (Dataset n) B) (hM : PrivateKernel n epsilon M)
(f : B → C) (hf : Measurable f) :
PrivateKernel n epsilon (M.mapOfMeasurable f hf) := by
let := hM.markov
refine ⟨?_, ?_⟩
· rw [Kernel.mapOfMeasurable_eq_map]
exact Kernel.IsMarkovKernel.map M hf
· intro D D' hadj E hE
change ((M D).map f E).toReal ≤ exp epsilon * ((M D').map f E).toReal
rw [Measure.map_apply hf hE, Measure.map_apply hf hE]
exact hM.pure_privacy D D' hadj (f ⁻¹' E) (hE.preimage hf)
theorem Thk_private
The clipped, floored ratio inherits pure privacy on all datasets. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
n k :
ℕ
epsilon h :
ℝ
he :
0 < epsilon
hh :
0 < h
hk :
0 < k
conclusion
PrivateKernel n epsilon (Thk n epsilon h k)
Proof (Lean source)
-- @node: Thk_private
lemma Thk_private (n k : ℕ) (epsilon h : ℝ)
(he : 0 < epsilon) (hh : 0 < h) (hk : 0 < k) :
PrivateKernel n epsilon (Thk n epsilon h k) := by
exact privateKernel_mapOfMeasurable _
(privateRatioRelease_private n k epsilon h he hh hk) _ (measurable_ratioMap n h k)
theorem Ihk_private
The conservative interval inherits pure privacy from the same joint release. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
n k :
ℕ
epsilon h :
ℝ
he :
0 < epsilon
hh :
0 < h
hk :
0 < k
conclusion
PrivateKernel n epsilon (Ihk n epsilon h k)
Proof (Lean source)
-- @node: Ihk_private
lemma Ihk_private (n k : ℕ) (epsilon h : ℝ)
(he : 0 < epsilon) (hh : 0 < h) (hk : 0 < k) :
PrivateKernel n epsilon (Ihk n epsilon h k) := by
exact privateKernel_mapOfMeasurable _
(privateRatioRelease_private n k epsilon h he hh hk) _
(measurable_intervalMap n epsilon h k)
theorem privateKernel_const
A data-independent probability release is private at every nonnegative budget. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
PrivateKernel n epsilon (Kernel.const (Dataset n) mu)
Proof (Lean source)
-- @node: privateKernel_const
lemma privateKernel_const {n : ℕ} {B : Type*} [MeasurableSpace B]
(epsilon : ℝ) (he : 0 ≤ epsilon) (mu : Measure B) [IsProbabilityMeasure mu] :
PrivateKernel n epsilon (Kernel.const (Dataset n) mu) := by
refine ⟨inferInstance, ?_⟩
intro D D' hadj E hE
change mu.real E ≤ exp epsilon * mu.real E
exact le_mul_of_one_le_left (measureReal_nonneg) (Real.one_le_exp_iff.mpr he)
theorem publicTunedRelease_private
Public tuning preserves privacy, including its data-independent fallback. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
n :
ℕ
epsilon :
ℝ
hn :
2 ≤ n
he :
0 < epsilon ∧ epsilon ≤ 1
conclusion
PrivateKernel n epsilon (publicTunedRelease n epsilon)
Proof (Lean source)
-- @node: publicTunedRelease_private
lemma publicTunedRelease_private (n : ℕ) (epsilon : ℝ)
(hn : 2 ≤ n) (he : 0 < epsilon ∧ epsilon ≤ 1) :
PrivateKernel n epsilon (publicTunedRelease n epsilon) := by
classical
unfold publicTunedRelease
split_ifs with hr
· exact privateKernel_const epsilon he.1.le _
· have hp := public_tuning_parameters n epsilon hn he (lt_of_not_ge hr)
exact Thk_private n _ epsilon _ he.1 hp.1 (by omega)
theorem publicTunedInterval_private
The tuned interval remains private in both public branches. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
n :
ℕ
epsilon :
ℝ
hn :
2 ≤ n
he :
0 < epsilon ∧ epsilon ≤ 1
conclusion
PrivateKernel n epsilon (publicTunedInterval n epsilon)
Proof (Lean source)
-- @node: publicTunedInterval_private
lemma publicTunedInterval_private (n : ℕ) (epsilon : ℝ)
(hn : 2 ≤ n) (he : 0 < epsilon ∧ epsilon ≤ 1) :
PrivateKernel n epsilon (publicTunedInterval n epsilon) := by
classical
unfold publicTunedInterval
split_ifs with hr
· exact privateKernel_const epsilon he.1.le _
· have hp := public_tuning_parameters n epsilon hn he (lt_of_not_ge hr)
exact Ihk_private n _ epsilon _ he.1 hp.1 (by omega)
Finite closed endpoint codes denote the usual real closed interval. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: closedInterval_toSet
lemma closedInterval_toSet (l u : ℝ) :
(closedInterval l u).toSet = Icc l u := by
ext x
simp [closedInterval, IntervalCode.toSet]
The length of a finite closed interval is its nonnegative endpoint difference. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: closedInterval_leb
lemma closedInterval_leb (l u : ℝ) :
(closedInterval l u).leb = ofReal (u - l) := by
rw [IntervalCode.leb, closedInterval_toSet, Real.volume_Icc]
The ratio center always belongs to the target range, even on arbitrary datasets. The displayed conclusion follows.
Formal statement
Proof (Lean source)
-- @node: ratioMap_mem_Icc
lemma ratioMap_mem_Icc (n k : ℕ) (h : ℝ) (v : Fin 2 → ℝ) :
ratioMap n h k v ∈ Icc (-1 : ℝ) 1 := by
unfold ratioMap clip
exact ⟨le_max_left _ _, max_le (by norm_num) (min_le_left _ _)⟩
A nonnegative radius around a center in the target range has ordered truncated endpoints. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
Proof (Lean source)
-- @node: truncated_interval_endpoints
lemma truncated_interval_endpoints (z r : ℝ) (hz : z ∈ Icc (-1 : ℝ) 1)
(hr : 0 ≤ r) : max (-1) (z - r) ≤ z ∧ z ≤ min 1 (z + r) := by
constructor
· exact max_le hz.1 (by linarith)
· exact le_min hz.2 (by linarith)
The released conservative interval is nonempty and contains its clipped center. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: ratioMap_mem_intervalMap
lemma ratioMap_mem_intervalMap (n k : ℕ) (epsilon h : ℝ) (v : Fin 2 → ℝ) :
ratioMap n h k v ∈ (intervalMap n epsilon h k v).toSet := by
rw [intervalMap, closedInterval_toSet]
exact truncated_interval_endpoints _ _ (ratioMap_mem_Icc n k h v)
(Real.sqrt_nonneg _)
theorem intervalMap_leb_le
Truncation bounds each interval by the target diameter and twice its public radius. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: intervalMap_leb_le
lemma intervalMap_leb_le (n k : ℕ) (epsilon h : ℝ) (v : Fin 2 → ℝ) :
(intervalMap n epsilon h k v).leb ≤
ofReal (min 2 (2 * sqrt (10 * Vbound n epsilon h k))) := by
rw [intervalMap, closedInterval_leb]
apply ENNReal.ofReal_le_ofReal
apply le_min
· have hl := le_max_left (-1 : ℝ)
(ratioMap n h k v - sqrt (10 * Vbound n epsilon h k))
have hu := min_le_left (1 : ℝ)
(ratioMap n h k v + sqrt (10 * Vbound n epsilon h k))
linarith
· have hl := le_max_right (-1 : ℝ)
(ratioMap n h k v - sqrt (10 * Vbound n epsilon h k))
have hu := min_le_right (1 : ℝ)
(ratioMap n h k v + sqrt (10 * Vbound n epsilon h k))
linarith
theorem Ihk_expectedLength_le
Integrating the deterministic interval-length bound gives the finite-sample guarantee. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: Ihk_expectedLength_le
lemma Ihk_expectedLength_le (n k : ℕ) (epsilon h : ℝ) (he : 0 < epsilon)
(P : CausalLaw) :
expectedLength n (Ihk n epsilon h k) P ≤
ofReal (min 2 (2 * sqrt (10 * Vbound n epsilon h k))) := by
letI : IsProbabilityMeasure (Pobs P) := by
unfold Pobs
exact Measure.isProbabilityMeasure_map measurable_observe.aemeasurable
letI : IsProbabilityMeasure (dataLaw n P) := by
unfold dataLaw
infer_instance
letI := privateRatioRelease_markov n k epsilon h he
unfold expectedLength Ihk
rw [Kernel.mapOfMeasurable_eq_map, ← Measure.map_comp _ _
(measurable_intervalMap n epsilon h k)]
exact (lintegral_map_le _ _).trans
(lintegral_le_const (Filter.Eventually.of_forall (intervalMap_leb_le n k epsilon h)))
Projection to the target range cannot increase distance to a target in that range. the asserted conclusion follows. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
Proof (Lean source)
-- @node: clip_abs_sub_le
lemma clip_abs_sub_le (z t : ℝ) (ht : t ∈ Icc (-1 : ℝ) 1) :
|clip (-1) 1 z - t| ≤ |z - t| := by
unfold clip
apply abs_le.mpr
constructor
· have hz := neg_abs_le (z - t)
have hl : t - |z - t| ≤ min 1 z := le_min (by linarith [abs_nonneg (z-t), ht.1, ht.2])
(by linarith)
have hm := hl.trans (le_max_right (-1 : ℝ) (min 1 z))
linarith
· have hz := le_abs_self (z - t)
have hu : max (-1) (min 1 z) ≤ t + |z - t| :=
max_le (by linarith [abs_nonneg (z-t), ht.1, ht.2])
((min_le_right 1 z).trans (by linarith))
linarith
The denominator floor fixes the population denominator and contracts its perturbation. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: floor_denominator_abs_sub_le
lemma floor_denominator_abs_sub_le (d y D : ℝ) (hD : d ≤ D) :
|max d y - D| ≤ |y - D| := by
have h := abs_max_sub_max_le_abs y D d
simpa only [max_comm y d, max_eq_left hD] using h
The raw floored ratio is controlled by the two coordinate perturbations over the floor. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
d x y N D :
ℝ
hd :
0 < d
hD :
d ≤ D
hN :
|N / D| ≤ 1
conclusion
|x / max d y - N / D| ≤ (|x - N| + |y - D|) / d
Proof (Lean source)
-- @node: floored_ratio_perturbation
lemma floored_ratio_perturbation (d x y N D : ℝ) (hd : 0 < d)
(hD : d ≤ D) (hN : |N / D| ≤ 1) :
|x / max d y - N / D| ≤ (|x - N| + |y - D|) / d := by
have hDp : 0 < D := hd.trans_le hD
have hBp : 0 < max d y := hd.trans_le (le_max_left _ _)
have hfloor := floor_denominator_abs_sub_le d y D hD
have hid : x / max d y - N / D =
((x - N) + (N / D) * (D - max d y)) / max d y := by
field_simp
<;> ring
calc
|x / max d y - N / D| =
|(x - N) + (N / D) * (D - max d y)| / max d y := by
rw [hid, abs_div, abs_of_pos hBp]
_ ≤ (|x - N| + |y - D|) / max d y := by
apply div_le_div_of_nonneg_right _ hBp.le
calc
|(x - N) + (N / D) * (D - max d y)| ≤
|x - N| + |N / D| * |D - max d y| := by
simpa only [abs_mul] using abs_add_le (x-N) ((N/D)*(D-max d y))
_ ≤ |x - N| + |D - max d y| := by
exact add_le_add_right (by
simpa using mul_le_mul_of_nonneg_right hN (abs_nonneg (D - max d y))) _
_ ≤ |x - N| + |y - D| := by
rw [abs_sub_comm D]
exact add_le_add_right hfloor _
_ ≤ (|x - N| + |y - D|) / d :=
div_le_div_of_nonneg_left (by positivity) hd (le_max_left _ _)
Bias plus the two released coordinate perturbations controls the clipped estimator error. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
Proof (Lean source)
-- @node: ratioMap_error_le
lemma ratioMap_error_le (n k : ℕ) (h N D t b : ℝ) (v : Fin 2 → ℝ)
(hd : 0 < d0 n h k) (hD : d0 n h k ≤ D) (hN : |N / D| ≤ 1)
(ht : t ∈ Icc (-1 : ℝ) 1) (hb : |N / D - t| ≤ b) :
|ratioMap n h k v - t| ≤ b + (|v 0 - N| + |v 1 - D|) / d0 n h k := by
calc
|ratioMap n h k v - t| ≤ |v 0 / max (d0 n h k) (v 1) - t| :=
clip_abs_sub_le _ _ ht
_ ≤ |v 0 / max (d0 n h k) (v 1) - N / D| + |N / D - t| :=
abs_sub_le _ _ _
_ ≤ (|v 0 - N| + |v 1 - D|) / d0 n h k + b :=
add_le_add (floored_ratio_perturbation _ _ _ _ _ hd hD hN) hb
_ = b + (|v 0 - N| + |v 1 - D|) / d0 n h k := add_comm _ _
theorem population_numerator_abs_le_denominator
Nonnegative occupancy weights preserve the binary pair numerator domination. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: population_numerator_abs_le_denominator
lemma population_numerator_abs_le_denominator (n k : ℕ) (h : ℝ)
(hn : 2 ≤ n) (hk : 2 ≤ k) (hh : 0 < h ∧ h ≤ 1 / 4)
(P : CausalLaw) (hP : CompleteModel P) :
|Nbar n h k (dataLaw n P)| ≤ Dbar n h k (dataLaw n P) := by
have hc := occupancy_cancellation n k h hn hk hh P (dataLaw n P) rfl hP
rw [hc.1, hc.2.1]
calc
|∑ j : Fin k, occupancyWeight n h k P j * nu h k P j| ≤
∑ j : Fin k, |occupancyWeight n h k P j * nu h k P j| :=
Finset.abs_sum_le_sum_abs _ _
_ ≤ ∑ j : Fin k, occupancyWeight n h k P j * dj h k P j := by
apply Finset.sum_le_sum
intro j hj
rw [abs_mul, abs_of_nonneg (occupancyWeight_nonneg n k h P j)]
exact mul_le_mul_of_nonneg_left
(conditional_cell_numerator_abs_le_denominator k h (by omega) hh P hP j)
(occupancyWeight_nonneg n k h P j)
theorem population_ratio_abs_le_one
The population comparison ratio belongs to the target range. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: population_ratio_abs_le_one
lemma population_ratio_abs_le_one (n k : ℕ) (h : ℝ)
(hn : 2 ≤ n) (hk : 2 ≤ k) (hh : 0 < h ∧ h ≤ 1 / 4)
(P : CausalLaw) (hP : CompleteModel P) :
|Nbar n h k (dataLaw n P) / Dbar n h k (dataLaw n P)| ≤ 1 := by
have hc := occupancy_cancellation n k h hn hk hh P (dataLaw n P) rfl hP
have hpos : 0 < Dbar n h k (dataLaw n P) :=
(info_d0_pos n k h (by omega) (by omega) hh.1).2.trans_le hc.2.2.2.2.2.1
rw [abs_div, abs_of_pos hpos, div_le_one hpos]
exact population_numerator_abs_le_denominator n k h hn hk hh P hP
Two applications of the quadratic sum bound give the roadmap's pointwise squared error. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
Proof (Lean source)
-- @node: ratioMap_squared_error_le
lemma ratioMap_squared_error_le (n k : ℕ) (h N D t b : ℝ) (v : Fin 2 → ℝ)
(hd : 0 < d0 n h k) (hD : d0 n h k ≤ D) (hN : |N / D| ≤ 1)
(ht : t ∈ Icc (-1 : ℝ) 1) (hb : |N / D - t| ≤ b) :
(ratioMap n h k v - t)^2 ≤
2*b^2 + 4/(d0 n h k)^2 * ((v 0-N)^2 + (v 1-D)^2) := by
have hbpos : 0 ≤ b := (abs_nonneg _).trans hb
have herr := ratioMap_error_le n k h N D t b v hd hD hN ht hb
have hsquare : (ratioMap n h k v - t)^2 ≤
(b + (|v 0-N| + |v 1-D|) / d0 n h k)^2 := by
simpa only [sq_abs] using pow_le_pow_left₀ (abs_nonneg _) herr 2
have hsum : (|v 0-N| + |v 1-D|)^2 ≤ 2*((v 0-N)^2 + (v 1-D)^2) := by
simpa only [sq_abs] using (add_sq_le (a := |v 0-N|) (b := |v 1-D|))
have hdiv : ((|v 0-N| + |v 1-D|) / d0 n h k)^2 ≤
2/(d0 n h k)^2 * ((v 0-N)^2 + (v 1-D)^2) := by
rw [div_pow]
calc
(|v 0-N| + |v 1-D|)^2 / (d0 n h k)^2 ≤
(2*((v 0-N)^2 + (v 1-D)^2)) / (d0 n h k)^2 :=
div_le_div_of_nonneg_right hsum (sq_nonneg _)
_ = _ := by ring
have hquad := add_sq_le (a := b) (b := (|v 0-N| + |v 1-D|) / d0 n h k)
calc
(ratioMap n h k v - t)^2 ≤
2 * (b^2 + ((|v 0-N| + |v 1-D|) / d0 n h k)^2) := hsquare.trans hquad
_ ≤ 2 * (b^2 + 2/(d0 n h k)^2 * ((v 0-N)^2 + (v 1-D)^2)) := by
gcongr
_ = _ := by ring
theorem ratioMap_model_squared_error_le
For each model law, only the public bias and released centered coordinate squares are needed. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: ratioMap_model_squared_error_le
lemma ratioMap_model_squared_error_le (n k : ℕ) (h : ℝ)
(hn : 2 ≤ n) (hk : 2 ≤ k) (hh : 0 < h ∧ h ≤ 1 / 4)
(P : CausalLaw) (hP : CompleteModel P) (v : Fin 2 → ℝ) :
(ratioMap n h k v - theta P)^2 ≤
2*(bias h k)^2 + 4/(d0 n h k)^2 *
((v 0-Nbar n h k (dataLaw n P))^2 + (v 1-Dbar n h k (dataLaw n P))^2) := by
have hc := occupancy_cancellation n k h hn hk hh P (dataLaw n P) rfl hP
exact ratioMap_squared_error_le n k h _ _ _ _ v
(info_d0_pos n k h (by omega) (by omega) hh.1).2 hc.2.2.2.2.2.1
(population_ratio_abs_le_one n k h hn hk hh P hP) (theta_mem_Icc P hP)
hc.2.2.2.2.2.2
Squaring the roughness term converts its fifth-root power to the two-fifths power. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
k :
ℕ
h :
ℝ
hh :
0 ≤ h
conclusion
((cellWidth h k)^(1/5 : ℝ))^2 = (cellWidth h k)^(2/5 : ℝ)
Proof (Lean source)
-- @node: cellWidth_fifth_power_sq
lemma cellWidth_fifth_power_sq (k : ℕ) (h : ℝ) (hh : 0 ≤ h) :
((cellWidth h k)^(1/5 : ℝ))^2 = (cellWidth h k)^(2/5 : ℝ) := by
have hw : 0 ≤ cellWidth h k := by unfold cellWidth; positivity
have hp := Real.rpow_mul_natCast hw (1/5 : ℝ) 2
norm_num at hp
exact hp.symm
The quadratic sum inequality bounds the two contributions to the public bias. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
hypotheses
k :
ℕ
h :
ℝ
hh :
0 ≤ h
conclusion
2*(bias h k)^2 ≤ 36*h^2 + 2304*(cellWidth h k)^(2/5 : ℝ)
Proof (Lean source)
-- @node: bias_squared_le
lemma bias_squared_le (k : ℕ) (h : ℝ) (hh : 0 ≤ h) :
2*(bias h k)^2 ≤ 36*h^2 + 2304*(cellWidth h k)^(2/5 : ℝ) := by
have hquad := add_sq_le (a := 3*h) (b := 24*(cellWidth h k)^(1/5 : ℝ))
have hp := cellWidth_fifth_power_sq k h hh
unfold bias
nlinarith [hp]
Substitution of the public floor converts the occupancy and noise terms to inverse scales. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: variance_floor_envelope_le
lemma variance_floor_envelope_le (n k : ℕ) (epsilon h W : ℝ)
(ha : 0 < info n h k) (he : 0 < epsilon)
(hW : W ≤ 9 / 2 * info n h k) :
144*W/(d0 n h k)^2 + 2304/(epsilon^2*(d0 n h k)^2) ≤
294912*(info n h k)⁻¹ + 1048576*(epsilon*info n h k)^(-2 : ℤ) := by
have hocc : 144*W/(d0 n h k)^2 ≤ 294912*(info n h k)⁻¹ := by
calc
144*W/(d0 n h k)^2 ≤ 144*(9/2*info n h k)/(d0 n h k)^2 :=
div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_left hW (by norm_num))
(sq_nonneg _)
_ = _ := by
unfold d0
field_simp [ha.ne']
ring
have hnoise : 2304/(epsilon^2*(d0 n h k)^2) =
1048576*(epsilon*info n h k)^(-2 : ℤ) := by
unfold d0
simp only [zpow_neg, zpow_ofNat]
field_simp [ha.ne', he.ne']
ring
rw [hnoise]
exact add_le_add hocc le_rfl
theorem model_variance_envelope_le
The roadmap's bias, occupancy variance and independent Laplace variance fit the public envelope. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: model_variance_envelope_le
lemma model_variance_envelope_le (n k : ℕ) (epsilon h : ℝ)
(hn : 2 ≤ n) (hk : 2 ≤ k) (he : 0 < epsilon)
(hh : 0 < h ∧ h ≤ 1 / 4) (P : CausalLaw) (hP : CompleteModel P) :
2*(bias h k)^2 + 144*Wocc n h k P/(d0 n h k)^2 +
2304/(epsilon^2*(d0 n h k)^2) ≤ Vbound n epsilon h k := by
have ha := (info_d0_pos n k h (by omega) (by omega) hh.1).1
have hc := occupancy_cancellation n k h hn hk hh P (dataLaw n P) rfl hP
have hv := variance_floor_envelope_le n k epsilon h (Wocc n h k P) ha he
(by linarith [hc.2.2.2.2.1])
have hb := bias_squared_le k h hh.1.le
have hcell : 0 ≤ (cellWidth h k)^(2/5 : ℝ) := Real.rpow_nonneg (by
unfold cellWidth; exact div_nonneg (by linarith) (Nat.cast_nonneg _)) _
have hi : 0 ≤ (info n h k)⁻¹ := inv_nonneg.mpr ha.le
have heps : 0 ≤ (epsilon*info n h k)^(-2 : ℤ) := by
simp only [zpow_neg, zpow_ofNat]
positivity
unfold Vbound
norm_num only [zpow_neg, zpow_ofNat, pow_succ, pow_zero, mul_one, OfNat.ofNat] at hv heps ⊢
nlinarith [sq_nonneg h]
theorem target_mem_intervalMap_of_error_le
Truncation retains any target in the model range within the public radius of the center. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
Proof (Lean source)
-- @node: target_mem_intervalMap_of_error_le
lemma target_mem_intervalMap_of_error_le (n k : ℕ) (epsilon h t : ℝ)
(v : Fin 2 → ℝ) (ht : t ∈ Icc (-1 : ℝ) 1)
(herr : |ratioMap n h k v - t| ≤ sqrt (10 * Vbound n epsilon h k)) :
t ∈ (intervalMap n epsilon h k v).toSet := by
rw [intervalMap, closedInterval_toSet]
have he := abs_le.mp herr
exact ⟨max_le ht.1 (by linarith), le_min ht.2 (by linarith)⟩
theorem Vbound_pos
The public variance envelope is strictly positive on admissible radii and cell counts. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
n k :
ℕ
epsilon h :
ℝ
hn :
0 < n
hk :
0 < k
hh :
0 < h
conclusion
0 < Vbound n epsilon h k
Proof (Lean source)
-- @node: Vbound_pos
lemma Vbound_pos (n k : ℕ) (epsilon h : ℝ) (hn : 0 < n) (hk : 0 < k)
(hh : 0 < h) : 0 < Vbound n epsilon h k := by
have ha := (info_d0_pos n k h hn hk hh).1
have hw : 0 ≤ cellWidth h k := by unfold cellWidth; positivity
unfold Vbound
have hc := Real.rpow_nonneg hw (2/5 : ℝ)
have hi := inv_pos.mpr ha
have he : 0 ≤ (epsilon*info n h k)^(-2 : ℤ) := by
simp only [zpow_neg, zpow_ofNat]
positivity
positivity
Markov's inequality turns a second-moment envelope into a one-tenth radius-tail bound. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
Proof (Lean source)
-- @node: squared_error_radius_tail_le
lemma squared_error_radius_tail_le {B : Type*} [MeasurableSpace B]
(Q : Measure B) (f : B → ℝ) (hf : Measurable f) (t V : ℝ) (hV : 0 < V)
(hmse : (∫⁻ x, ofReal ((f x-t)^2) ∂Q) ≤ ofReal V) :
Q {x | sqrt (10*V) < |f x-t|} ≤ ofReal (1/10 : ℝ) := by
have hmeas : Measurable (fun x => ofReal ((f x-t)^2)) := by fun_prop
have ht := meas_ge_le_lintegral_div (μ := Q) hmeas.aemeasurable
(ne_of_gt (by positivity : 0 < ofReal (10*V))) ENNReal.ofReal_ne_top
have hsub : {x | sqrt (10*V) < |f x-t|} ⊆
{x | ofReal (10*V) ≤ ofReal ((f x-t)^2)} := by
intro x hx
apply ENNReal.ofReal_le_ofReal
have hs := Real.sq_sqrt (by positivity : 0 ≤ 10*V)
have hr := Real.sqrt_nonneg (10*V)
have ha := abs_nonneg (f x-t)
have habs := sq_abs (f x-t)
change sqrt (10*V) < |f x-t| at hx
nlinarith
calc
Q {x | sqrt (10*V) < |f x-t|} ≤
Q {x | ofReal (10*V) ≤ ofReal ((f x-t)^2)} := measure_mono hsub
_ ≤ (∫⁻ x, ofReal ((f x-t)^2) ∂Q) / ofReal (10*V) := ht
_ ≤ ofReal V / ofReal (10*V) := ENNReal.div_le_div_right hmse _
_ = ofReal (1/10 : ℝ) := by
rw [← ENNReal.ofReal_div_of_pos (by positivity : 0 < 10*V)]
congr 1
field_simp
theorem intervalMap_miss_probability_le
The truncated interval misses a model-range target only on the radius-tail event. The result uses the stated assumptions and establishes the displayed conclusion.
Formal statement
Proof (Lean source)
-- @node: intervalMap_miss_probability_le
lemma intervalMap_miss_probability_le (n k : ℕ) (epsilon h t : ℝ)
(Q : Measure (Fin 2 → ℝ)) (ht : t ∈ Icc (-1 : ℝ) 1)
(hV : 0 < Vbound n epsilon h k)
(hmse : (∫⁻ v, ofReal ((ratioMap n h k v - t) ^ 2) ∂Q) ≤
ofReal (Vbound n epsilon h k)) :
Q {v | t ∉ (intervalMap n epsilon h k v).toSet} ≤ ofReal (1/10 : ℝ) := by
refine (measure_mono ?_).trans
(squared_error_radius_tail_le Q _ (measurable_ratioMap n h k) t _ hV hmse)
intro v hv
exact lt_of_not_ge (fun herr => hv
(target_mem_intervalMap_of_error_le n k epsilon h t v ht herr))
theorem Thk_squared_error_eq
Scalar postprocessing expresses the mean-square error as an integral over the joint release. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: Thk_squared_error_eq
lemma Thk_squared_error_eq (n k : ℕ) (epsilon h t : ℝ) (P : CausalLaw) :
(∫⁻ u, ofReal ((u-t)^2) ∂(Thk n epsilon h k ∘ₘ dataLaw n P)) =
∫⁻ v, ofReal ((ratioMap n h k v-t)^2)
∂(privateRatioRelease n epsilon h k ∘ₘ dataLaw n P) := by
unfold Thk
rw [Kernel.mapOfMeasurable_eq_map, ← Measure.map_comp _ _ (measurable_ratioMap n h k)]
exact lintegral_map (by fun_prop) (measurable_ratioMap n h k)
theorem Ihk_coverage_of_squared_error_le
The finite-sample second-moment guarantee implies coverage for the interval from the same joint release, including targets on the boundary of the model range. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: Ihk_coverage_of_squared_error_le
lemma Ihk_coverage_of_squared_error_le (n k : ℕ) (epsilon h : ℝ)
(hn : 0 < n) (hk : 0 < k) (he : 0 < epsilon) (hh : 0 < h)
(P : CausalLaw) (hP : CompleteModel P)
(hmse : (∫⁻ u, ofReal ((u-theta P)^2)
∂(Thk n epsilon h k ∘ₘ dataLaw n P)) ≤ ofReal (Vbound n epsilon h k)) :
9/10 ≤ coverage n (Ihk n epsilon h k) P := by
letI : IsProbabilityMeasure (Pobs P) := by
unfold Pobs
exact Measure.isProbabilityMeasure_map measurable_observe.aemeasurable
letI : IsProbabilityMeasure (dataLaw n P) := by unfold dataLaw; infer_instance
letI := privateRatioRelease_markov n k epsilon h he
let Q := privateRatioRelease n epsilon h k ∘ₘ dataLaw n P
let E : Set (Fin 2 → ℝ) := {v | theta P ∈ (intervalMap n epsilon h k v).toSet}
have hE : MeasurableSet E :=
(measurableSet_interval_contains (theta P)).preimage (measurable_intervalMap n epsilon h k)
have hmiss : Q Eᶜ ≤ ofReal (1/10 : ℝ) := by
exact intervalMap_miss_probability_le n k epsilon h (theta P) Q
(theta_mem_Icc P hP) (Vbound_pos n k epsilon h hn hk hh)
((Thk_squared_error_eq n k epsilon h (theta P) P) ▸ hmse)
have hmissReal : Q.real Eᶜ ≤ 1/10 := by
simpa [Measure.real] using ENNReal.toReal_mono ENNReal.ofReal_ne_top hmiss
have hsum := probReal_compl_eq_one_sub (μ := Q) hE
have hcov : coverage n (Ihk n epsilon h k) P = Q.real E := by
unfold coverage Ihk Measure.real
rw [Kernel.mapOfMeasurable_eq_map, ← Measure.map_comp _ _
(measurable_intervalMap n epsilon h k), Measure.map_apply
(measurable_intervalMap n epsilon h k) (measurableSet_interval_contains (theta P))]
rfl
rw [hcov]
linarith
theorem full_interval_coverage
The public fallback interval covers every law in the complete model with probability one. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: full_interval_coverage
lemma full_interval_coverage (n : ℕ) (P : CausalLaw) (hP : CompleteModel P) :
coverage n (Kernel.const (Dataset n) (Measure.dirac (closedInterval (-1) 1))) P = 1 := by
letI : IsProbabilityMeasure (Pobs P) := by
unfold Pobs
exact Measure.isProbabilityMeasure_map measurable_observe.aemeasurable
letI : IsProbabilityMeasure (dataLaw n P) := by unfold dataLaw; infer_instance
have ht : theta P ∈ (closedInterval (-1) 1).toSet := by
rw [closedInterval_toSet]
exact theta_mem_Icc P hP
unfold coverage Measure.real
change ((dataLaw n P).bind (fun _ => Measure.dirac (closedInterval (-1) 1))
{c | theta P ∈ c.toSet}).toReal = 1
rw [Measure.bind_const]
simp [Measure.dirac_apply' _ (measurableSet_interval_contains (theta P)), ht]
theorem publicTunedInterval_coverage_of_squared_error_le
Coverage of the tuned interval follows from the tuned second moment on its active branch and full-range coverage on its fallback branch. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
n :
ℕ
epsilon :
ℝ
hn :
2 ≤ n
he :
0 < epsilon ∧ epsilon ≤ 1
P :
hP :
conclusion
9/10 ≤ coverage n (publicTunedInterval n epsilon) P
Proof (Lean source)
-- @node: publicTunedInterval_coverage_of_squared_error_le
lemma publicTunedInterval_coverage_of_squared_error_le (n : ℕ) (epsilon : ℝ)
(hn : 2 ≤ n) (he : 0 < epsilon ∧ epsilon ≤ 1) (P : CausalLaw) (hP : CompleteModel P)
(hmse : rate n epsilon < 1/8 →
(∫⁻ u, ofReal ((u-theta P)^2)
∂(Thk n epsilon (tunedH n epsilon) (tunedK n epsilon) ∘ₘ dataLaw n P)) ≤
ofReal (Vbound n epsilon (tunedH n epsilon) (tunedK n epsilon))) :
9/10 ≤ coverage n (publicTunedInterval n epsilon) P := by
classical
unfold publicTunedInterval
split_ifs with hr
· rw [full_interval_coverage n P hP]
norm_num
· have hp := public_tuning_parameters n epsilon hn he (lt_of_not_ge hr)
exact Ihk_coverage_of_squared_error_le n _ epsilon _ (by omega) (by omega)
he.1 hp.1 P hP (hmse (lt_of_not_ge hr))
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.TMatchedRiskFrontier 6 declarations Matched private absolute-risk frontier over all total randomized mechanisms in the same observation experiment, attained above by the explicit publicly tuned release.
Matched private absolute-risk frontier over all total randomized mechanisms in the same observation experiment, attained above by the explicit publicly tuned release.
theorem finiteProductMixture_probability
A nonempty finite uniform prior is a probability measure. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: finiteProductMixture_probability
lemma finiteProductMixture_probability (n s : ℕ) (F : Fin s → CausalLaw)
(hs : 0 < s) : IsProbabilityMeasure (finiteProductMixture n s F) := by
have hprob : ∀ i, IsProbabilityMeasure (dataLaw n (F i)) := by
intro i
let : IsProbabilityMeasure (Pobs (F i)) :=
Measure.isProbabilityMeasure_map measurable_observe.aemeasurable
unfold dataLaw
infer_instance
constructor
simp only [finiteProductMixture, Measure.smul_apply, smul_eq_mul,
Measure.finsetSum_apply, measure_univ, Finset.sum_const, Finset.card_univ,
Fintype.card_fin, nsmul_eq_mul, mul_one]
exact ENNReal.inv_mul_cancel (by exact_mod_cast (Nat.ne_of_gt hs)) (by simp)
Kernel composition distributes over a finite sum of input laws. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: prior_comp_finsetSum
lemma prior_comp_finsetSum {ι B : Type*} [MeasurableSpace B]
(n : ℕ) (M : Kernel (Dataset n) B) (s : Finset ι) (Q : ι → Measure (Dataset n)) :
M ∘ₘ (∑ i ∈ s, Q i) = ∑ i ∈ s, M ∘ₘ Q i := by
classical
induction s using Finset.induction_on with
| empty => simp
| @insert i s hi ih => simp [Finset.sum_insert hi, Measure.comp_add, ih]
theorem finite_prior_loss_le_worstRisk
Prior-averaged loss at a common target is bounded by the maximal class risk, including when the latter is infinite. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
(∫⁻ u, ofReal |u - target| ∂(M ∘ₘ finiteProductMixture n s F)) ≤ worstRisk n M
Proof (Lean source)
-- @node: finite_prior_loss_le_worstRisk
lemma finite_prior_loss_le_worstRisk (n s : ℕ) (F : Fin s → CausalLaw)
(hs : 0 < s) (M : Kernel (Dataset n) ℝ) (target : ℝ)
(hF : ∀ i, CompleteModel (F i) ∧ theta (F i) = target) :
(∫⁻ u, ofReal |u - target| ∂(M ∘ₘ finiteProductMixture n s F)) ≤
worstRisk n M := by
rw [finiteProductMixture, Measure.comp_smul, prior_comp_finsetSum,
lintegral_smul_measure, lintegral_finsetSum_measure]
have hbound : ∀ i : Fin s,
(∫⁻ u, ofReal |u - target| ∂(M ∘ₘ dataLaw n (F i))) ≤ worstRisk n M := by
intro i
rw [← (hF i).2]
exact le_iSup_of_le (F i) (le_iSup_of_le (hF i).1 le_rfl)
calc
_ ≤ (s : ℝ≥0∞)⁻¹ * ∑ _i : Fin s, worstRisk n M :=
mul_le_mul_of_nonneg_left (Finset.sum_le_sum fun i _ => hbound i) (zero_le)
_ = worstRisk n M := by
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul]
rw [← mul_assoc, ENNReal.inv_mul_cancel
(by exact_mod_cast (Nat.ne_of_gt hs)) (by simp), one_mul]
A real error event of probability at least three eighths forces absolute loss. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
L :
target a :
ℝ
ha :
0 ≤ a
hp :
3 / 8 ≤ L.real {u | a ≤ dist u target}
Proof (Lean source)
-- @node: absolute_loss_of_error_probability
lemma absolute_loss_of_error_probability (L : Measure ℝ) [IsProbabilityMeasure L]
(target a : ℝ) (ha : 0 ≤ a)
(hp : 3 / 8 ≤ L.real {u | a ≤ dist u target}) :
ofReal (a * (3 / 8)) ≤ ∫⁻ u, ofReal |u - target| ∂L := by
have hm : Measurable (fun u : ℝ => ofReal |u - target|) := by fun_prop
have hmarkov := mul_meas_ge_le_lintegral (μ := L) hm (ofReal a)
have hset : {u : ℝ | ofReal a ≤ ofReal |u - target|} =
{u | a ≤ dist u target} := by
ext u
simp only [Set.mem_setOf_eq, Real.dist_eq]
exact ENNReal.ofReal_le_ofReal_iff (abs_nonneg _)
rw [hset] at hmarkov
have hmass : ofReal (3 / 8 : ℝ) ≤ L {u | a ≤ dist u target} := by
rw [← ENNReal.ofReal_toReal (measure_ne_top L _)]
exact ENNReal.ofReal_le_ofReal hp
rw [ENNReal.ofReal_mul ha]
exact (mul_le_mul_of_nonneg_left hmass (zero_le)).trans hmarkov
theorem finite_priors_scalar_lower
Le Cam's two-error inequality and finite-prior averaging give the scalar converse. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
n s0 s1 :
ℕ
hs0 :
0 < s0
hs1 :
0 < s1
M :
Delta :
ℝ
hd :
0 ≤ Delta
hF0 :
∀ i, CompleteModel (F0 i) ∧ theta (F0 i) = 0
hF1 :
∀ i, CompleteModel (F1 i) ∧ theta (F1 i) = Delta
htv :
conclusion
ofReal (3 * Delta / 16) ≤ worstRisk n M
Proof (Lean source)
-- @node: finite_priors_scalar_lower
lemma finite_priors_scalar_lower (n s0 s1 : ℕ)
(F0 : Fin s0 → CausalLaw) (F1 : Fin s1 → CausalLaw)
(hs0 : 0 < s0) (hs1 : 0 < s1) (M : Kernel (Dataset n) ℝ) [IsMarkovKernel M]
(Delta : ℝ) (hd : 0 ≤ Delta)
(hF0 : ∀ i, CompleteModel (F0 i) ∧ theta (F0 i) = 0)
(hF1 : ∀ i, CompleteModel (F1 i) ∧ theta (F1 i) = Delta)
(htv : TV (M ∘ₘ finiteProductMixture n s0 F0)
(M ∘ₘ finiteProductMixture n s1 F1) ≤ 1 / 4) :
ofReal (3 * Delta / 16) ≤ worstRisk n M := by
let L0 := M ∘ₘ finiteProductMixture n s0 F0
let L1 := M ∘ₘ finiteProductMixture n s1 F1
let : IsProbabilityMeasure (finiteProductMixture n s0 F0) :=
finiteProductMixture_probability n s0 F0 hs0
let : IsProbabilityMeasure (finiteProductMixture n s1 F1) :=
finiteProductMixture_probability n s1 F1 hs1
have hsep : 2 * (Delta / 2) ≤ dist (0 : ℝ) Delta := by
rw [Real.dist_eq, zero_sub, abs_neg, abs_of_nonneg hd]
linarith
have htest := one_sub_tvDist_le_error_sum
(P₀ := L0) (P₁ := L1) measurable_id hsep
have hsum : 3 / 4 ≤ L0.real {u | Delta / 2 ≤ dist u 0} +
L1.real {u | Delta / 2 ≤ dist u Delta} := by
change TV L0 L1 ≤ 1 / 4 at htv
simp only [id_eq] at htest
linarith
have hcases : 3 / 8 ≤ L0.real {u | Delta / 2 ≤ dist u 0} ∨
3 / 8 ≤ L1.real {u | Delta / 2 ≤ dist u Delta} := by
by_contra h
push_neg at h
linarith
have heq : 3 * Delta / 16 = (Delta / 2) * (3 / 8) := by ring
rw [heq]
rcases hcases with h0 | h1
· exact (absolute_loss_of_error_probability L0 0 (Delta / 2) (by positivity) h0).trans
(finite_prior_loss_le_worstRisk n s0 F0 hs0 M 0 hF0)
· exact (absolute_loss_of_error_probability L1 Delta (Delta / 2) (by positivity) h1).trans
(finite_prior_loss_le_worstRisk n s1 F1 hs1 M Delta hF1)
theorem matched_risk_frontier
The minimax risk lies between the specified fixed multiples of the benchmark. The total publicly tuned kernel attains the upper bound across the entire complete model. The public parameters and their domains and the Efron--Stein gate, bounded-differences gate, and Cayley gate specify the conditional scope.
Formal statement
hypotheses
efronStein_of_gate :
boundedDifferences_of_gate :
cayley_of_gate :
n :
ℕ
epsilon :
ℝ
hn :
2 ≤ n
he :
0 < epsilon ∧ epsilon ≤ 1
conclusion
conclusion 1
conclusion 2
conclusion 3
IsMarkovKernel (publicTunedRelease n epsilon)
conclusion 4
PrivateKernel n epsilon (publicTunedRelease n epsilon)
conclusion 5
Proof (Lean source)
-- @node: thm:matched-risk-frontier
theorem matched_risk_frontier (efronStein_of_gate : EfronSteinReplacement)
(boundedDifferences_of_gate : BoundedDifferencesMGF) (cayley_of_gate : CayleyLabeledTreeCount)
(n : ℕ) (epsilon : ℝ) (hn : 2 ≤ n) (he : 0 < epsilon ∧ epsilon ≤ 1) :
ofReal ((2 : ℝ)^(-20 : ℤ)*rate n epsilon) ≤ privateMinimaxRisk n epsilon ∧
privateMinimaxRisk n epsilon ≤ ofReal (2^18*rate n epsilon) ∧
IsMarkovKernel (publicTunedRelease n epsilon) ∧
PrivateKernel n epsilon (publicTunedRelease n epsilon) ∧
worstRisk n (publicTunedRelease n epsilon) ≤ ofReal (2^18*rate n epsilon) := by
obtain ⟨s0, s1, F0, F1, Delta, hs0, hs1, hd, hF0, hF1, _, htv⟩ :=
frontier_testing_priors boundedDifferences_of_gate cayley_of_gate n epsilon hn he
have hr : 0 < rate n epsilon := rate_pos n epsilon (by omega)
have hd0 : 0 ≤ Delta := le_trans (by positivity) hd
have hlower : ofReal ((2 : ℝ)^(-20 : ℤ) * rate n epsilon) ≤
privateMinimaxRisk n epsilon := by
unfold privateMinimaxRisk
refine le_iInf fun M => le_iInf fun hM => ?_
let : IsMarkovKernel M := hM.1
have htest := finite_priors_scalar_lower n s0 s1 F0 F1 hs0 hs1 M Delta hd0
hF0 hF1 (htv ℝ M hM.2)
apply le_trans (ENNReal.ofReal_le_ofReal ?_) htest
have hscale : (2 : ℝ)^(-20 : ℤ) ≤ (3 / 16) * (2 : ℝ)^(-14 : ℤ) := by norm_num
calc
_ ≤ ((3 / 16) * (2 : ℝ)^(-14 : ℤ)) * rate n epsilon :=
mul_le_mul_of_nonneg_right hscale hr.le
_ = (3 / 16) * ((2 : ℝ)^(-14 : ℤ) * rate n epsilon) := by ring
_ ≤ (3 / 16) * Delta := mul_le_mul_of_nonneg_left hd (by norm_num)
_ = _ := by ring
obtain ⟨_, _, _, _, _, _, _, hmarkov, hprivate, _, _, hrisk, _, _⟩ :=
uniform_private_upper efronStein_of_gate n 2 epsilon (1 / 4) hn (by omega) he
⟨by norm_num, le_rfl⟩
have hupper : privateMinimaxRisk n epsilon ≤ worstRisk n (publicTunedRelease n epsilon) :=
iInf_le_of_le (publicTunedRelease n epsilon)
(iInf_le_of_le ⟨hmarkov, hprivate⟩ le_rfl)
exact ⟨hlower, hupper.trans hrisk, hmarkov, hprivate, hrisk⟩
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.TRegimeAndSanity 5 declarations Exact public-budget regimes, the unrestricted nonprivate order, and the exact condition for consistency along budget sequences.
Exact public-budget regimes, the unrestricted nonprivate order, and the exact condition for consistency along budget sequences.
theorem private_risk_consistency_iff
The matched risk frontier transfers the benchmark's exact consistency criterion to risk. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
efronStein_of_gate :
boundedDifferences_of_gate :
cayley_of_gate :
budgets :
ℕ → ℝ
he :
∀ j
if
2 ≤ j
then
0 < budgets j ∧ budgets j ≤ 1
Proof (Lean source)
-- @node: private_risk_consistency_iff
lemma private_risk_consistency_iff (efronStein_of_gate : EfronSteinReplacement)
(boundedDifferences_of_gate : BoundedDifferencesMGF) (cayley_of_gate : CayleyLabeledTreeCount)
(budgets : ℕ → ℝ) (he : ∀ j, 2 ≤ j → 0 < budgets j ∧ budgets j ≤ 1) :
Tendsto (fun j => privateMinimaxRisk j (budgets j)) atTop (𝓝 0) ↔
Tendsto (fun j : ℕ => (j : ℝ)*budgets j) atTop atTop := by
have hfront := fun j hj => matched_risk_frontier efronStein_of_gate
boundedDifferences_of_gate cayley_of_gate j (budgets j) hj (he j hj)
rw [← rate_consistency_iff budgets (fun j hj => (he j hj).1)]
constructor
· intro hrisk
apply tendsto_order.2
constructor
· intro a ha
filter_upwards [eventually_ge_atTop 2] with j hj
exact ha.trans (rate_pos j (budgets j) (by omega))
· intro a ha
have hc : (0 : ℝ) < (2 : ℝ)^(-20 : ℤ) := by positivity
have hca : 0 < (2 : ℝ)^(-20 : ℤ)*a := mul_pos hc ha
have hbound := (tendsto_order.1 hrisk).2 _ (ENNReal.ofReal_pos.mpr hca)
filter_upwards [hbound, eventually_ge_atTop 2] with j hj hj2
have hlt := lt_of_le_of_lt (hfront j hj2).1 hj
have hre := (ENNReal.ofReal_lt_ofReal_iff hca).mp hlt
exact (mul_lt_mul_iff_right₀ hc).mp hre
· intro hr
have hu : Tendsto (fun j => ofReal (2^18*rate j (budgets j))) atTop (𝓝 0) := by
simpa only [mul_zero, ENNReal.ofReal_zero] using
ENNReal.tendsto_ofReal (hr.const_mul (2^18 : ℝ))
apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hu
· exact Filter.Eventually.of_forall (fun _ => bot_le)
· filter_upwards [eventually_ge_atTop 2] with j hj
exact (hfront j hj).2.1
theorem nonprivate_sampling_upper
The unit-budget release also attains the sampling upper bound without privacy constraints. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
efronStein_of_gate :
boundedDifferences_of_gate :
cayley_of_gate :
n :
ℕ
hn :
2 ≤ n
conclusion
nonprivateMinimaxRisk n ≤ ofReal (2^18*(n : ℝ)^(-1/4 : ℝ))
Proof (Lean source)
-- @node: nonprivate_sampling_upper
lemma nonprivate_sampling_upper (efronStein_of_gate : EfronSteinReplacement)
(boundedDifferences_of_gate : BoundedDifferencesMGF) (cayley_of_gate : CayleyLabeledTreeCount)
(n : ℕ) (hn : 2 ≤ n) :
nonprivateMinimaxRisk n ≤ ofReal (2^18*(n : ℝ)^(-1/4 : ℝ)) := by
obtain ⟨_, _, hM, _, hR⟩ := matched_risk_frontier efronStein_of_gate
boundedDifferences_of_gate cayley_of_gate n 1 hn ⟨zero_lt_one, le_rfl⟩
rw [rate_unit_budget n hn] at hR
exact (iInf_le_of_le (publicTunedRelease n 1) (iInf_le_of_le hM le_rfl)).trans hR
theorem nonprivate_sampling_lower
The explicit sign mixture at one quarter of the sampling radius gives the unrestricted nonprivate lower bound, through chi-square comparison and Le Cam testing. the theorem's stated inputs and assumptions, and the asserted conclusion follows.
Formal statement
hypotheses
boundedDifferences_of_gate :
cayley_of_gate :
n :
ℕ
hn :
2 ≤ n
conclusion
ofReal ((2 : ℝ)^(-20 : ℤ)*(n : ℝ)^(-1/4 : ℝ)) ≤ nonprivateMinimaxRisk n
Proof (Lean source)
-- @node: nonprivate_sampling_lower
lemma nonprivate_sampling_lower
(boundedDifferences_of_gate : BoundedDifferencesMGF) (cayley_of_gate : CayleyLabeledTreeCount)
(n : ℕ) (hn : 2 ≤ n) :
ofReal ((2 : ℝ)^(-20 : ℤ)*(n : ℝ)^(-1/4 : ℝ)) ≤
nonprivateMinimaxRisk n := by
obtain ⟨s0, s1, F0, F1, Delta, hs0, hs1, hd, hF0, hF1,
hrepr, _⟩ := frontier_testing_priors.{0} boundedDifferences_of_gate
cayley_of_gate n 1 hn ⟨zero_lt_one, le_rfl⟩
obtain ⟨hL, hhL, hradius, hnull, hsign, _⟩ := hrepr
have hx : 0 < (n : ℝ) := by exact_mod_cast (show 0 < n by omega)
have hx1 : 1 ≤ (n : ℝ) := by exact_mod_cast (show 1 ≤ n by omega)
have hr := rate_unit_budget n hn
have huses : testingUsesSigns n 1 := by
dsimp [testingUsesSigns]
refine ⟨by simpa using (show (1 : ℝ) < n from by exact_mod_cast (show 1 < n by omega)), inl ?_⟩
constructor
· exact (sparse_le_sampling_iff (n : ℝ) 1 hx zero_lt_one).2
(Real.rpow_le_one_of_one_le_of_nonpos hx1 (by norm_num))
· exact (dense_le_sampling_iff (n : ℝ) 1 hx zero_lt_one).2
(Real.rpow_le_one_of_one_le_of_nonpos hx1 (by norm_num))
obtain ⟨_, hmix⟩ := hsign huses
have hnullmix : finiteProductMixture n s0 F0 = dataLaw n fairNull := by
ext E hE
simp only [finiteProductMixture, Measure.smul_apply, smul_eq_mul,
Measure.finsetSum_apply, hnull, Finset.sum_const, Finset.card_univ,
Fintype.card_fin, nsmul_eq_mul]
rw [← mul_assoc, ENNReal.inv_mul_cancel
(by exact_mod_cast hs0.ne') (by simp), one_mul]
obtain ⟨_, _, _, _, _, hchi, hac, hint⟩ :=
positive_family_certificate hL hhL boundedDifferences_of_gate n
have hrad : hL = (n : ℝ)^(-1/4 : ℝ) / 4 := by simpa [hr] using hradius
have hpow : (n : ℝ)^2 * ((n : ℝ)^(-1/4 : ℝ))^8 = 1 := by
rw [← Real.rpow_mul_natCast hx.le]
norm_num
exact mul_inv_cancel₀ (pow_ne_zero _ hx.ne')
have hexponent : 160*(n : ℝ)^2*(separation hL)^2*hL*deltaL hL =
160 * (2 : ℝ)^(-40 : ℤ) := by
rw [hrad]
dsimp [separation, kappa, deltaL]
norm_num
nlinarith [hpow]
have hexp : exp (160 * (2 : ℝ)^(-40 : ℤ)) - 1 ≤ 1/4 := by
have h := exp_privacy_increment_le (160 * (2 : ℝ)^(-40 : ℤ))
⟨by norm_num, by norm_num⟩
norm_num at h ⊢
linarith
let : IsProbabilityMeasure (finiteProductMixture n s0 F0) :=
finiteProductMixture_probability n s0 F0 hs0
let : IsProbabilityMeasure (finiteProductMixture n s1 F1) :=
finiteProductMixture_probability n s1 F1 hs1
have hprob0 : IsProbabilityMeasure (dataLaw n fairNull) := hnullmix ▸ inferInstance
have hprob1 : IsProbabilityMeasure (signMixture hL hhL n) := hmix ▸ inferInstance
let : IsProbabilityMeasure (dataLaw n fairNull) := hprob0
let : IsProbabilityMeasure (signMixture hL hhL n) := hprob1
have hdata : TV (finiteProductMixture n s0 F0) (finiteProductMixture n s1 F1) ≤ 1/4 := by
rw [hnullmix, hmix]
change tvDist _ _ ≤ _
rw [tvDist_symm]
have htv := tvDist_le_half_sqrt_chiSqDiv
(signMixture hL hhL n) (dataLaw n fairNull) hac hint
have hb : chiSqDiv (signMixture hL hhL n)
(dataLaw n fairNull) ≤ 1/4 := by
rw [hexponent] at hchi
exact hchi.trans hexp
have hsqrt : sqrt (chiSqDiv
(signMixture hL hhL n) (dataLaw n fairNull)) ≤ 1/2 :=
(Real.sqrt_le_iff).2 ⟨by norm_num, by nlinarith [hb]⟩
linarith
have hrpos : 0 < rate n 1 := rate_pos n 1 (by omega)
unfold nonprivateMinimaxRisk
refine le_iInf fun M => le_iInf fun hM => ?_
let : IsMarkovKernel M := hM
have htest := finite_priors_scalar_lower n s0 s1 F0 F1 hs0 hs1 M Delta
(le_trans (by positivity) hd) hF0 hF1
((kernel_TV_contraction M _ _).trans hdata)
apply (ENNReal.ofReal_le_ofReal ?_).trans htest
rw [hr] at hd
have hc : (2 : ℝ)^(-20 : ℤ) ≤ (3/16) * (2 : ℝ)^(-14 : ℤ) := by norm_num
have ha : 0 ≤ (n : ℝ)^(-1/4 : ℝ) := by positivity
nlinarith [mul_le_mul_of_nonneg_right hc ha]
theorem nonprivate_sampling_bounds
Given the Efron--Stein gate, bounded-differences gate, and Cayley gate, the unrestricted minimax risk at every sample size at least two lies between the explicit constants 2⁻²⁰ and 2¹⁸ times the fourth-root rate.
Formal statement
hypotheses
efronStein_of_gate :
boundedDifferences_of_gate :
cayley_of_gate :
n :
ℕ
hn :
2 ≤ n
conclusion
conclusion 1
ofReal ((2 : ℝ)^(-20 : ℤ)*(n : ℝ)^(-1/4 : ℝ)) ≤ nonprivateMinimaxRisk n
conclusion 2
nonprivateMinimaxRisk n ≤ ofReal (2^18*(n : ℝ)^(-1/4 : ℝ))
Proof (Lean source)
-- keep: separate unrestricted two-sided minimax benchmark required by the resolved R1 scope
lemma nonprivate_sampling_bounds (efronStein_of_gate : EfronSteinReplacement)
(boundedDifferences_of_gate : BoundedDifferencesMGF) (cayley_of_gate : CayleyLabeledTreeCount)
(n : ℕ) (hn : 2 ≤ n) :
ofReal ((2 : ℝ)^(-20 : ℤ)*(n : ℝ)^(-1/4 : ℝ)) ≤ nonprivateMinimaxRisk n ∧
nonprivateMinimaxRisk n ≤ ofReal (2^18*(n : ℝ)^(-1/4 : ℝ)) := by
exact ⟨nonprivate_sampling_lower boundedDifferences_of_gate cayley_of_gate n hn,
nonprivate_sampling_upper efronStein_of_gate boundedDifferences_of_gate cayley_of_gate n hn⟩
theorem regime_and_sanity
The benchmark has four exact regimes, its unit-budget scale is exactly the nonprivate fourth-root order, and private minimax consistency holds exactly when sample size times the privacy budget diverges. The Efron--Stein gate, bounded-differences gate, and Cayley gate specify the conditional scope.
Formal statement
hypotheses
efronStein_of_gate :
boundedDifferences_of_gate :
cayley_of_gate :
conclusion
conclusion 1
n :
ℕ
epsilon :
ℝ
2 ≤ n
0 < epsilon ∧ epsilon ≤ 1
((n : ℝ)^(-1/4 : ℝ) ≤ epsilon)
rate n epsilon = (n : ℝ)^(-1/4 : ℝ)
((n : ℝ)^(-3/5 : ℝ) ≤ epsilon ∧ epsilon ≤ (n : ℝ)^(-1/4 : ℝ))
rate n epsilon = ((n : ℝ)^2*epsilon)^(-1/7 : ℝ)
((n : ℝ)^(-1 : ℤ) ≤ epsilon ∧ epsilon ≤ (n : ℝ)^(-3/5 : ℝ))
rate n epsilon = ((n : ℝ)*epsilon)^(-1/2 : ℝ)
epsilon ≤ (n : ℝ)^(-1 : ℤ)
rate n epsilon = 1
rate n 1 = (n : ℝ)^(-1/4 : ℝ)
Proof (Lean source)
-- @node: prop:regime-and-sanity
theorem regime_and_sanity (efronStein_of_gate : EfronSteinReplacement)
(boundedDifferences_of_gate : BoundedDifferencesMGF) (cayley_of_gate : CayleyLabeledTreeCount) :
(∀ (n : ℕ) (epsilon : ℝ), 2 ≤ n → 0 < epsilon ∧ epsilon ≤ 1 →
(((n : ℝ)^(-1/4 : ℝ) ≤ epsilon) → rate n epsilon = (n : ℝ)^(-1/4 : ℝ)) ∧
(((n : ℝ)^(-3/5 : ℝ) ≤ epsilon ∧ epsilon ≤ (n : ℝ)^(-1/4 : ℝ)) →
rate n epsilon = ((n : ℝ)^2*epsilon)^(-1/7 : ℝ)) ∧
(((n : ℝ)^(-1 : ℤ) ≤ epsilon ∧ epsilon ≤ (n : ℝ)^(-3/5 : ℝ)) →
rate n epsilon = ((n : ℝ)*epsilon)^(-1/2 : ℝ)) ∧
(epsilon ≤ (n : ℝ)^(-1 : ℤ) → rate n epsilon = 1) ∧
rate n 1 = (n : ℝ)^(-1/4 : ℝ)) ∧
(∀ budgets : ℕ → ℝ, (∀ j, 2 ≤ j → 0 < budgets j ∧ budgets j ≤ 1) →
(Tendsto (fun j => privateMinimaxRisk j (budgets j)) atTop (𝓝 0) ↔
Tendsto (fun j : ℕ => (j : ℝ)*budgets j) atTop atTop)) := by
constructor
· intro n epsilon hn he
exact ⟨rate_sampling_regime n epsilon hn he.1,
rate_sparse_regime n epsilon hn he.1,
rate_dense_regime n epsilon hn he.1,
rate_capped_regime n epsilon hn he.1,
rate_unit_budget n hn⟩
· intro budgets he
exact private_risk_consistency_iff efronStein_of_gate boundedDifferences_of_gate
cayley_of_gate budgets he
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.TSharpIntervalFrontier 9 declarations Honest private expected-length frontier, explicit inversion identity, and uniform equivalence to the scalar absolute-risk frontier.
Honest private expected-length frontier, explicit inversion identity, and uniform equivalence to the scalar absolute-risk frontier.
Clipping a numerator commutes with division by a positive denominator. the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
b z :
ℝ
hb :
0 < b
conclusion
clip (-b) b z / b = clip (-1) 1 (z / b)
Proof (Lean source)
-- @node: clip_div_positive
lemma clip_div_positive (b z : ℝ) (hb : 0 < b) :
clip (-b) b z / b = clip (-1) 1 (z / b) := by
unfold clip
rw [← max_div_div_right hb.le, ← min_div_div_right hb.le]
simp [hb.ne']
At legal public parameters the acceptance hull is the released interval. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
n k :
ℕ
epsilon h :
ℝ
hn :
0 < n
hk :
0 < k
hh :
0 < h
conclusion
inversionMap n epsilon h k = intervalMap n epsilon h k
Proof (Lean source)
-- @node: inversionMap_eq_intervalMap
lemma inversionMap_eq_intervalMap (n k : ℕ) (epsilon h : ℝ)
(hn : 0 < n) (hk : 0 < k) (hh : 0 < h) :
inversionMap n epsilon h k = intervalMap n epsilon h k := by
funext v
have hb : 0 < max (d0 n h k) (v 1) :=
lt_of_lt_of_le (info_d0_pos n k h hn hk hh).2 (le_max_left _ _)
rw [inversionMap_eq_endpoints, if_pos hb, clip_div_positive _ _ hb]
rfl
theorem optimalIntervalHandle_eq_publicTunedInterval
Public tuning uses the same interval on both release branches. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
n :
ℕ
epsilon :
ℝ
hn :
2 ≤ n
he :
0 < epsilon ∧ epsilon ≤ 1
conclusion
optimalIntervalHandle n epsilon = publicTunedInterval n epsilon
Proof (Lean source)
-- @node: optimalIntervalHandle_eq_publicTunedInterval
lemma optimalIntervalHandle_eq_publicTunedInterval (n : ℕ) (epsilon : ℝ)
(hn : 2 ≤ n) (he : 0 < epsilon ∧ epsilon ≤ 1) :
optimalIntervalHandle n epsilon = publicTunedInterval n epsilon := by
classical
by_cases hr : 1 / 8 ≤ rate n epsilon
· simp only [optimalIntervalHandle, publicTunedInterval, if_pos hr]
· have hp := public_tuning_parameters n epsilon hn he (lt_of_not_ge hr)
simp only [optimalIntervalHandle, publicTunedInterval, hr, ↓reduceIte, Ihk]
congr 1
exact inversionMap_eq_intervalMap n (tunedK n epsilon) epsilon
(tunedH n epsilon) (by omega) (by omega) hp.1
An interval containing two ordered points contains the open span between them. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
Proof (Lean source)
-- @node: interval_span_subset
lemma interval_span_subset (c : IntervalCode) (a b : ℝ)
(ha : a ∈ c.toSet) (hb : b ∈ c.toSet) : Ioo a b ⊆ c.toSet := by
cases c with
| inl u => simp [IntervalCode.toSet] at ha
| inr p =>
rcases p with ⟨l, u, lc, uc⟩
intro x hx
simp only [IntervalCode.toSet, Set.mem_ofPred_eq] at ha hb ⊢
have hax := EReal.coe_lt_coe hx.1
have hxb := EReal.coe_lt_coe hx.2
constructor
· cases lc <;> simp only [Bool.false_eq_true, ↓reduceIte] at ha ⊢
· exact ha.1.trans hax
· exact (ha.1.trans_lt hax).le
· cases uc <;> simp only [Bool.false_eq_true, ↓reduceIte] at hb ⊢
· exact hxb.trans hb.2
· exact (hxb.trans_le hb.2).le
Simultaneous containment forces Lebesgue length even for unbounded codes. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
ofReal Delta ≤ c.leb
Proof (Lean source)
-- @node: interval_leb_ge_span
lemma interval_leb_ge_span (c : IntervalCode) (Delta : ℝ)
(h0 : 0 ∈ c.toSet) (h1 : Delta ∈ c.toSet) :
ofReal Delta ≤ c.leb := by
have h := measure_mono (μ := (volume : Measure ℝ))
(interval_span_subset c 0 Delta h0 h1)
simpa [IntervalCode.leb, Real.volume_Ioo] using h
theorem finite_prior_coverage_lower
A finite class prior inherits honesty at its common target. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
n s :
ℕ
hs :
0 < s
I :
target :
ℝ
hF :
∀ i, CompleteModel (F i) ∧ theta (F i) = target
hhonest :
∀ P
if
then
9 / 10 ≤ coverage n I P
conclusion
9 / 10 ≤ (I ∘ₘ finiteProductMixture n s F).real {c | target ∈ c.toSet}
Proof (Lean source)
-- @node: finite_prior_coverage_lower
lemma finite_prior_coverage_lower (n s : ℕ) (F : Fin s → CausalLaw)
(hs : 0 < s) (I : Kernel (Dataset n) IntervalCode) [IsMarkovKernel I]
(target : ℝ) (hF : ∀ i, CompleteModel (F i) ∧ theta (F i) = target)
(hhonest : ∀ P, CompleteModel P → 9 / 10 ≤ coverage n I P) :
9 / 10 ≤ (I ∘ₘ finiteProductMixture n s F).real {c | target ∈ c.toSet} := by
let : IsProbabilityMeasure (finiteProductMixture n s F) :=
finiteProductMixture_probability n s F hs
have hmass : ofReal (9 / 10 : ℝ) ≤
(I ∘ₘ finiteProductMixture n s F) {c | target ∈ c.toSet} := by
rw [finiteProductMixture, Measure.comp_smul, prior_comp_finsetSum,
Measure.smul_apply, smul_eq_mul, Measure.finsetSum_apply]
have hrow : ∀ i : Fin s, ofReal (9 / 10 : ℝ) ≤
(I ∘ₘ dataLaw n (F i)) {c | target ∈ c.toSet} := by
intro i
let : IsProbabilityMeasure (Pobs (F i)) :=
Measure.isProbabilityMeasure_map measurable_observe.aemeasurable
let : IsProbabilityMeasure (dataLaw n (F i)) := by unfold dataLaw; infer_instance
have hc := hhonest (F i) (hF i).1
unfold coverage at hc
rw [(hF i).2] at hc
rw [← ENNReal.ofReal_toReal (measure_ne_top (I ∘ₘ dataLaw n (F i)) _)]
exact ENNReal.ofReal_le_ofReal hc
calc
_ = (s : ℝ≥0∞)⁻¹ * ∑ _i : Fin s, ofReal (9 / 10 : ℝ) := by
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul]
rw [← mul_assoc, ENNReal.inv_mul_cancel
(by exact_mod_cast (Nat.ne_of_gt hs)) (by simp), one_mul]
_ ≤ _ := mul_le_mul_of_nonneg_left (Finset.sum_le_sum fun i _ => hrow i) (zero_le)
exact (ENNReal.ofReal_le_iff_le_toReal (measure_ne_top _ _)).mp hmass
theorem finite_prior_length_le_worstLength
Prior expected length is no greater than maximal class expected length. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
conclusion
(∫⁻ c, c.leb ∂(I ∘ₘ finiteProductMixture n s F)) ≤ worstLength n I
Proof (Lean source)
-- @node: finite_prior_length_le_worstLength
lemma finite_prior_length_le_worstLength (n s : ℕ) (F : Fin s → CausalLaw)
(hs : 0 < s) (I : Kernel (Dataset n) IntervalCode)
(hF : ∀ i, CompleteModel (F i)) :
(∫⁻ c, c.leb ∂(I ∘ₘ finiteProductMixture n s F)) ≤ worstLength n I := by
rw [finiteProductMixture, Measure.comp_smul, prior_comp_finsetSum,
lintegral_smul_measure, lintegral_finsetSum_measure]
have hbound : ∀ i : Fin s, (∫⁻ c, c.leb ∂(I ∘ₘ dataLaw n (F i))) ≤
worstLength n I := fun i => le_iSup_of_le (F i) (le_iSup_of_le (hF i) le_rfl)
calc
_ ≤ (s : ℝ≥0∞)⁻¹ * ∑ _i : Fin s, worstLength n I :=
mul_le_mul_of_nonneg_left (Finset.sum_le_sum fun i _ => hbound i) (zero_le)
_ = worstLength n I := by
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul]
rw [← mul_assoc, ENNReal.inv_mul_cancel
(by exact_mod_cast (Nat.ne_of_gt hs)) (by simp), one_mul]
theorem finite_priors_interval_lower
Two honest finite priors force simultaneous containment under the null output law. the theorem's stated inputs and assumptions, and the asserted conclusion follows. the theorem's stated inputs and assumptions.
Formal statement
hypotheses
n s0 s1 :
ℕ
hs0 :
0 < s0
hs1 :
0 < s1
I :
Delta :
ℝ
hd :
0 ≤ Delta
hF0 :
∀ i, CompleteModel (F0 i) ∧ theta (F0 i) = 0
hF1 :
∀ i, CompleteModel (F1 i) ∧ theta (F1 i) = Delta
hhonest :
∀ P
if
then
9 / 10 ≤ coverage n I P
htv :
conclusion
ofReal ((11 / 20) * Delta) ≤ worstLength n I
Proof (Lean source)
-- @node: finite_priors_interval_lower
lemma finite_priors_interval_lower (n s0 s1 : ℕ)
(F0 : Fin s0 → CausalLaw) (F1 : Fin s1 → CausalLaw)
(hs0 : 0 < s0) (hs1 : 0 < s1) (I : Kernel (Dataset n) IntervalCode)
[IsMarkovKernel I] (Delta : ℝ) (hd : 0 ≤ Delta)
(hF0 : ∀ i, CompleteModel (F0 i) ∧ theta (F0 i) = 0)
(hF1 : ∀ i, CompleteModel (F1 i) ∧ theta (F1 i) = Delta)
(hhonest : ∀ P, CompleteModel P → 9 / 10 ≤ coverage n I P)
(htv : TV (I ∘ₘ finiteProductMixture n s0 F0)
(I ∘ₘ finiteProductMixture n s1 F1) ≤ 1 / 4) :
ofReal ((11 / 20) * Delta) ≤ worstLength n I := by
let L0 := I ∘ₘ finiteProductMixture n s0 F0
let L1 := I ∘ₘ finiteProductMixture n s1 F1
let : IsProbabilityMeasure (finiteProductMixture n s0 F0) :=
finiteProductMixture_probability n s0 F0 hs0
let : IsProbabilityMeasure (finiteProductMixture n s1 F1) :=
finiteProductMixture_probability n s1 F1 hs1
let E0 : Set IntervalCode := {c | 0 ∈ c.toSet}
let E1 : Set IntervalCode := {c | Delta ∈ c.toSet}
have hm0 : MeasurableSet E0 := measurableSet_interval_contains 0
have hm1 : MeasurableSet E1 := measurableSet_interval_contains Delta
have hc0 : 9 / 10 ≤ L0.real E0 :=
finite_prior_coverage_lower n s0 F0 hs0 I 0 hF0 hhonest
have hc1 : 9 / 10 ≤ L1.real E1 :=
finite_prior_coverage_lower n s1 F1 hs1 I Delta hF1 hhonest
have hgap := measureReal_sub_le_tvDist (μ := L0) (ν := L1) hm1
change TV L0 L1 ≤ 1 / 4 at htv
have hboth : 11 / 20 ≤ L0.real (E0 ∩ E1) := by
have hu := measureReal_union_add_inter (μ := L0) (s := E0) hm1
have hb : L0.real (E0 ∪ E1) ≤ 1 := measureReal_le_one
linarith
have hmass : ofReal (11 / 20 : ℝ) ≤ L0 (E0 ∩ E1) := by
rw [← ENNReal.ofReal_toReal (measure_ne_top L0 _)]
exact ENNReal.ofReal_le_ofReal hboth
have hind : (∫⁻ c, (E0 ∩ E1).indicator (fun _ => ofReal Delta) c ∂L0) =
ofReal Delta * L0 (E0 ∩ E1) := by
rw [lintegral_indicator (hm0.inter hm1)]
simp
calc
_ = ofReal Delta * ofReal (11 / 20 : ℝ) := by
rw [← ENNReal.ofReal_mul hd]
congr 1
ring
_ ≤ ofReal Delta * L0 (E0 ∩ E1) :=
mul_le_mul_of_nonneg_left hmass (zero_le)
_ = ∫⁻ c, (E0 ∩ E1).indicator (fun _ => ofReal Delta) c ∂L0 := hind.symm
_ ≤ ∫⁻ c, c.leb ∂L0 := by
apply lintegral_mono
intro c
by_cases hc : c ∈ E0 ∩ E1
· rw [Set.indicator_of_mem hc]
exact interval_leb_ge_span c Delta hc.1 hc.2
· rw [Set.indicator_of_notMem hc]
exact zero_le
_ ≤ worstLength n I := finite_prior_length_le_worstLength n s0 F0 hs0 I
(fun i => (hF0 i).1)
theorem sharp_interval_frontier
The explicit inversion kernel equals the conservative tuned interval and attains honest coverage and optimal maximal expected length. The scalar and interval frontiers are uniformly comparable with the stated numerical constants. The public parameters and their domains and the Efron--Stein gate, bounded-differences gate, and Cayley gate specify the conditional scope.
Formal statement
hypotheses
efronStein_of_gate :
boundedDifferences_of_gate :
cayley_of_gate :
n :
ℕ
epsilon :
ℝ
hn :
2 ≤ n
he :
0 < epsilon ∧ epsilon ≤ 1
conclusion
conclusion 2
cI = (2 : ℝ)^(-20 : ℤ)
conclusion 3
CI = (2 : ℝ)^22
conclusion 4
0 < ell n epsilon
conclusion 5
0 < cI
conclusion 6
0 < CI
conclusion 7
optimalIntervalHandle n epsilon = publicTunedInterval n epsilon
conclusion 8
IsMarkovKernel (optimalIntervalHandle n epsilon)
conclusion 9
PrivateKernel n epsilon (optimalIntervalHandle n epsilon)
conclusion 10
conclusion 11
conclusion 12
conclusion 14
ofReal ((2 : ℝ)^(-38 : ℤ))*privateMinimaxRisk n epsilon
≤ privateHonestLength n epsilon
conclusion 15
Proof (Lean source)
-- @node: thm:sharp-interval-frontier
theorem sharp_interval_frontier (efronStein_of_gate : EfronSteinReplacement)
(boundedDifferences_of_gate : BoundedDifferencesMGF) (cayley_of_gate : CayleyLabeledTreeCount)
(n : ℕ) (epsilon : ℝ) (hn : 2 ≤ n) (he : 0 < epsilon ∧ epsilon ≤ 1) :
ell n epsilon = rate n epsilon ∧ cI = (2 : ℝ)^(-20 : ℤ) ∧ CI = (2 : ℝ)^22 ∧
0 < ell n epsilon ∧ 0 < cI ∧ 0 < CI ∧
optimalIntervalHandle n epsilon = publicTunedInterval n epsilon ∧
IsMarkovKernel (optimalIntervalHandle n epsilon) ∧
PrivateKernel n epsilon (optimalIntervalHandle n epsilon) ∧
ofReal ((2 : ℝ)^(-20 : ℤ)*rate n epsilon) ≤ privateHonestLength n epsilon ∧
privateHonestLength n epsilon ≤ worstLength n (optimalIntervalHandle n epsilon) ∧
worstLength n (optimalIntervalHandle n epsilon) ≤ ofReal (2^22*rate n epsilon) ∧
(∀ P, CompleteModel P → 9/10 ≤ coverage n (optimalIntervalHandle n epsilon) P) ∧
ofReal ((2 : ℝ)^(-38 : ℤ))*privateMinimaxRisk n epsilon ≤
privateHonestLength n epsilon ∧
privateHonestLength n epsilon ≤ ofReal (2^42)*privateMinimaxRisk n epsilon := by
have hr : 0 < rate n epsilon := rate_pos n epsilon (by omega)
have hhandle := optimalIntervalHandle_eq_publicTunedInterval n epsilon hn he
obtain ⟨s0, s1, F0, F1, Delta, hs0, hs1, hd, hF0, hF1, _, htv⟩ :=
frontier_testing_priors boundedDifferences_of_gate cayley_of_gate n epsilon hn he
have hd0 : 0 ≤ Delta := le_trans (by positivity) hd
have hlower : ofReal ((2 : ℝ)^(-20 : ℤ) * rate n epsilon) ≤
privateHonestLength n epsilon := by
unfold privateHonestLength
refine le_iInf fun I => le_iInf fun hI => ?_
let : IsMarkovKernel I := hI.1
have htest := finite_priors_interval_lower n s0 s1 F0 F1 hs0 hs1 I Delta hd0
hF0 hF1 hI.2.2 (htv IntervalCode I hI.2.1)
apply le_trans (ENNReal.ofReal_le_ofReal ?_) htest
have hscale : (2 : ℝ)^(-20 : ℤ) ≤ (11 / 20) * (2 : ℝ)^(-14 : ℤ) := by norm_num
calc
_ ≤ ((11 / 20) * (2 : ℝ)^(-14 : ℤ)) * rate n epsilon :=
mul_le_mul_of_nonneg_right hscale hr.le
_ = (11 / 20) * ((2 : ℝ)^(-14 : ℤ) * rate n epsilon) := by ring
_ ≤ (11 / 20) * Delta := mul_le_mul_of_nonneg_left hd (by norm_num)
obtain ⟨_, _, _, _, _, _, _, _, _, hmarkov, hprivate, _, hlength, hcoverage⟩ :=
uniform_private_upper efronStein_of_gate n 2 epsilon (1 / 4) hn (by omega) he
⟨by norm_num, le_rfl⟩
have hfeasible : privateHonestLength n epsilon ≤
worstLength n (publicTunedInterval n epsilon) :=
iInf_le_of_le (publicTunedInterval n epsilon)
(iInf_le_of_le ⟨hmarkov, hprivate, hcoverage⟩ le_rfl)
have hfront := matched_risk_frontier efronStein_of_gate boundedDifferences_of_gate
cayley_of_gate n epsilon hn he
have hcompareLower : ofReal ((2 : ℝ)^(-38 : ℤ)) *
privateMinimaxRisk n epsilon ≤ privateHonestLength n epsilon := by
calc
_ ≤ ofReal ((2 : ℝ)^(-38 : ℤ)) *
ofReal (2^18 * rate n epsilon) :=
mul_le_mul_of_nonneg_left hfront.2.1 (zero_le)
_ = ofReal ((2 : ℝ)^(-20 : ℤ) * rate n epsilon) := by
rw [← ENNReal.ofReal_mul (by positivity)]
congr 1
norm_num
ring
_ ≤ _ := hlower
have hcompareUpper : privateHonestLength n epsilon ≤
ofReal (2^42) * privateMinimaxRisk n epsilon := by
calc
_ ≤ ofReal (2^22 * rate n epsilon) := hfeasible.trans hlength
_ = ofReal (2^42 : ℝ) *
ofReal ((2 : ℝ)^(-20 : ℤ) * rate n epsilon) := by
rw [← ENNReal.ofReal_mul (by positivity)]
congr 1
norm_num
ring
_ ≤ _ := mul_le_mul_of_nonneg_left hfront.1 (zero_le)
rw [hhandle]
exact ⟨rfl, rfl, rfl, hr, by norm_num [cI], by norm_num [CI], rfl,
hmarkov, hprivate, hlower, hfeasible, hlength, hcoverage, hcompareLower, hcompareUpper⟩
CausalSmith.Stat.STAT_PrivateCateRoughdesignFrontier_Research.TUniformPrivateUpper 1 declarations Assembly of the uniform private scalar-risk, coverage and length guarantees.
Assembly of the uniform private scalar-risk, coverage and length guarantees.
theorem uniform_private_upper
The constructed kernels are Markov and pure private on every input. Their second moments, coverage and interval lengths obey the public envelopes throughout the complete model, and public tuning attains the stated scalar-risk and interval-length constants. The public parameters and their domains and published substrate gates specify the conditional scope.
Formal statement
hypotheses
efronStein_of_gate :
n k :
ℕ
epsilon h :
ℝ
hn :
2 ≤ n
@realizes n(sample size at least two)
hk :
2 ≤ k
@realizes k(cell count at least two)
he :
0 < epsilon ∧ epsilon ≤ 1
@realizes epsilon(budget in (0,1])
hh :
0 < h ∧ h ≤ 1 / 4
conclusion
conclusion 1
IsMarkovKernel (privateRatioRelease n epsilon h k)
conclusion 2
PrivateKernel n epsilon (privateRatioRelease n epsilon h k)
conclusion 3
IsMarkovKernel (Thk n epsilon h k)
conclusion 4
PrivateKernel n epsilon (Thk n epsilon h k)
conclusion 5
IsMarkovKernel (Ihk n epsilon h k)
conclusion 6
PrivateKernel n epsilon (Ihk n epsilon h k)
conclusion 7
P :
conclusion 8
IsMarkovKernel (publicTunedRelease n epsilon)
conclusion 9
PrivateKernel n epsilon (publicTunedRelease n epsilon)
conclusion 10
IsMarkovKernel (publicTunedInterval n epsilon)
conclusion 11
PrivateKernel n epsilon (publicTunedInterval n epsilon)
conclusion 12
conclusion 13
Proof (Lean source)
-- @node: thm:uniform-private-upper
theorem uniform_private_upper (efronStein_of_gate : EfronSteinReplacement)
(n k : ℕ) (epsilon h : ℝ)
(hn : 2 ≤ n) -- @realizes n(sample size at least two)
(hk : 2 ≤ k) -- @realizes k(cell count at least two)
(he : 0 < epsilon ∧ epsilon ≤ 1) -- @realizes epsilon(budget in (0,1])
(hh : 0 < h ∧ h ≤ 1 / 4) : -- @realizes h(radius in (0,1/4])
IsMarkovKernel (privateRatioRelease n epsilon h k) ∧
PrivateKernel n epsilon (privateRatioRelease n epsilon h k) ∧
IsMarkovKernel (Thk n epsilon h k) ∧ PrivateKernel n epsilon (Thk n epsilon h k) ∧
IsMarkovKernel (Ihk n epsilon h k) ∧ PrivateKernel n epsilon (Ihk n epsilon h k) ∧
(∀ P : CausalLaw, CompleteModel P →
(∫⁻ u, ofReal ((u-theta P)^2) ∂(Thk n epsilon h k ∘ₘ dataLaw n P)) ≤
ofReal (Vbound n epsilon h k) ∧
9/10 ≤ coverage n (Ihk n epsilon h k) P ∧
expectedLength n (Ihk n epsilon h k) P ≤
ofReal (min 2 (2*sqrt (10*Vbound n epsilon h k)))) ∧
IsMarkovKernel (publicTunedRelease n epsilon) ∧
PrivateKernel n epsilon (publicTunedRelease n epsilon) ∧
IsMarkovKernel (publicTunedInterval n epsilon) ∧
PrivateKernel n epsilon (publicTunedInterval n epsilon) ∧
worstRisk n (publicTunedRelease n epsilon) ≤ ofReal (2^18*rate n epsilon) ∧
worstLength n (publicTunedInterval n epsilon) ≤ ofReal (2^22*rate n epsilon) ∧
(∀ P, CompleteModel P → 9/10 ≤ coverage n (publicTunedInterval n epsilon) P) := by
have hpair := privateRatioRelease_private n k epsilon h he.1 hh.1 (by omega)
have hscalar := Thk_private n k epsilon h he.1 hh.1 (by omega)
have hinterval := Ihk_private n k epsilon h he.1 hh.1 (by omega)
refine ⟨hpair.markov, hpair, hscalar.markov, hscalar, hinterval.markov, hinterval, ?_⟩
have htuned := publicTunedRelease_private n epsilon hn he
have htunedInterval := publicTunedInterval_private n epsilon hn he
suffices hstats :
∀ (k' : ℕ) (h' : ℝ), 2 ≤ k' → (0 < h' ∧ h' ≤ 1/4) →
∀ P : CausalLaw, CompleteModel P →
(∫⁻ u, ofReal ((u-theta P)^2) ∂(Thk n epsilon h' k' ∘ₘ dataLaw n P)) ≤
ofReal (Vbound n epsilon h' k') by
refine ⟨?_, htuned.markov, htuned, htunedInterval.markov, htunedInterval,
publicTunedRelease_worstRisk_of_squared_error_le n epsilon hn he hstats,
publicTunedInterval_worstLength_le n epsilon hn he, ?_⟩
· intro P hP
have hmse := hstats k h hk hh P hP
exact ⟨hmse, Ihk_coverage_of_squared_error_le n k epsilon h
(by omega) (by omega) he.1 hh.1 P hP hmse,
Ihk_expectedLength_le n k epsilon h he.1 P⟩
· intro P hP
apply publicTunedInterval_coverage_of_squared_error_le n epsilon hn he P hP
intro hr
have hp := public_tuning_parameters n epsilon hn he hr
exact hstats _ _ hp.2.2.1 ⟨hp.1, hp.2.1⟩ P hP
intro k' h' hk' hh' P hP
exact Thk_model_squared_error_le efronStein_of_gate n k' epsilon h' hn hk' he.1 hh' P hP