Formalization: One Intervention per Latent Variable: Generic Identification of Nonlinear Causal Representations on Fixed-sign Compact Strata
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.ExactID.EID_CrlCoverratioMmdGenericity_Research.Basic 29 declarations This file fixes the latent finite-DAG mechanism, its observed environment family, and the population assumptions shared by the paper's results.
Cover-ratio causal representation model
This file fixes the latent finite-DAG mechanism, its observed environment family, and the population assumptions shared by the paper's results.
For a finite dimension, the latent state space is the real coordinate space.
Definition (Lean source)
def latentCube
The closed latent cube.
Definition (Lean source)
A sign vector with every coordinate equal to -1 or 1.
Definition (Lean source)
A family of observational conditional densities and parent-independent intervention densities. The proof field makes the parent scope part of the carrier.
Definition (Lean source)
parameters
fields
p :
Fin n → LatentState n → ℝ
q :
Fin n → ℝ → ℝ
parent_local :
∀ i v w
if
v i = w i
and
(∀ j ∈ G.parents i, v j = w j)
then
p i v = p i w
def mechanismC2Coordinates
Values and the first two derivatives within the closed domain of every mechanism slot, each equipped with uniform convergence there. Pulling this topology back gives the paper's finite relative product C² topology, including its one-sided boundary derivatives, rather than Lean's default pointwise function topology.
Definition (Lean source)
parameters
def
mechanismC2Coordinates θ :
(Fin n → UniformOnFun (LatentState n) ℝ {latentCube n}) × (Fin n → UniformOnFun (LatentState n) (LatentState n →L[ℝ] ℝ) {latentCube n}) × (Fin n → UniformOnFun (LatentState n) (LatentState n [×2]→L[ℝ] ℝ) {latentCube n}) × (Fin n → UniformOnFun ℝ ℝ {Icc 0 1}) × (Fin n → UniformOnFun ℝ (ℝ →L[ℝ] ℝ) {Icc 0 1}) × (Fin n → UniformOnFun ℝ (ℝ [×2]→L[ℝ] ℝ) {Icc 0 1})
given by
((fun i => ofFun {latentCube n} (θ.p i)), (fun i => ofFun {latentCube n} (fderivWithin ℝ (θ.p i) (latentCube n))), (fun i => ofFun {latentCube n} (iteratedFDerivWithin ℝ 2 (θ.p i) (latentCube n))), (fun i => ofFun {Icc (0 : ℝ) 1} (θ.q i)), (fun i => ofFun {Icc (0 : ℝ) 1} (fderivWithin ℝ (θ.q i) (Icc (0 : ℝ) 1))), (fun i => ofFun {Icc (0 : ℝ) 1} (iteratedFDerivWithin ℝ 2 (θ.q i) (Icc (0 : ℝ) 1))))
instance instTopologicalSpaceMechanism
For a finite latent dimension and DAG, the mechanism space carries the relative product C² topology.
Definition (Lean source)
instance {n : ℕ} {G : DAG (Fin n)} : TopologicalSpace (Mechanism n G) :=
TopologicalSpace.induced mechanismC2Coordinates inferInstance
The intervention distribution function Q_i(v) = ∫₀ᵛ q_i(u) du.
The observational product density.
Definition (Lean source)
parameters
def
observationalDensity θ v :
ℝ
given by
∏ i, θ.p i v
def interventionalDensity
The density after replacing exactly the target mechanism by q_i.
Definition (Lean source)
parameters
def
interventionalDensity θ i v :
ℝ
given by
θ.q i (v i) * ∏ l ∈ Finset.univ.erase i, θ.p l v
def observationalLaw
The observational law on the compact latent cube.
Definition (Lean source)
parameters
def
given by
(volume.restrict (latentCube n)).withDensity (fun v => ofReal (observationalDensity θ v))
def interventionalLaw
The target-i perfect-intervention law on the latent cube.
Definition (Lean source)
parameters
def
given by
def coordinateProjection
Finite-coordinate projection, used to state conditional independence.
Definition (Lean source)
parameters
def
coordinateProjection S v j :
ℝ
given by
fun j => v j
Conditional independence of latent coordinate blocks under the observational law.
Definition (Lean source)
parameters
def
CondIndepCoordinates θ X Y Z :
Prop
given by
∃ hμ : IsFiniteMeasure (observationalLaw θ),
letI := hμ ∃ _hX : Measurable (coordinateProjection X), ∃ _hY : Measurable (coordinateProjection Y), ∃ hZ : Measurable (coordinateProjection Z), CondIndepFun (comap (coordinateProjection Z) inferInstance) hZ.comap_le (coordinateProjection X) (coordinateProjection Y) (observationalLaw θ)
def ownLogRatioDerivative
The own-coordinate derivative of the latent log density ratio.
Definition (Lean source)
parameters
def
ownLogRatioDerivative θ i v :
ℝ
given by
For a finite node set and DAG, the ancestral cover relation consists of cover pairs in the DAG's ancestor order.
def PositiveNormalizedSmoothMechanisms
Every observational and intervention mechanism is a positive normalized C³ density on its closed cube, with each conditional normalized in its own coordinate.
Definition (Lean source)
parameters
def
PositiveNormalizedSmoothMechanisms G θ :
Prop
given by
def FixedOwnDerivativeSign
Each latent log ratio has the prescribed strict own-coordinate derivative sign.
Definition (Lean source)
parameters
def
FixedOwnDerivativeSign G s θ :
Prop
given by
0 < s.value i * ownLogRatioDerivative θ i v
def CausalMinimality
Every direct causal edge remains conditionally dependent given the other parents.
def Faithfulness
Every observational conditional independence is graphically entailed by d-separation.
structure ModelStratum
The positive, normalized, smooth, causal-minimal, fixed-sign mechanism stratum.
Definition (Lean source)
parameters
assumes
def StratumPoint
The mechanism subtype carrying the relative product C² topology.
Definition (Lean source)
The shared observed-space representation and the supplied family of observed laws and ratios.
Definition (Lean source)
parameters
fields
mix :
LatentState n → LatentState n
unmix :
LatentState n → LatentState n
law :
ratio :
Fin n → LatentState n → ℝ
The identity-mixing environment family generated directly by a mechanism and target permutation. This is used for the explicit witness calculations.
Definition (Lean source)
parameters
def
given by
clause 1
mix := id
clause 2
unmix := id
clause 3
targetPerm := π
clause 4
law := Fin.cases (observationalLaw θ) (fun e => interventionalLaw θ (π e))
clause 5
ratio := fun e v => θ.q (π e) (v (π e)) / θ.p (π e) v
def observedSupport
The observed support is the image of the latent cube.
Definition (Lean source)
def observation
The observed random vector obtained from a latent state.
Definition (Lean source)
parameters
def
given by
W.mix v
The environment-label DAG obtained by pulling G back through the target permutation.
Definition (Lean source)
parameters
def
permutedGraph G W j i :
Prop
given by
G.edge (W.targetPerm j) (W.targetPerm i)
def logRatio
The observable log-ratio coordinate.
Definition (Lean source)
The observed laws are the shared pushforwards of one observational and one distinct single-target intervention law, and the supplied ratios are their Radon--Nikodym derivatives.
Definition (Lean source)
parameters
def
OnePerfectInterventionPerNode G θ W :
Prop
given by
clause 1
W.law 0 = Measure.map W.mix (observationalLaw θ)
clause 4
e v :
v ∈ latentCube n
W.ratio e (W.mix v) = θ.q (W.targetPerm e) (v (W.targetPerm e)) / θ.p (W.targetPerm e) v
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.AffinePathTopology 10 declarations This file records compact-uniform continuity of all six value and within-derivative coordinates of the affine mechanism path, and hence continuity in the induced relative product C² topology.
Topology of the affine witness path
This file records compact-uniform continuity of all six value and within-derivative coordinates of
the affine mechanism path, and hence continuity in the induced relative product C² topology.
theorem derivWithin_coordinateSection_eq_fderivWithin_apply
Restricting a differentiable function on the product cube to one coordinate turns its within Fréchet derivative into evaluation on the corresponding coordinate basis vector. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
n :
ℕ
(Fin n → ℝ) → ℝ
hF :
DifferentiableOn ℝ F (latentCube n)
v :
Fin n → ℝ
hv :
v ∈ latentCube n
i :
Fin n
z :
ℝ
hz :
z ∈ Icc (0 : ℝ) 1
conclusion
Proof (Lean source)
lemma derivWithin_coordinateSection_eq_fderivWithin_apply
{n : ℕ} {F : (Fin n → ℝ) → ℝ}
(hF : DifferentiableOn ℝ F (latentCube n))
{v : Fin n → ℝ} (hv : v ∈ latentCube n) (i : Fin n)
{z : ℝ} (hz : z ∈ Icc (0 : ℝ) 1) :
derivWithin (fun y ↦ F (update v i y)) (Icc (0 : ℝ) 1) z =
fderivWithin ℝ F (latentCube n) (update v i z) (single i 1) := by
have hmap : Set.MapsTo (update v i) (Icc (0 : ℝ) 1) (latentCube n) := by
intro y hy k
by_cases hki : k = i
· subst k
simpa using hy
· simpa [update, hki] using hv k
have hcomp := (hF (update v i z) (hmap hz)).hasFDerivWithinAt.comp z
(hasDerivAt_update v i z).hasFDerivAt.hasFDerivWithinAt hmap
simpa [Function.comp_def] using hcomp.hasDerivWithinAt.derivWithin
((uniqueDiffOn_Icc (by norm_num)).uniqueDiffWithinAt hz)
theorem continuous_affinePathExtension_p_values
Given the selected directed edge, the observational-factor value coordinate of the unrestricted affine path varies continuously in the compact-uniform topology.
Formal statement
hypotheses
conclusion
Proof (Lean source)
@[fun_prop] lemma continuous_affinePathExtension_p_values
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(θ : StratumPoint G s) {j i : Fin n} (hji : G.edge j i) (l : Fin n) :
Continuous (fun t : ℝ => ofFun {latentCube n}
((affinePathExtension s θ hji t).p l)) := by
apply continuous_uniformOnFun_affine_of_compact
· rw [latentCube]
exact isCompact_univ_pi fun _ => isCompact_Icc
· exact ⟨0, fun _ _ => ⟨by norm_num, by norm_num⟩⟩
· simpa only [smul_eq_mul] using (θ.property.positiveSmooth.2.2.1 l).continuousOn
· exact ((embeddedSparseWitness_positive_normalized_smooth s hji).2.2.1 l).continuousOn
theorem continuous_affinePathExtension_q_values
Given the selected directed edge, the intervention-factor value coordinate of the unrestricted affine path varies continuously in the compact-uniform topology.
Formal statement
hypotheses
conclusion
Proof (Lean source)
@[fun_prop] lemma continuous_affinePathExtension_q_values
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(θ : StratumPoint G s) {j i : Fin n} (hji : G.edge j i) (l : Fin n) :
Continuous (fun t : ℝ => ofFun {Icc (0 : ℝ) 1}
((affinePathExtension s θ hji t).q l)) := by
apply continuous_uniformOnFun_affine_of_compact isCompact_Icc
(nonempty_Icc.mpr (by norm_num))
· simpa only [smul_eq_mul] using (θ.property.positiveSmooth.2.2.2.1 l).continuousOn
· exact ((embeddedSparseWitness_positive_normalized_smooth s hji).2.2.2.1 l).continuousOn
theorem continuous_affinePathExtension_p_fderivWithin
Given the selected directed edge, the observational-factor first within-derivative coordinate varies continuously along the unrestricted affine path.
Formal statement
hypotheses
conclusion
Continuous (fun t : ℝ => ofFun {latentCube n} (fderivWithin ℝ ((affinePathExtension s θ hji t).p l) (latentCube n)))
Proof (Lean source)
@[fun_prop] lemma continuous_affinePathExtension_p_fderivWithin
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(θ : StratumPoint G s) {j i : Fin n} (hji : G.edge j i) (l : Fin n) :
Continuous (fun t : ℝ => ofFun {latentCube n}
(fderivWithin ℝ ((affinePathExtension s θ hji t).p l) (latentCube n))) := by
let hu : UniqueDiffOn ℝ (latentCube n) := by
rw [latentCube]
exact UniqueDiffOn.univ_pi fun _ => uniqueDiffOn_Icc (by norm_num)
let hθ := θ.property.positiveSmooth.2.2.1 l
let hstar := (embeddedSparseWitness_positive_normalized_smooth s hji).2.2.1 l
apply continuous_uniformOnFun_of_eq_affine_on_compact
(K := latentCube n)
(by rw [latentCube]; exact isCompact_univ_pi fun _ => isCompact_Icc)
(show (latentCube n).Nonempty from
⟨0, fun _ _ => ⟨by norm_num, by norm_num⟩⟩)
(hθ.continuousOn_fderivWithin hu (by norm_num))
(hstar.continuousOn_fderivWithin hu (by norm_num))
intro t x hx
change fderivWithin ℝ
(fun y => (1 - t) • θ.1.p l y + t • (embeddedSparseWitness s hji).p l y)
(latentCube n) x = _
rw [fderivWithin_fun_add (hu.uniqueDiffWithinAt hx)
(((hθ.const_smul (1 - t)) x hx).differentiableWithinAt (by norm_num))
(((hstar.const_smul t) x hx).differentiableWithinAt (by norm_num)),
fderivWithin_fun_const_smul (hu.uniqueDiffWithinAt hx)
((hθ x hx).differentiableWithinAt (by norm_num)) (1 - t),
fderivWithin_fun_const_smul (hu.uniqueDiffWithinAt hx)
((hstar x hx).differentiableWithinAt (by norm_num)) t]
theorem continuous_affinePathExtension_p_iteratedFDerivWithin_two
Given the selected directed edge, the observational-factor second within-derivative coordinate varies continuously along the unrestricted affine path.
Formal statement
hypotheses
conclusion
Continuous (fun t : ℝ => ofFun {latentCube n} (iteratedFDerivWithin ℝ 2 ((affinePathExtension s θ hji t).p l) (latentCube n)))
Proof (Lean source)
@[fun_prop] lemma continuous_affinePathExtension_p_iteratedFDerivWithin_two
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(θ : StratumPoint G s) {j i : Fin n} (hji : G.edge j i) (l : Fin n) :
Continuous (fun t : ℝ => ofFun {latentCube n}
(iteratedFDerivWithin ℝ 2 ((affinePathExtension s θ hji t).p l)
(latentCube n))) := by
let hu : UniqueDiffOn ℝ (latentCube n) := by
rw [latentCube]
exact UniqueDiffOn.univ_pi fun _ => uniqueDiffOn_Icc (by norm_num)
let hθ := θ.property.positiveSmooth.2.2.1 l
let hstar := (embeddedSparseWitness_positive_normalized_smooth s hji).2.2.1 l
let hθ2 : ContDiffOn ℝ 2 (θ.1.p l) (latentCube n) :=
hθ.of_le (by norm_num)
let hstar2 : ContDiffOn ℝ 2 ((embeddedSparseWitness s hji).p l) (latentCube n) :=
hstar.of_le (by norm_num)
apply continuous_uniformOnFun_of_eq_affine_on_compact
(K := latentCube n)
(by rw [latentCube]; exact isCompact_univ_pi fun _ => isCompact_Icc)
(show (latentCube n).Nonempty from
⟨0, fun _ _ => ⟨by norm_num, by norm_num⟩⟩)
(hθ.continuousOn_iteratedFDerivWithin (by norm_num) hu)
(hstar.continuousOn_iteratedFDerivWithin (by norm_num) hu)
intro t x hx
change iteratedFDerivWithin ℝ 2
(fun y => (1 - t) • θ.1.p l y + t • (embeddedSparseWitness s hji).p l y)
(latentCube n) x = _
have hscaleθ : iteratedFDerivWithin ℝ 2
(fun y => (1 - t) • θ.1.p l y) (latentCube n) x =
(1 - t) • iteratedFDerivWithin ℝ 2 (θ.1.p l) (latentCube n) x := by
change iteratedFDerivWithin ℝ 2 ((1 - t) • θ.1.p l) (latentCube n) x = _
exact iteratedFDerivWithin_const_smul_apply
(hθ2 x hx) hu hx
have hscaleStar : iteratedFDerivWithin ℝ 2
(fun y => t • (embeddedSparseWitness s hji).p l y) (latentCube n) x =
t • iteratedFDerivWithin ℝ 2 ((embeddedSparseWitness s hji).p l)
(latentCube n) x := by
change iteratedFDerivWithin ℝ 2
(t • (embeddedSparseWitness s hji).p l) (latentCube n) x = _
exact iteratedFDerivWithin_const_smul_apply
(hstar2 x hx) hu hx
rw [fun_iteratedFDerivWithin_add_apply
((hθ2.const_smul (1 - t)) x hx)
((hstar2.const_smul t) x hx) hu hx,
hscaleθ, hscaleStar]
theorem continuous_affinePathExtension_q_fderivWithin
Given the selected directed edge, the intervention-factor first within-derivative coordinate varies continuously along the unrestricted affine path.
Formal statement
hypotheses
conclusion
Continuous (fun t : ℝ => ofFun {Icc (0 : ℝ) 1} (fderivWithin ℝ ((affinePathExtension s θ hji t).q l) (Icc (0 : ℝ) 1)))
Proof (Lean source)
@[fun_prop] lemma continuous_affinePathExtension_q_fderivWithin
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(θ : StratumPoint G s) {j i : Fin n} (hji : G.edge j i) (l : Fin n) :
Continuous (fun t : ℝ => ofFun {Icc (0 : ℝ) 1}
(fderivWithin ℝ ((affinePathExtension s θ hji t).q l) (Icc (0 : ℝ) 1))) := by
let K := Icc (0 : ℝ) 1
let hu : UniqueDiffOn ℝ K := uniqueDiffOn_Icc (by norm_num)
let hθ := θ.property.positiveSmooth.2.2.2.1 l
let hstar := (embeddedSparseWitness_positive_normalized_smooth s hji).2.2.2.1 l
apply continuous_uniformOnFun_of_eq_affine_on_compact isCompact_Icc
(nonempty_Icc.mpr (by norm_num))
(hθ.continuousOn_fderivWithin hu (by norm_num))
(hstar.continuousOn_fderivWithin hu (by norm_num))
intro t x hx
change fderivWithin ℝ
(fun y => (1 - t) • θ.1.q l y + t • (embeddedSparseWitness s hji).q l y) K x = _
rw [fderivWithin_fun_add (hu.uniqueDiffWithinAt hx)
(((hθ.const_smul (1 - t)) x hx).differentiableWithinAt (by norm_num))
(((hstar.const_smul t) x hx).differentiableWithinAt (by norm_num)),
fderivWithin_fun_const_smul (hu.uniqueDiffWithinAt hx)
((hθ x hx).differentiableWithinAt (by norm_num)) (1 - t),
fderivWithin_fun_const_smul (hu.uniqueDiffWithinAt hx)
((hstar x hx).differentiableWithinAt (by norm_num)) t]
theorem continuous_affinePathExtension_q_iteratedFDerivWithin_two
Given the selected directed edge, the intervention-factor second within-derivative coordinate varies continuously along the unrestricted affine path.
Formal statement
hypotheses
conclusion
Continuous (fun t : ℝ => ofFun {Icc (0 : ℝ) 1} (iteratedFDerivWithin ℝ 2 ((affinePathExtension s θ hji t).q l) (Icc (0 : ℝ) 1)))
Proof (Lean source)
@[fun_prop] lemma continuous_affinePathExtension_q_iteratedFDerivWithin_two
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(θ : StratumPoint G s) {j i : Fin n} (hji : G.edge j i) (l : Fin n) :
Continuous (fun t : ℝ => ofFun {Icc (0 : ℝ) 1}
(iteratedFDerivWithin ℝ 2 ((affinePathExtension s θ hji t).q l)
(Icc (0 : ℝ) 1))) := by
let K := Icc (0 : ℝ) 1
let hu : UniqueDiffOn ℝ K := uniqueDiffOn_Icc (by norm_num)
let hθ := θ.property.positiveSmooth.2.2.2.1 l
let hstar := (embeddedSparseWitness_positive_normalized_smooth s hji).2.2.2.1 l
let hθ2 : ContDiffOn ℝ 2 (θ.1.q l) K :=
hθ.of_le (by norm_num)
let hstar2 : ContDiffOn ℝ 2 ((embeddedSparseWitness s hji).q l) K :=
hstar.of_le (by norm_num)
apply continuous_uniformOnFun_of_eq_affine_on_compact isCompact_Icc
(nonempty_Icc.mpr (by norm_num))
(hθ.continuousOn_iteratedFDerivWithin (by norm_num) hu)
(hstar.continuousOn_iteratedFDerivWithin (by norm_num) hu)
intro t x hx
change iteratedFDerivWithin ℝ 2
(fun y => (1 - t) • θ.1.q l y + t • (embeddedSparseWitness s hji).q l y) K x = _
have hscaleθ : iteratedFDerivWithin ℝ 2
(fun y => (1 - t) • θ.1.q l y) K x =
(1 - t) • iteratedFDerivWithin ℝ 2 (θ.1.q l) K x := by
change iteratedFDerivWithin ℝ 2 ((1 - t) • θ.1.q l) K x = _
exact iteratedFDerivWithin_const_smul_apply
(hθ2 x hx) hu hx
have hscaleStar : iteratedFDerivWithin ℝ 2
(fun y => t • (embeddedSparseWitness s hji).q l y) K x =
t • iteratedFDerivWithin ℝ 2 ((embeddedSparseWitness s hji).q l) K x := by
change iteratedFDerivWithin ℝ 2 (t • (embeddedSparseWitness s hji).q l) K x = _
exact iteratedFDerivWithin_const_smul_apply
(hstar2 x hx) hu hx
rw [fun_iteratedFDerivWithin_add_apply
((hθ2.const_smul (1 - t)) x hx)
((hstar2.const_smul t) x hx) hu hx,
hscaleθ, hscaleStar]
theorem continuous_affinePathExtension
Given the selected directed edge, the unrestricted affine mechanism path is continuous in the induced relative product C² topology.
Formal statement
hypotheses
conclusion
Continuous (fun t : ℝ => affinePathExtension s θ hji t)
Proof (Lean source)
@[fun_prop] lemma continuous_affinePathExtension
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(θ : StratumPoint G s) {j i : Fin n} (hji : G.edge j i) :
Continuous (fun t : ℝ => affinePathExtension s θ hji t) := by
rw [continuous_induced_rng]
unfold mechanismC2Coordinates
fun_prop
theorem tendsto_affinePathExtension_zero
The unrestricted affine mechanism path tends to its initial stratum point at parameter zero. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
Tendsto (fun t : ℝ => affinePathExtension s θ hji t) (𝓝 0) (𝓝 θ.1)
Proof (Lean source)
lemma tendsto_affinePathExtension_zero
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(θ : StratumPoint G s) {j i : Fin n} (hji : G.edge j i) :
Tendsto (fun t : ℝ => affinePathExtension s θ hji t) (𝓝 0) (𝓝 θ.1) := by
have hzero : affinePathExtension s θ hji 0 = θ.1 := by
rcases θ with ⟨⟨p, q, hlocal⟩, hprop⟩
simp [affinePathExtension]
have hc : Tendsto (fun t : ℝ => affinePathExtension s θ hji t) (𝓝 0)
(𝓝 (affinePathExtension s θ hji 0)) :=
(continuous_affinePathExtension θ hji).continuousAt
rwa [hzero] at hc
theorem affinePathExtension_p_eventually_uniform_close
The observational factors of the affine path are uniformly close to their initial values for all nodes and cube points when the parameter is sufficiently small. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
∃ δ > 0,
∀ t : ℝ,
|t| < δ → ∀ l v, v ∈ latentCube n → |(affinePathExtension s θ hji t).p l v
- θ.1.p l v| < ε
Proof (Lean source)
lemma affinePathExtension_p_eventually_uniform_close
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(θ : StratumPoint G s) {j i : Fin n} (hji : G.edge j i)
{ε : ℝ} (hε : 0 < ε) :
∃ δ > 0, ∀ t : ℝ, |t| < δ → ∀ l v, v ∈ latentCube n →
|(affinePathExtension s θ hji t).p l v - θ.1.p l v| < ε := by
let θstar := embeddedSparseWitness s hji
let K : Set (Fin n × LatentState n) := Set.univ ×ˢ latentCube n
have hK : IsCompact K := isCompact_univ.prod (by
rw [latentCube]
exact isCompact_univ_pi fun _ => isCompact_Icc)
have hKne : K.Nonempty := ⟨(j, 0), ⟨Set.mem_univ _, fun _ _ => ⟨by norm_num, by norm_num⟩⟩⟩
have hcont : ContinuousOn
(fun z : Fin n × LatentState n => |θstar.p z.1 z.2 - θ.1.p z.1 z.2|) K := by
rw [continuousOn_prod_of_discrete_left]
intro l
simpa only [K, Set.mem_prod, Set.mem_univ, true_and, Set.ofPred_mem_eq, θstar,
Pi.sub_apply] using
(((embeddedSparseWitness_positive_normalized_smooth s hji).2.2.1 l).continuousOn.sub
(θ.property.positiveSmooth.2.2.1 l).continuousOn).abs
rcases hK.exists_isMaxOn hKne hcont with ⟨z₀, hz₀, hzmax⟩
let B := |θstar.p z₀.1 z₀.2 - θ.1.p z₀.1 z₀.2| + 1
have hB : 0 < B := by dsimp [B]; positivity
refine ⟨ε / B, div_pos hε hB, ?_⟩
intro t ht l v hv
have hbound0 := hzmax (show (l, v) ∈ K from ⟨Set.mem_univ _, hv⟩)
change |θstar.p l v - θ.1.p l v| ≤
|θstar.p z₀.1 z₀.2 - θ.1.p z₀.1 z₀.2| at hbound0
have hbound : |θstar.p l v - θ.1.p l v| < B := by dsimp [B]; linarith
change |((1 - t) * θ.1.p l v + t * θstar.p l v) - θ.1.p l v| < ε
rw [show ((1 - t) * θ.1.p l v + t * θstar.p l v) - θ.1.p l v =
t * (θstar.p l v - θ.1.p l v) by ring, abs_mul]
calc
|t| * |θstar.p l v - θ.1.p l v| ≤ |t| * B :=
mul_le_mul_of_nonneg_left hbound.le (abs_nonneg _)
_ < (ε / B) * B := mul_lt_mul_of_pos_right ht hB
_ = ε := div_mul_cancel₀ ε hB.ne'
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.AnalyticEdgePerturbation 52 declarations The lemma states the path integral identity, analytic nonidentity certificate, isolated-zero property, and arbitrarily small stratum-preserving perturbation.
Analytic edge perturbation
The lemma states the path integral identity, analytic nonidentity certificate, isolated-zero property, and arbitrarily small stratum-preserving perturbation.
theorem isOpen_affine_forall_gt
A strict uniform lower bound for finitely many affine families on a compact space defines an open set of parameters. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
IsOpen {t : ℝ | ∀ i x, c < (1 - t) * a i x + t * b i x}
Proof (Lean source)
lemma isOpen_affine_forall_gt
{ι X : Type*} [Finite ι] [TopologicalSpace ι] [DiscreteTopology ι]
[TopologicalSpace X] [CompactSpace X]
(a b : ι → X → ℝ) (c : ℝ)
(ha : ∀ i, Continuous (a i)) (hb : ∀ i, Continuous (b i)) :
IsOpen {t : ℝ | ∀ i x, c < (1 - t) * a i x + t * b i x} := by
rw [isOpen_iff_mem_nhds]
intro t ht
have hcompact : IsCompact (Set.univ : Set (ι × X)) := isCompact_univ
have ha' : Continuous (fun z : ι × X ↦ a z.1 z.2) :=
continuous_prod_of_discrete_left.mpr ha
have hb' : Continuous (fun z : ι × X ↦ b z.1 z.2) :=
continuous_prod_of_discrete_left.mpr hb
have hevent : ∀ᶠ u in 𝓝 t, ∀ z : ι × X,
c < (1 - u) * a z.1 z.2 + u * b z.1 z.2 := by
simpa only [Set.mem_univ, forall_const] using
(hcompact.eventually_forall_of_forall_eventually (x₀ := t)
(P := fun u z ↦ c < (1 - u) * a z.1 z.2 + u * b z.1 z.2) (by
intro z _
apply (show ContinuousAt (fun w : ℝ × (ι × X) ↦
(1 - w.1) * a w.2.1 w.2.2 + w.1 * b w.2.1 w.2.2) (t, z) by
exact (((continuous_const.sub continuous_fst).mul
(ha'.comp continuous_snd)).add
(continuous_fst.mul (hb'.comp continuous_snd))).continuousAt).tendsto
exact Ioi_mem_nhds (ht z.1 z.2)))
filter_upwards [hevent] with u hu
exact fun i x ↦ hu (i, x)
theorem convex_affine_forall_gt
Strict affine lower-bound constraints are convex in the scalar parameter. the stated conclusion follows.
Formal statement
hypotheses
iota X :
Type*
a b :
iota → X → ℝ
c :
ℝ
conclusion
Convex ℝ {t : ℝ | ∀ i x, c < (1 - t) * a i x + t * b i x}
Proof (Lean source)
lemma convex_affine_forall_gt
{iota X : Type*} (a b : iota → X → ℝ) (c : ℝ) :
Convex ℝ {t : ℝ | ∀ i x, c < (1 - t) * a i x + t * b i x} := by
intro x hx y hy u v hu hv huv i z
change c < (1 - (u * x + v * y)) * a i z + (u * x + v * y) * b i z
have hx' := hx i z
have hy' := hy i z
have hvEq : v = 1 - u := by linarith
rw [show (1 - (u * x + v * y)) * a i z + (u * x + v * y) * b i z =
u * ((1 - x) * a i z + x * b i z) +
v * ((1 - y) * a i z + y * b i z) by
rw [hvEq]
ring,
show c = u * c + v * c by rw [hvEq]; ring]
by_cases hupos : 0 < u
· exact add_lt_add_of_lt_of_le (mul_lt_mul_of_pos_left hx' hupos)
(mul_le_mul_of_nonneg_left hy'.le hv)
· have hvpos : 0 < v := by linarith
exact add_lt_add_of_le_of_lt (mul_le_mul_of_nonneg_left hx'.le hu)
(mul_lt_mul_of_pos_left hy' hvpos)
The ordered two-point finset is equivalent to Fin 2.
Definition (Lean source)
parameters
α :
hab :
a ≠ b
given by
clause 1
toFun k := if hk : k = 0 then ⟨a, by simp⟩ else ⟨b, by simp⟩
clause 2
invFun x := if x.1 = a then 0 else 1
clause 3
left_inv k := by fin_cases k · simp · simp [Ne.symm hab]
clause 4
right_inv x := by apply Subtype.ext simp only by_cases hx : x.1
= a · simp [hx] · have hxmem := x.2 simp only [Finset.mem_insert, mem_singleton] at hxmem have hxb : x.1 = b := hxmem.resolve_left hx simp [hxb, Ne.symm hab]
Fubini's theorem for a two-coordinate product measure in coordinate order. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
(∫ v, f v ∂Measure.pi fun _ : Fin 2 => μ) = ∫ x, (∫ y, f ![x, y] ∂μ) ∂μ
Proof (Lean source)
lemma integral_fin_two_pi_eq_iterated (μ : Measure ℝ) [SigmaFinite μ]
(f : (Fin 2 → ℝ) → ℝ) (hf : Integrable f (Measure.pi fun _ : Fin 2 => μ)) :
(∫ v, f v ∂Measure.pi fun _ : Fin 2 => μ) =
∫ x, (∫ y, f ![x, y] ∂μ) ∂μ := by
let e2 := MeasurableEquiv.piFinSuccAbove (fun _ : Fin 2 => ℝ) 0
have he2 := (measurePreserving_piFinSuccAbove (fun _ : Fin 2 => μ) 0).symm
have hf2 : Integrable (f ∘ e2.symm) (μ.prod (Measure.pi fun _ : Fin 1 => μ)) :=
(he2.integrable_comp_emb e2.symm.measurableEmbedding).2 hf
rw [← he2.integral_comp e2.symm.measurableEmbedding]
have hf2' := hf2
change Integrable (fun z => f (e2.symm z))
(μ.prod (Measure.pi fun _ : Fin 1 => μ)) at hf2'
rw [integral_prod _ hf2']
apply integral_congr_ae
filter_upwards [hf2'.prod_right_ae] with x hfx
let e1 := MeasurableEquiv.piUnique (fun _ : Fin 1 => ℝ)
have he1 := (measurePreserving_piUnique (fun _ : Fin 1 => μ)).symm
rw [← he1.integral_comp e1.symm.measurableEmbedding]
apply integral_congr_ae
filter_upwards with y
congr 1
funext k
fin_cases k <;> rfl
def affinePathContrast
Direct-edge contrast along the identity-target affine path.
Definition (Lean source)
def affinePathContrastIntegral
The explicit rational-integral expression for the direct-edge path contrast.
Definition (Lean source)
parameters
def
affinePathContrastIntegral s θ hji t :
ℝ
given by
θt :
affinePathExtension s θ hji t
∫ v in latentCube n, (θt.q i (v i)) ^ 2 * (θt.q j (v j) - θt.p j v) * (∏ l ∈ (Finset.univ.erase i).erase j, θt.p l v) / θt.p i v
def affineFactorPolynomial
The degree-one polynomial whose value at t is the affine interpolation from a to b.
Definition (Lean source)
parameters
a b :
ℝ
def
given by
Polynomial.C a + Polynomial.X * Polynomial.C (b - a)
theorem affineFactorPolynomial_eval
The degree-one factor polynomial evaluates to affine interpolation.
Formal statement
Proof (Lean source)
@[simp] lemma affineFactorPolynomial_eval (a b t : ℝ) :
(affineFactorPolynomial a b).eval t = (1 - t) * a + t * b := by
simp [affineFactorPolynomial]
ring
theorem affineFactorPolynomial_natDegree_le_one
The affine factor polynomial has degree at most one.
Formal statement
Proof (Lean source)
lemma affineFactorPolynomial_natDegree_le_one (a b : ℝ) :
(affineFactorPolynomial a b).natDegree ≤ 1 := by
unfold affineFactorPolynomial
compute_degree
def affinePathNumeratorPolynomial
Polynomial encoding of the complete numerator in affinePathContrastIntegral.
Definition (Lean source)
parameters
def
given by
θstar :
embeddedSparseWitness s hji
(affineFactorPolynomial (θs.1.q i (v i)) (θstar.q i (v i))) ^ 2 * (affineFactorPolynomial (θs.1.q j (v j)) (θstar.q j (v j)) - affineFactorPolynomial (θs.1.p j v) (θstar.p j v)) * ∏ l ∈ (Finset.univ.erase i).erase j, affineFactorPolynomial (θs.1.p l v) (θstar.p l v)
theorem affinePathNumeratorPolynomial_eval
Evaluating the numerator polynomial recovers exactly the affine-path numerator. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
(affinePathNumeratorPolynomial s θs hji v).eval t
= ((affinePathExtension s θs hji t).q i (v i)) ^ 2 * ((affinePathExtension s θs hji t).q j (v j) - (affinePathExtension s θs hji t).p j v) * (∏ l ∈ (Finset.univ.erase i).erase j, (affinePathExtension s θs hji t).p l v)
Proof (Lean source)
lemma affinePathNumeratorPolynomial_eval
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(θs : StratumPoint G s) {j i : Fin n} (hji : G.edge j i)
(t : ℝ) (v : LatentState n) :
(affinePathNumeratorPolynomial s θs hji v).eval t =
((affinePathExtension s θs hji t).q i (v i)) ^ 2 *
((affinePathExtension s θs hji t).q j (v j) -
(affinePathExtension s θs hji t).p j v) *
(∏ l ∈ (Finset.univ.erase i).erase j,
(affinePathExtension s θs hji t).p l v) := by
unfold affinePathNumeratorPolynomial
simp only [Polynomial.eval_mul, Polynomial.eval_sub, Polynomial.eval_pow]
simp_rw [Polynomial.eval_prod]
simp [affinePathExtension]
def affinePathNumeratorCoefficient
Coefficients of the affine-path numerator, padded to the uniform degree bound n + 3.
Definition (Lean source)
theorem continuousOn_polynomial_mul_coeff
If every coefficient of the first polynomial family is continuous and every coefficient of the second is continuous, then each product coefficient is continuous.
Formal statement
hypotheses
α :
hP :
∀ k, ContinuousOn (fun x ↦ (P x).coeff k) K
hQ :
∀ k, ContinuousOn (fun x ↦ (Q x).coeff k) K
k :
ℕ
conclusion
ContinuousOn (fun x ↦ (P x * Q x).coeff k) K
Proof (Lean source)
lemma continuousOn_polynomial_mul_coeff
{α : Type*} [TopologicalSpace α] {K : Set α}
{P Q : α → Polynomial ℝ}
(hP : ∀ k, ContinuousOn (fun x ↦ (P x).coeff k) K)
(hQ : ∀ k, ContinuousOn (fun x ↦ (Q x).coeff k) K)
(k : ℕ) : ContinuousOn (fun x ↦ (P x * Q x).coeff k) K := by
simp only [Polynomial.coeff_mul]
exact continuousOn_finsetSum _ fun ij _ ↦ (hP ij.1).mul (hQ ij.2)
theorem continuousOn_affineFactorPolynomial_coeff
If the first endpoint varies continuously and the second endpoint varies continuously, then every affine-factor coefficient varies continuously.
Formal statement
hypotheses
conclusion
ContinuousOn (fun x ↦ (affineFactorPolynomial (a x) (b x)).coeff k) K
Proof (Lean source)
lemma continuousOn_affineFactorPolynomial_coeff
{α : Type*} [TopologicalSpace α] {K : Set α} {a b : α → ℝ}
(ha : ContinuousOn a K) (hb : ContinuousOn b K) (k : ℕ) :
ContinuousOn (fun x ↦ (affineFactorPolynomial (a x) (b x)).coeff k) K := by
rcases k with _ | k
· simpa [affineFactorPolynomial] using ha
· rcases k with _ | k
· have h := hb.sub ha
change ContinuousOn (fun x ↦ b x - a x) K at h
simpa [affineFactorPolynomial, Polynomial.coeff_X_mul] using h
· simp [affineFactorPolynomial]
fun_prop
theorem continuousOn_polynomial_finsetProd_coeff
If every coefficient in a finite polynomial family is continuous, then every coefficient of its finite product is continuous.
Formal statement
hypotheses
α ι :
S :
Finset ι
P :
ι → α → Polynomial ℝ
hP :
∀ i ∈ S, ∀ k, ContinuousOn (fun x ↦ (P i x).coeff k) K
k :
ℕ
conclusion
ContinuousOn (fun x ↦ (∏ i ∈ S, P i x).coeff k) K
Proof (Lean source)
lemma continuousOn_polynomial_finsetProd_coeff
{α ι : Type*} [TopologicalSpace α] [DecidableEq ι]
{K : Set α} (S : Finset ι) (P : ι → α → Polynomial ℝ)
(hP : ∀ i ∈ S, ∀ k, ContinuousOn (fun x ↦ (P i x).coeff k) K)
(k : ℕ) : ContinuousOn (fun x ↦ (∏ i ∈ S, P i x).coeff k) K := by
classical
revert hP k
induction S using Finset.induction_on with
| empty =>
intro _ k
simp
fun_prop
| @insert a S ha ih =>
intro hP k
simp only [Finset.prod_insert ha]
exact continuousOn_polynomial_mul_coeff
(fun m ↦ hP a (by simp) m)
(fun m ↦ ih (fun i hi k ↦ hP i (by simp [hi]) k) m) k
theorem affinePathNumeratorPolynomial_natDegree_lt
The uniform padding bound really contains every numerator coefficient. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
(affinePathNumeratorPolynomial s θs hji v).natDegree < n + 3 + 1
Proof (Lean source)
lemma affinePathNumeratorPolynomial_natDegree_lt
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(θs : StratumPoint G s) {j i : Fin n} (hji : G.edge j i)
(v : LatentState n) :
(affinePathNumeratorPolynomial s θs hji v).natDegree < n + 3 + 1 := by
unfold affinePathNumeratorPolynomial
calc
_ ≤ 2 * 1 + 1 + ∑ l ∈ (Finset.univ.erase i).erase j, 1 := by
refine Polynomial.natDegree_mul_le.trans ?_
refine Nat.add_le_add ?_ ?_
· exact Polynomial.natDegree_mul_le.trans
(Nat.add_le_add
(Polynomial.natDegree_pow_le.trans
(Nat.mul_le_mul_left 2 (affineFactorPolynomial_natDegree_le_one _ _)))
((Polynomial.natDegree_sub_le _ _).trans
(max_le (affineFactorPolynomial_natDegree_le_one _ _)
(affineFactorPolynomial_natDegree_le_one _ _))))
· calc
_ ≤ ∑ l ∈ (Finset.univ.erase i).erase j,
(affineFactorPolynomial (θs.1.p l v)
((embeddedSparseWitness s hji).p l v)).natDegree :=
Polynomial.natDegree_prod_le ((Finset.univ.erase i).erase j)
(fun l ↦ affineFactorPolynomial (θs.1.p l v)
((embeddedSparseWitness s hji).p l v))
_ ≤ ∑ l ∈ (Finset.univ.erase i).erase j, 1 :=
Finset.sum_le_sum fun l _ ↦ affineFactorPolynomial_natDegree_le_one _ _
_ ≤ n + 3 := by
have hc : ((Finset.univ.erase i).erase j).card ≤ n := by
calc
_ ≤ (Finset.univ : Finset (Fin n)).card :=
Finset.card_le_card (by simp)
_ = n := Finset.card_fin n
simpa [Finset.sum_const, Nat.add_comm] using Nat.add_le_add_left hc 3
_ < n + 3 + 1 := Nat.lt_succ_self _
theorem affinePathNumerator_polynomialNumerator_eq
The padded coefficient family evaluates to the complete affine numerator. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
polynomialNumerator (n + 3) (affinePathNumeratorCoefficient s θs hji) t v
= (affinePathNumeratorPolynomial s θs hji v).eval t
Proof (Lean source)
lemma affinePathNumerator_polynomialNumerator_eq
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(θs : StratumPoint G s) {j i : Fin n} (hji : G.edge j i)
(t : ℝ) (v : LatentState n) :
polynomialNumerator
(n + 3) (affinePathNumeratorCoefficient s θs hji) t v =
(affinePathNumeratorPolynomial s θs hji v).eval t := by
rw [Polynomial.eval_eq_sum_range'
(affinePathNumeratorPolynomial_natDegree_lt θs hji v)]
unfold polynomialNumerator
simpa [affinePathNumeratorCoefficient] using
(Fin.sum_univ_eq_sum_range (fun k : ℕ ↦
(affinePathNumeratorPolynomial s θs hji v).coeff k * t ^ k) (n + 3 + 1))
theorem affinePathNumeratorCoefficient_continuousOn
Every padded numerator coefficient is continuous on the compact latent cube. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma affinePathNumeratorCoefficient_continuousOn
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(θs : StratumPoint G s) {j i : Fin n} (hji : G.edge j i)
(k : Fin (n + 3 + 1)) :
ContinuousOn (affinePathNumeratorCoefficient s θs hji k) (latentCube n) := by
let θstar := embeddedSparseWitness s hji
have hθ := θs.property.positiveSmooth
have hstar := embeddedSparseWitness_positive_normalized_smooth s hji
have hpθ (l : Fin n) : ContinuousOn (θs.1.p l) (latentCube n) :=
(hθ.2.2.1 l).continuousOn
have hpstar (l : Fin n) : ContinuousOn (θstar.p l) (latentCube n) :=
(hstar.2.2.1 l).continuousOn
have hqθ (l : Fin n) : ContinuousOn (fun v : LatentState n ↦ θs.1.q l (v l))
(latentCube n) :=
(hθ.2.2.2.1 l).continuousOn.comp ((continuous_apply l).continuousOn)
(fun v hv ↦ hv l (Set.mem_univ l))
have hqstar (l : Fin n) : ContinuousOn (fun v : LatentState n ↦ θstar.q l (v l))
(latentCube n) :=
(hstar.2.2.2.1 l).continuousOn.comp ((continuous_apply l).continuousOn)
(fun v hv ↦ hv l (Set.mem_univ l))
have hpfac (l : Fin n) (m : ℕ) : ContinuousOn (fun v ↦
(affineFactorPolynomial (θs.1.p l v) (θstar.p l v)).coeff m)
(latentCube n) :=
continuousOn_affineFactorPolynomial_coeff (hpθ l) (hpstar l) m
have hqfac (l : Fin n) (m : ℕ) : ContinuousOn (fun v ↦
(affineFactorPolynomial (θs.1.q l (v l)) (θstar.q l (v l))).coeff m)
(latentCube n) :=
continuousOn_affineFactorPolynomial_coeff (hqθ l) (hqstar l) m
have hqpow (m : ℕ) : ContinuousOn (fun v ↦
((affineFactorPolynomial (θs.1.q i (v i)) (θstar.q i (v i))) ^ 2).coeff m)
(latentCube n) := by
simpa only [pow_two] using continuousOn_polynomial_mul_coeff (hqfac i) (hqfac i) m
have hdiff (m : ℕ) : ContinuousOn (fun v ↦
(affineFactorPolynomial (θs.1.q j (v j)) (θstar.q j (v j)) -
affineFactorPolynomial (θs.1.p j v) (θstar.p j v)).coeff m)
(latentCube n) := by
have h := (hqfac j m).sub (hpfac j m)
change ContinuousOn (fun v ↦
(affineFactorPolynomial (θs.1.q j (v j)) (θstar.q j (v j))).coeff m -
(affineFactorPolynomial (θs.1.p j v) (θstar.p j v)).coeff m) (latentCube n) at h
simpa only [Polynomial.coeff_sub] using h
have hprod (m : ℕ) : ContinuousOn (fun v ↦
(∏ l ∈ (Finset.univ.erase i).erase j,
affineFactorPolynomial (θs.1.p l v) (θstar.p l v)).coeff m)
(latentCube n) :=
continuousOn_polynomial_finsetProd_coeff ((Finset.univ.erase i).erase j)
(fun l v ↦ affineFactorPolynomial (θs.1.p l v) (θstar.p l v))
(fun l _ m ↦ hpfac l m) m
have hleft (m : ℕ) := continuousOn_polynomial_mul_coeff hqpow hdiff m
have hall (m : ℕ) := continuousOn_polynomial_mul_coeff hleft hprod m
change ContinuousOn (fun v ↦
(affinePathNumeratorPolynomial s θs hji v).coeff k) (latentCube n)
simpa [affinePathNumeratorPolynomial, θstar] using hall k
theorem latentCube_isClosed
A globally continuous extension of each cube coefficient, needed because the generic parametric-integral API states measurability on the ambient sample space. the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma latentCube_isClosed (n : ℕ) : IsClosed (latentCube n) := by
rw [latentCube]
exact (isCompact_univ_pi fun _ ↦ isCompact_Icc).isClosed
def affinePathNumeratorCoefficientExtension
For a finite dimension, DAG, sign pattern, stratum point, directed edge endpoints, edge certificate, and coefficient index, the globally continuous extension of the affine numerator coefficient agrees on the cube.
Definition (Lean source)
parameters
def
given by
f :
⟨fun v ↦ affinePathNumeratorCoefficient s θs hji k v.1, (affinePathNumeratorCoefficient_continuousOn θs hji k).restrict⟩ (: C({v : LatentState n // v ∈ latentCube n}, ℝ))
⇑(choose (ContinuousMap.exists_restrict_eq (latentCube_isClosed n) f))
theorem affinePathNumeratorCoefficientExtension_continuous
Given the selected directed edge, the extended affine numerator coefficient is continuous on the ambient latent space.
Formal statement
hypotheses
conclusion
Continuous (affinePathNumeratorCoefficientExtension s θs hji k)
Proof (Lean source)
lemma affinePathNumeratorCoefficientExtension_continuous
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(θs : StratumPoint G s) {j i : Fin n} (hji : G.edge j i)
(k : Fin (n + 3 + 1)) :
Continuous (affinePathNumeratorCoefficientExtension s θs hji k) := by
unfold affinePathNumeratorCoefficientExtension
exact (choose (ContinuousMap.exists_restrict_eq (latentCube_isClosed n)
(show C({v : LatentState n // v ∈ latentCube n}, ℝ) from
⟨fun v ↦ affinePathNumeratorCoefficient s θs hji k v.1,
(affinePathNumeratorCoefficient_continuousOn θs hji k).restrict⟩))).continuous
theorem affinePathNumeratorCoefficientExtension_eq
Given the selected directed edge and a point in the latent cube, the coefficient extension agrees with the original coefficient.
Formal statement
hypotheses
n :
θs :
StratumPoint G s
j i :
Fin n
hji :
G.edge j i
k :
Fin (n + 3 + 1)
v :
hv :
v ∈ latentCube n
conclusion
affinePathNumeratorCoefficientExtension s θs hji k v
= affinePathNumeratorCoefficient s θs hji k v
Proof (Lean source)
lemma affinePathNumeratorCoefficientExtension_eq
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(θs : StratumPoint G s) {j i : Fin n} (hji : G.edge j i)
(k : Fin (n + 3 + 1)) {v : LatentState n} (hv : v ∈ latentCube n) :
affinePathNumeratorCoefficientExtension s θs hji k v =
affinePathNumeratorCoefficient s θs hji k v := by
let f : C({v : LatentState n // v ∈ latentCube n}, ℝ) :=
⟨fun v ↦ affinePathNumeratorCoefficient s θs hji k v.1,
(affinePathNumeratorCoefficient_continuousOn θs hji k).restrict⟩
have hs := Classical.choose_spec
(ContinuousMap.exists_restrict_eq (latentCube_isClosed n) f)
change (choose
(ContinuousMap.exists_restrict_eq (latentCube_isClosed n) f)) v = _
exact DFunLike.congr_fun hs ⟨v, hv⟩
theorem affinePathNumerator_extension_polynomialNumerator_eq
Given the selected directed edge and a point in the latent cube, the extended polynomial numerator equals the original numerator polynomial evaluation.
Formal statement
hypotheses
conclusion
polynomialNumerator (n + 3) (affinePathNumeratorCoefficientExtension s θs hji) t v
= (affinePathNumeratorPolynomial s θs hji v).eval t
Proof (Lean source)
lemma affinePathNumerator_extension_polynomialNumerator_eq
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(θs : StratumPoint G s) {j i : Fin n} (hji : G.edge j i)
(t : ℝ) {v : LatentState n} (hv : v ∈ latentCube n) :
polynomialNumerator
(n + 3) (affinePathNumeratorCoefficientExtension s θs hji) t v =
(affinePathNumeratorPolynomial s θs hji v).eval t := by
rw [← affinePathNumerator_polynomialNumerator_eq θs hji t v]
unfold polynomialNumerator
apply Finset.sum_congr rfl
intro k _
rw [affinePathNumeratorCoefficientExtension_eq θs hji k hv]
theorem affinePathNumeratorCoefficientExtension_measurable
Given the selected directed edge, the extended affine numerator coefficient is measurable.
Formal statement
hypotheses
conclusion
Measurable (affinePathNumeratorCoefficientExtension s θs hji k)
Proof (Lean source)
lemma affinePathNumeratorCoefficientExtension_measurable
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(θs : StratumPoint G s) {j i : Fin n} (hji : G.edge j i)
(k : Fin (n + 3 + 1)) :
Measurable (affinePathNumeratorCoefficientExtension s θs hji k) :=
(affinePathNumeratorCoefficientExtension_continuous θs hji k).measurable
theorem affinePathNumeratorCoefficientExtension_compact_bounds
Given the selected directed edge, the extended numerator coefficients have uniform bounds on the compact latent cube.
Formal statement
hypotheses
conclusion
∃ C : Fin (n + 3 + 1) → ℝ,
∀ k v, v ∈ latentCube n → |affinePathNumeratorCoefficientExtension s θs hji k v| ≤ C k
Proof (Lean source)
lemma affinePathNumeratorCoefficientExtension_compact_bounds
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(θs : StratumPoint G s) {j i : Fin n} (hji : G.edge j i) :
∃ C : Fin (n + 3 + 1) → ℝ, ∀ k v, v ∈ latentCube n →
|affinePathNumeratorCoefficientExtension s θs hji k v| ≤ C k := by
have hK : IsCompact (latentCube n) := by
rw [latentCube]
exact isCompact_univ_pi fun _ ↦ isCompact_Icc
choose C hC using fun k : Fin (n + 3 + 1) ↦
hK.exists_bound_of_continuousOn
(affinePathNumeratorCoefficientExtension_continuous θs hji k).continuousOn
exact ⟨C, fun k v hv ↦ hC k v hv⟩
def affinePathDenominatorStartExtension
A globally continuous representative of the initial distinguished denominator factor.
Definition (Lean source)
parameters
def
given by
f :
⟨fun v ↦ theta.1.p i v.1, (theta.property.positiveSmooth.2.2.1 i).continuousOn.restrict⟩ (: C({v : LatentState n // v ∈ latentCube n}, ℝ))
⇑(choose (ContinuousMap.exists_restrict_eq (latentCube_isClosed n) f))
def affinePathDenominatorEndExtension
A globally continuous representative of the terminal distinguished denominator factor.
Definition (Lean source)
parameters
def
given by
thetaStar :
embeddedSparseWitness s hji
f :
⟨fun v ↦ thetaStar.p i v.1, ((embeddedSparseWitness_positive_normalized_smooth s hji).2.2.1 i).continuousOn.restrict⟩ (: C({v : LatentState n // v ∈ latentCube n}, ℝ))
⇑(choose (ContinuousMap.exists_restrict_eq (latentCube_isClosed n) f))
theorem affinePathDenominatorStartExtension_continuous
The initial denominator extension is continuous on the ambient latent space.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma affinePathDenominatorStartExtension_continuous
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(theta : StratumPoint G s) (i : Fin n) :
Continuous (affinePathDenominatorStartExtension theta i) := by
unfold affinePathDenominatorStartExtension
exact (choose (ContinuousMap.exists_restrict_eq (latentCube_isClosed n)
(show C({v : LatentState n // v ∈ latentCube n}, ℝ) from
⟨fun v ↦ theta.1.p i v.1,
(theta.property.positiveSmooth.2.2.1 i).continuousOn.restrict⟩))).continuous
theorem affinePathDenominatorEndExtension_continuous
Given the selected directed edge, the terminal denominator extension is continuous on the ambient latent space.
Formal statement
hypotheses
n :
hji :
G.edge j i
conclusion
Proof (Lean source)
lemma affinePathDenominatorEndExtension_continuous
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
{j i : Fin n} (hji : G.edge j i) :
Continuous (affinePathDenominatorEndExtension s hji) := by
unfold affinePathDenominatorEndExtension
exact (choose (ContinuousMap.exists_restrict_eq (latentCube_isClosed n)
(show C({v : LatentState n // v ∈ latentCube n}, ℝ) from
⟨fun v ↦ (embeddedSparseWitness s hji).p i v.1,
((embeddedSparseWitness_positive_normalized_smooth s hji).2.2.1 i).continuousOn.restrict⟩))).continuous
theorem affinePathDenominatorStartExtension_eq
For a point in the latent cube, the initial denominator extension agrees with the stratum mechanism factor.
Formal statement
hypotheses
conclusion
affinePathDenominatorStartExtension theta i v = theta.1.p i v
Proof (Lean source)
lemma affinePathDenominatorStartExtension_eq
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(theta : StratumPoint G s) (i : Fin n) {v : LatentState n}
(hv : v ∈ latentCube n) :
affinePathDenominatorStartExtension theta i v = theta.1.p i v := by
let f : C({v : LatentState n // v ∈ latentCube n}, ℝ) :=
⟨fun v ↦ theta.1.p i v.1, (theta.property.positiveSmooth.2.2.1 i).continuousOn.restrict⟩
have hs := Classical.choose_spec
(ContinuousMap.exists_restrict_eq (latentCube_isClosed n) f)
change (choose
(ContinuousMap.exists_restrict_eq (latentCube_isClosed n) f)) v = _
exact DFunLike.congr_fun hs ⟨v, hv⟩
theorem affinePathDenominatorEndExtension_eq
Given the selected directed edge and a point in the latent cube, the terminal denominator extension agrees with the sparse-witness factor.
Formal statement
hypotheses
conclusion
affinePathDenominatorEndExtension s hji v = (embeddedSparseWitness s hji).p i v
Proof (Lean source)
lemma affinePathDenominatorEndExtension_eq
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
{j i : Fin n} (hji : G.edge j i) {v : LatentState n}
(hv : v ∈ latentCube n) :
affinePathDenominatorEndExtension s hji v =
(embeddedSparseWitness s hji).p i v := by
let f : C({v : LatentState n // v ∈ latentCube n}, ℝ) :=
⟨fun v ↦ (embeddedSparseWitness s hji).p i v.1,
((embeddedSparseWitness_positive_normalized_smooth s hji).2.2.1 i).continuousOn.restrict⟩
have hs := Classical.choose_spec
(ContinuousMap.exists_restrict_eq (latentCube_isClosed n) f)
change (choose
(ContinuousMap.exists_restrict_eq (latentCube_isClosed n) f)) v = _
exact DFunLike.congr_fun hs ⟨v, hv⟩
theorem affinePathExtension_zero
The unrestricted affine extension starts at the supplied stratum point. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
affinePathExtension s θ hji 0 = θ.1
Proof (Lean source)
lemma affinePathExtension_zero
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(θ : StratumPoint G s) {j i : Fin n} (hji : G.edge j i) :
affinePathExtension s θ hji 0 = θ.1 := by
rcases θ with ⟨⟨p, q, hlocal⟩, hprop⟩
simp [affinePathExtension]
theorem affinePathExtension_one
The unrestricted affine extension ends at the edge-specific sparse witness. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
affinePathExtension s θ hji 1 = embeddedSparseWitness s hji
Proof (Lean source)
lemma affinePathExtension_one
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(θ : StratumPoint G s) {j i : Fin n} (hji : G.edge j i) :
affinePathExtension s θ hji 1 = embeddedSparseWitness s hji := by
rcases θ with ⟨⟨p, q, hlocal⟩, hprop⟩
simp [affinePathExtension]
theorem affinePathExtension_normalized_smooth
The unrestricted affine extension is normalized and C³ for every real path parameter; only positivity requires restricting the parameter to a neighborhood of the closed unit interval. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma affinePathExtension_normalized_smooth
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(θ : StratumPoint G s) {j i : Fin n} (hji : G.edge j i) (t : ℝ) :
(∀ l, ContDiffOn ℝ 3 ((affinePathExtension s θ hji t).p l) (latentCube n)) ∧
(∀ l, ContDiffOn ℝ 3 ((affinePathExtension s θ hji t).q l) (Icc (0 : ℝ) 1)) ∧
(∀ l v, v ∈ latentCube n →
∫ z in Icc (0 : ℝ) 1,
(affinePathExtension s θ hji t).p l (update v l z) = 1) ∧
(∀ l, ∫ z in Icc (0 : ℝ) 1,
(affinePathExtension s θ hji t).q l z = 1) := by
let θstar := embeddedSparseWitness s hji
have hθ := θ.property.positiveSmooth
have hstar := embeddedSparseWitness_positive_normalized_smooth s hji
refine ⟨?_, ?_, ?_, ?_⟩
· intro l
exact (hθ.2.2.1 l).const_smul (1 - t) |>.add
((hstar.2.2.1 l).const_smul t)
· intro l
exact (hθ.2.2.2.1 l).const_smul (1 - t) |>.add
((hstar.2.2.2.1 l).const_smul t)
· intro l v hv
have hupdate : Continuous (fun z : ℝ => update v l z) := by fun_prop
have hmaps : MapsTo (fun z : ℝ => update v l z)
(Icc (0 : ℝ) 1) (latentCube n) := by
intro z hz k _
by_cases hkl : k = l
· subst k
simpa using hz
· simpa only [update, dif_neg hkl] using hv k (Set.mem_univ k)
have hpInt : IntegrableOn (fun z : ℝ =>
(1 - t) * θ.1.p l (update v l z)) (Icc (0 : ℝ) 1) := by
exact ((hθ.2.2.1 l).continuousOn.comp hupdate.continuousOn hmaps |>.const_mul _)
|>.integrableOn_Icc
have hpstarInt : IntegrableOn (fun z : ℝ =>
t * θstar.p l (update v l z)) (Icc (0 : ℝ) 1) := by
exact ((hstar.2.2.1 l).continuousOn.comp hupdate.continuousOn hmaps |>.const_mul _)
|>.integrableOn_Icc
change ∫ z in Icc (0 : ℝ) 1,
((1 - t) * θ.1.p l (update v l z) +
t * θstar.p l (update v l z)) = 1
rw [MeasureTheory.integral_add,
MeasureTheory.integral_const_mul, MeasureTheory.integral_const_mul,
hθ.2.2.2.2.1 l v hv, hstar.2.2.2.2.1 l v hv]
· ring
· exact hpInt
· exact hpstarInt
· intro l
change ∫ z in Icc (0 : ℝ) 1,
((1 - t) * θ.1.q l z + t * θstar.q l z) = 1
rw [MeasureTheory.integral_add,
MeasureTheory.integral_const_mul, MeasureTheory.integral_const_mul,
hθ.2.2.2.2.2 l, hstar.2.2.2.2.2 l]
· ring
· exact ((hθ.2.2.2.1 l).continuousOn.const_mul _).integrableOn_Icc
· exact ((hstar.2.2.2.1 l).continuousOn.const_mul _).integrableOn_Icc
theorem mechanism_p_uniform_lower
Positivity and compactness give one strict lower bound valid for every observational mechanism slot on the latent cube. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma mechanism_p_uniform_lower
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(hθ : PositiveNormalizedSmoothMechanisms G θ) :
∃ ε > 0, ∀ i v, v ∈ latentCube n → ε ≤ θ.p i v := by
let K : Set (Fin n × LatentState n) := Set.univ ×ˢ latentCube n
have hK : IsCompact K := by
exact isCompact_univ.prod (by
rw [latentCube]
exact isCompact_univ_pi fun _ => isCompact_Icc)
have hcont : ContinuousOn (fun z : Fin n × LatentState n => θ.p z.1 z.2) K := by
rw [continuousOn_prod_of_discrete_left]
intro i
simpa only [K, Set.mem_prod, Set.mem_univ, true_and, Set.ofPred_mem_eq] using
(hθ.2.2.1 i).continuousOn
have hpos : ∀ z ∈ K, (0 : ℝ) < θ.p z.1 z.2 := by
intro z hz
exact hθ.1 z.1 z.2 hz.2
rcases hK.exists_forall_le' hcont hpos with ⟨ε, hε, hbound⟩
exact ⟨ε, hε, fun i v hv => hbound (i, v) ⟨Set.mem_univ i, hv⟩⟩
theorem mechanism_q_uniform_lower
Positivity and compactness give one strict lower bound valid for every intervention mechanism slot on the unit interval. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma mechanism_q_uniform_lower
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(hθ : PositiveNormalizedSmoothMechanisms G θ) :
∃ ε > 0, ∀ i z, z ∈ Icc (0 : ℝ) 1 → ε ≤ θ.q i z := by
let K : Set (Fin n × ℝ) := univ ×ˢ Icc (0 : ℝ) 1
have hK : IsCompact K := isCompact_univ.prod isCompact_Icc
have hcont : ContinuousOn (fun z : Fin n × ℝ => θ.q z.1 z.2) K := by
rw [continuousOn_prod_of_discrete_left]
intro i
simpa only [K, Set.mem_prod, Set.mem_univ, true_and, Set.ofPred_mem_eq] using
(hθ.2.2.2.1 i).continuousOn
have hpos : ∀ z ∈ K, (0 : ℝ) < θ.q z.1 z.2 := by
intro z hz
exact hθ.2.1 z.1 z.2 hz.2
rcases hK.exists_forall_le' hcont hpos with ⟨ε, hε, hbound⟩
exact ⟨ε, hε, fun i z hz => hbound (i, z) ⟨Set.mem_univ i, hz⟩⟩
theorem affinePathExtension_open_positive_uniformDenominator
The closed affine path admits an open parameter enlargement on which all factors remain positive and the distinguished observational denominator has one uniform separation margin. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
conclusion 1
t ∈ O :
PositiveNormalizedSmoothMechanisms G (affinePathExtension s θs hji t)
v ∈ latentCube n :
ε
≤ |affineDenominator (θs.1.p i) ((embeddedSparseWitness s hji).p i) t v|
Proof (Lean source)
lemma affinePathExtension_open_positive_uniformDenominator
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(θs : StratumPoint G s) {j i : Fin n} (hji : G.edge j i) :
∃ O : Set ℝ, ∃ ε > 0, IsOpen O ∧ Icc (0 : ℝ) 1 ⊆ O ∧ Convex ℝ O ∧
∀ t ∈ O,
PositiveNormalizedSmoothMechanisms G (affinePathExtension s θs hji t) ∧
∀ v ∈ latentCube n,
ε ≤ |affineDenominator
(θs.1.p i) ((embeddedSparseWitness s hji).p i) t v| := by
let θstar := embeddedSparseWitness s hji
have hθ := θs.property.positiveSmooth
have hstar := embeddedSparseWitness_positive_normalized_smooth s hji
rcases mechanism_p_uniform_lower hθ with ⟨mpθ, hmpθ, hpθ⟩
rcases mechanism_p_uniform_lower hstar with ⟨mpstar, hmpstar, hpstar⟩
rcases mechanism_q_uniform_lower hθ with ⟨mqθ, hmqθ, hqθ⟩
rcases mechanism_q_uniform_lower hstar with ⟨mqstar, hmqstar, hqstar⟩
let m := min mpθ (min mpstar (min mqθ mqstar))
have hm : 0 < m := by dsimp [m]; positivity
let Cube := {v : LatentState n // v ∈ latentCube n}
let Unit := {z : ℝ // z ∈ Icc (0 : ℝ) 1}
letI : CompactSpace Cube := isCompact_iff_compactSpace.mp (by
rw [latentCube]
exact isCompact_univ_pi fun _ ↦ isCompact_Icc)
letI : CompactSpace Unit := isCompact_iff_compactSpace.mp isCompact_Icc
let Op : Set ℝ := {t | ∀ l (v : Cube),
m / 2 < (1 - t) * θs.1.p l v.1 + t * θstar.p l v.1}
let Oq : Set ℝ := {t | ∀ l (z : Unit),
m / 2 < (1 - t) * θs.1.q l z.1 + t * θstar.q l z.1}
have hOp : IsOpen Op := isOpen_affine_forall_gt
(fun l (v : Cube) ↦ θs.1.p l v.1) (fun l (v : Cube) ↦ θstar.p l v.1) (m / 2)
(fun l ↦ (hθ.2.2.1 l).continuousOn.restrict)
(fun l ↦ (hstar.2.2.1 l).continuousOn.restrict)
have hOq : IsOpen Oq := isOpen_affine_forall_gt
(fun l (z : Unit) ↦ θs.1.q l z.1) (fun l (z : Unit) ↦ θstar.q l z.1) (m / 2)
(fun l ↦ (hθ.2.2.2.1 l).continuousOn.restrict)
(fun l ↦ (hstar.2.2.2.1 l).continuousOn.restrict)
have hOpConvex : Convex ℝ Op := convex_affine_forall_gt
(fun l (v : Cube) ↦ θs.1.p l v.1) (fun l (v : Cube) ↦ θstar.p l v.1) (m / 2)
have hOqConvex : Convex ℝ Oq := convex_affine_forall_gt
(fun l (z : Unit) ↦ θs.1.q l z.1) (fun l (z : Unit) ↦ θstar.q l z.1) (m / 2)
refine ⟨Op ∩ Oq, m / 2, half_pos hm, hOp.inter hOq, ?_,
hOpConvex.inter hOqConvex, ?_⟩
· intro t ht
constructor
· intro l v
have ha : m ≤ θs.1.p l v.1 :=
le_trans (min_le_left _ _) (hpθ l v.1 v.2)
have hb : m ≤ θstar.p l v.1 :=
le_trans (min_le_right _ _ |>.trans (min_le_left _ _)) (hpstar l v.1 v.2)
calc
m / 2 < m := half_lt_self hm
_ = (1 - t) * m + t * m := by ring
_ ≤ (1 - t) * θs.1.p l v.1 + t * θstar.p l v.1 :=
add_le_add (mul_le_mul_of_nonneg_left ha (sub_nonneg.mpr ht.2))
(mul_le_mul_of_nonneg_left hb ht.1)
· intro l z
have ha : m ≤ θs.1.q l z.1 :=
le_trans (min_le_right _ _ |>.trans (min_le_right _ _ |>.trans
(min_le_left _ _))) (hqθ l z.1 z.2)
have hb : m ≤ θstar.q l z.1 :=
le_trans (min_le_right _ _ |>.trans (min_le_right _ _ |>.trans
(min_le_right _ _))) (hqstar l z.1 z.2)
calc
m / 2 < m := half_lt_self hm
_ = (1 - t) * m + t * m := by ring
_ ≤ (1 - t) * θs.1.q l z.1 + t * θstar.q l z.1 :=
add_le_add (mul_le_mul_of_nonneg_left ha (sub_nonneg.mpr ht.2))
(mul_le_mul_of_nonneg_left hb ht.1)
· intro t ht
have hnorm := affinePathExtension_normalized_smooth θs hji t
have hposp : ∀ l v, v ∈ latentCube n →
0 < (affinePathExtension s θs hji t).p l v := by
intro l v hv
exact (half_pos hm).trans (ht.1 l ⟨v, hv⟩)
have hposq : ∀ l z, z ∈ Icc (0 : ℝ) 1 →
0 < (affinePathExtension s θs hji t).q l z := by
intro l z hz
exact (half_pos hm).trans (ht.2 l ⟨z, hz⟩)
refine ⟨⟨hposp, hposq, hnorm.1, hnorm.2.1, hnorm.2.2.1, hnorm.2.2.2⟩, ?_⟩
intro v hv
change m / 2 ≤ |(1 - t) * θs.1.p i v +
t * (embeddedSparseWitness s hji).p i v|
rw [abs_of_pos ((half_pos hm).trans (ht.1 i ⟨v, hv⟩))]
exact (ht.1 i ⟨v, hv⟩).le
theorem affinePathContrastIntegral_eq_extendedIntegral
The ambient continuous extensions used by the generic analytic theorem give exactly the paper's rational integral, because the integration measure is restricted to the latent cube. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
affinePathContrastIntegral s theta hji t
= ∫ v in latentCube n, polynomialNumerator (n + 3) (affinePathNumeratorCoefficientExtension s theta hji) t v / affineDenominator (affinePathDenominatorStartExtension theta i) (affinePathDenominatorEndExtension s hji) t v
Proof (Lean source)
lemma affinePathContrastIntegral_eq_extendedIntegral
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(theta : StratumPoint G s) {j i : Fin n} (hji : G.edge j i) (t : ℝ) :
affinePathContrastIntegral s theta hji t =
∫ v in latentCube n,
polynomialNumerator
(n + 3) (affinePathNumeratorCoefficientExtension s theta hji) t v /
affineDenominator
(affinePathDenominatorStartExtension theta i)
(affinePathDenominatorEndExtension s hji) t v := by
have hK : MeasurableSet (latentCube n) := by
rw [latentCube]
exact MeasurableSet.univ_pi fun _ ↦ measurableSet_Icc
unfold affinePathContrastIntegral
apply integral_congr_ae
filter_upwards [ae_restrict_mem hK] with v hv
rw [affinePathNumerator_extension_polynomialNumerator_eq theta hji t hv,
affinePathNumeratorPolynomial_eval theta hji t v]
unfold affineDenominator
rw [
affinePathDenominatorStartExtension_eq theta i hv,
affinePathDenominatorEndExtension_eq hji hv]
simp [affinePathExtension]
theorem affinePathContrastIntegral_analyticOnNhd
The affine-path rational integral is analytic on the common positive enlargement. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
conclusion 2
AnalyticOnNhd ℝ (affinePathContrastIntegral s theta hji) O
Proof (Lean source)
lemma affinePathContrastIntegral_analyticOnNhd
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(theta : StratumPoint G s) {j i : Fin n} (hji : G.edge j i) :
∃ O : Set ℝ, IsOpen O ∧ Icc (0 : ℝ) 1 ⊆ O ∧ Convex ℝ O ∧
(∀ t ∈ O, PositiveNormalizedSmoothMechanisms G
(affinePathExtension s theta hji t)) ∧
AnalyticOnNhd ℝ (affinePathContrastIntegral s theta hji) O := by
rcases affinePathExtension_open_positive_uniformDenominator theta hji with
⟨O, epsilon, hepsilon, hO, hIcc, hconvex, hpositive⟩
rcases affinePathNumeratorCoefficientExtension_compact_bounds theta hji with
⟨C, hC⟩
have hK : MeasurableSet (latentCube n) := by
rw [latentCube]
exact MeasurableSet.univ_pi fun _ ↦ measurableSet_Icc
have hmu : volume (latentCube n) ≠ ⊤ := by
exact (show IsCompact (latentCube n) by
rw [latentCube]
exact isCompact_univ_pi fun _ ↦ isCompact_Icc).measure_lt_top.ne
have hden' : ∀ t ∈ O, ∀ v ∈ latentCube n,
epsilon ≤
|affineDenominator
(affinePathDenominatorStartExtension theta i)
(affinePathDenominatorEndExtension s hji) t v| := by
intro t ht v hv
unfold affineDenominator
rw [affinePathDenominatorStartExtension_eq theta i hv,
affinePathDenominatorEndExtension_eq hji hv]
exact (hpositive t ht).2 v hv
have han :=
analyticOnNhd_setIntegral_polynomial_div_affine_of_uniform_nonzero
volume hK hmu (n + 3) (affinePathNumeratorCoefficientExtension s theta hji)
(affinePathDenominatorStartExtension theta i)
(affinePathDenominatorEndExtension s hji) O epsilon C hO hepsilon
(affinePathNumeratorCoefficientExtension_measurable theta hji)
(affinePathDenominatorStartExtension_continuous theta i).measurable
(affinePathDenominatorEndExtension_continuous hji).measurable hC hden'
refine ⟨O, hO, hIcc, hconvex, fun t ht ↦ (hpositive t ht).1, ?_⟩
simpa only [← affinePathContrastIntegral_eq_extendedIntegral theta hji] using han
theorem ownLogRatioDerivative_eq_fderivWithin
The paper's scalar own-coordinate log-ratio derivative is the difference of the intervention logarithmic derivative and the full-cube observational Fréchet derivative in the own-coordinate direction. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
ownLogRatioDerivative θ i v
= fderivWithin ℝ (θ.q i) (Icc (0 : ℝ) 1) (v i) 1 / θ.q i (v i)
Proof (Lean source)
lemma ownLogRatioDerivative_eq_fderivWithin
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(hθ : PositiveNormalizedSmoothMechanisms G θ)
(i : Fin n) (v : LatentState n) (hv : v ∈ latentCube n) :
ownLogRatioDerivative θ i v =
fderivWithin ℝ (θ.q i) (Icc (0 : ℝ) 1) (v i) 1 / θ.q i (v i) -
fderivWithin ℝ (θ.p i) (latentCube n) v (single i 1) / θ.p i v := by
have hvi : v i ∈ Icc (0 : ℝ) 1 := hv i (Set.mem_univ i)
have hpsection : DifferentiableOn ℝ
(fun z ↦ θ.p i (update v i z)) (Icc (0 : ℝ) 1) := by
intro z hz
apply ((hθ.2.2.1 i).differentiableOn (by norm_num)
(update v i z) (by
intro k _
by_cases hki : k = i
· subst k
simpa using hz
· simpa [update, hki] using hv k (Set.mem_univ k))).comp z
(hasDerivAt_update v i z).differentiableAt.differentiableWithinAt
intro y hy k _
by_cases hki : k = i
· subst k
simpa using hy
· simpa [update, hki] using hv k (Set.mem_univ k)
unfold ownLogRatioDerivative
change derivWithin (logRatio (θ.q i)
(fun z ↦ θ.p i (update v i z))) (Icc (0 : ℝ) 1) (v i) = _
rw [derivWithin_logRatio (by norm_num)
((hθ.2.2.2.1 i).differentiableOn (by norm_num)) hpsection
(hθ.2.1 i) (fun z hz ↦ hθ.1 i (update v i z) (by
intro k
by_cases hki : k = i
· subst k
simpa using hz
· simpa [update, hki] using hv k (Set.mem_univ k))) hvi]
rw [derivWithin_coordinateSection_eq_fderivWithin_apply
((hθ.2.2.1 i).differentiableOn (by norm_num)) hv i hvi]
simp [derivWithin]
theorem affinePathContrast_zero
At the initial endpoint, the affine-path contrast is the original mechanism's contrast. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
affinePathContrast s θ hji 0
Proof (Lean source)
lemma affinePathContrast_zero
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(θ : StratumPoint G s) {j i : Fin n} (hji : G.edge j i) :
affinePathContrast s θ hji 0 =
secondMomentContrast
(canonicalObservedWorld G θ.1 (refl (Fin n))) j i := by
unfold affinePathContrast
rw [affinePathExtension_zero]
theorem affinePathContrast_one
At the terminal endpoint, the affine-path contrast is the embedded sparse contrast. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
affinePathContrast s θ hji 1
Proof (Lean source)
lemma affinePathContrast_one
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(θ : StratumPoint G s) {j i : Fin n} (hji : G.edge j i) :
affinePathContrast s θ hji 1 =
secondMomentContrast
(canonicalObservedWorld G (embeddedSparseWitness s hji)
(refl (Fin n))) j i := by
unfold affinePathContrast
rw [affinePathExtension_one]
theorem canonical_secondMomentContrast_eq_integral
For a positive normalized mechanism and distinct intervention and ratio targets, the canonical second-moment contrast equals the rational mechanism integral obtained by cancelling the observational child-density factor. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
= ∫ v in latentCube n, (θ.q i (v i)) ^ 2 * (θ.q j (v j) - θ.p j v) * (∏ l ∈ (Finset.univ.erase i).erase j, θ.p l v) / θ.p i v
Proof (Lean source)
lemma canonical_secondMomentContrast_eq_integral
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(hpos : PositiveNormalizedSmoothMechanisms G θ) {j i : Fin n}
(hji : j ≠ i) :
secondMomentContrast (canonicalObservedWorld G θ (refl (Fin n))) j i =
∫ v in latentCube n,
(θ.q i (v i)) ^ 2 * (θ.q j (v j) - θ.p j v) *
(∏ l ∈ (Finset.univ.erase i).erase j, θ.p l v) / θ.p i v := by
let μ : Measure (LatentState n) := volume.restrict (latentCube n)
let r : LatentState n → ℝ := fun v => θ.q i (v i) / θ.p i v
have hcube : MeasurableSet (latentCube n) := by
rw [latentCube]
exact MeasurableSet.univ_pi fun _ => measurableSet_Icc
have hcubeCompact : IsCompact (latentCube n) := by
rw [latentCube]
exact isCompact_univ_pi fun _ => isCompact_Icc
have hpcont (l : Fin n) : ContinuousOn (θ.p l) (latentCube n) :=
(hpos.2.2.1 l).continuousOn
have hqcont (l : Fin n) : ContinuousOn (fun v : LatentState n => θ.q l (v l))
(latentCube n) :=
(hpos.2.2.2.1 l).continuousOn.comp ((continuous_apply l).continuousOn)
(fun v hv => hv l (Set.mem_univ l))
have hrcont : ContinuousOn r (latentCube n) := by
exact (hqcont i).div (hpcont i) fun v hv => ne_of_gt (hpos.1 i v hv)
have hobscont : ContinuousOn (observationalDensity θ) (latentCube n) := by
unfold observationalDensity
exact continuousOn_finsetProd _ fun l _ => hpcont l
have hintcont : ContinuousOn (interventionalDensity θ j) (latentCube n) := by
unfold interventionalDensity
exact (hqcont j).mul (continuousOn_finsetProd _ fun l _ => hpcont l)
have hobsint : Integrable (fun v => observationalDensity θ v * r v ^ 2) μ := by
exact (hobscont.mul (hrcont.pow 2)).integrableOn_compact hcubeCompact
have hintint : Integrable (fun v => interventionalDensity θ j v * r v ^ 2) μ := by
exact (hintcont.mul (hrcont.pow 2)).integrableOn_compact hcubeCompact
have hobs : (∫ v, r v ^ 2 ∂observationalLaw θ) =
∫ v, observationalDensity θ v * r v ^ 2 ∂μ := by
unfold observationalLaw
rw [integral_withDensity_eq_integral_toReal_smul₀]
· apply integral_congr_ae
filter_upwards [ae_restrict_mem hcube] with v hv
rw [ENNReal.toReal_ofReal (le_of_lt (by
unfold observationalDensity
exact Finset.prod_pos fun l _ => hpos.1 l v hv))]
rfl
· exact (hobscont.aestronglyMeasurable hcube).aemeasurable.ennreal_ofReal
· filter_upwards with v
exact ENNReal.ofReal_lt_top
have hint : (∫ v, r v ^ 2 ∂interventionalLaw θ j) =
∫ v, interventionalDensity θ j v * r v ^ 2 ∂μ := by
unfold interventionalLaw
rw [integral_withDensity_eq_integral_toReal_smul₀]
· apply integral_congr_ae
filter_upwards [ae_restrict_mem hcube] with v hv
rw [ENNReal.toReal_ofReal (le_of_lt (by
unfold interventionalDensity
exact mul_pos (hpos.2.1 j (v j) (hv j (Set.mem_univ j)))
(Finset.prod_pos fun l _ => hpos.1 l v hv)))]
rfl
· exact (hintcont.aestronglyMeasurable hcube).aemeasurable.ennreal_ofReal
· filter_upwards with v
exact ENNReal.ofReal_lt_top
unfold secondMomentContrast
change (∫ v, r v ^ 2 ∂interventionalLaw θ j) -
(∫ v, r v ^ 2 ∂observationalLaw θ) = _
rw [hint, hobs, ← integral_sub hintint hobsint]
apply integral_congr_ae
filter_upwards [ae_restrict_mem hcube] with v hv
have hpi : θ.p i v ≠ 0 := ne_of_gt (hpos.1 i v hv)
have hinterDensity : interventionalDensity θ j v =
θ.q j (v j) * θ.p i v *
(∏ l ∈ (Finset.univ.erase i).erase j, θ.p l v) := by
unfold interventionalDensity
have hprod : (∏ l ∈ Finset.univ.erase j, θ.p l v) =
θ.p i v * (∏ l ∈ (Finset.univ.erase j).erase i, θ.p l v) := by
rw [Finset.prod_eq_mul_prod_sdiff_singleton_of_mem
(Finset.mem_erase.mpr ⟨Ne.symm hji, Finset.mem_univ i⟩)]
simp only [Finset.sdiff_singleton_eq_erase]
rw [hprod, Finset.erase_right_comm]
ring
have hobsDensity : observationalDensity θ v =
θ.p i v * θ.p j v *
(∏ l ∈ (Finset.univ.erase i).erase j, θ.p l v) := by
unfold observationalDensity
rw [Finset.prod_eq_mul_prod_sdiff_singleton_of_mem (Finset.mem_univ i)]
simp only [Finset.sdiff_singleton_eq_erase]
rw [Finset.prod_eq_mul_prod_sdiff_singleton_of_mem
(Finset.mem_erase.mpr ⟨hji, Finset.mem_univ j⟩)]
simp only [Finset.sdiff_singleton_eq_erase]
ring
unfold r
rw [hinterDensity, hobsDensity]
field_simp
theorem affinePathContrast_eq_integral_of_mem_Icc
On the closed affine path, the contrast is exactly its rational mechanism integral. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
affinePathContrast s θs hji t = affinePathContrastIntegral s θs hji t
Proof (Lean source)
lemma affinePathContrast_eq_integral_of_mem_Icc
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(θs : StratumPoint G s) {j i : Fin n} (hji : G.edge j i)
{t : ℝ} (ht : t ∈ Icc (0 : ℝ) 1) :
affinePathContrast s θs hji t = affinePathContrastIntegral s θs hji t := by
have hne : j ≠ i := by
intro h
subst j
exact G.irrefl i hji
unfold affinePathContrast affinePathContrastIntegral
exact canonical_secondMomentContrast_eq_integral
(affinePathExtension_positive_normalized_smooth θs hji ht) hne
theorem analyticOnNhd_isolated_zeros_of_nonzero Lemma analyticOnNhd_isolated_zeros_of_nonzero in the paper ↗
A real analytic function on a preconnected set that is nonzero somewhere has an isolated zero at every point of the set. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
f :
ℝ → ℝ
O :
Set ℝ
hf :
AnalyticOnNhd ℝ f O
hO :
z t :
ℝ
hz :
z ∈ O
hfz :
f z ≠ 0
ht :
t ∈ O
hft :
f t = 0
conclusion
∃ ε > 0, ∀ u ∈ Ioo (t - ε) (t + ε), u ≠ t → f u ≠ 0
Proof (Lean source)
lemma analyticOnNhd_isolated_zeros_of_nonzero {f : ℝ → ℝ} {O : Set ℝ}
(hf : AnalyticOnNhd ℝ f O) (hO : IsPreconnected O)
{z t : ℝ} (hz : z ∈ O) (hfz : f z ≠ 0) (ht : t ∈ O) (hft : f t = 0) :
∃ ε > 0, ∀ u ∈ Ioo (t - ε) (t + ε), u ≠ t → f u ≠ 0 := by
rcases (hf t ht).eventually_eq_zero_or_eventually_ne_zero with hzero | hne
· exact (hfz (hf.eqOn_zero_of_preconnected_of_eventuallyEq_zero hO ht hzero hz)).elim
· change {u | f u ≠ 0} ∈ nhdsWithin t {t}ᶜ at hne
rw [Metric.mem_nhdsWithin_iff] at hne
rcases hne with ⟨ε, hε, hball⟩
refine ⟨ε, hε, fun u hu hut => hball ?_⟩
refine ⟨?_, ?_⟩
· simp only [Metric.mem_ball, Real.dist_eq]
rw [abs_lt]
constructor <;> linarith [hu.1, hu.2]
· simpa only [Set.mem_compl_iff, Set.mem_singleton_iff]
theorem threeNode_affinePathContrast_one_ne_zero
On the explicit three-node edge, the affine path ends at the quantitatively separated sparse witness. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
affinePathContrast s θs hji 1 ≠ 0
Proof (Lean source)
lemma threeNode_affinePathContrast_one_ne_zero (s : SignVector 3)
(θs : StratumPoint threeNodeDAG s)
(hji : threeNodeDAG.edge (0 : Fin 3) 1) :
affinePathContrast s θs hji 1 ≠ 0 := by
rw [affinePathContrast_one]
have heq : embeddedSparseWitness s hji = sparseWitness s := by
rfl
rw [heq]
have hgap := (sparse_witness_certificate s).2.2.2.2.2.2.2.2.2.2.1
have hc : secondMomentContrast
(canonicalObservedWorld threeNodeDAG (sparseWitness s) (refl (Fin 3))) 0 1 <
-(3 / 10000 : ℝ) := by
unfold secondMomentContrast
linarith
exact ne_of_lt (lt_of_lt_of_le hc (by norm_num))
theorem affinePathContrast_one_ne_zero
Every edge-specific affine path ends at a sparse mechanism with nonzero contrast. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
affinePathContrast s θs hji 1 ≠ 0
Proof (Lean source)
lemma affinePathContrast_one_ne_zero
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(θs : StratumPoint G s) {j i : Fin n} (hji : G.edge j i) :
affinePathContrast s θs hji 1 ≠ 0 := by
have hne : j ≠ i := by
intro h
subst j
exact G.irrefl i hji
rw [affinePathContrast_one]
rw [canonical_secondMomentContrast_eq_integral
(embeddedSparseWitness_positive_normalized_smooth s hji) hne]
let μ : Measure ℝ := volume.restrict (Icc (0 : ℝ) 1)
let f : LatentState n → ℝ := fun v =>
((embeddedSparseWitness s hji).q i (v i)) ^ 2 *
((embeddedSparseWitness s hji).q j (v j) -
(embeddedSparseWitness s hji).p j v) *
(∏ l ∈ (Finset.univ.erase i).erase j,
(embeddedSparseWitness s hji).p l v) /
(embeddedSparseWitness s hji).p i v
have hpos := embeddedSparseWitness_positive_normalized_smooth s hji
have hfcont : ContinuousOn f (latentCube n) := by
apply ContinuousOn.div
· apply ContinuousOn.mul
· apply ContinuousOn.mul
· exact ((hpos.2.2.2.1 i).continuousOn.comp
((continuous_apply i).continuousOn)
(fun (v : LatentState n) (hv : v ∈ latentCube n) =>
hv i (Set.mem_univ i))).pow 2
· exact ((hpos.2.2.2.1 j).continuousOn.comp
((continuous_apply j).continuousOn)
(fun (v : LatentState n) (hv : v ∈ latentCube n) =>
hv j (Set.mem_univ j))).sub (hpos.2.2.1 j).continuousOn
· exact continuousOn_finsetProd _ fun l _ => (hpos.2.2.1 l).continuousOn
· exact (hpos.2.2.1 i).continuousOn
· intro v hv
exact ne_of_gt (hpos.1 i v hv)
have hf : Integrable f (volume.restrict (latentCube n)) :=
hfcont.integrableOn_compact (by
rw [latentCube]
exact isCompact_univ_pi fun _ => isCompact_Icc)
have hμ : volume.restrict (latentCube n) = Measure.pi (fun _ : Fin n => μ) := by
change volume.restrict (Set.univ.pi fun _ : Fin n => Icc (0 : ℝ) 1) = _
rw [MeasureTheory.volume_pi, Measure.restrict_pi_pi]
change (∫ v in latentCube n, f v) ≠ 0
rw [hμ]
letI : IsProbabilityMeasure μ := ⟨by simp [μ, Real.volume_Icc]⟩
let S : Finset (Fin n) := {j, i}
let jj : {l : Fin n // l ∈ S} := ⟨j, by simp [S]⟩
let ii : {l : Fin n // l ∈ S} := ⟨i, by simp [S]⟩
let g : ({l : Fin n // l ∈ S} → ℝ) → ℝ := fun z =>
exponentialInterventionDensity (reflectedCoordinate s i (z ii)) ^ 2 *
(exponentialInterventionDensity (reflectedCoordinate s j (z jj)) - 1) /
(1 + (1 / 10 : ℝ) *
centeredCoordinate (reflectedCoordinate s j (z jj)) *
centeredCoordinate (reflectedCoordinate s i (z ii)))
have hfg : f = fun v => g (fun l : {l : Fin n // l ∈ S} => v l.1) := by
funext v
simp only [f, g, ii, jj, embeddedSparseWitness, embeddedSparseP]
simp [hne]
rw [hfg, integral_comp_pi_restrict_finset μ S g]
let e : Fin 2 ≃ {l : Fin n // l ∈ S} := pairFinEquiv hne
have he := measurePreserving_piCongrLeft (fun _ : {l : Fin n // l ∈ S} => μ) e
rw [← he.integral_comp']
have hgint : Integrable (g ∘ MeasurableEquiv.piCongrLeft
(fun _ : {l : Fin n // l ∈ S} => ℝ) e)
(Measure.pi fun _ : Fin 2 => μ) := by
apply (he.integrable_comp_emb
(MeasurableEquiv.piCongrLeft
(fun _ : {l : Fin n // l ∈ S} => ℝ) e).measurableEmbedding).2
have hp := Causalean.Stat.measurePreserving_pi_restrict_finset μ S
have hgmeas : Measurable g := by
rcases s.signed i with hi | hi <;> rcases s.signed j with hj | hj <;>
simp [g, exponentialInterventionDensity, reflectedCoordinate, reflect,
centeredCoordinate, hi, hj, show (-1 : ℝ) ≠ 1 by norm_num] <;> fun_prop
apply (hp.integrable_comp hgmeas.aestronglyMeasurable).1
rw [show g ∘ (fun (v : Fin n → ℝ) (l : {l : Fin n // l ∈ S}) => v l.1) = f by
simpa only [Function.comp_def] using hfg.symm]
simpa only [← hμ] using hf
change (∫ v : Fin 2 → ℝ, (g ∘ MeasurableEquiv.piCongrLeft
(fun _ : {l : Fin n // l ∈ S} => ℝ) e) v
∂Measure.pi fun _ : Fin 2 => μ) ≠ 0
rw [integral_fin_two_pi_eq_iterated μ _ hgint]
have hj_eval (z : Fin 2 → ℝ) :
(MeasurableEquiv.piCongrLeft
(fun _ : {l : Fin n // l ∈ S} => ℝ) e z) jj = z 0 := by
have hjj : jj = e 0 := by
apply Subtype.ext
simp [e, pairFinEquiv, jj, S]
rw [hjj, MeasurableEquiv.piCongrLeft_apply_apply]
have hi_eval (z : Fin 2 → ℝ) :
(MeasurableEquiv.piCongrLeft
(fun _ : {l : Fin n // l ∈ S} => ℝ) e z) ii = z 1 := by
have hii : ii = e 1 := by
apply Subtype.ext
simp [e, pairFinEquiv, ii, S]
rw [hii, MeasurableEquiv.piCongrLeft_apply_apply]
simp_rw [Function.comp_apply, g, hj_eval, hi_eval,
Matrix.cons_val_zero, Matrix.cons_val_one]
simp only [Matrix.cons_val_zero]
have hrefY (x : ℝ) :
(∫ y in Icc (0 : ℝ) 1,
exponentialInterventionDensity (reflectedCoordinate s i y) ^ 2 *
(exponentialInterventionDensity (reflectedCoordinate s j x) - 1) /
(1 + (10 : ℝ)⁻¹ * centeredCoordinate (reflectedCoordinate s j x) *
centeredCoordinate (reflectedCoordinate s i y))) =
(exponentialInterventionDensity (reflectedCoordinate s j x) - 1) *
sparseA (reflectedCoordinate s j x) := by
change (∫ y in Icc (0 : ℝ) 1,
(fun y => exponentialInterventionDensity y ^ 2 *
(exponentialInterventionDensity (reflectedCoordinate s j x) - 1) /
(1 + (10 : ℝ)⁻¹ * centeredCoordinate (reflectedCoordinate s j x) *
centeredCoordinate y)) (reflectedCoordinate s i y)) = _
have href := integral_reflectedCoordinate s i
(fun y => exponentialInterventionDensity y ^ 2 *
(exponentialInterventionDensity (reflectedCoordinate s j x) - 1) /
(1 + (10 : ℝ)⁻¹ * centeredCoordinate (reflectedCoordinate s j x) *
centeredCoordinate y))
rw [href]
unfold sparseA
rw [show (fun y : ℝ =>
exponentialInterventionDensity y ^ 2 *
(exponentialInterventionDensity (reflectedCoordinate s j x) - 1) /
(1 + (10 : ℝ)⁻¹ * centeredCoordinate (reflectedCoordinate s j x) *
centeredCoordinate y)) =
fun y => (exponentialInterventionDensity (reflectedCoordinate s j x) - 1) *
(exponentialInterventionDensity y ^ 2 /
(1 + centeredCoordinate (reflectedCoordinate s j x) / 10 *
centeredCoordinate y)) by funext y; ring,
MeasureTheory.integral_const_mul]
dsimp only [μ]
simp only [one_div]
simp_rw [hrefY]
change (∫ x in Icc (0 : ℝ) 1,
(fun x => (exponentialInterventionDensity x - 1) * sparseA x)
(reflectedCoordinate s j x)) ≠ 0
rw [show (∫ x in Icc (0 : ℝ) 1,
(fun x => (exponentialInterventionDensity x - 1) * sparseA x)
(reflectedCoordinate s j x)) =
∫ x in Icc (0 : ℝ) 1,
(exponentialInterventionDensity x - 1) * sparseA x by
simpa only using integral_reflectedCoordinate s j
(fun x => (exponentialInterventionDensity x - 1) * sparseA x)]
have hgap := sparse_unreflected_moment_gap
intro hzero
have hneg : (∫ x in Icc (0 : ℝ) 1,
(1 - exponentialInterventionDensity x) * sparseA x) = 0 := by
rw [show (fun x : ℝ => (1 - exponentialInterventionDensity x) * sparseA x) =
fun x => -((exponentialInterventionDensity x - 1) * sparseA x) by
funext x
ring,
MeasureTheory.integral_neg, hzero, neg_zero]
linarith
theorem affinePathExtension_eventually_causalMinimal
Causal minimality persists for sufficiently small positive parameters along the affine path. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
∃ δ > 0,
∀ t : {t : ℝ // t ∈ Icc (0 : ℝ) 1},
t.1 < δ → CausalMinimality G (affinePathExtension s θs hji t.1)
Proof (Lean source)
lemma affinePathExtension_eventually_causalMinimal
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(θs : StratumPoint G s) {j i : Fin n} (hji : G.edge j i) :
∃ δ > 0, ∀ t : {t : ℝ // t ∈ Icc (0 : ℝ) 1}, t.1 < δ →
CausalMinimality G (affinePathExtension s θs hji t.1) := by
rcases mechanismCompact_all_edge_witnesses_open θs with ⟨ε, hε, hopen⟩
rcases affinePathExtension_p_eventually_uniform_close θs hji hε with
⟨δ, hδ, hclose⟩
refine ⟨δ, hδ, ?_⟩
intro t htδ a b hab
let η := affinePathExtension s θs hji t.1
have hη : PositiveNormalizedSmoothMechanisms G η :=
affinePathExtension_positive_normalized_smooth θs hji t.2
have htAbs : |t.1| < δ := by simpa only [abs_of_nonneg t.2.1] using htδ
have hfactor :
(mechanismCompactPositiveFactorization θs.property.positiveSmooth).FactorSupClose
(mechanismCompactPositiveFactorization hη) ε :=
mechanismCompact_factorSupClose_of_p_close θs.property.positiveSmooth hη
(hclose t.1 htAbs)
have hnotCompact := hopen (mechanismCompactPositiveFactorization hη) hfactor b a hab
intro hpaper
apply hnotCompact
exact (condIndepCoordinates_iff_compactPositiveFactorization_mechanism η
(mechanismCompactPositiveFactorization hη)
(mechanismCompactPositiveFactorization_observationalMeasure hη)
b a ((G.parents b).erase a)).mp hpaper
theorem affinePathExtension_eventually_fixedOwnDerivativeSign
The prescribed own-coordinate derivative signs persist uniformly for small affine-path parameters. Compactness is used only in the latent-state variable; finiteness then combines the nodewise neighborhoods. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
∃ δ > 0,
∀ t : {t : ℝ // t ∈ Icc (0 : ℝ) 1},
t.1 < δ → FixedOwnDerivativeSign G s (affinePathExtension s θs hji t.1)
Proof (Lean source)
lemma affinePathExtension_eventually_fixedOwnDerivativeSign
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(θs : StratumPoint G s) {j i : Fin n} (hji : G.edge j i) :
∃ δ > 0, ∀ t : {t : ℝ // t ∈ Icc (0 : ℝ) 1}, t.1 < δ →
FixedOwnDerivativeSign G s (affinePathExtension s θs hji t.1) := by
let θstar := embeddedSparseWitness s hji
let Cube := {v : LatentState n // v ∈ latentCube n}
letI : CompactSpace Cube := isCompact_iff_compactSpace.mp (by
rw [latentCube]
exact isCompact_univ_pi fun _ ↦ isCompact_Icc)
let score (l : Fin n) (t : ℝ) (v : Cube) : ℝ := s.value l *
(((1 - t) * fderivWithin ℝ (θs.1.q l) (Icc (0 : ℝ) 1) (v.1 l) 1 +
t * fderivWithin ℝ (θstar.q l) (Icc (0 : ℝ) 1) (v.1 l) 1) /
((1 - t) * θs.1.q l (v.1 l) + t * θstar.q l (v.1 l)) -
((1 - t) * fderivWithin ℝ (θs.1.p l) (latentCube n) v.1 (single l 1) +
t * fderivWithin ℝ (θstar.p l) (latentCube n) v.1 (single l 1)) /
((1 - t) * θs.1.p l v.1 + t * θstar.p l v.1))
have hnode : ∀ l : Fin n, ∀ᶠ t in 𝓝 (0 : ℝ), ∀ v : Cube, 0 < score l t v := by
intro l
have huCube : UniqueDiffOn ℝ (latentCube n) := by
rw [latentCube]
exact UniqueDiffOn.univ_pi fun _ ↦ uniqueDiffOn_Icc (by norm_num)
have huIcc : UniqueDiffOn ℝ (Icc (0 : ℝ) 1) :=
uniqueDiffOn_Icc (by norm_num)
have hcoord : Continuous (fun v : Cube ↦ v.1 l) :=
(continuous_apply l).comp continuous_subtype_val
have hcoordMaps : Set.MapsTo (fun v : Cube ↦ v.1 l) Set.univ
(Icc (0 : ℝ) 1) := fun v _ ↦ v.2 l (Set.mem_univ l)
have hqθd : Continuous (fun v : Cube ↦
fderivWithin ℝ (θs.1.q l) (Icc (0 : ℝ) 1) (v.1 l)) :=
((θs.property.positiveSmooth.2.2.2.1 l).continuousOn_fderivWithin huIcc
(by norm_num)).comp_continuous hcoord (fun v ↦ hcoordMaps (Set.mem_univ v))
have hstarSmooth := embeddedSparseWitness_positive_normalized_smooth s hji
have hqstard : Continuous (fun v : Cube ↦
fderivWithin ℝ (θstar.q l) (Icc (0 : ℝ) 1) (v.1 l)) :=
(hstarSmooth.2.2.2.1 l).continuousOn_fderivWithin huIcc (by norm_num)
|>.comp_continuous hcoord (fun v ↦ hcoordMaps (Set.mem_univ v))
have hpθd : Continuous (fun v : Cube ↦
fderivWithin ℝ (θs.1.p l) (latentCube n) v.1) :=
((θs.property.positiveSmooth.2.2.1 l).continuousOn_fderivWithin huCube
(by norm_num)).comp_continuous continuous_subtype_val (fun v ↦ v.2)
have hpstard : Continuous (fun v : Cube ↦
fderivWithin ℝ (θstar.p l) (latentCube n) v.1) :=
(hstarSmooth.2.2.1 l).continuousOn_fderivWithin huCube (by norm_num)
|>.comp_continuous continuous_subtype_val (fun v ↦ v.2)
have hqθ : Continuous (fun v : Cube ↦ θs.1.q l (v.1 l)) :=
((θs.property.positiveSmooth.2.2.2.1 l).continuousOn.comp_continuous
hcoord (fun v ↦ hcoordMaps (Set.mem_univ v)))
have hqstar : Continuous (fun v : Cube ↦ θstar.q l (v.1 l)) :=
(hstarSmooth.2.2.2.1 l).continuousOn.comp_continuous hcoord
(fun v ↦ hcoordMaps (Set.mem_univ v))
have hpθ : Continuous (fun v : Cube ↦ θs.1.p l v.1) :=
((θs.property.positiveSmooth.2.2.1 l).continuousOn.comp_continuous
continuous_subtype_val (fun v ↦ v.2))
have hpstar : Continuous (fun v : Cube ↦ θstar.p l v.1) :=
(hstarSmooth.2.2.1 l).continuousOn.comp_continuous continuous_subtype_val
(fun v ↦ v.2)
have hqθd1 : Continuous (fun v : Cube ↦
fderivWithin ℝ (θs.1.q l) (Icc (0 : ℝ) 1) (v.1 l) 1) := by
fun_prop
have hqstard1 : Continuous (fun v : Cube ↦
fderivWithin ℝ (θstar.q l) (Icc (0 : ℝ) 1) (v.1 l) 1) := by
fun_prop
have hpθd1 : Continuous (fun v : Cube ↦
fderivWithin ℝ (θs.1.p l) (latentCube n) v.1 (single l 1)) := by
fun_prop
have hpstard1 : Continuous (fun v : Cube ↦
fderivWithin ℝ (θstar.p l) (latentCube n) v.1 (single l 1)) := by
fun_prop
have hnumq : Continuous (fun z : ℝ × Cube ↦
(1 - z.1) * fderivWithin ℝ (θs.1.q l) (Icc (0 : ℝ) 1) (z.2.1 l) 1 +
z.1 * fderivWithin ℝ (θstar.q l) (Icc (0 : ℝ) 1) (z.2.1 l) 1) := by
exact ((continuous_const.sub continuous_fst).mul (hqθd1.comp continuous_snd)).add
(continuous_fst.mul (hqstard1.comp continuous_snd))
have hdenq : Continuous (fun z : ℝ × Cube ↦
(1 - z.1) * θs.1.q l (z.2.1 l) + z.1 * θstar.q l (z.2.1 l)) := by
exact ((continuous_const.sub continuous_fst).mul (hqθ.comp continuous_snd)).add
(continuous_fst.mul (hqstar.comp continuous_snd))
have hnump : Continuous (fun z : ℝ × Cube ↦
(1 - z.1) * fderivWithin ℝ (θs.1.p l) (latentCube n) z.2.1
(single l 1) +
z.1 * fderivWithin ℝ (θstar.p l) (latentCube n) z.2.1
(single l 1)) := by
exact ((continuous_const.sub continuous_fst).mul (hpθd1.comp continuous_snd)).add
(continuous_fst.mul (hpstard1.comp continuous_snd))
have hdenp : Continuous (fun z : ℝ × Cube ↦
(1 - z.1) * θs.1.p l z.2.1 + z.1 * θstar.p l z.2.1) := by
exact ((continuous_const.sub continuous_fst).mul (hpθ.comp continuous_snd)).add
(continuous_fst.mul (hpstar.comp continuous_snd))
have hcont : ∀ v : Cube, ContinuousAt
(fun z : ℝ × Cube ↦ score l z.1 z.2) (0, v) := by
intro v
dsimp [score, θstar]
apply ContinuousAt.mul continuousAt_const
apply ContinuousAt.sub
· apply ContinuousAt.div
· exact hnumq.continuousAt
· exact hdenq.continuousAt
· simpa using (θs.property.positiveSmooth.2.1 l (v.1 l)
(v.2 l (Set.mem_univ l))).ne'
· apply ContinuousAt.div
· exact hnump.continuousAt
· exact hdenp.continuousAt
· simpa using (θs.property.positiveSmooth.1 l v.1 v.2).ne'
simpa only [Set.mem_univ, forall_const] using
(isCompact_univ.eventually_forall_of_forall_eventually (x₀ := (0 : ℝ))
(P := fun t v ↦ 0 < score l t v) (by
intro v _
have hformula := ownLogRatioDerivative_eq_fderivWithin
θs.property.positiveSmooth l v.1 v.2
have hpos := θs.property.fixedSign l v.1 v.2
apply (hcont v).tendsto
exact Ioi_mem_nhds (by simpa [score, θstar, hformula] using hpos)))
have hall : ∀ᶠ t in 𝓝 (0 : ℝ), ∀ l : Fin n, ∀ v : Cube, 0 < score l t v := by
simpa only [Set.mem_univ, forall_const] using
((Filter.eventually_all_finite (Set.toFinite (Set.univ : Set (Fin n)))).2
(fun l _ ↦ hnode l))
rcases Metric.eventually_nhds_iff.mp hall with ⟨δ, hδ, hδall⟩
refine ⟨δ, hδ, ?_⟩
intro t htδ l v hv
have hscore : 0 < score l t.1 ⟨v, hv⟩ :=
hδall (by simpa [Real.dist_eq, abs_of_nonneg t.2.1] using htδ) l ⟨v, hv⟩
have hη := affinePathExtension_positive_normalized_smooth θs hji t.2
rw [ownLogRatioDerivative_eq_fderivWithin hη l v hv]
have huCube : UniqueDiffOn ℝ (latentCube n) := by
rw [latentCube]
exact UniqueDiffOn.univ_pi fun _ ↦ uniqueDiffOn_Icc (by norm_num)
have huIcc : UniqueDiffOn ℝ (Icc (0 : ℝ) 1) :=
uniqueDiffOn_Icc (by norm_num)
let hθq := θs.property.positiveSmooth.2.2.2.1 l
let hstarq := (embeddedSparseWitness_positive_normalized_smooth s hji).2.2.2.1 l
let hθp := θs.property.positiveSmooth.2.2.1 l
let hstarp := (embeddedSparseWitness_positive_normalized_smooth s hji).2.2.1 l
have hqfd : fderivWithin ℝ ((affinePathExtension s θs hji t.1).q l)
(Icc (0 : ℝ) 1) (v l) =
(1 - t.1) • fderivWithin ℝ (θs.1.q l) (Icc (0 : ℝ) 1) (v l) +
t.1 • fderivWithin ℝ ((embeddedSparseWitness s hji).q l)
(Icc (0 : ℝ) 1) (v l) := by
change fderivWithin ℝ
(fun z ↦ (1 - t.1) • θs.1.q l z +
t.1 • (embeddedSparseWitness s hji).q l z)
(Icc (0 : ℝ) 1) (v l) = _
rw [fderivWithin_fun_add (huIcc.uniqueDiffWithinAt (hv l (Set.mem_univ l)))
(((hθq.const_smul (1 - t.1)) (v l) (hv l (Set.mem_univ l))).differentiableWithinAt
(by norm_num))
(((hstarq.const_smul t.1) (v l) (hv l (Set.mem_univ l))).differentiableWithinAt
(by norm_num)),
fderivWithin_fun_const_smul (huIcc.uniqueDiffWithinAt (hv l (Set.mem_univ l)))
((hθq (v l)
(hv l (Set.mem_univ l))).differentiableWithinAt (by norm_num)) (1 - t.1),
fderivWithin_fun_const_smul (huIcc.uniqueDiffWithinAt (hv l (Set.mem_univ l)))
((hstarq (v l) (hv l (Set.mem_univ l))).differentiableWithinAt
(by norm_num)) t.1]
have hpfd : fderivWithin ℝ ((affinePathExtension s θs hji t.1).p l)
(latentCube n) v =
(1 - t.1) • fderivWithin ℝ (θs.1.p l) (latentCube n) v +
t.1 • fderivWithin ℝ ((embeddedSparseWitness s hji).p l) (latentCube n) v := by
change fderivWithin ℝ
(fun w ↦ (1 - t.1) • θs.1.p l w +
t.1 • (embeddedSparseWitness s hji).p l w) (latentCube n) v = _
rw [fderivWithin_fun_add (huCube.uniqueDiffWithinAt hv)
(((hθp.const_smul (1 - t.1)) v hv).differentiableWithinAt (by norm_num))
(((hstarp.const_smul t.1) v hv).differentiableWithinAt (by norm_num)),
fderivWithin_fun_const_smul (huCube.uniqueDiffWithinAt hv)
((hθp v hv).differentiableWithinAt
(by norm_num)) (1 - t.1),
fderivWithin_fun_const_smul (huCube.uniqueDiffWithinAt hv)
((hstarp v hv).differentiableWithinAt (by norm_num)) t.1]
rw [hqfd, hpfd]
simpa [score, θstar, affinePathExtension, smul_eq_mul] using hscore
theorem affinePathExtension_eventually_modelStratum
All three paper-local stratum conditions hold simultaneously along a sufficiently short positive initial segment of the affine path. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
∃ δ > 0,
∀ t : {t : ℝ // t ∈ Icc (0 : ℝ) 1},
t.1 < δ → ModelStratum G s (affinePathExtension s θs hji t.1)
Proof (Lean source)
lemma affinePathExtension_eventually_modelStratum
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(θs : StratumPoint G s) {j i : Fin n} (hji : G.edge j i) :
∃ δ > 0, ∀ t : {t : ℝ // t ∈ Icc (0 : ℝ) 1}, t.1 < δ →
ModelStratum G s (affinePathExtension s θs hji t.1) := by
rcases affinePathExtension_eventually_causalMinimal θs hji with
⟨δmin, hδmin, hmin⟩
rcases affinePathExtension_eventually_fixedOwnDerivativeSign θs hji with
⟨δsign, hδsign, hsign⟩
refine ⟨min δmin δsign, lt_min hδmin hδsign, ?_⟩
intro t ht
exact ⟨affinePathExtension_positive_normalized_smooth θs hji t.2,
hmin t (ht.trans_le (min_le_left _ _)),
hsign t (ht.trans_le (min_le_right _ _))⟩
theorem affinePath_exists_small_stratum_nonzero
Once the analytic package is available, isolated zeros, affine-path continuity, and local stratum preservation produce the arbitrarily small nonzero perturbation used by the headline. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
n :
θs :
StratumPoint G s
j i :
Fin n
hji :
G.edge j i
O :
Set ℝ
hO :
Icc (0 : ℝ) 1 ⊆ O
hanalytic :
AnalyticOnNhd ℝ (affinePathContrast s θs hji) O
hisolated :
∀ t ∈ O,
affinePathContrast s θs hji t
= 0 → ∃ ε > 0, ∀ u ∈ Ioo (t - ε) (t + ε), u ≠ t → affinePathContrast s θs hji u ≠ 0
hN :
N ∈ 𝓝 θs.1
conclusion
∃ t : {t : ℝ // t ∈ Ioo (0 : ℝ) 1},
conclusion 1
affinePath s θs hji ⟨t.1, ⟨le_of_lt t.2.1, le_of_lt t.2.2⟩⟩ ∈ N
conclusion 2
ModelStratum G s (affinePath s θs hji ⟨t.1, ⟨le_of_lt t.2.1, le_of_lt t.2.2⟩⟩)
conclusion 3
affinePathContrast s θs hji t.1 ≠ 0
Proof (Lean source)
lemma affinePath_exists_small_stratum_nonzero
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(θs : StratumPoint G s) {j i : Fin n} (hji : G.edge j i)
{O : Set ℝ} (hO : Icc (0 : ℝ) 1 ⊆ O)
(hanalytic : AnalyticOnNhd ℝ (affinePathContrast s θs hji) O)
(hisolated : ∀ t ∈ O, affinePathContrast s θs hji t = 0 →
∃ ε > 0, ∀ u ∈ Ioo (t - ε) (t + ε), u ≠ t →
affinePathContrast s θs hji u ≠ 0)
(N : Set (Mechanism n G)) (hN : N ∈ 𝓝 θs.1) :
∃ t : {t : ℝ // t ∈ Ioo (0 : ℝ) 1},
affinePath s θs hji ⟨t.1, ⟨le_of_lt t.2.1, le_of_lt t.2.2⟩⟩ ∈ N ∧
ModelStratum G s (affinePath s θs hji
⟨t.1, ⟨le_of_lt t.2.1, le_of_lt t.2.2⟩⟩) ∧
affinePathContrast s θs hji t.1 ≠ 0 := by
rcases affinePathExtension_eventually_modelStratum θs hji with ⟨δs, hδs, hs⟩
have heventN : ∀ᶠ t in 𝓝 (0 : ℝ), affinePathExtension s θs hji t ∈ N :=
(tendsto_affinePathExtension_zero θs hji) hN
rcases Metric.eventually_nhds_iff.mp heventN with ⟨δN, hδN, hnearN⟩
have hzeroNhd : ∃ δ > 0, ∀ u, |u| < δ → u ≠ 0 →
affinePathContrast s θs hji u ≠ 0 := by
by_cases hz : affinePathContrast s θs hji 0 = 0
· rcases hisolated 0 (hO ⟨by norm_num, by norm_num⟩) hz with ⟨δ, hδ, hiso⟩
exact ⟨δ, hδ, fun u hu hu0 ↦ hiso u (by simpa [abs_lt] using hu) hu0⟩
· have hc := (hanalytic 0 (hO ⟨by norm_num, by norm_num⟩)).continuousAt
have hev : ∀ᶠ u in 𝓝 (0 : ℝ), affinePathContrast s θs hji u ≠ 0 :=
hc (isOpen_compl_singleton.mem_nhds (by simpa using hz))
rcases Metric.eventually_nhds_iff.mp hev with ⟨δ, hδ, hδev⟩
exact ⟨δ, hδ, fun u hu _ ↦ hδev (by simpa [Real.dist_eq] using hu)⟩
rcases hzeroNhd with ⟨δz, hδz, hz⟩
let r := min 1 (min δs (min δN δz))
have hr : 0 < r := by
dsimp [r]
positivity
let u : ℝ := r / 2
have hu0 : 0 < u := div_pos hr (by norm_num)
have hu1 : u < 1 := by
dsimp [u, r]
have := min_le_left (1 : ℝ) (min δs (min δN δz))
linarith
let t : {t : ℝ // t ∈ Ioo (0 : ℝ) 1} := ⟨u, hu0, hu1⟩
refine ⟨t, ?_, ?_, ?_⟩
· exact hnearN (by
rw [Real.dist_eq, sub_zero, abs_of_pos hu0]
dsimp [u, r]
have := min_le_right (1 : ℝ) (min δs (min δN δz))
have := min_le_right δs (min δN δz)
have := min_le_left δN δz
linarith)
· apply hs ⟨u, ⟨hu0.le, hu1.le⟩⟩
dsimp [u, r]
have := min_le_right (1 : ℝ) (min δs (min δN δz))
have := min_le_left δs (min δN δz)
linarith
· apply hz u
· rw [abs_of_pos hu0]
dsimp [u, r]
have := min_le_right (1 : ℝ) (min δs (min δN δz))
have := min_le_right δs (min δN δz)
have := min_le_right δN δz
linarith
· exact ne_of_gt hu0
Every edge admits arbitrarily small stratum-preserving affine perturbations with nonzero second-moment contrast; the contrast has the stated analytic integral and isolated zeros. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
conclusion 1
∃ O : Set ℝ,
IsOpen O
Icc (0 : ℝ) 1 ⊆ O
t ∈ O :
affinePathContrast s θs hji t = affinePathContrastIntegral s θs hji t
AnalyticOnNhd ℝ (affinePathContrast s θs hji) O
t ∈ O :
affinePathContrast s θs hji t = 0
∃ ε > 0, ∀ u ∈ Ioo (t - ε) (t + ε), u ≠ t → affinePathContrast s θs hji u ≠ 0
conclusion 2
affinePathContrast s θs hji 1 ≠ 0
conclusion 3
N ∈ 𝓝 θs.1 :
∃ t : {t : ℝ // t ∈ Ioo (0 : ℝ) 1},
affinePath s θs hji ⟨t.1, ⟨le_of_lt t.2.1, le_of_lt t.2.2⟩⟩ ∈ N
ModelStratum G s (affinePath s θs hji ⟨t.1, ⟨le_of_lt t.2.1, le_of_lt t.2.2⟩⟩)
affinePathContrast s θs hji t.1 ≠ 0
Proof (Lean source)
lemma analytic_edge_perturbation
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(θs : StratumPoint G s) {j i : Fin n} (hji : G.edge j i)
(hIntervention : OnePerfectInterventionPerNode G θs.1
(canonicalObservedWorld G θs.1 (refl (Fin n)))) :
(∃ O : Set ℝ, IsOpen O ∧ Icc (0 : ℝ) 1 ⊆ O ∧
(∀ t ∈ O, affinePathContrast s θs hji t =
affinePathContrastIntegral s θs hji t) ∧
AnalyticOnNhd ℝ (affinePathContrast s θs hji) O ∧
∀ t ∈ O, affinePathContrast s θs hji t = 0 →
∃ ε > 0, ∀ u ∈ Ioo (t - ε) (t + ε), u ≠ t →
affinePathContrast s θs hji u ≠ 0) ∧
affinePathContrast s θs hji 1 ≠ 0 ∧
(∀ N ∈ 𝓝 θs.1, ∃ t : {t : ℝ // t ∈ Ioo (0 : ℝ) 1},
affinePath s θs hji ⟨t.1, ⟨le_of_lt t.2.1, le_of_lt t.2.2⟩⟩ ∈ N ∧
ModelStratum G s (affinePath s θs hji
⟨t.1, ⟨le_of_lt t.2.1, le_of_lt t.2.2⟩⟩) ∧
affinePathContrast s θs hji t.1 ≠ 0) := by
rcases affinePathContrastIntegral_analyticOnNhd θs hji with
⟨O, hO, hIcc, hconvex, hpositive, hintegralAnalytic⟩
have hneji : j ≠ i := by
intro h
subst j
exact G.irrefl i hji
have heq : ∀ t ∈ O, affinePathContrast s θs hji t =
affinePathContrastIntegral s θs hji t := by
intro t ht
unfold affinePathContrast affinePathContrastIntegral
exact canonical_secondMomentContrast_eq_integral (hpositive t ht) hneji
have hanalytic : AnalyticOnNhd ℝ (affinePathContrast s θs hji) O :=
hintegralAnalytic.congr hO (fun t ht ↦ (heq t ht).symm)
have hone : affinePathContrast s θs hji 1 ≠ 0 :=
affinePathContrast_one_ne_zero θs hji
have hisolated : ∀ t ∈ O, affinePathContrast s θs hji t = 0 →
∃ epsilon > 0, ∀ u ∈ Ioo (t - epsilon) (t + epsilon), u ≠ t →
affinePathContrast s θs hji u ≠ 0 := by
intro t ht hzero
exact analyticOnNhd_isolated_zeros_of_nonzero hanalytic hconvex.isPreconnected
(hIcc ⟨by norm_num, by norm_num⟩) hone ht hzero
refine ⟨⟨O, hO, hIcc, heq, hanalytic, hisolated⟩, hone, ?_⟩
intro N hN
exact affinePath_exists_small_stratum_nonzero θs hji hIcc hanalytic hisolated N hN
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.BoundedSubclass 21 declarations This file records the determinate bounded statistical regime and its uniform first-stage ratio event.
Bounded Hölder subclass and first-stage contract
This file records the determinate bounded statistical regime and its uniform first-stage ratio event. The unresolved generated-rank estimator remains a descriptive payload rather than a fabricated construction.
Parameters of the bounded generated-rank regime.
Definition (Lean source)
parameters
W :
ObservedWorld G θ
def ModelGeneratedObservedWorld
An observed world generated by the latent mechanism through a shared C² diffeomorphism and one relabeled perfect intervention per latent node.
Definition (Lean source)
parameters
def
ModelGeneratedObservedWorld G θ :
Type
given by
def DecoderSelectedBoundedRegime
A bounded regime whose predecessor sets use exactly the ordering selected by the law-only population decoder.
Definition (Lean source)
parameters
W :
def
DecoderSelectedBoundedRegime W :
Type
given by
{R : BoundedRegime W.1 // R.order = selectedTopologicalOrder (observedProbabilityLawFamily W.1.law)}
def conditioningDomain
Conditioning-coordinate domain induced by the log-ratios on K.
Definition (Lean source)
parameters
K :
Set (LatentState n)
i :
Fin n
def
given by
familyProjection (logRatio G W) (predecessorSet R.order i) '' K
Maximum conditioning dimension of the selected ordering.
Definition (Lean source)
parameters
def
maximumConditioningDimension R :
ℕ
given by
Finset.univ.sup (fun i : Fin n => (predecessorSet R.order i).card)
def regimeMinimumSampleSize
Minimum environment sample size at global index N.
Definition (Lean source)
parameters
def
regimeMinimumSampleSize R N :
ℕ
given by
Finset.univ.inf' ⟨0, Finset.mem_univ 0⟩ (R.sampleSizeSequence N)
def HolderOrderBound
Hölder-ball predicate of real order β: every derivative through ⌊β⌋ is bounded by M, and the top derivative has the corresponding fractional Hölder modulus.
Definition (Lean source)
parameters
def
HolderOrderBound β M f K :
Prop
given by
k :
⌊β⌋₊
ContDiffOn ℝ k f K
m :
ℕ
m ≤ k
x ∈ K :
‖iteratedFDerivWithin ℝ m f K x‖ ≤ M
x ∈ K :
y ∈ K :
‖iteratedFDerivWithin ℝ k f K x - iteratedFDerivWithin ℝ k f K y‖ ≤ M * ‖x - y‖ ^ (β - k)
All observational mechanisms, intervention densities, and unmixed log ratios lie in the common Hölder ball of order β and radius M.
Definition (Lean source)
parameters
def
BoundedHolderRadius _R β M :
Prop
given by
clause 1
2 < β
clause 2
1 < M
Every observational and intervention density is uniformly bounded below by c.
Definition (Lean source)
Every signed own-coordinate observed log-ratio derivative is at least c.
Definition (Lean source)
The latent preimage of K is at least ρ from the boundary of the latent cube.
Definition (Lean source)
parameters
def
CommonInteriorDomain _R K ρ :
Prop
def jacobianDeterminantWithin
The Jacobian determinant of a Euclidean self-map relative to its stated domain.
Definition (Lean source)
parameters
def
jacobianDeterminantWithin f S x :
ℝ
given by
LinearMap.det (fderivWithin ℝ f S x).toLinearMap
Every observed log ratio has uniformly bounded value and first two derivatives on K.
Definition (Lean source)
parameters
def
ObservedLogRatioRegularity _R K M :
Prop
given by
clause 2
i x :
x ∈ K
|logRatio G W i x| ≤ M
‖fderivWithin ℝ (logRatio G W i) K x‖ ≤ M
‖iteratedFDerivWithin ℝ 2 (logRatio G W i) K x‖ ≤ M
For each environment, some density version of the law of its conditioning coordinates obeys the empty-predecessor convention and lies between c and M on its induced domain.
Definition (Lean source)
parameters
def
ConditionalDesignEnvelope R c M K :
Prop
given by
∃ g : (((j : {j // j ∈ predecessorSet R.order i}) → ℝ) → ℝ),
clause 2
Measure.map (hmeas.mk (familyProjection (logRatio G W) (predecessorSet R.order i))) (W.law i.succ)
= volume.withDensity (fun z => ofReal (g z))
structure BoundedHolderSubclass
Membership evidence for the bounded Hölder subclass, indexed by a point of the full model stratum. Its extension is boundedSubclassSet.
Definition (Lean source)
parameters
n :
ℕ
world :
∀ θs : StratumPoint G s, ModelGeneratedObservedWorld G θs.1
regime :
∀ θs : StratumPoint G s, DecoderSelectedBoundedRegime (world θs)
β c M :
ℝ
K :
Set (LatentState n)
ρ :
ℝ
θs :
StratumPoint G s
assumes
holderRadius :
BoundedHolderRadius (regime θs).1 β M
densityLower :
DensityLowerEnvelope (regime θs).1 c
derivativeMargin :
OwnDerivativeMargin (s := s) (regime θs).1 c
interiorDomain :
CommonInteriorDomain (regime θs).1 K ρ
mixingGeometry :
BoundedSharedMixingGeometry (regime θs).1 M
ratioRegularity :
ObservedLogRatioRegularity (regime θs).1 K M
designEnvelope :
ConditionalDesignEnvelope (regime θs).1 c M K
def boundedSubclassSet
The bounded Hölder subclass as a subset of the population stratum.
Definition (Lean source)
parameters
world :
∀ θs : StratumPoint G s, ModelGeneratedObservedWorld G θs.1
regime :
∀ θs : StratumPoint G s, DecoderSelectedBoundedRegime (world θs)
β c M :
ℝ
K :
Set (LatentState n)
ρ :
ℝ
def
given by
{θs | BoundedHolderSubclass world regime β c M K ρ θs}
def uniformC1RatioEvent
The uniform C¹ log-ratio error event, including C¹ membership of the fitted and true log-ratios rather than only pointwise derivative inequalities.
Definition (Lean source)
parameters
def
given by
{ω | ∀ i, ContDiffOn ℝ 1 (fun y => log (S.ratioEstimate ω i y)) (observedSupport G W) ∧ ContDiffOn ℝ 1 (logRatio G W i) (observedSupport G W) ∧ ∀ x ∈ observedSupport G W, |log (S.ratioEstimate ω i x) - logRatio G W i x| ≤ b ∧ ‖fderivWithin ℝ (fun y => log (S.ratioEstimate ω i y)) (observedSupport G W) x - fderivWithin ℝ (logRatio G W i) (observedSupport G W) x‖ ≤ b}
def firstStageRatioContract
The common N-indexed first-stage contract over the bounded subclass. Each sampling world uses the regime's environment sample-size sequence, and the event is explicitly measurable.
Definition (Lean source)
parameters
Ω :
Type*
world :
∀ θs : StratumPoint G s, ModelGeneratedObservedWorld G θs.1
regime :
∀ θs : StratumPoint G s, DecoderSelectedBoundedRegime (world θs)
sampling :
∀ (θs : StratumPoint G s) (_N : ℕ), SampleSplitWorld (world θs).1 Ω
sampleSizeSequence :
ℕ → Fin (n + 1) → ℕ
bN etaN :
ℕ → ℝ
β c M :
ℝ
K :
Set (LatentState n)
ρ :
ℝ
def
firstStageRatioContract world regime sampling sampleSizeSequence bN etaN β c M K ρ :
Prop
given by
clause 1
N :
0 < bN N
clause 3
N :
0 < etaN N
clause 5
N θs :
θs ∈ boundedSubclassSet world regime β c M K ρ
(regime θs).1.sampleSizeSequence = sampleSizeSequence
(regime θs).1.ratioRate = bN
(regime θs).1.failureRate = etaN
IndependentEnvironmentSampling (sampling θs N)
(sampling θs N).sampleSize = sampleSizeSequence N
MeasurableSet (uniformC1RatioEvent (sampling θs N) (bN N))
1 - etaN N ≤ (sampling θs N).probability.real (uniformC1RatioEvent (sampling θs N) (bN N))
def coordinateRate
The proposed uniform coordinate-rate expression.
Definition (Lean source)
parameters
def
coordinateRate R N :
ℝ
given by
R.bandwidth N ^ 2
+ sqrt (log (regimeMinimumSampleSize R N) / ((regimeMinimumSampleSize R N : ℝ) * R.bandwidth N ^ maximumConditioningDimension R))
+ R.ratioRate N / R.bandwidth N
@realizes (open generated-rank proof-strategy handle) @realizes (open second-stage cross-fitted rank estimator)
Definition (Lean source)
def
generatedRankHandle :
_root_.String
given by
"Nonassertive generated-rank handle: adapt Xie's local-linear conditional-CDF estimator to \ cross-fitted log-ratio responses and covariates; decompose the second stage into the oracle \ conditional-CDF process, indicator boundary crossings from the generated threshold, and \ local-design perturbations from generated conditioning arguments; then linearize that last \ component using Mammen convention, bandwidth selector, influence representation, attainment claim, or sharpness claim \ is supplied here."
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.CitedGates 3 declarations These closed string payloads record cited scope facts.
Bibliographic comparator-scope records
These closed string payloads record cited scope facts. They are metadata only and are not logical hypotheses of any theorem.
Wendong, Kekić, von Kügelgen, Buchholz, Besserve, Gresele, and Schölkopf (2023), "Causal Component Analysis," Definition 3.2, Theorem 4.2, and Appendix E.2, NeurIPS paper handle WendongEtAl2023CauCA.
Definition (Lean source)
def
caucaComparatorScope :
_root_.String
given by
"Wendong et al. (2023), Causal Component Analysis, Definition 3.2, Theorem 4.2, and Appendix E.2 (WendongEtAl2023CauCA; https://papers.nips.cc/paper_files/paper/2023/file/67089958e98b243d5cc1881ad60418b8-Paper-Conference.pdf): the graph G is assumed known, intervention targets are observed and fixed across candidate models, and one perfect stochastic intervention per node, each satisfying Assumption 4.1, gives identification up to componentwise scaling."
von Kügelgen, Besserve, Wendong, Gresele, Kekić, Bareinboim, Blei, and Schölkopf (2023), "Nonparametric Identifiability of Causal Representations from Unknown Interventions," Theorems 3.2 and 3.4 and Section 7, handle vonKugelgenEtAl2023UnknownInterventions.
Definition (Lean source)
def
vonKugelgenComparatorScope :
_root_.String
given by
"von Kugelgen et al. (2023), Nonparametric Identifiability of Causal Representations from Unknown Interventions, Theorems 3.2 and 3.4 and Section 7 (vonKugelgenEtAl2023UnknownInterventions; https://papers.nips.cc/paper_files/paper/2023/file/97fe251c25b6f99a2a23b330a75b11d4-Paper-Conference.pdf): Theorem 3.2 is bivariate with one unknown-target perfect intervention per node and a continuous witness genericity condition; Theorem 3.4 uses two paired perfect interventions per node in arbitrary dimension; the one-intervention extension for n greater than two is stated as a conjecture."
Yao, Rancati, Cadei, Fumero, and Locatello (2025), "Unifying Causal Representation Learning with the Invariance Principle," Assumption D.1, Corollary D.1, and the following remark, arXiv handle 2409.02772v2.
Definition (Lean source)
def
yaoComparatorScope :
_root_.String
given by
"Yao et al. (2025), Unifying Causal Representation Learning with the Invariance Principle, Assumption D.1, Corollary D.1, and the immediately following remark (YaoEtAl2025InvariancePrinciple; arXiv:2409.02772v2; https://arxiv.org/abs/2409.02772v2): exactly one imperfect intervention is supplied per node, the target labels preserve a supplied topological order, componentwise identification follows from marginal and score invariances, and identifying the order is explicitly treated as a separate subproblem."
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.CompactCondIndepBridge 12 declarations This file transports the paper's ambient, cube-supported coordinate conditional independence to the compact coordinate-product presentation used by positive finite-DAG factorizations.
Conditional-independence bridge to compact cube factorizations
This file transports the paper's ambient, cube-supported coordinate conditional independence to the compact coordinate-product presentation used by positive finite-DAG factorizations.
theorem measurable_paperCoordinateProjection
The finite-coordinate projection from latent states is measurable.
Formal statement
Proof (Lean source)
@[fun_prop]
lemma measurable_paperCoordinateProjection {n : ℕ} (S : Finset (Fin n)) :
Measurable (coordinateProjection S) := by
exact measurable_pi_lambda _ fun j ↦ measurable_pi_apply j.1
theorem comap_compact_realProjection_eq
On compact coordinates, the real-valued coordinate projection generates the same sigma-algebra as the subtype-valued coordinate projection. the stated conclusion follows.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma comap_compact_realProjection_eq (n : ℕ) (S : Finset (Fin n)) :
comap
((coordinateProjection S) ∘ compactCubeInclude n) inferInstance =
comap
(coordinateProjection
(X := fun _ : Fin n ↦ Icc (0 : ℝ) 1) S) inferInstance := by
let compactProjection :=
coordinateProjection
(X := fun _ : Fin n ↦ Icc (0 : ℝ) 1) S
let coeProjection : ((j : {j // j ∈ S}) → Icc (0 : ℝ) 1) →
((j : {j // j ∈ S}) → ℝ) := fun x j ↦ x j
let retractProjection : ((j : {j // j ∈ S}) → ℝ) →
((j : {j // j ∈ S}) → Icc (0 : ℝ) 1) :=
fun x j ↦ projIcc 0 1 (by norm_num) (x j)
have hcoe : Measurable coeProjection := by
exact measurable_pi_lambda _ fun j ↦
measurable_subtype_coe.comp (measurable_pi_apply j)
have hretract : Measurable retractProjection := by
exact measurable_pi_lambda _ fun j ↦
continuous_projIcc.measurable.comp (measurable_pi_apply j)
have hreal : ((coordinateProjection S) ∘ compactCubeInclude n) =
coeProjection ∘ compactProjection := rfl
have hback : compactProjection = retractProjection ∘
((coordinateProjection S) ∘ compactCubeInclude n) := by
funext x j
apply Subtype.ext
change (x j : ℝ) =
((projIcc 0 1 (by norm_num) (x j) : Icc (0 : ℝ) 1) : ℝ)
exact congrArg val
(Set.projIcc_of_mem (show (0 : ℝ) ≤ 1 by norm_num) (x j).property).symm
apply le_antisymm
· rw [hreal]
change comap (coeProjection ∘ compactProjection) inferInstance ≤
comap compactProjection inferInstance
simpa only [MeasurableSpace.comap_comp] using
comap_mono hcoe.comap_le
· change comap compactProjection inferInstance ≤
comap
((coordinateProjection S) ∘ compactCubeInclude n) inferInstance
rw [hback]
simpa only [MeasurableSpace.comap_comp] using
comap_mono hretract.comap_le
theorem comap_compact_realCoordinate_eq
On compact coordinates, coercing one interval-valued coordinate to a real generates the same sigma-algebra as the original interval-valued coordinate. the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma comap_compact_realCoordinate_eq (n : ℕ) (i : Fin n) :
comap
((fun v : LatentState n ↦ v i) ∘ compactCubeInclude n) inferInstance =
comap
(fun x : (k : Fin n) → Icc (0 : ℝ) 1 ↦ x i) inferInstance := by
let compactCoordinate : ((k : Fin n) → Icc (0 : ℝ) 1) →
Icc (0 : ℝ) 1 := fun x ↦ x i
let coeCoordinate : Icc (0 : ℝ) 1 → ℝ := Subtype.val
let retractCoordinate : ℝ → Icc (0 : ℝ) 1 :=
projIcc 0 1 (by norm_num)
have hcoe : Measurable coeCoordinate := measurable_subtype_coe
have hretract : Measurable retractCoordinate := continuous_projIcc.measurable
have hreal : ((fun v : LatentState n ↦ v i) ∘ compactCubeInclude n) =
coeCoordinate ∘ compactCoordinate := rfl
have hback : compactCoordinate = retractCoordinate ∘
((fun v : LatentState n ↦ v i) ∘ compactCubeInclude n) := by
funext x
apply Subtype.ext
exact congrArg Subtype.val
(Set.projIcc_of_mem (show (0 : ℝ) ≤ 1 by norm_num) (x i).property).symm
apply le_antisymm
· rw [hreal]
change comap (coeCoordinate ∘ compactCoordinate) inferInstance ≤
comap compactCoordinate inferInstance
simpa only [MeasurableSpace.comap_comp] using
comap_mono hcoe.comap_le
· change comap compactCoordinate inferInstance ≤
comap
((fun v : LatentState n ↦ v i) ∘ compactCubeInclude n) inferInstance
rw [hback]
simpa only [MeasurableSpace.comap_comp] using
comap_mono hretract.comap_le
theorem compactRealCondIndep_iff
The real-coordinate presentation on the compact cube is equivalent to the native subtype-coordinate presentation of a compact positive factorization. the stated conclusion follows.
Formal statement
hypotheses
conclusion
CondIndepFun (comap ((coordinateProjection C) ∘ compactCubeInclude n) inferInstance) ((measurable_paperCoordinateProjection C).comp (measurable_compactCubeInclude n)).comap_le ((fun v : LatentState n ↦ v i) ∘ compactCubeInclude n) ((fun v : LatentState n ↦ v j) ∘ compactCubeInclude n) M.observationalMeasure
↔ M.CondIndepCoordinates i j C
Proof (Lean source)
lemma compactRealCondIndep_iff
{n : ℕ} {G : DAG (Fin n)}
{mu : Fin n → Measure (Icc (0 : ℝ) 1)} [∀ k, SigmaFinite (mu k)]
(M : CompactPositiveFactorization G
(fun _ : Fin n ↦ Icc (0 : ℝ) 1) mu)
(i j : Fin n) (C : Finset (Fin n)) :
CondIndepFun
(comap
((coordinateProjection C) ∘ compactCubeInclude n) inferInstance)
((measurable_paperCoordinateProjection C).comp
(measurable_compactCubeInclude n)).comap_le
((fun v : LatentState n ↦ v i) ∘ compactCubeInclude n)
((fun v : LatentState n ↦ v j) ∘ compactCubeInclude n)
M.observationalMeasure ↔
M.CondIndepCoordinates i j C := by
unfold CompactPositiveFactorization.CondIndepCoordinates
rw [ProbabilityTheory.condIndepFun_iff_condIndep,
ProbabilityTheory.condIndepFun_iff_condIndep]
have hZ := comap_compact_realProjection_eq n C
have hX := comap_compact_realCoordinate_eq n i
have hY := comap_compact_realCoordinate_eq n j
simpa only [hZ, hX, hY]
theorem condIndepCoordinates_iff_compactPositiveFactorization
If a compact positive factorization's observational measure includes to the paper's ambient observational law, then the two singleton-coordinate conditional-independence encodings are equivalent. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
∀ k, SigmaFinite (mu k)
theta :
StratumPoint G s
M :
hobs :
Measure.map (compactCubeInclude n) M.observationalMeasure = observationalLaw theta.1
i j :
Fin n
conclusion
CondIndepCoordinates theta.1 {i} {j} C ↔ M.CondIndepCoordinates i j C
Proof (Lean source)
lemma condIndepCoordinates_iff_compactPositiveFactorization
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
{mu : Fin n → Measure (Icc (0 : ℝ) 1)} [∀ k, SigmaFinite (mu k)]
(theta : StratumPoint G s)
(M : CompactPositiveFactorization G
(fun _ : Fin n ↦ Icc (0 : ℝ) 1)
mu)
(hobs : Measure.map (compactCubeInclude n) M.observationalMeasure =
observationalLaw theta.1)
(i j : Fin n) (C : Finset (Fin n)) :
CondIndepCoordinates theta.1 {i} {j} C ↔ M.CondIndepCoordinates i j C := by
letI : IsFiniteMeasure (observationalLaw theta.1) := by
rw [← hobs]
infer_instance
have hmapRetract : Measure.map (compactCubeRetract n) (observationalLaw theta.1) =
M.observationalMeasure := by
rw [← hobs]
exact map_compactCubeRetract_map_compactCubeInclude M.observationalMeasure
have hcompact := condIndepFun_comp_aeEquiv_iff
(compactCubeInclude n) (compactCubeRetract n)
(measurable_compactCubeInclude n) (measurable_compactCubeRetract n)
hobs hmapRetract
(compactCubeInclude_retract_ae_observationalLaw theta.1)
(fun v : LatentState n ↦ v i) (fun v : LatentState n ↦ v j)
(coordinateProjection C)
(measurable_pi_apply i) (measurable_pi_apply j)
(measurable_paperCoordinateProjection C)
rw [condIndepCoordinates_singletons_iff_condIndepGiven i j C]
constructor
· rintro ⟨_, _, _, _, hambient⟩
exact (compactRealCondIndep_iff M i j C).mp (hcompact.mpr hambient)
· intro hM
have hreal := (compactRealCondIndep_iff M i j C).mpr hM
refine ⟨inferInstance, measurable_pi_apply i, measurable_pi_apply j,
measurable_paperCoordinateProjection C, ?_⟩
exact hcompact.mp hreal
theorem condIndepCoordinates_iff_compactPositiveFactorization_mechanism
The paper and compact-factorization singleton conditional-independence encodings agree for an arbitrary mechanism whenever their observational measures agree. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
∀ k, SigmaFinite (mu k)
theta :
Mechanism n G
M :
hobs :
Measure.map (compactCubeInclude n) M.observationalMeasure = observationalLaw theta
i j :
Fin n
conclusion
CondIndepCoordinates theta {i} {j} C ↔ M.CondIndepCoordinates i j C
Proof (Lean source)
lemma condIndepCoordinates_iff_compactPositiveFactorization_mechanism
{n : ℕ} {G : DAG (Fin n)}
{mu : Fin n → Measure (Icc (0 : ℝ) 1)} [∀ k, SigmaFinite (mu k)]
(theta : Mechanism n G)
(M : CompactPositiveFactorization G
(fun _ : Fin n ↦ Icc (0 : ℝ) 1) mu)
(hobs : Measure.map (compactCubeInclude n) M.observationalMeasure =
observationalLaw theta)
(i j : Fin n) (C : Finset (Fin n)) :
CondIndepCoordinates theta {i} {j} C ↔ M.CondIndepCoordinates i j C := by
letI : IsFiniteMeasure (observationalLaw theta) := by
rw [← hobs]
infer_instance
have hmapRetract : Measure.map (compactCubeRetract n) (observationalLaw theta) =
M.observationalMeasure := by
rw [← hobs]
exact map_compactCubeRetract_map_compactCubeInclude M.observationalMeasure
have hcompact := condIndepFun_comp_aeEquiv_iff
(compactCubeInclude n) (compactCubeRetract n)
(measurable_compactCubeInclude n) (measurable_compactCubeRetract n)
hobs hmapRetract
(compactCubeInclude_retract_ae_observationalLaw theta)
(fun v : LatentState n ↦ v i) (fun v : LatentState n ↦ v j)
(coordinateProjection C)
(measurable_pi_apply i) (measurable_pi_apply j)
(measurable_paperCoordinateProjection C)
rw [condIndepCoordinates_singletons_iff_condIndepGiven i j C]
constructor
· rintro ⟨_, _, _, _, hambient⟩
exact (compactRealCondIndep_iff M i j C).mp (hcompact.mpr hambient)
· intro hM
have hreal := (compactRealCondIndep_iff M i j C).mpr hM
refine ⟨inferInstance, measurable_pi_apply i, measurable_pi_apply j,
measurable_paperCoordinateProjection C, ?_⟩
exact hcompact.mp hreal
def mechanismCompactPositiveFactorization
A positive normalized smooth paper mechanism restricts to a compact positive factorization on the product of unit-interval coordinate subtypes.
Definition (Lean source)
parameters
hpos :
def
given by
by let B := mechanismUnitCubeFactorization hpos have hcube (x : (k : Fin n) → Icc (0 : ℝ) 1) : compactCubeInclude n x ∈ latentCube n := fun k _ ↦ (x k).property have hfactor (i : Fin n) (x : (k : Fin n) → Icc (0 : ℝ) 1) : B.factor i (compactCubeInclude n x)
= ofReal (theta.p i (compactCubeInclude n x)) := by simpa only [B, mechanismUnitCubeFactorization, latentCube] using unitCubeFactorizationOfRealCubeFactors_factor_eq theta.p (measurableOnSet_of_continuousOn_cube theta.p (fun k => (hpos.2.2.1 k).continuousOn)) (fun k v hv => (hpos.1 k v hv).le) (fun k v w _ _ hvw => theta.parent_local k v w (hvw k (mem_insert_self _ _)) (fun l hl => hvw l (mem_insert_of_mem hl))) hpos.2.2.2.2.1 i (hcube x) have hlower_exists : ∃ lower > 0, ∀ z ∈ univ ×ˢ latentCube n, lower ≤ theta.p z.1 z.2 := (isCompact_univ.prod (by rw [latentCube] exact isCompact_univ_pi fun _ => isCompact_Icc)).exists_forall_le' (by rw [continuousOn_prod_of_discrete_left] intro i simpa only [Set.mem_prod, Set.mem_univ, true_and, Set.ofPred_mem_eq] using (hpos.2.2.1 i).continuousOn) (fun z hz => hpos.1 z.1 z.2 hz.2) let lower := choose hlower_exists have hlower_spec := Classical.choose_spec hlower_exists have hlower : 0 < lower := hlower_spec.1 have hlower_le : ∀ z ∈ univ ×ˢ latentCube n, lower ≤ theta.p z.1 z.2 := hlower_spec.2 refine { toFactorization := { factor := fun i x ↦ B.factor i (compactCubeInclude n x) measurable_factor := fun i ↦ (B.measurable_factor i).comp (measurable_compactCubeInclude n) local_factor := ?_ normalized_factor := ?_ } lower := lower lower_pos := hlower factor_ne_top := ?_ factor_continuous := ?_ lower_le_factor := ?_ } · intro i x y hxy apply B.local_factor i intro k hk exact congrArg Subtype.val (hxy k hk) · intro i x have hfun : (fun z : Icc (0 : ℝ) 1 ↦ B.factor i (compactCubeInclude n (update x i z))) = (fun z : ℝ ↦ B.factor i (update (compactCubeInclude n x) i z)) ∘ Subtype.val := by funext z congr 2 funext k by_cases hki : k = i · subst k simp [compactCubeInclude] · simp [update, hki, compactCubeInclude] have hf : Measurable (fun z : ℝ ↦ B.factor i (update (compactCubeInclude n x) i z)) := (B.measurable_factor i).comp (measurable_update (compactCubeInclude n x)) rw [hfun, unitInterval.volume_def] let f : ℝ → ℝ≥0∞ := fun z ↦ B.factor i (update (compactCubeInclude n x) i z) have he : MeasurePreserving (Subtype.val : Icc (0 : ℝ) 1 → ℝ) (Measure.comap Subtype.val volume) (volume.restrict (Icc (0 : ℝ) 1)) := measurePreserving_subtype_coe measurableSet_Icc calc (∫⁻ z : Icc (0 : ℝ) 1, f z ∂Measure.comap Subtype.val volume) = ∫⁻ z : ℝ, f z ∂volume.restrict (Icc (0 : ℝ) 1) := he.lintegral_comp hf _ = 1 := B.normalized_factor i (compactCubeInclude n x) · intro i x rw [hfactor] exact ENNReal.ofReal_ne_top · intro i have hinclude : Continuous (compactCubeInclude n) := by exact continuous_pi fun k ↦ continuous_subtype_val.comp (continuous_apply k) have hmaps : MapsTo (compactCubeInclude n) univ (latentCube n) := fun x _ ↦ hcube x have hcont : Continuous (fun x => theta.p i (compactCubeInclude n x)) := (hpos.2.2.1 i).continuousOn.comp_continuous hinclude (fun x => hcube x) convert hcont using 1 funext x rw [hfactor, ENNReal.toReal_ofReal (hpos.1 i _ (hcube x)).le] · intro i x rw [hfactor, ENNReal.toReal_ofReal (hpos.1 i _ (hcube x)).le] exact hlower_le (i, compactCubeInclude n x) ⟨Set.mem_univ i, hcube x⟩
theorem mechanismCompactPositiveFactorization_factor_eq
The compact factorization's local factor is the paper mechanism factor evaluated at the included compact assignment. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
hpos :
i :
Fin n
conclusion
(mechanismCompactPositiveFactorization hpos).toFactorization.factor i x
= ofReal (theta.p i (compactCubeInclude n x))
Proof (Lean source)
lemma mechanismCompactPositiveFactorization_factor_eq
{n : ℕ} {G : DAG (Fin n)} {theta : Mechanism n G}
(hpos : PositiveNormalizedSmoothMechanisms G theta)
(i : Fin n) (x : (k : Fin n) → Icc (0 : ℝ) 1) :
(mechanismCompactPositiveFactorization hpos).toFactorization.factor i x =
ofReal (theta.p i (compactCubeInclude n x)) := by
change (mechanismUnitCubeFactorization hpos).factor i (compactCubeInclude n x) = _
change (mechanismUnitCubeFactorization hpos).factor i (fun k => (x k : ℝ)) =
ofReal (theta.p i (fun k => (x k : ℝ)))
simpa only [mechanismUnitCubeFactorization, latentCube] using
unitCubeFactorizationOfRealCubeFactors_factor_eq theta.p
(measurableOnSet_of_continuousOn_cube theta.p
(fun k => (hpos.2.2.1 k).continuousOn))
(fun k v hv => (hpos.1 k v hv).le)
(fun k v w _ _ hvw => theta.parent_local k v w
(hvw k (mem_insert_self _ _))
(fun l hl => hvw l (mem_insert_of_mem hl)))
hpos.2.2.2.2.1 i (fun k _ => (x k).property)
theorem mechanismCompact_factorSupClose_of_p_close
Uniform closeness of paper observational factors implies FactorSupClose for their compact positive factorizations. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
(mechanismCompactPositiveFactorization hθ).FactorSupClose (mechanismCompactPositiveFactorization hη) ε
Proof (Lean source)
lemma mechanismCompact_factorSupClose_of_p_close
{n : ℕ} {G : DAG (Fin n)} {theta eta : Mechanism n G}
(hθ : PositiveNormalizedSmoothMechanisms G theta)
(hη : PositiveNormalizedSmoothMechanisms G eta) {ε : ℝ}
(hclose : ∀ i v, v ∈ latentCube n → |eta.p i v - theta.p i v| < ε) :
(mechanismCompactPositiveFactorization hθ).FactorSupClose
(mechanismCompactPositiveFactorization hη) ε := by
intro i x
rw [mechanismCompactPositiveFactorization_factor_eq,
mechanismCompactPositiveFactorization_factor_eq]
have hx : compactCubeInclude n x ∈ latentCube n := fun k _ => (x k).property
rw [ENNReal.toReal_ofReal (hη.1 i _ hx).le,
ENNReal.toReal_ofReal (hθ.1 i _ hx).le]
exact hclose i _ hx
theorem mechanismCompactPositiveFactorization_observationalMeasure
Including the compact factorization's observational measure into the ambient latent space recovers the paper's cube-supported observational law. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
hpos :
conclusion
Measure.map (compactCubeInclude n) (mechanismCompactPositiveFactorization hpos).observationalMeasure
Proof (Lean source)
lemma mechanismCompactPositiveFactorization_observationalMeasure
{n : ℕ} {G : DAG (Fin n)} {theta : Mechanism n G}
(hpos : PositiveNormalizedSmoothMechanisms G theta) :
Measure.map (compactCubeInclude n)
(mechanismCompactPositiveFactorization hpos).observationalMeasure =
observationalLaw theta := by
let M := mechanismCompactPositiveFactorization hpos
let e := compactCubeEquiv n
let d : {v : LatentState n // v ∈ latentCube n} → ℝ≥0∞ :=
fun v => ofReal (observationalDensity theta v.1)
have hdensity : M.toFactorization.observationalDensity = d ∘ e := by
funext x
unfold observationalDensity
partialDensity
change (∏ i, (mechanismCompactPositiveFactorization hpos).toFactorization.factor i x) =
ofReal (∏ i, theta.p i (compactCubeInclude n x))
simp_rw [mechanismCompactPositiveFactorization_factor_eq hpos]
rw [← ENNReal.ofReal_prod_of_nonneg]
intro i hi
exact (hpos.1 i (compactCubeInclude n x) (fun k _ => (x k).property)).le
have heWeighted : Measure.map e M.observationalMeasure =
(volume : Measure {v : LatentState n // v ∈ latentCube n}).withDensity d := by
unfold CompactPositiveFactorization.observationalMeasure
observationalMeasure
rw [hdensity]
exact map_withDensity_comp_measurableEquiv e
(compactCubeEquiv_measurePreserving n) d
have hcube : MeasurableSet (latentCube n) := by
rw [latentCube]
exact MeasurableSet.univ_pi fun _ => measurableSet_Icc
have hcoe : MeasurePreserving
(Subtype.val : {v : LatentState n // v ∈ latentCube n} → LatentState n)
volume (volume.restrict (latentCube n)) :=
measurePreserving_subtype_coe hcube
have hcoeWeighted : Measure.map
(Subtype.val : {v : LatentState n // v ∈ latentCube n} → LatentState n)
((volume : Measure {v : LatentState n // v ∈ latentCube n}).withDensity d) =
observationalLaw theta := by
ext s hs
rw [Measure.map_apply measurable_subtype_coe hs,
withDensity_apply _ (measurable_subtype_coe hs)]
unfold observationalLaw
rw [withDensity_apply _ hs]
exact hcoe.setLIntegral_comp_preimage_emb
(MeasurableEmbedding.subtype_coe hcube)
(fun v => ofReal (observationalDensity theta v)) s
rw [← heWeighted] at hcoeWeighted
rw [Measure.map_map measurable_subtype_coe e.measurable] at hcoeWeighted
exact hcoeWeighted
theorem condIndepCoordinates_iff_mechanismCompactPositiveFactorization
For a positive smooth mechanism, the paper's singleton-block conditional independence is exactly the compact positive factorization's coordinate conditional independence. the stated conclusion follows.
Formal statement
hypotheses
conclusion
CondIndepCoordinates theta.1 {i} {j} C
↔ CompactPositiveFactorization.CondIndepCoordinates (mechanismCompactPositiveFactorization theta.property.positiveSmooth) i j C
Proof (Lean source)
lemma condIndepCoordinates_iff_mechanismCompactPositiveFactorization
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(theta : StratumPoint G s) (i j : Fin n) (C : Finset (Fin n)) :
CondIndepCoordinates theta.1 {i} {j} C ↔
CompactPositiveFactorization.CondIndepCoordinates
(mechanismCompactPositiveFactorization theta.property.positiveSmooth) i j C := by
exact condIndepCoordinates_iff_compactPositiveFactorization theta
(mechanismCompactPositiveFactorization theta.property.positiveSmooth)
(mechanismCompactPositiveFactorization_observationalMeasure
theta.property.positiveSmooth) i j C
theorem mechanismCompact_all_edge_witnesses_open
Every paper stratum point has one uniform compact-factor neighborhood in which causal minimality persists on all directed edges. the stated conclusion follows.
Formal statement
hypotheses
n :
theta :
StratumPoint G s
conclusion
M :
mechanismCompactPositiveFactorization theta.property.positiveSmooth
Proof (Lean source)
lemma mechanismCompact_all_edge_witnesses_open
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(theta : StratumPoint G s) :
let M := mechanismCompactPositiveFactorization theta.property.positiveSmooth
∃ ε > 0, ∀ N : CompactPositiveFactorization G
(fun _ : Fin n ↦ Icc (0 : ℝ) 1)
(fun _ : Fin n ↦ (volume : Measure (Icc (0 : ℝ) 1))),
M.FactorSupClose N ε →
∀ i j (hji : G.edge j i),
¬ N.CondIndepCoordinates i j ((G.parents i).erase j) := by
classical
dsimp only
let M := mechanismCompactPositiveFactorization theta.property.positiveSmooth
have hwitness : ∀ i j, G.edge j i → Nonempty (M.EdgeWitness i j) := by
intro i j hji
have hnot : ¬ M.CondIndepCoordinates i j ((G.parents i).erase j) := by
rw [← condIndepCoordinates_iff_mechanismCompactPositiveFactorization theta]
exact theta.property.causalMinimal hji
have hnotAll : ¬ ∀ (x : ∀ k : Fin n, Icc (0 : ℝ) 1)
(xi xi' : Icc (0 : ℝ) 1) (xj xj' : Icc (0 : ℝ) 1),
M.localContrast i j x xi xi' xj xj' = 0 := by
exact fun hall => hnot ((M.edge_condIndep_iff_localContrast_zero hji).mpr hall)
push_neg at hnotAll
rcases hnotAll with ⟨x, xi, xi', xj, xj', hne⟩
exact ⟨
{ base := x
child₀ := xi
child₁ := xi'
parent₀ := xj
parent₁ := xj'
nonzero := hne }⟩
exact M.all_edge_witnesses_open fun i j hji => choice (hwitness i j hji)
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.CompactCubeBridge 11 declarations This file identifies the ambient closed cube used by the paper with the product of compact interval coordinate types used by finite positive-density factorizations.
Compact cube carrier bridge
This file identifies the ambient closed cube used by the paper with the product of compact interval coordinate types used by finite positive-density factorizations.
The product of unit-interval coordinate types is measurably equivalent to the subtype of ambient latent vectors lying in the closed cube.
Definition (Lean source)
parameters
n :
ℕ
def
given by
clause 1
toFun x := ⟨fun i ↦ x i, fun i _ ↦ (x i).property⟩
clause 2
invFun v := fun i ↦ ⟨v.1 i, v.2 i (Set.mem_univ i)⟩
clause 5
measurable_toFun := by apply Measurable.subtype_mk exact measurable_pi_iff.mpr fun i ↦ measurable_subtype_coe.comp (measurable_pi_apply i)
clause 6
measurable_invFun := by apply measurable_pi_iff.mpr intro i change Measurable (fun v : {v : LatentState n // v ∈ latentCube n} ↦ (⟨v.1 i, v.2 i (Set.mem_univ i)⟩ : Icc (0 : ℝ) 1)) exact Measurable.subtype_mk ((measurable_pi_apply i).comp measurable_subtype_coe)
theorem compactCubeEquiv_measurePreserving
The cube equivalence carries the product of restricted coordinate volumes to the restricted ambient volume on the closed cube. the stated conclusion follows.
Formal statement
hypotheses
n :
ℕ
conclusion
MeasurePreserving (compactCubeEquiv n) (Measure.pi fun _ : Fin n ↦ (volume : Measure (Icc (0 : ℝ) 1))) (volume : Measure {v : LatentState n // v ∈ latentCube n})
Proof (Lean source)
lemma compactCubeEquiv_measurePreserving (n : ℕ) :
MeasurePreserving (compactCubeEquiv n)
(Measure.pi fun _ : Fin n ↦ (volume : Measure (Icc (0 : ℝ) 1)))
(volume : Measure {v : LatentState n // v ∈ latentCube n}) := by
let e := compactCubeEquiv n
let c : ((i : Fin n) → Icc (0 : ℝ) 1) → LatentState n :=
fun x i ↦ x i
have hc : MeasurePreserving c
(Measure.pi fun _ : Fin n ↦ (volume : Measure (Icc (0 : ℝ) 1)))
(Measure.pi fun _ : Fin n ↦ volume.restrict (Icc (0 : ℝ) 1)) := by
exact measurePreserving_pi _ _ fun _ ↦
measurePreserving_subtype_coe measurableSet_Icc
have hcube : MeasurableSet (latentCube n) := by
rw [latentCube]
exact MeasurableSet.univ_pi fun _ ↦ measurableSet_Icc
have hcoe : MeasurePreserving
(Subtype.val : {v : LatentState n // v ∈ latentCube n} → LatentState n)
volume (volume.restrict (latentCube n)) :=
measurePreserving_subtype_coe hcube
refine ⟨e.measurable, ?_⟩
apply (MeasurableEmbedding.subtype_coe hcube).map_injective
rw [Measure.map_map measurable_subtype_coe e.measurable, hcoe.map_eq]
change Measure.map c
(Measure.pi fun _ : Fin n ↦ (volume : Measure (Icc (0 : ℝ) 1))) =
volume.restrict (latentCube n)
rw [hc.map_eq]
change (Measure.pi fun _ : Fin n ↦ volume.restrict (Icc (0 : ℝ) 1)) =
volume.restrict (Set.univ.pi fun _ : Fin n ↦ Icc (0 : ℝ) 1)
rw [MeasureTheory.volume_pi, Measure.restrict_pi_pi]
Compact coordinate assignments include into the ambient latent vector space.
Definition (Lean source)
parameters
n :
ℕ
def
given by
fun x i ↦ x i
Coordinatewise projection onto the unit interval retracts the ambient latent vector space onto compact coordinate assignments.
Definition (Lean source)
theorem measurable_compactCubeInclude
Inclusion of compact coordinate assignments into ambient latent vectors is measurable. the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma measurable_compactCubeInclude (n : ℕ) : Measurable (compactCubeInclude n) := by
exact measurable_pi_iff.mpr fun i ↦
measurable_subtype_coe.comp (measurable_pi_apply i)
theorem measurable_compactCubeRetract
The coordinatewise compact-cube retraction is measurable. the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma measurable_compactCubeRetract (n : ℕ) : Measurable (compactCubeRetract n) := by
exact measurable_pi_iff.mpr fun i ↦
continuous_projIcc.measurable.comp (measurable_pi_apply i)
theorem compactCubeRetract_include
Retraction after inclusion is exactly the identity on compact coordinate assignments. the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma compactCubeRetract_include (n : ℕ) :
compactCubeRetract n ∘ compactCubeInclude n = id := by
funext x
funext i
apply Subtype.ext
exact congrArg val
(Set.projIcc_of_mem (show (0 : ℝ) ≤ 1 by norm_num) (x i).property)
theorem compactCubeInclude_retract_of_mem
Inclusion after retraction fixes every ambient point belonging to the latent cube. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma compactCubeInclude_retract_of_mem {n : ℕ} {v : LatentState n}
(hv : v ∈ latentCube n) :
compactCubeInclude n (compactCubeRetract n v) = v := by
funext i
exact congrArg val
(Set.projIcc_of_mem (show (0 : ℝ) ≤ 1 by norm_num)
(hv i (Set.mem_univ i)))
theorem compactCubeInclude_retract_ae_observationalLaw
Under a positive smooth mechanism's cube-supported observational law, inclusion after the compact-cube retraction is almost everywhere the ambient identity. the stated conclusion follows.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma compactCubeInclude_retract_ae_observationalLaw
{n : ℕ} {G : DAG (Fin n)} (θ : Mechanism n G) :
(compactCubeInclude n ∘ compactCubeRetract n) =ᵐ[observationalLaw θ] id := by
have hcube : MeasurableSet (latentCube n) := by
rw [latentCube]
exact MeasurableSet.univ_pi fun _ ↦ measurableSet_Icc
have hbase : (compactCubeInclude n ∘ compactCubeRetract n)
=ᵐ[volume.restrict (latentCube n)] id := by
filter_upwards [ae_restrict_mem hcube] with v hv
simpa only [Function.comp_apply, id_eq] using compactCubeInclude_retract_of_mem hv
exact (withDensity_absolutelyContinuous _ _).ae_le hbase
theorem map_compactCubeInclude_map_compactCubeRetract_observationalLaw
Pushing a cube-supported observational law to compact coordinates and including it back recovers the original ambient law. the stated conclusion follows.
Formal statement
hypotheses
conclusion
= observationalLaw θ
Proof (Lean source)
lemma map_compactCubeInclude_map_compactCubeRetract_observationalLaw
{n : ℕ} {G : DAG (Fin n)} (θ : Mechanism n G) :
Measure.map (compactCubeInclude n)
(Measure.map (compactCubeRetract n) (observationalLaw θ)) =
observationalLaw θ := by
rw [Measure.map_map (measurable_compactCubeInclude n)
(measurable_compactCubeRetract n)]
rw [Measure.map_congr (compactCubeInclude_retract_ae_observationalLaw θ)]
exact Measure.map_id
theorem map_compactCubeRetract_map_compactCubeInclude
For every compact-coordinate measure, inclusion followed by retraction also recovers the original measure. the stated conclusion follows.
Formal statement
hypotheses
conclusion
Measure.map (compactCubeRetract n) (Measure.map (compactCubeInclude n) μ) = μ
Proof (Lean source)
lemma map_compactCubeRetract_map_compactCubeInclude
{n : ℕ} (μ : Measure ((i : Fin n) → Icc (0 : ℝ) 1)) :
Measure.map (compactCubeRetract n) (Measure.map (compactCubeInclude n) μ) = μ := by
rw [Measure.map_map (measurable_compactCubeRetract n)
(measurable_compactCubeInclude n)]
rw [compactCubeRetract_include]
exact Measure.map_id
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.CondIndepIntersection 10 declarations This file scaffolds the positivity-based graphoid intersection step and the finite-coordinate product bridge used with Causalean's generic weak-union lemma.
Conditional-independence bridges for parent pruning
This file scaffolds the positivity-based graphoid intersection step and the finite-coordinate product bridge used with Causalean's generic weak-union lemma.
def observedLatentCoordinate
The latent coordinate targeted by an environment label, expressed on observed space.
Definition (Lean source)
parameters
def
given by
by classical exact (observedSupport G W).piecewise (fun x => W.unmix x (W.targetPerm e)) (fun _ => 0)
theorem measurable_observedLatentCoordinate
The support-restricted version of an observed latent coordinate is measurable; on the observed support it is exactly the corresponding coordinate of the inverse mixing map. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma measurable_observedLatentCoordinate
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) (hmix : SharedDiffeomorphicMixing G θ W) (e : Fin n) :
Measurable (observedLatentCoordinate W e) := by
classical
have hcube : IsCompact (latentCube n) := by
rw [latentCube]
exact isCompact_univ_pi fun _ => isCompact_Icc
have hs : IsClosed (observedSupport G W) :=
(hcube.image_of_continuousOn hmix.1.continuousOn).isClosed
unfold observedLatentCoordinate
exact ((continuous_apply (W.targetPerm e)).comp_continuousOn hmix.2.1.continuousOn)
|>.measurable_piecewise continuous_const.continuousOn hs.measurableSet
theorem observedLatentCoordinate_mix
On the latent cube, the measurable observed coordinate version recovers the targeted latent coordinate after applying the mixing map. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
observedLatentCoordinate W e (W.mix v) = v (W.targetPerm e)
Proof (Lean source)
lemma observedLatentCoordinate_mix
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) (hmix : SharedDiffeomorphicMixing G θ W)
(e : Fin n) (v : LatentState n) (hv : v ∈ latentCube n) :
observedLatentCoordinate W e (W.mix v) = v (W.targetPerm e) := by
classical
rw [observedLatentCoordinate, Set.piecewise_eq_of_mem]
· exact congrFun (hmix.2.2.1 v hv) (W.targetPerm e)
· exact ⟨v, hv, rfl⟩
Conditional independence is equivalent before and after transporting the ambient law along a measurable map. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
A B X Y Z :
Type*
Nonempty X
Nonempty Y
A → B
hphi :
Measurable phi
f :
B → X
hf :
g :
B → Y
hg :
k :
B → Z
hk :
mu :
IsFiniteMeasure (mu.map phi)
conclusion
Proof (Lean source)
lemma condIndepFun_map_iff
{A B X Y Z : Type*}
[MeasurableSpace A] [StandardBorelSpace A]
[MeasurableSpace B] [StandardBorelSpace B]
[MeasurableSpace X] [StandardBorelSpace X] [Nonempty X]
[MeasurableSpace Y] [StandardBorelSpace Y] [Nonempty Y]
[MeasurableSpace Z]
{phi : A → B} (hphi : Measurable phi)
{f : B → X} (hf : Measurable f) {g : B → Y} (hg : Measurable g)
{k : B → Z} (hk : Measurable k)
{mu : Measure A} [IsFiniteMeasure mu] [IsFiniteMeasure (mu.map phi)] :
CondIndepFun (comap (k ∘ phi) inferInstance)
(hk.comp hphi).comap_le (f ∘ phi) (g ∘ phi) mu ↔
CondIndepFun (comap k inferInstance)
hk.comap_le f g (mu.map phi) := by
have hcd1 : condDistrib (g ∘ phi) (k ∘ phi) mu =
condDistrib g k (mu.map phi) := by
simp only [condDistrib]
congr 1
exact (Measure.map_map (hk.prodMk hg) hphi).symm
have hcd2 : condDistrib (g ∘ phi) (fun a ↦ ((k ∘ phi) a, (f ∘ phi) a)) mu =
condDistrib g (fun b ↦ (k b, f b)) (mu.map phi) := by
simp only [condDistrib]
congr 1
exact (Measure.map_map ((hk.prodMk hf).prodMk hg) hphi).symm
have hmap : mu.map (fun a ↦ ((k ∘ phi) a, (f ∘ phi) a)) =
(mu.map phi).map (fun b ↦ (k b, f b)) :=
(Measure.map_map (hk.prodMk hf) hphi).symm
rw [condIndepFun_iff_condDistrib_prod_ae_eq_prodMkRight
(hg.comp hphi) (hf.comp hphi) (hk.comp hphi),
condIndepFun_iff_condDistrib_prod_ae_eq_prodMkRight hg hf hk,
hcd2, hcd1, hmap]
theorem condIndepGiven_map_iff
Pulling three measurable variables back along a measurable map preserves and reflects conditional independence when the target law is the corresponding pushforward. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
A B X Y Z :
Type*
Nonempty X
Nonempty Y
A → B
hphi :
Measurable phi
f :
B → X
hf :
g :
B → Y
hg :
k :
B → Z
hk :
mu :
IsFiniteMeasure (mu.map phi)
conclusion
CondIndepGiven mu (f ∘ phi) (g ∘ phi) (k ∘ phi) ↔ CondIndepGiven (mu.map phi) f g k
Proof (Lean source)
lemma condIndepGiven_map_iff
{A B X Y Z : Type*}
[MeasurableSpace A] [StandardBorelSpace A]
[MeasurableSpace B] [StandardBorelSpace B]
[MeasurableSpace X] [StandardBorelSpace X] [Nonempty X]
[MeasurableSpace Y] [StandardBorelSpace Y] [Nonempty Y]
[MeasurableSpace Z]
{phi : A → B} (hphi : Measurable phi)
{f : B → X} (hf : Measurable f) {g : B → Y} (hg : Measurable g)
{k : B → Z} (hk : Measurable k)
{mu : Measure A} [IsFiniteMeasure mu] [IsFiniteMeasure (mu.map phi)] :
CondIndepGiven mu (f ∘ phi) (g ∘ phi) (k ∘ phi) ↔
CondIndepGiven (mu.map phi) f g k := by
constructor
· rintro ⟨_, _, _, _, hCI⟩
refine ⟨inferInstance, hf, hg, hk, ?_⟩
exact (condIndepFun_map_iff hphi hf hg hk).1 hCI
· rintro ⟨_, _, _, _, hCI⟩
refine ⟨inferInstance, hf.comp hphi, hg.comp hphi, hk.comp hphi, ?_⟩
exact (condIndepFun_map_iff hphi hf hg hk).2 hCI
theorem condIndepGiven_measurableEquiv_comp
Applying bimeasurable bijections separately to the two variables and the conditioning variable preserves conditional independence. the stated conclusion follows.
Formal statement
hypotheses
Ω X Y Z X' Y' Z' :
UX :
Ω → X
UY :
Ω → Y
UZ :
Ω → Z
eX :
X ≃ᵐ X'
eY :
Y ≃ᵐ Y'
eZ :
Z ≃ᵐ Z'
conclusion
CondIndepGiven μ (eX ∘ UX) (eY ∘ UY) (eZ ∘ UZ) ↔ CondIndepGiven μ UX UY UZ
Proof (Lean source)
lemma condIndepGiven_measurableEquiv_comp
{Ω X Y Z X' Y' Z' : Type*}
[MeasurableSpace Ω] [StandardBorelSpace Ω]
[MeasurableSpace X] [MeasurableSpace Y] [MeasurableSpace Z]
[MeasurableSpace X'] [MeasurableSpace Y'] [MeasurableSpace Z']
{μ : Measure Ω} (UX : Ω → X) (UY : Ω → Y) (UZ : Ω → Z)
(eX : X ≃ᵐ X') (eY : Y ≃ᵐ Y') (eZ : Z ≃ᵐ Z') :
CondIndepGiven μ (eX ∘ UX) (eY ∘ UY) (eZ ∘ UZ) ↔
CondIndepGiven μ UX UY UZ := by
constructor
· rintro ⟨hμ, hX, hY, hZ, hCI⟩
letI := hμ
have hX' : Measurable UX := by
convert eX.symm.measurable.comp hX using 1
funext ω
exact (eX.symm_apply_apply (UX ω)).symm
have hY' : Measurable UY := by
convert eY.symm.measurable.comp hY using 1
funext ω
exact (eY.symm_apply_apply (UY ω)).symm
have hZ' : Measurable UZ := by
convert eZ.symm.measurable.comp hZ using 1
funext ω
exact (eZ.symm_apply_apply (UZ ω)).symm
refine ⟨hμ, hX', hY', hZ', ?_⟩
have hraw := hCI.comp eX.symm.measurable eY.symm.measurable
have hs : CondIndepFun (comap (eZ ∘ UZ) inferInstance)
hZ.comap_le UX UY μ := by
simpa only [Function.comp_def, eX.symm_apply_apply,
eY.symm_apply_apply] using hraw
simpa only [← MeasurableSpace.comap_comp,
eZ.measurableEmbedding.comap_eq] using hs
· rintro ⟨hμ, hX, hY, hZ, hCI⟩
letI := hμ
refine ⟨hμ, eX.measurable.comp hX, eY.measurable.comp hY,
eZ.measurable.comp hZ, ?_⟩
have hraw := hCI.comp eX.measurable eY.measurable
simpa only [← MeasurableSpace.comap_comp,
eZ.measurableEmbedding.comap_eq] using hraw
theorem condIndepGiven_congr_ae_forward
Replacing all three measurable coordinates by almost-everywhere equal versions preserves conditional independence. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
Ω X Y Z :
hX₁ :
Measurable X₁
hX₂ :
Measurable X₂
hY₁ :
Measurable Y₁
hY₂ :
Measurable Y₂
hZ₁ :
Measurable Z₁
hZ₂ :
Measurable Z₂
hX :
X₁ =ᵐ[μ] X₂
hY :
Y₁ =ᵐ[μ] Y₂
hZ :
Z₁ =ᵐ[μ] Z₂
CondIndepGiven μ X₁ Y₁ Z₁
conclusion
CondIndepGiven μ X₂ Y₂ Z₂
Proof (Lean source)
lemma condIndepGiven_congr_ae_forward
{Ω X Y Z : Type*}
[MeasurableSpace Ω] [StandardBorelSpace Ω]
[MeasurableSpace X] [StandardBorelSpace X] [Nonempty X]
[MeasurableSpace Y] [StandardBorelSpace Y] [Nonempty Y]
[MeasurableSpace Z]
{μ : Measure Ω} {X₁ X₂ : Ω → X} {Y₁ Y₂ : Ω → Y} {Z₁ Z₂ : Ω → Z}
(hX₁ : Measurable X₁) (hX₂ : Measurable X₂)
(hY₁ : Measurable Y₁) (hY₂ : Measurable Y₂)
(hZ₁ : Measurable Z₁) (hZ₂ : Measurable Z₂)
(hX : X₁ =ᵐ[μ] X₂) (hY : Y₁ =ᵐ[μ] Y₂) (hZ : Z₁ =ᵐ[μ] Z₂) :
CondIndepGiven μ X₁ Y₁ Z₁ → CondIndepGiven μ X₂ Y₂ Z₂ := by
rintro ⟨hμ, _, _, _, hCI⟩
letI := hμ
refine ⟨hμ, hX₂, hY₂, hZ₂, ?_⟩
rw [condIndepFun_iff_condDistrib_prod_ae_eq_prodMkRight hY₂ hX₂ hZ₂]
rw [condIndepFun_iff_condDistrib_prod_ae_eq_prodMkRight hY₁ hX₁ hZ₁] at hCI
have hZY : (fun ω => (Z₁ ω, Y₁ ω)) =ᵐ[μ] fun ω => (Z₂ ω, Y₂ ω) :=
hZ.prodMk hY
have hZX : (fun ω => (Z₁ ω, X₁ ω)) =ᵐ[μ] fun ω => (Z₂ ω, X₂ ω) :=
hZ.prodMk hX
have hZXY := (hZ.prodMk hX).prodMk hY
simpa only [condDistrib, Measure.map_congr hZY, Measure.map_congr hZX,
Measure.map_congr hZXY] using hCI
theorem condIndepGiven_congr_ae
Conditional independence is invariant under almost-everywhere replacement of each of its three measurable coordinates. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
Ω X Y Z :
hX₁ :
Measurable X₁
hX₂ :
Measurable X₂
hY₁ :
Measurable Y₁
hY₂ :
Measurable Y₂
hZ₁ :
Measurable Z₁
hZ₂ :
Measurable Z₂
hX :
X₁ =ᵐ[μ] X₂
hY :
Y₁ =ᵐ[μ] Y₂
hZ :
Z₁ =ᵐ[μ] Z₂
conclusion
CondIndepGiven μ X₁ Y₁ Z₁ ↔ CondIndepGiven μ X₂ Y₂ Z₂
Proof (Lean source)
lemma condIndepGiven_congr_ae
{Ω X Y Z : Type*}
[MeasurableSpace Ω] [StandardBorelSpace Ω]
[MeasurableSpace X] [StandardBorelSpace X] [Nonempty X]
[MeasurableSpace Y] [StandardBorelSpace Y] [Nonempty Y]
[MeasurableSpace Z]
{μ : Measure Ω} {X₁ X₂ : Ω → X} {Y₁ Y₂ : Ω → Y} {Z₁ Z₂ : Ω → Z}
(hX₁ : Measurable X₁) (hX₂ : Measurable X₂)
(hY₁ : Measurable Y₁) (hY₂ : Measurable Y₂)
(hZ₁ : Measurable Z₁) (hZ₂ : Measurable Z₂)
(hX : X₁ =ᵐ[μ] X₂) (hY : Y₁ =ᵐ[μ] Y₂) (hZ : Z₁ =ᵐ[μ] Z₂) :
CondIndepGiven μ X₁ Y₁ Z₁ ↔ CondIndepGiven μ X₂ Y₂ Z₂ := by
constructor
· exact condIndepGiven_congr_ae_forward hX₁ hX₂ hY₁ hY₂ hZ₁ hZ₂ hX hY hZ
· exact condIndepGiven_congr_ae_forward hX₂ hX₁ hY₂ hY₁ hZ₂ hZ₁
hX.symm hY.symm hZ.symm
theorem condIndep_intersection_of_pos
Positivity-based graphoid intersection for four disjoint latent-coordinate blocks on the paper's full product support. The disjointness hypotheses are the internal DAG bookkeeping used in equations (15)--(18); they are not assumptions of the delivered decoder theorem. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
W :
ObservedWorld G θ
hpos :
hmix :
hone :
i b :
Fin n
hib :
i ≠ b
hiC :
i ∉ C
hiZ :
i ∉ Z
hbC :
b ∉ C
hbZ :
b ∉ Z
hCZ :
Disjoint C Z
h₁ :
CondIndepGiven (W.law 0) (observedLatentCoordinate W i) (observedLatentCoordinate W b) (fun x => (familyProjection (observedLatentCoordinate W) Z x, familyProjection (observedLatentCoordinate W) C x))
h₂ :
CondIndepGiven (W.law 0) (observedLatentCoordinate W i) (familyProjection (observedLatentCoordinate W) C) (fun x => (familyProjection (observedLatentCoordinate W) Z x, observedLatentCoordinate W b x))
conclusion
conclusion 1
CondIndepGiven (W.law 0) (observedLatentCoordinate W i) (observedLatentCoordinate W b) (familyProjection (observedLatentCoordinate W) Z)
conclusion 2
CondIndepGiven (W.law 0) (observedLatentCoordinate W i) (fun x => (observedLatentCoordinate W b x, familyProjection (observedLatentCoordinate W) C x)) (familyProjection (observedLatentCoordinate W) Z)
Proof (Lean source)
-- @node: condIndep_intersection_of_pos
lemma condIndep_intersection_of_pos
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W)
(i b : Fin n) (C Z : Finset (Fin n))
(hib : i ≠ b) (hiC : i ∉ C) (hiZ : i ∉ Z)
(hbC : b ∉ C) (hbZ : b ∉ Z) (hCZ : Disjoint C Z)
(h₁ : CondIndepGiven (W.law 0)
(observedLatentCoordinate W i)
(observedLatentCoordinate W b)
(fun x =>
(familyProjection (observedLatentCoordinate W) Z x,
familyProjection (observedLatentCoordinate W) C x)))
(h₂ : CondIndepGiven (W.law 0)
(observedLatentCoordinate W i)
(familyProjection (observedLatentCoordinate W) C)
(fun x =>
(familyProjection (observedLatentCoordinate W) Z x,
observedLatentCoordinate W b x))) :
CondIndepGiven (W.law 0)
(observedLatentCoordinate W i)
(observedLatentCoordinate W b)
(familyProjection (observedLatentCoordinate W) Z) ∧
CondIndepGiven (W.law 0)
(observedLatentCoordinate W i)
(fun x =>
(observedLatentCoordinate W b x,
familyProjection (observedLatentCoordinate W) C x))
(familyProjection (observedLatentCoordinate W) Z) := by
classical
let unit : Measure ℝ := volume.restrict (Icc (0 : ℝ) 1)
letI : IsProbabilityMeasure unit := ⟨by simp [unit, Real.volume_Icc]⟩
let ref := CausalSmith.Substrate.PositiveDensityCondindepIntersection.fourBlockReference
unit unit (Measure.pi fun _ : C ↦ unit) (Measure.pi fun _ : Z ↦ unit)
let Fobs : LatentState n →
CausalSmith.Substrate.PositiveDensityCondindepIntersection.FourBlock
ℝ ℝ (ValuesOn C (fun _ ↦ ℝ)) (ValuesOn Z (fun _ ↦ ℝ)) :=
fun x ↦ (observedLatentCoordinate W i x, (observedLatentCoordinate W b x,
(familyProjection (observedLatentCoordinate W) C x,
familyProjection (observedLatentCoordinate W) Z x)))
let Flat : LatentState n →
CausalSmith.Substrate.PositiveDensityCondindepIntersection.FourBlock
ℝ ℝ (ValuesOn C (fun _ ↦ ℝ)) (ValuesOn Z (fun _ ↦ ℝ)) :=
fun v ↦ (v (W.targetPerm i), (v (W.targetPerm b),
(familyProjection (fun e w ↦ w (W.targetPerm e)) C v,
familyProjection (fun e w ↦ w (W.targetPerm e)) Z v)))
rcases h₁ with ⟨hfinite, hUi, hUb, hZC, hCI₁⟩
letI : IsFiniteMeasure (W.law 0) := hfinite
rcases h₂ with ⟨_, _, hUC, hZb, hCI₂⟩
have hUZ : Measurable (familyProjection (observedLatentCoordinate W) Z) :=
measurable_fst.comp hZC
have hUC' : Measurable (familyProjection (observedLatentCoordinate W) C) :=
measurable_snd.comp hZC
have hFobs : Measurable Fobs := hUi.prodMk (hUb.prodMk (hUC'.prodMk hUZ))
have hFlat : Measurable Flat :=
(measurable_pi_apply _).prodMk ((measurable_pi_apply _).prodMk
((measurable_pi_lambda _ fun _ : C ↦ measurable_pi_apply _).prodMk
(measurable_pi_lambda _ fun _ : Z ↦ measurable_pi_apply _)))
let fullRef : Measure (LatentState n) := Measure.pi fun _ : Fin n ↦ unit
have hcubeRef : volume.restrict (latentCube n) = fullRef := by
change volume.restrict (Set.univ.pi fun _ : Fin n ↦ Icc (0 : ℝ) 1) = _
rw [MeasureTheory.volume_pi, Measure.restrict_pi_pi]
let dens : LatentState n → ENNReal := fun v ↦ ofReal (observationalDensity θ v)
have hdensOn : ContinuousOn dens (latentCube n) := by
apply ENNReal.continuous_ofReal.comp_continuousOn
unfold observationalDensity
exact continuousOn_finset_prod _ fun k _ ↦ (hpos.2.2.1 k).continuousOn
have hcube : MeasurableSet (latentCube n) := by
rw [latentCube]
measurability
have hdensAE : AEMeasurable dens (volume.restrict (latentCube n)) :=
aemeasurable_restrict_of_measurable_subtype hcube hdensOn.restrict.measurable
have hdens_ne : ∀ᵐ v ∂volume.restrict (latentCube n), dens v ≠ 0 := by
filter_upwards [ae_restrict_mem hcube] with v hv
have hp : 0 < observationalDensity θ v := by
unfold observationalDensity
exact Finset.prod_pos fun k _ ↦ hpos.1 k v hv
exact (ENNReal.ofReal_pos.mpr hp).ne'
have hobs_forward : observationalLaw θ ≪ fullRef := by
rw [← hcubeRef]
exact withDensity_absolutelyContinuous _ _
have hobs_reverse : fullRef ≪ observationalLaw θ := by
rw [← hcubeRef]
exact withDensity_absolutelyContinuous' hdensAE hdens_ne
have hFlatRef : Measure.map Flat fullRef = ref := by
simpa only [Flat, fullRef, ref] using
(map_permuted_scalar_fourBlockProjection_pi i b C Z hib hiC hiZ hbC hbZ hCZ
W.targetPerm unit)
have hlatent_forward : Measure.map Flat (observationalLaw θ) ≪ ref := by
rw [← hFlatRef]
exact @Measure.AbsolutelyContinuous.map _ _ _ _ _ _ hobs_forward Flat hFlat
have hlatent_reverse : ref ≪ Measure.map Flat (observationalLaw θ) := by
rw [← hFlatRef]
exact @Measure.AbsolutelyContinuous.map _ _ _ _ _ _ hobs_reverse Flat hFlat
have hobs_cube : observationalLaw θ (latentCube n)ᶜ = 0 :=
hobs_forward (by
rw [← hcubeRef, Measure.restrict_apply hcube.compl]
simp)
have hcube_ae : ∀ᵐ v ∂observationalLaw θ, v ∈ latentCube n := by
rw [ae_iff]
change observationalLaw θ (latentCube n)ᶜ = 0
exact hobs_cube
let mix' : LatentState n → LatentState n :=
(latentCube n).piecewise W.mix (fun _ ↦ 0)
have hmix' : Measurable mix' := by
exact hmix.1.continuousOn.measurable_piecewise continuous_const.continuousOn hcube
have hmix_eq : W.mix =ᵐ[observationalLaw θ] mix' := by
filter_upwards [hcube_ae] with v hv
simp [mix', Set.piecewise, hv]
have hcomp : Fobs ∘ mix' =ᵐ[observationalLaw θ] Flat := by
filter_upwards [hmix_eq, hcube_ae] with v hmv hv
change Fobs (mix' v) = Flat v
rw [← hmv]
have hinv := hmix.2.2.1 v hv
simp only [Fobs, Flat, Function.comp_apply, familyProjection]
apply Prod.ext
· exact observedLatentCoordinate_mix W hmix i v hv
apply Prod.ext
· exact observedLatentCoordinate_mix W hmix b v hv
apply Prod.ext
· funext a
exact observedLatentCoordinate_mix W hmix a v hv
· funext a
exact observedLatentCoordinate_mix W hmix a v hv
have hlaw : W.law 0 = Measure.map mix' (observationalLaw θ) := by
rw [hone.1]
exact Measure.map_congr hmix_eq
have hjoint : Measure.map Fobs (W.law 0) = Measure.map Flat (observationalLaw θ) := by
rw [hlaw]
calc
Measure.map Fobs (Measure.map mix' (observationalLaw θ)) =
Measure.map (Fobs ∘ mix') (observationalLaw θ) :=
Measure.map_map hFobs hmix'
_ = Measure.map Flat (observationalLaw θ) := Measure.map_congr hcomp
have hforward : Measure.map Fobs (W.law 0) ≪ ref := by
rw [hjoint]
exact hlatent_forward
have hreverse : ref ≪ Measure.map Fobs (W.law 0) := by
rw [hjoint]
exact hlatent_reverse
apply condIndepGiven_intersection_of_equivalent_fourBlock
(observedLatentCoordinate W i) (observedLatentCoordinate W b)
(familyProjection (observedLatentCoordinate W) C)
(familyProjection (observedLatentCoordinate W) Z)
hUi hUb hUC' hUZ unit unit (Measure.pi fun _ : C ↦ unit)
(Measure.pi fun _ : Z ↦ unit) hforward hreverse
· exact ⟨hfinite, hUi, hUb, hZC, hCI₁⟩
· exact ⟨hfinite, hUi, hUC, hZb, hCI₂⟩
theorem condIndep_coordSplit_prodMk
Coordinate-pair presentation of generic weak union for a finite family of measurable maps. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
Ω :
n :
ℕ
U :
Fin n → Ω → ℝ
i b :
Fin n
hU :
∀ c, Measurable (U c)
hA :
A ⊆ S
hb :
b ∈ S
hbA :
b ∉ A
hCI :
conclusion
conclusion 1
comap (familyProjection U A) inferInstance ⊔ comap (familyProjection U ((S \ A).erase b)) inferInstance
conclusion 2
CondIndepGiven μ (U i) (U b) (familyProjection U (S.erase b))
Proof (Lean source)
-- @node: condIndep_coordSplit_prodMk
lemma condIndep_coordSplit_prodMk
{Ω : Type*} [MeasurableSpace Ω] [StandardBorelSpace Ω]
{μ : Measure Ω} [IsFiniteMeasure μ]
{n : ℕ} (U : Fin n → Ω → ℝ) (i b : Fin n) (A S : Finset (Fin n))
(hU : ∀ c, Measurable (U c))
(hA : A ⊆ S) (hb : b ∈ S) (hbA : b ∉ A)
(hCI : CondIndepGiven μ (U i)
(familyProjection U (S \ A)) (familyProjection U A)) :
(comap (familyProjection U A) inferInstance ⊔
comap (familyProjection U ((S \ A).erase b)) inferInstance =
comap (familyProjection U (S.erase b)) inferInstance) ∧
CondIndepGiven μ (U i) (U b) (familyProjection U (S.erase b)) := by
have hproj (T : Finset (Fin n)) : Measurable (familyProjection U T) := by
apply measurable_pi_lambda
intro j
exact hU j
have hbDiff : b ∈ S \ A := Finset.mem_sdiff.mpr ⟨hb, hbA⟩
let split : ((j : {j // j ∈ S \ A}) → ℝ) →
ℝ × ((j : {j // j ∈ (S \ A).erase b}) → ℝ) :=
fun x => (x ⟨b, hbDiff⟩, fun j => x ⟨j, Finset.mem_sdiff.mpr
(Finset.mem_sdiff.mp (Finset.mem_erase.mp j.2).2)⟩)
have hsplit : Measurable split := by
apply Measurable.prod
· exact measurable_pi_apply _
· apply measurable_pi_lambda
intro j
exact measurable_pi_apply _
have hCIpair : CondIndepGiven μ (U i)
(fun ω => (U b ω, familyProjection U ((S \ A).erase b) ω))
(familyProjection U A) := by
rcases hCI with ⟨hμ, hXi, hY, hZA, hCI⟩
refine ⟨hμ, hXi, (hU b).prod (hproj _), hZA, ?_⟩
convert hCI.comp measurable_id hsplit using 1 <;>
ext ω j <;> rfl
have hsigma :
comap (familyProjection U A) inferInstance ⊔
comap (familyProjection U ((S \ A).erase b)) inferInstance =
comap (familyProjection U (S.erase b)) inferInstance := by
apply le_antisymm
· apply sup_le
· have hm : @Measurable Ω _
(comap (familyProjection U (S.erase b)) inferInstance)
inferInstance (familyProjection U A) := by
letI : MeasurableSpace Ω :=
comap (familyProjection U (S.erase b)) inferInstance
refine measurable_pi_lambda _ (fun j => ?_)
have hjS : (j : Fin n) ∈ S.erase b := Finset.mem_erase.mpr
⟨by intro h; subst b; exact hbA j.2, hA j.2⟩
have heval : Measurable (fun x : ((j : {j // j ∈ S.erase b}) → ℝ) =>
x ⟨j, hjS⟩) := measurable_pi_apply _
convert heval.comp (Measurable.of_comap_le le_rfl) using 1
funext c
rfl
exact hm.comap_le
· have hm : @Measurable Ω _
(comap (familyProjection U (S.erase b)) inferInstance)
inferInstance (familyProjection U ((S \ A).erase b)) := by
letI : MeasurableSpace Ω :=
comap (familyProjection U (S.erase b)) inferInstance
refine measurable_pi_lambda _ (fun j => ?_)
have hjS : (j : Fin n) ∈ S.erase b := by
rcases Finset.mem_erase.mp j.2 with ⟨hjb, hjSA⟩
exact Finset.mem_erase.mpr ⟨hjb, (Finset.mem_sdiff.mp hjSA).1⟩
have heval : Measurable (fun x : ((j : {j // j ∈ S.erase b}) → ℝ) =>
x ⟨j, hjS⟩) := measurable_pi_apply _
convert heval.comp (Measurable.of_comap_le le_rfl) using 1
funext c
rfl
exact hm.comap_le
· have hm : @Measurable Ω _
(comap (familyProjection U A) inferInstance ⊔
comap (familyProjection U ((S \ A).erase b)) inferInstance)
inferInstance (familyProjection U (S.erase b)) := by
letI : MeasurableSpace Ω :=
comap (familyProjection U A) inferInstance ⊔
comap (familyProjection U ((S \ A).erase b)) inferInstance
refine measurable_pi_lambda _ (fun j => ?_)
rcases Finset.mem_erase.mp j.2 with ⟨hjb, hjS⟩
by_cases hjA : (j : Fin n) ∈ A
· have heval : Measurable (fun x : ((j : {j // j ∈ A}) → ℝ) =>
x ⟨j, hjA⟩) := measurable_pi_apply _
convert heval.comp (Measurable.of_comap_le le_sup_left) using 1
funext c
rfl
· have hjRest : (j : Fin n) ∈ (S \ A).erase b :=
Finset.mem_erase.mpr ⟨hjb, Finset.mem_sdiff.mpr ⟨hjS, hjA⟩⟩
have heval : Measurable
(fun x : ((j : {j // j ∈ (S \ A).erase b}) → ℝ) =>
x ⟨j, hjRest⟩) := measurable_pi_apply _
convert heval.comp (Measurable.of_comap_le le_sup_right) using 1
funext c
rfl
exact hm.comap_le
refine ⟨hsigma, ?_⟩
rcases hCIpair with ⟨hμ, hXi, hPair, hZA, hCIpair⟩
refine ⟨hμ, hXi, hU b, hproj _, ?_⟩
have hraw := condIndepFun_weak_union_of_prodMk hZA.comap_le
(hU i) (hU b) (hproj ((S \ A).erase b)) hCIpair
simpa only [hsigma] using hraw
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.ContrastIntegral 1 declarations This file isolates the density cancellation that rewrites a canonical second-moment contrast as the rational mechanism integral used by both the explicit sparse witness and the analytic perturbation argument.
Canonical second-moment contrast integral
This file isolates the density cancellation that rewrites a canonical second-moment contrast as the rational mechanism integral used by both the explicit sparse witness and the analytic perturbation argument.
theorem canonical_secondMomentContrast_eq_integral_for_witness
For a positive normalized mechanism and distinct intervention and ratio targets, the canonical second-moment contrast equals the rational mechanism integral obtained by cancelling the observational child-density factor. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
= ∫ v in latentCube n, (θ.q i (v i)) ^ 2 * (θ.q j (v j) - θ.p j v) * (∏ l ∈ (Finset.univ.erase i).erase j, θ.p l v) / θ.p i v
Proof (Lean source)
-- @node: canonical_secondMomentContrast_eq_integral_for_witness
lemma canonical_secondMomentContrast_eq_integral_for_witness
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(hpos : PositiveNormalizedSmoothMechanisms G θ) {j i : Fin n}
(hji : j ≠ i) :
secondMomentContrast (canonicalObservedWorld G θ (refl (Fin n))) j i =
∫ v in latentCube n,
(θ.q i (v i)) ^ 2 * (θ.q j (v j) - θ.p j v) *
(∏ l ∈ (Finset.univ.erase i).erase j, θ.p l v) / θ.p i v := by
let μ : Measure (LatentState n) := volume.restrict (latentCube n)
let r : LatentState n → ℝ := fun v => θ.q i (v i) / θ.p i v
have hcube : MeasurableSet (latentCube n) := by
rw [latentCube]
exact MeasurableSet.univ_pi fun _ => measurableSet_Icc
have hcubeCompact : IsCompact (latentCube n) := by
rw [latentCube]
exact isCompact_univ_pi fun _ => isCompact_Icc
have hpcont (l : Fin n) : ContinuousOn (θ.p l) (latentCube n) :=
(hpos.2.2.1 l).continuousOn
have hqcont (l : Fin n) : ContinuousOn (fun v : LatentState n => θ.q l (v l))
(latentCube n) :=
(hpos.2.2.2.1 l).continuousOn.comp ((continuous_apply l).continuousOn)
(fun v hv => hv l (Set.mem_univ l))
have hrcont : ContinuousOn r (latentCube n) := by
exact (hqcont i).div (hpcont i) fun v hv => ne_of_gt (hpos.1 i v hv)
have hobscont : ContinuousOn (observationalDensity θ) (latentCube n) := by
unfold observationalDensity
exact continuousOn_finsetProd _ fun l _ => hpcont l
have hintcont : ContinuousOn (interventionalDensity θ j) (latentCube n) := by
unfold interventionalDensity
exact (hqcont j).mul (continuousOn_finsetProd _ fun l _ => hpcont l)
have hobsint : Integrable (fun v => observationalDensity θ v * r v ^ 2) μ := by
exact (hobscont.mul (hrcont.pow 2)).integrableOn_compact hcubeCompact
have hintint : Integrable (fun v => interventionalDensity θ j v * r v ^ 2) μ := by
exact (hintcont.mul (hrcont.pow 2)).integrableOn_compact hcubeCompact
have hobs : (∫ v, r v ^ 2 ∂observationalLaw θ) =
∫ v, observationalDensity θ v * r v ^ 2 ∂μ := by
unfold observationalLaw
rw [integral_withDensity_eq_integral_toReal_smul₀]
· apply integral_congr_ae
filter_upwards [ae_restrict_mem hcube] with v hv
rw [ENNReal.toReal_ofReal (le_of_lt (by
unfold observationalDensity
exact Finset.prod_pos fun l _ => hpos.1 l v hv))]
rfl
· exact (hobscont.aestronglyMeasurable hcube).aemeasurable.ennreal_ofReal
· filter_upwards with v
exact ENNReal.ofReal_lt_top
have hint : (∫ v, r v ^ 2 ∂interventionalLaw θ j) =
∫ v, interventionalDensity θ j v * r v ^ 2 ∂μ := by
unfold interventionalLaw
rw [integral_withDensity_eq_integral_toReal_smul₀]
· apply integral_congr_ae
filter_upwards [ae_restrict_mem hcube] with v hv
rw [ENNReal.toReal_ofReal (le_of_lt (by
unfold interventionalDensity
exact mul_pos (hpos.2.1 j (v j) (hv j (Set.mem_univ j)))
(Finset.prod_pos fun l _ => hpos.1 l v hv)))]
rfl
· exact (hintcont.aestronglyMeasurable hcube).aemeasurable.ennreal_ofReal
· filter_upwards with v
exact ENNReal.ofReal_lt_top
unfold secondMomentContrast
change (∫ v, r v ^ 2 ∂interventionalLaw θ j) -
(∫ v, r v ^ 2 ∂observationalLaw θ) = _
rw [hint, hobs, ← integral_sub hintint hobsint]
apply integral_congr_ae
filter_upwards [ae_restrict_mem hcube] with v hv
have hpi : θ.p i v ≠ 0 := ne_of_gt (hpos.1 i v hv)
have hinterDensity : interventionalDensity θ j v =
θ.q j (v j) * θ.p i v *
(∏ l ∈ (Finset.univ.erase i).erase j, θ.p l v) := by
unfold interventionalDensity
have hprod : (∏ l ∈ Finset.univ.erase j, θ.p l v) =
θ.p i v * (∏ l ∈ (Finset.univ.erase j).erase i, θ.p l v) := by
rw [Finset.prod_eq_mul_prod_sdiff_singleton_of_mem
(Finset.mem_erase.mpr ⟨Ne.symm hji, Finset.mem_univ i⟩)]
simp only [Finset.sdiff_singleton_eq_erase]
rw [hprod, Finset.erase_right_comm]
ring
have hobsDensity : observationalDensity θ v =
θ.p i v * θ.p j v *
(∏ l ∈ (Finset.univ.erase i).erase j, θ.p l v) := by
unfold observationalDensity
rw [Finset.prod_eq_mul_prod_sdiff_singleton_of_mem (Finset.mem_univ i)]
simp only [Finset.sdiff_singleton_eq_erase]
rw [Finset.prod_eq_mul_prod_sdiff_singleton_of_mem
(Finset.mem_erase.mpr ⟨hji, Finset.mem_univ j⟩)]
simp only [Finset.sdiff_singleton_eq_erase]
ring
unfold r
rw [hinterDensity, hobsDensity]
field_simp
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.Decoder 45 declarations The decoder is a function only of the observed probability laws.
Population ratio and conditional-rank decoder
The decoder is a function only of the observed probability laws. It selects a
topological ordering internally, constructs [0,1]-valued conditional ranks,
and prunes to the unique minimal admissible parent sets on the model domain.
A numerical linear extension of a directed relation. Injectivity excludes tied labels.
def predecessorSet
Labels preceding i in the explicitly selected ordering.
def familyProjection
Projection of a coordinate family onto a finite index set.
def ObservedProbabilityLawFamily
A decoder input consists of genuine probability laws in all n+1 environments.
Definition (Lean source)
parameters
n :
ℕ
def
ObservedProbabilityLawFamily n :
Type
given by
{laws : ObservedLawFamily n // ∀ e, IsProbabilityMeasure (laws e)}
def observedProbabilityLawFamily
Turn a measure family into a probability-law family when it is one, using a fixed Dirac probability family only outside that domain. Model hypotheses prove that this fallback is never used by the exact-decoder theorem.
Definition (Lean source)
parameters
n :
ℕ
laws :
def
given by
by classical exact if h : ∀ e, IsProbabilityMeasure (laws e) then ⟨laws, h⟩ else ⟨fun _
=> Measure.dirac 0, by intro e; infer_instance⟩
A continuous observed-ratio version for law family laws and environment i agrees almost everywhere with the canonical Radon--Nikodym ratio and is continuous on the support of the observational law.
Definition (Lean source)
parameters
def
IsContinuousObservedRatioVersion laws i R :
Prop
given by
clause 1
R =ᵐ[laws 0] observedLawRatio laws i
clause 2
ContinuousOn R (Measure.support (laws 0))
def observedContinuousRatio
The law-only continuous ratio selector for laws and environment i chooses a continuous version when one exists and otherwise retains the canonical Radon--Nikodym ratio.
Definition (Lean source)
parameters
def
given by
by classical exact if h : ∃ R, IsContinuousObservedRatioVersion laws i R then (Measure.support (laws 0)).piecewise (choose h) (fun _ ↦ 0) else observedLawRatio laws i
theorem observedContinuousRatio_isVersion
Whenever a continuous observed-ratio version exists, the law-only selector returns one. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
IsContinuousObservedRatioVersion laws i (observedContinuousRatio laws i)
Proof (Lean source)
lemma observedContinuousRatio_isVersion
(laws : ObservedLawFamily n) (i : Fin n)
(hex : ∃ R, IsContinuousObservedRatioVersion laws i R) :
IsContinuousObservedRatioVersion laws i (observedContinuousRatio laws i) := by
classical
rw [observedContinuousRatio, dif_pos hex]
let R := Classical.choose hex
have hR : IsContinuousObservedRatioVersion laws i R := Classical.choose_spec hex
constructor
· filter_upwards [hR.1, Measure.support_mem_ae (μ := laws 0)] with x hx hxs
rw [Set.piecewise_eq_of_mem (Measure.support (laws 0))
(Classical.choose hex) (fun _ ↦ 0) hxs]
simpa only [R] using hx
· exact hR.2.congr fun x hx ↦
Set.piecewise_eq_of_mem (Measure.support (laws 0)) R (fun _ ↦ 0) hx
theorem measurable_observedContinuousRatio
The law-selected continuous ratio is globally measurable: outside the observational support the selector uses the fixed zero extension. the stated conclusion follows.
Formal statement
hypotheses
conclusion
Measurable (observedContinuousRatio laws i)
Proof (Lean source)
lemma measurable_observedContinuousRatio
(laws : ObservedLawFamily n) (i : Fin n) :
Measurable (observedContinuousRatio laws i) := by
classical
rw [observedContinuousRatio]
split
· rename_i hex
let R := Classical.choose hex
have hR : IsContinuousObservedRatioVersion laws i R := Classical.choose_spec hex
exact hR.2.measurable_piecewise continuous_const.continuousOn
(laws 0).isClosed_support.measurableSet
· exact measurable_observedLawRatio laws i
theorem eqOn
Two continuous versions of the same almost-everywhere function agree throughout the measure's support. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
hR :
IsContinuousObservedRatioVersion laws i R
hS :
IsContinuousObservedRatioVersion laws i S
conclusion
EqOn R S (Measure.support (laws 0))
Proof (Lean source)
lemma IsContinuousObservedRatioVersion.eqOn
{laws : ObservedLawFamily n} {i : Fin n} {R S : LatentState n → ℝ}
(hR : IsContinuousObservedRatioVersion laws i R)
(hS : IsContinuousObservedRatioVersion laws i S) :
EqOn R S (Measure.support (laws 0)) := by
exact continuousOn_support_eq_of_ae_eq (hR.1.trans hS.1.symm) hR.2 hS.2
def observedLawLogRatio
The observable log ratio constructed from the law-selected continuous ratio version.
Definition (Lean source)
theorem measurable_observedLawLogRatio
The law-selected logarithmic ratio is globally measurable. the stated conclusion follows.
Formal statement
hypotheses
conclusion
Measurable (observedLawLogRatio laws i)
Proof (Lean source)
lemma measurable_observedLawLogRatio (laws : ObservedLawFamily n) (i : Fin n) :
Measurable (observedLawLogRatio laws i) :=
(measurable_observedContinuousRatio laws i).log
def observedLawDiscrepancy
Population MMD computed solely from the observed environment laws.
Definition (Lean source)
parameters
def
observedLawDiscrepancy U laws j i :
ℝ
given by
‖meanEmbedding U (Measure.map (observedLawRatio laws i) (laws 0)) - meanEmbedding U (Measure.map (observedLawRatio laws i) (laws j.succ))‖
The ratio graph constructed solely from observed laws.
Definition (Lean source)
parameters
n :
ℕ
H :
U :
laws :
given by
fun j i => j ≠ i ∧ 0 < observedLawDiscrepancy U laws j i
def PredecessorLogRatios
The predecessor log-ratio vector used as the conditioning variable.
Definition (Lean source)
parameters
def
PredecessorLogRatios order i :
Type
given by
(j : {j // j ∈ predecessorSet order i}) → ℝ
def conditionalRatioArgument
The joint observed log-ratio and predecessor-log-ratio argument.
Definition (Lean source)
parameters
def
given by
(observedLawLogRatio laws i x, familyProjection (observedLawLogRatio laws) (predecessorSet order i) x)
theorem measurable_conditionalRatioArgument
The joint log-ratio and predecessor-log-ratio argument is measurable. the stated conclusion follows.
Formal statement
hypotheses
conclusion
Measurable (conditionalRatioArgument laws order i)
Proof (Lean source)
lemma measurable_conditionalRatioArgument
(laws : ObservedLawFamily n) (order : Fin n → ℕ) (i : Fin n) :
Measurable (conditionalRatioArgument laws order i) := by
apply prodMk (measurable_observedLawLogRatio laws i)
apply measurable_pi_lambda
intro j
exact measurable_observedLawLogRatio laws j
theorem measurable_kernel_Iic_uncurry
Evaluation of an s-finite real kernel on a varying lower interval is jointly measurable in the endpoint and kernel parameter. the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma measurable_kernel_Iic_uncurry
{B : Type*} [MeasurableSpace B] (κ : Kernel B ℝ) [IsSFiniteKernel κ] :
Measurable (fun z : ℝ × B ↦ κ z.2 (Iic z.1)) := by
let η : Kernel (ℝ × B) ℝ := κ.comap Prod.snd measurable_snd
have hs : MeasurableSet {q : (ℝ × B) × ℝ | q.2 ≤ q.1.1} :=
measurableSet_le measurable_snd measurable_fst.fst
have hf : Measurable (uncurry (fun z : ℝ × B ↦ fun y : ℝ ↦
indicator {q : (ℝ × B) × ℝ | q.2 ≤ q.1.1}
(fun _ ↦ (1 : ENNReal)) (z, y))) :=
measurable_const.indicator hs
have hi := hf.lintegral_kernel_prod_right (κ := η)
convert hi using 1
funext z
rw [Kernel.lintegral_comap]
simp only [indicator, Set.mem_ofPred_eq]
change (κ z.2) (Iic z.1) =
∫⁻ b, (Iic z.1).indicator (fun _ ↦ (1 : ENNReal)) b ∂κ z.2
rw [MeasureTheory.lintegral_indicator measurableSet_Iic, lintegral_one]
simp
def observedConditionalRatioSupport
The law-derived domain on which a conditional-CDF version is required to be continuous: all thresholds over the support of the conditioning predecessor scores.
Definition (Lean source)
parameters
def
given by
univ ×ˢ Measure.support (Measure.map (familyProjection (observedLawLogRatio laws) (predecessorSet order i)) (laws i.succ))
def rawObservedConditionalRatioCDF
The raw regular-conditional-distribution representative, with the original zero fallback when the supplied environment laws are not finite.
Definition (Lean source)
parameters
def
rawObservedConditionalRatioCDF laws order i t ell :
ℝ
given by
by classical exact if hfinite : ∀ e, IsFiniteMeasure (laws.1 e) then letI := hfinite i.succ (condDistrib (observedLawLogRatio laws.1 i) (familyProjection (observedLawLogRatio laws.1) (predecessorSet order i)) (laws.1 i.succ) ell (Iic t)).toReal else 0
theorem measurable_rawObservedConditionalRatioCDF
The raw regular-conditional-distribution CDF is jointly measurable in threshold and conditioning argument. the stated conclusion follows.
Formal statement
hypotheses
conclusion
Measurable (fun z : ℝ × PredecessorLogRatios order i ↦ rawObservedConditionalRatioCDF laws order i z.1 z.2)
Proof (Lean source)
lemma measurable_rawObservedConditionalRatioCDF
(laws : ObservedProbabilityLawFamily n) (order : Fin n → ℕ) (i : Fin n) :
Measurable (fun z : ℝ × PredecessorLogRatios order i ↦
rawObservedConditionalRatioCDF laws order i z.1 z.2) := by
classical
unfold rawObservedConditionalRatioCDF
split
· rename_i hfinite
letI := hfinite i.succ
exact ENNReal.measurable_toReal.comp (measurable_kernel_Iic_uncurry
(condDistrib (observedLawLogRatio laws.1 i)
(familyProjection (observedLawLogRatio laws.1) (predecessorSet order i))
(laws.1 i.succ)))
· exact measurable_const
def IsContinuousConditionalRatioCDFVersion Definition aux_IsContinuousConditionalRatioCDFVersion in the paper ↗
A law-only conditional-CDF version: it is jointly measurable, agrees almost everywhere with the regular conditional distribution both at every fixed threshold and under the joint ratio/predecessor law, and is continuous on the joint model support.
Definition (Lean source)
parameters
n :
ℕ
laws :
order :
Fin n → ℕ
i :
Fin n
C :
ℝ → PredecessorLogRatios order i → Icc (0 : ℝ) 1
def
IsContinuousConditionalRatioCDFVersion laws order i C :
Prop
given by
clause 1
Measurable (fun z : ℝ × PredecessorLogRatios order i => (C z.1 z.2 : ℝ))
clause 2
t :
(fun ell => (C t ell : ℝ))
=ᵐ[Measure.map (familyProjection (observedLawLogRatio laws.1) (predecessorSet order i)) (laws.1 i.succ)] rawObservedConditionalRatioCDF laws order i t
clause 3
(fun z : ℝ × PredecessorLogRatios order i => (C z.1 z.2 : ℝ))
=ᵐ[ Measure.map (conditionalRatioArgument laws.1 order i) (laws.1 i.succ)] (fun z => rawObservedConditionalRatioCDF laws order i z.1 z.2)
clause 4
ContinuousOn (fun z : ℝ × PredecessorLogRatios order i => (C z.1 z.2 : ℝ)) (observedConditionalRatioSupport laws.1 order i)
def rawObservedConditionalRatioCDFUnit
The raw conditional-CDF representative as a unit-interval value, retaining it when it has the required range and using zero only as the range-check fallback.
Definition (Lean source)
parameters
def
given by
by classical exact if h : rawObservedConditionalRatioCDF laws order i t ell ∈ Icc (0 : ℝ) 1 then ⟨rawObservedConditionalRatioCDF laws order i t ell, h⟩ else ⟨0, by constructor <;> norm_num⟩
theorem measurable_rawObservedConditionalRatioCDFUnit
The range-checked raw conditional CDF is jointly measurable. the stated conclusion follows.
Formal statement
hypotheses
conclusion
Measurable (fun z : ℝ × PredecessorLogRatios order i ↦ rawObservedConditionalRatioCDFUnit laws order i z.1 z.2)
Proof (Lean source)
lemma measurable_rawObservedConditionalRatioCDFUnit
(laws : ObservedProbabilityLawFamily n) (order : Fin n → ℕ) (i : Fin n) :
Measurable (fun z : ℝ × PredecessorLogRatios order i ↦
rawObservedConditionalRatioCDFUnit laws order i z.1 z.2) := by
have hraw := measurable_rawObservedConditionalRatioCDF laws order i
have hs : MeasurableSet {z : ℝ × PredecessorLogRatios order i |
rawObservedConditionalRatioCDF laws order i z.1 z.2 ∈ Icc (0 : ℝ) 1} :=
measurableSet_Icc.preimage hraw
have hreal : Measurable (fun z : ℝ × PredecessorLogRatios order i ↦
if rawObservedConditionalRatioCDF laws order i z.1 z.2 ∈ Icc (0 : ℝ) 1
then rawObservedConditionalRatioCDF laws order i z.1 z.2 else 0) := by
exact Measurable.ite hs hraw measurable_const
have hval : ∀ z : ℝ × PredecessorLogRatios order i,
(if rawObservedConditionalRatioCDF laws order i z.1 z.2 ∈ Icc (0 : ℝ) 1
then rawObservedConditionalRatioCDF laws order i z.1 z.2 else 0) ∈ Icc (0 : ℝ) 1 := by
intro z
split
· assumption
· constructor <;> norm_num
convert hreal.subtype_mk (h := hval) using 1
funext z
unfold rawObservedConditionalRatioCDFUnit
split <;> rfl
The continuous conditional-CDF version selected from the observed laws alone. If no continuous version exists, this retains the raw condDistrib/zero fallback.
Definition (Lean source)
parameters
def
given by
by classical exact if h : ∃ C, IsContinuousConditionalRatioCDFVersion laws order i C then choose h else rawObservedConditionalRatioCDFUnit laws order i
theorem observedConditionalRatioCDF_isContinuousVersion
Whenever a continuous conditional-CDF version exists, the law-only selector returns one. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
n :
ℕ
laws :
order :
Fin n → ℕ
i :
Fin n
hex :
∃ C, IsContinuousConditionalRatioCDFVersion laws order i C
conclusion
IsContinuousConditionalRatioCDFVersion laws order i (observedConditionalRatioCDF laws order i)
Proof (Lean source)
lemma observedConditionalRatioCDF_isContinuousVersion
(laws : ObservedProbabilityLawFamily n) (order : Fin n → ℕ) (i : Fin n)
(hex : ∃ C, IsContinuousConditionalRatioCDFVersion laws order i C) :
IsContinuousConditionalRatioCDFVersion laws order i
(observedConditionalRatioCDF laws order i) := by
classical
rw [observedConditionalRatioCDF, dif_pos hex]
exact Classical.choose_spec hex
theorem measurable_observedConditionalRatioCDF_unconditional
The selected conditional CDF is jointly measurable whether or not a continuous version exists: the fallback is the measurable range-checked raw conditional distribution. the stated conclusion follows.
Formal statement
hypotheses
conclusion
Measurable (fun z : ℝ × PredecessorLogRatios order i ↦ observedConditionalRatioCDF laws order i z.1 z.2)
Proof (Lean source)
lemma measurable_observedConditionalRatioCDF_unconditional
(laws : ObservedProbabilityLawFamily n) (order : Fin n → ℕ) (i : Fin n) :
Measurable (fun z : ℝ × PredecessorLogRatios order i ↦
observedConditionalRatioCDF laws order i z.1 z.2) := by
classical
unfold observedConditionalRatioCDF
split
· rename_i hex
exact (Classical.choose_spec hex).1.subtype_mk
· exact measurable_rawObservedConditionalRatioCDFUnit laws order i
theorem eqOn
Two continuous conditional-CDF versions agree everywhere on the support of the observed joint ratio/predecessor law. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
C D :
ℝ → PredecessorLogRatios order i → Icc (0 : ℝ) 1
hC :
IsContinuousConditionalRatioCDFVersion laws order i C
hD :
IsContinuousConditionalRatioCDFVersion laws order i D
conclusion
EqOn (fun z : ℝ × PredecessorLogRatios order i ↦ (C z.1 z.2 : ℝ)) (fun z ↦ (D z.1 z.2 : ℝ)) (observedConditionalRatioSupport laws.1 order i)
Proof (Lean source)
lemma IsContinuousConditionalRatioCDFVersion.eqOn
{laws : ObservedProbabilityLawFamily n} {order : Fin n → ℕ} {i : Fin n}
{C D : ℝ → PredecessorLogRatios order i → Icc (0 : ℝ) 1}
(hC : IsContinuousConditionalRatioCDFVersion laws order i C)
(hD : IsContinuousConditionalRatioCDFVersion laws order i D) :
EqOn (fun z : ℝ × PredecessorLogRatios order i ↦ (C z.1 z.2 : ℝ))
(fun z ↦ (D z.1 z.2 : ℝ)) (observedConditionalRatioSupport laws.1 order i) := by
intro z hz
let μ := Measure.map
(familyProjection (observedLawLogRatio laws.1) (predecessorSet order i))
(laws.1 i.succ)
have hmap : Set.MapsTo (fun ell : PredecessorLogRatios order i => (z.1, ell))
(Measure.support μ) (observedConditionalRatioSupport laws.1 order i) := by
intro ell hell
exact ⟨Set.mem_univ _, hell⟩
have hCcont : ContinuousOn (fun ell => (C z.1 ell : ℝ)) (Measure.support μ) :=
hC.2.2.2.comp (continuousOn_const.prodMk continuousOn_id) hmap
have hDcont : ContinuousOn (fun ell => (D z.1 ell : ℝ)) (Measure.support μ) :=
hD.2.2.2.comp (continuousOn_const.prodMk continuousOn_id) hmap
exact continuousOn_support_eq_of_ae_eq
((hC.2.1 z.1).trans (hD.2.1 z.1).symm) hCcont hDcont hz.2
theorem observedConditionalRatioCDF_eq_on_support
Every continuous conditional-CDF version agrees on joint support with the law-only selected version. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
n :
ℕ
laws :
order :
Fin n → ℕ
i :
Fin n
hex :
∃ C, IsContinuousConditionalRatioCDFVersion laws order i C
C :
ℝ → PredecessorLogRatios order i → Icc (0 : ℝ) 1
hC :
IsContinuousConditionalRatioCDFVersion laws order i C
z ∈ observedConditionalRatioSupport laws.1 order i :
conclusion
C z.1 z.2 = observedConditionalRatioCDF laws order i z.1 z.2
Proof (Lean source)
lemma observedConditionalRatioCDF_eq_on_support
(laws : ObservedProbabilityLawFamily n) (order : Fin n → ℕ) (i : Fin n)
(hex : ∃ C, IsContinuousConditionalRatioCDFVersion laws order i C)
(C : ℝ → PredecessorLogRatios order i → Icc (0 : ℝ) 1)
(hC : IsContinuousConditionalRatioCDFVersion laws order i C) :
∀ z ∈ observedConditionalRatioSupport laws.1 order i,
C z.1 z.2 = observedConditionalRatioCDF laws order i z.1 z.2 := by
intro z hz
apply Subtype.ext
exact hC.eqOn (observedConditionalRatioCDF_isContinuousVersion laws order i hex) hz
def observedLawRankCoordinate
The identified rank coordinate, constructed only from observed laws and an explicit ordering.
Definition (Lean source)
parameters
def
given by
observedConditionalRatioCDF laws order i (observedLawLogRatio laws.1 i x) (familyProjection (observedLawLogRatio laws.1) (predecessorSet order i) x)
theorem measurable_observedLawRankCoordinate
Every law-selected rank coordinate is measurable, including on the selector's raw fallback branch. the stated conclusion follows.
Formal statement
hypotheses
conclusion
Measurable (observedLawRankCoordinate laws order i)
Proof (Lean source)
lemma measurable_observedLawRankCoordinate
(laws : ObservedProbabilityLawFamily n) (order : Fin n → ℕ) (i : Fin n) :
Measurable (observedLawRankCoordinate laws order i) := by
exact (measurable_observedConditionalRatioCDF_unconditional laws order i).comp
(measurable_conditionalRatioArgument laws.1 order i)
theorem measurable_rankFamilyProjection
Every finite projection of the law-selected rank family is measurable. the stated conclusion follows.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma measurable_rankFamilyProjection
(laws : ObservedProbabilityLawFamily n) (order : Fin n → ℕ)
(S : Finset (Fin n)) :
Measurable (familyProjection (observedLawRankCoordinate laws order) S) := by
apply measurable_pi_lambda
intro j
exact measurable_observedLawRankCoordinate laws order j
Conditional independence of two measurable functions given a third.
Definition (Lean source)
parameters
def
CondIndepGiven μ X Y Z :
Prop
given by
∃ hμ : IsFiniteMeasure μ,
letI := hμ ∃ _hX : Measurable X, ∃ _hY : Measurable Y, ∃ hZ : Measurable Z, CondIndepFun (comap Z inferInstance) hZ.comap_le X Y μ
A candidate parent set satisfies the observed-law conditional-independence test.
Definition (Lean source)
parameters
def
AdmissibleParentSet laws order i A :
Prop
given by
clause 1
A ⊆ predecessorSet order i
clause 2
CondIndepGiven (laws.1 0) (observedLawRankCoordinate laws order i) (familyProjection (observedLawRankCoordinate laws order) ((predecessorSet order i) \ A)) (familyProjection (observedLawRankCoordinate laws order) A)
Inclusion-minimal admissibility for parent pruning.
Definition (Lean source)
parameters
def
MinimalAdmissibleParentSet laws order i A :
Prop
given by
clause 1
AdmissibleParentSet laws order i A
def selectedParentSet
The uniquely inclusion-minimal admissible set, defined only when it is genuinely unique. none records that the observed laws lie outside the decoder's parent-pruning domain.
Definition (Lean source)
The parent relation carried by a successful unique-minimum selection.
Definition (Lean source)
parameters
given by
fun j i => ∃ A, selectedParentSet laws order i = some A ∧ j ∈ A
theorem selectedParentSet_precedes
Every selected edge points forward in the supplied ordering. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
order j < order i
Proof (Lean source)
lemma selectedParentSet_precedes
(laws : ObservedProbabilityLawFamily n) (order : Fin n → ℕ) {j i : Fin n}
(hj : selectedParentRelation laws order j i) : order j < order i := by
rcases hj with ⟨A, hA, hjA⟩
unfold selectedParentSet at hA
split at hA
· rename_i h_unique
injection hA with hA'
subst A
simpa [predecessorSet] using (Classical.choose_spec h_unique.exists).1.1 hjA
· simp at hA
theorem selectedParentRelation_acyclic
The selected-parent relation is acyclic because every edge points forward in order. the stated conclusion follows.
Formal statement
hypotheses
conclusion
¬ TransGen (selectedParentRelation laws order) i i
Proof (Lean source)
lemma selectedParentRelation_acyclic
(laws : ObservedProbabilityLawFamily n) (order : Fin n → ℕ) (i : Fin n) :
¬ TransGen (selectedParentRelation laws order) i i := by
intro h
have path_lt : ∀ {a b : Fin n},
TransGen (selectedParentRelation laws order) a b → order a < order b := by
intro a b hab
induction hab with
| single hji => exact selectedParentSet_precedes laws order hji
| tail hab hbc ih => exact lt_trans ih (selectedParentSet_precedes laws order hbc)
have hi : order i < order i := path_lt h
exact (lt_irrefl _ hi)
def selectedParentDAG
The acyclic graph induced by all successful unique-minimum parent selections.
Definition (Lean source)
parameters
given by
clause 1
edge := selectedParentRelation laws order
clause 2
decEdge := Classical.decRel _
clause 3
acyclic := selectedParentRelation_acyclic laws order
Compatibility restricts law-coherence comparisons to representations of the same observed law family on the same observed support.
Definition (Lean source)
parameters
W₁ :
ObservedWorld G₁ θ₁
W₂ :
ObservedWorld G₂ θ₂
def
CompatibleObservedRepresentation W₁ W₂ :
Prop
Coherence uses almost-everywhere Radon--Nikodym representatives, but compares the selected continuous ranks pointwise on the full common observed support.
Definition (Lean source)
parameters
W :
ObservedWorld G θ
def
ObservedWorldLawCoherent W :
Prop
given by
clause 2
observedLawRankCoordinate (observedProbabilityLawFamily W'.law) order i x
= observedLawRankCoordinate (observedProbabilityLawFamily W.law) order i x
def selectedTopologicalOrder
A topological ordering selected from the ratio graph using only the observed laws. The numeric label order is a total fallback outside the acyclic model domain.
Definition (Lean source)
parameters
n :
ℕ
laws :
def
given by
by classical exact if h : ∃ order, IsTopologicalOrdering (observedLawRatioGraph gaussianFeatureMap laws.1) order then choose h else fun i
=> i.val
theorem selectedTopologicalOrder_isTopologicalOrdering
If the observed ratio graph has a topological ordering, its law-only selected ordering is itself topological. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
n :
ℕ
laws :
hex :
conclusion
Proof (Lean source)
lemma selectedTopologicalOrder_isTopologicalOrdering
(laws : ObservedProbabilityLawFamily n)
(hex : ∃ order, IsTopologicalOrdering
(observedLawRatioGraph gaussianFeatureMap laws.1) order) :
IsTopologicalOrdering (observedLawRatioGraph gaussianFeatureMap laws.1)
(selectedTopologicalOrder laws) := by
classical
rw [selectedTopologicalOrder, dif_pos hex]
exact Classical.choose_spec hex
The law-only population recovery map. It forms the Gaussian-MMD graph, selects its own topological ordering, constructs unit-interval ranks, and returns the parent-pruned DAG.
Definition (Lean source)
parameters
n :
ℕ
laws :
def
given by
order :
(observedLawRatioGraph gaussianFeatureMap laws.1, observedLawRankCoordinate laws order, selectedParentDAG laws order)
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.DecoderCompProd 14 declarations This file separates one distinguished coordinate from a finite product-density measure.
Product-law assembly for decoder coordinates
This file separates one distinguished coordinate from a finite product-density measure. It supplies the measure-product step needed to assemble the equation-(11) conditional kernel.
Transporting a weighted measure through a measurable equivalence transports its density by the inverse equivalence. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
Measure.map e (μ.withDensity f) = ν.withDensity (f ∘ e.symm)
Proof (Lean source)
lemma map_withDensity_measurableEquiv
{α β : Type*} [MeasurableSpace α] [MeasurableSpace β]
(e : α ≃ᵐ β) {μ : Measure α} {ν : Measure β}
(h : MeasurePreserving e μ ν) (f : α → ℝ≥0∞) :
Measure.map e (μ.withDensity f) = ν.withDensity (f ∘ e.symm) := by
classical
ext s hs
rw [Measure.map_apply e.measurable hs,
withDensity_apply _ (e.measurable hs), withDensity_apply _ hs]
rw [← lintegral_indicator (e.measurable hs), ← lintegral_indicator hs]
rw [h.lintegral_map_equiv]
apply lintegral_congr
intro a
change (if e a ∈ s then f a else 0) =
if e a ∈ s then f (e.symm (e a)) else 0
rw [e.symm_apply_apply]
If a finite product density factors into a normalized density at one coordinate and a factor depending only on a disjoint coordinate set, that coordinate is independent of the projected set under the weighted measure. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
V :
μ :
j :
V
S :
Finset V
hjS :
j ∉ S
q :
ℝ → ℝ≥0∞
hq :
g :
(V → ℝ) → ℝ≥0∞
hg :
hgS :
DependsOn S g
IsProbabilityMeasure ((μ j).withDensity q)
conclusion
Measure.map (fun x => (coordinateProjection S x, x j)) ((Measure.pi μ).withDensity (fun x => q (x j) * g x))
= (Measure.map (coordinateProjection S) ((Measure.pi μ).withDensity (fun x => q (x j) * g x))).prod ((μ j).withDensity q)
Proof (Lean source)
lemma weightedCoordinate_independence
{V : Type*} [Fintype V] [DecidableEq V]
(μ : V → Measure ℝ) [∀ k, SigmaFinite (μ k)] [∀ k, IsProbabilityMeasure (μ k)]
(j : V) (S : Finset V) (hjS : j ∉ S)
(q : ℝ → ℝ≥0∞) (hq : Measurable q)
(g : (V → ℝ) → ℝ≥0∞) (hg : Measurable g)
(hgS : DependsOn S g)
[IsProbabilityMeasure ((μ j).withDensity q)] :
Measure.map (fun x =>
(coordinateProjection S x, x j))
((Measure.pi μ).withDensity (fun x => q (x j) * g x)) =
(Measure.map (coordinateProjection S)
((Measure.pi μ).withDensity (fun x => q (x j) * g x))).prod
((μ j).withDensity q) := by
classical
let p : V → Prop := fun k => k = j
let E0 := MeasurableEquiv.piEquivPiSubtypeProd (fun _ : V => ℝ) p
let μone : Measure ((k : {k // p k}) → ℝ) :=
@Measure.pi (Subtype p) (fun _ => ℝ) (fintype p) (fun _ => inferInstance)
(fun k => μ k)
let μrest : Measure ((k : {k // ¬ p k}) → ℝ) :=
@Measure.pi (Subtype fun k => ¬ p k) (fun _ => ℝ)
(fintype fun k => ¬ p k) (fun _ => inferInstance) (fun k => μ k)
let qone : ((k : {k // p k}) → ℝ) → ℝ≥0∞ := fun z => q (z ⟨j, rfl⟩)
let zeroOne : (k : {k // p k}) → ℝ := fun _ => 0
let grest : ((k : {k // ¬ p k}) → ℝ) → ℝ≥0∞ :=
fun z => g (E0.symm (zeroOne, z))
have hsplit : MeasurePreserving E0 (Measure.pi μ) (μone.prod μrest) := by
simpa [E0, μone, μrest] using
(MeasureTheory.measurePreserving_piEquivPiSubtypeProd μ p)
have hqone : Measurable qone := by
exact hq.comp (measurable_pi_apply (⟨j, rfl⟩ : {k // p k}))
have hgrest : Measurable grest :=
hg.comp (E0.symm.measurable.comp (measurable_const.prodMk measurable_id))
have hdensity : (fun z => q (E0.symm z j) * g (E0.symm z)) =
fun z => qone z.1 * grest z.2 := by
funext z
congr 1
· change q (E0.symm z j) = q (z.1 ⟨j, rfl⟩)
congr 1
simp [E0, p]
· apply hgS
intro k hk
have hkj : k ≠ j := fun h => hjS (h ▸ hk)
change E0.symm z k = E0.symm (zeroOne, z.2) k
simp [E0, p, hkj]
have hweighted : Measure.map E0
((Measure.pi μ).withDensity (fun x => q (x j) * g x)) =
(μone.withDensity qone).prod (μrest.withDensity grest) := by
rw [map_withDensity_measurableEquiv E0 hsplit]
rw [MeasureTheory.prod_withDensity hqone hgrest]
congr 1
let evalOne : ((k : {k // p k}) → ℝ) → ℝ := fun z => z ⟨j, rfl⟩
let projectRest : ((k : {k // ¬ p k}) → ℝ) → ((k : S) → ℝ) :=
fun z k => z ⟨k, by
intro hkj
apply hjS
rw [← hkj]
exact k.property⟩
have heval : Measurable evalOne :=
measurable_pi_apply (⟨j, rfl⟩ : {k // p k})
have hproject : Measurable projectRest := by
apply measurable_pi_lambda
intro k
exact measurable_pi_apply _
have htarget : Measure.map evalOne (μone.withDensity qone) =
(μ j).withDensity q := by
let Eone := MeasurableEquiv.piUnique (fun _ : {k // p k} => ℝ)
have hone : MeasurePreserving Eone μone (μ j) := by
have hraw := @MeasureTheory.measurePreserving_piUnique
(Subtype p) (fintype p) (fun _ => ℝ) inferInstance
(fun _ => inferInstance) (fun k => μ k)
have hdef : (default : {k // p k}).1 = j := (default : {k // p k}).2
simpa [Eone, μone, p, hdef] using hraw
rw [show evalOne = Eone by rfl]
rw [map_withDensity_measurableEquiv Eone hone]
congr 1
have hdown : Measurable (fun z : ((k : {k // p k}) → ℝ) ×
((k : {k // ¬ p k}) → ℝ) => (projectRest z.2, evalOne z.1)) :=
(hproject.comp measurable_snd).prodMk (heval.comp measurable_fst)
have hpair : (fun x : V → ℝ =>
(coordinateProjection S x, x j)) =
(fun z => (projectRest z.2, evalOne z.1)) ∘ E0 := by
funext x
apply Prod.ext
· funext k
simp [projectRest, E0, p, coordinateProjection]
· simp [evalOne, E0, p]
have hjoint : Measure.map (fun x : V → ℝ =>
(coordinateProjection S x, x j))
((Measure.pi μ).withDensity (fun x => q (x j) * g x)) =
(Measure.map projectRest (μrest.withDensity grest)).prod
(Measure.map evalOne (μone.withDensity qone)) := by
rw [hpair, ← Measure.map_map hdown E0.measurable, hweighted]
calc
Measure.map (fun z => (projectRest z.2, evalOne z.1))
((μone.withDensity qone).prod (μrest.withDensity grest)) =
Measure.map swap
(Measure.map (map evalOne projectRest)
((μone.withDensity qone).prod (μrest.withDensity grest))) := by
rw [Measure.map_map measurable_swap (heval.prodMap hproject)]
rfl
_ = Measure.map swap
((Measure.map evalOne (μone.withDensity qone)).prod
(Measure.map projectRest (μrest.withDensity grest))) := by
rw [← Measure.map_prod_map _ _ heval hproject]
_ = _ := Measure.prod_swap
have hpred : Measure.map (coordinateProjection S)
((Measure.pi μ).withDensity (fun x => q (x j) * g x)) =
Measure.map projectRest (μrest.withDensity grest) := by
calc
Measure.map (coordinateProjection S)
((Measure.pi μ).withDensity (fun x => q (x j) * g x)) =
Measure.map fst
(Measure.map (fun x : V → ℝ =>
(coordinateProjection S x, x j))
((Measure.pi μ).withDensity (fun x => q (x j) * g x))) := by
rw [Measure.map_map measurable_fst (by fun_prop)]
rfl
_ = Measure.map fst
((Measure.map projectRest (μrest.withDensity grest)).prod
((μ j).withDensity q)) := by rw [hjoint, htarget]
_ = Measure.map projectRest (μrest.withDensity grest) := by
rw [Measure.map_fst_prod]
simp
rw [hjoint, htarget, hpred]
theorem equationTen_predecessorCoordinates_independent_target
Equation (10) as a product law: under intervention i, the target coordinate has its replacement-density law and is independent of all latent coordinates preceding i. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
W :
ObservedWorld G θ
hpos :
order :
Fin n → ℕ
horder :
hgraphOrder :
PermutedGraphOrdered W order
i :
Fin n
conclusion
A :
decoderRetainedLatentSet W order i
S :
A.erase (W.targetPerm i)
pred :
coordinateProjection (X := fun _ : Fin n => ℝ) S
Measure.map (fun v => (pred v, v (W.targetPerm i))) (interventionalLaw θ (W.targetPerm i))
= (Measure.map pred (interventionalLaw θ (W.targetPerm i))).prod (decoderInterventionCoordinateMeasure W hpos i)
Proof (Lean source)
lemma equationTen_predecessorCoordinates_independent_target
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) (hpos : PositiveNormalizedSmoothMechanisms G θ)
{order : Fin n → ℕ}
(horder : IsTopologicalOrdering (observedLawRatioGraph gaussianFeatureMap W.law) order)
(hgraphOrder : PermutedGraphOrdered W order) (i : Fin n) :
let A := decoderRetainedLatentSet W order i
let S := A.erase (W.targetPerm i)
let pred := coordinateProjection
(X := fun _ : Fin n => ℝ) S
Measure.map (fun v => (pred v, v (W.targetPerm i)))
(interventionalLaw θ (W.targetPerm i)) =
(Measure.map pred (interventionalLaw θ (W.targetPerm i))).prod
(decoderInterventionCoordinateMeasure W hpos i) := by
dsimp only
let B := mechanismUnitCubeFactorization hpos
let q := mechanismInterventionDensity W hpos i
let A := decoderRetainedLatentSet W order i
let S := A.erase (W.targetPerm i)
let μ : Fin n → Measure ℝ := fun _ =>
Causalean.Graph.FiniteDensity.unitIntervalReference
let ν := (Measure.pi μ).withDensity
(fun v => q.density (v (W.targetPerm i)) * B.partialDensity S v)
have hσ : ∀ k, SigmaFinite (μ k) := fun _ => by
dsimp only [μ]
unfold Causalean.Graph.FiniteDensity.unitIntervalReference
infer_instance
letI : ∀ k, SigmaFinite (μ k) := hσ
have hprob : ∀ k, IsProbabilityMeasure (μ k) := fun _ => by
constructor
dsimp only [μ]
unfold Causalean.Graph.FiniteDensity.unitIntervalReference
simp [Real.volume_Icc]
letI : ∀ k, IsProbabilityMeasure (μ k) := hprob
have htargetNotMem : W.targetPerm i ∉ S := Finset.notMem_erase _ _
have hg : Measurable (B.partialDensity S) :=
B.measurable_partialDensity S
have hgS : DependsOn S (B.partialDensity S) := by
intro x y hxy
unfold partialDensity
apply Finset.prod_congr rfl
intro k hk
apply B.local_factor k
intro a ha
apply hxy a
rcases Finset.mem_insert.mp ha with rfl | ha
· exact hk
have hkA : k ∈ A := (Finset.mem_erase.mp hk).2
have haA : a ∈ A :=
decoderRetainedLatentSet_parentClosed W horder hgraphOrder i hkA ha
have hane : a ≠ W.targetPerm i := by
intro hat
subst a
change k ∈ decoderRetainedLatentSet W order i at hkA
rw [decoderRetainedLatentSet, Finset.mem_insert] at hkA
rcases hkA with hkt | hkpred
· exact (Finset.mem_erase.mp hk).1 hkt
· rcases Finset.mem_map.mp hkpred with ⟨b, hb, rfl⟩
exact target_not_parent_of_mem_predecessorSet W horder hgraphOrder hb ha
exact Finset.mem_erase.mpr ⟨hane, haA⟩
have hνprod := @weightedCoordinate_independence (Fin n) inferInstance inferInstance
μ hσ hprob (W.targetPerm i) S htargetNotMem q.density q.measurable_density
(B.partialDensity S) hg hgS (by
simpa [μ, q, decoderInterventionCoordinateMeasure] using
decoderInterventionCoordinateMeasure_isProbability W hpos i)
have hAeq : A = decoderRetainedLatentSet W order i := rfl
have hretained := equationTen_retainedMarginalMeasure W hpos horder hgraphOrder i
change Measure.map (coordinateProjection A)
(interventionalLaw θ (W.targetPerm i)) =
Measure.map (coordinateProjection A) ν at hretained
have hpairMeas : Measurable (fun v : LatentState n =>
(coordinateProjection S v, v (W.targetPerm i))) :=
(Causalean.Mathlib.MeasureTheory.FiniteCoordinate.measurable_coordinateProjection S).prodMk
(measurable_pi_apply _)
have hpairDepends : DependsOn A
(fun v : LatentState n =>
(coordinateProjection S v, v (W.targetPerm i))) := by
intro x y hxy
apply Prod.ext
· funext k
exact hxy k (mem_of_mem_erase k.property)
· exact hxy _ (by simp [A, decoderRetainedLatentSet])
have hpredMeas : Measurable (coordinateProjection
(X := fun _ : Fin n => ℝ) S) :=
Causalean.Mathlib.MeasureTheory.FiniteCoordinate.measurable_coordinateProjection S
have hpredDepends : DependsOn A
(coordinateProjection
(X := fun _ : Fin n => ℝ) S) := by
intro x y hxy
funext k
exact hxy k (mem_of_mem_erase k.property)
have hpairMap := map_eq_of_map_coordinateProjection_eq
(fun _ : Fin n => (0 : ℝ)) hpairMeas hpairDepends hretained
have hpredMap := map_eq_of_map_coordinateProjection_eq
(fun _ : Fin n => (0 : ℝ)) hpredMeas hpredDepends hretained
calc
Measure.map (fun v : LatentState n =>
(coordinateProjection S v, v (W.targetPerm i)))
(interventionalLaw θ (W.targetPerm i)) =
Measure.map (fun v : LatentState n =>
(coordinateProjection S v, v (W.targetPerm i))) ν :=
hpairMap
_ = (Measure.map (coordinateProjection S) ν).prod
(decoderInterventionCoordinateMeasure W hpos i) := by
simpa [μ, q, decoderInterventionCoordinateMeasure] using hνprod
_ = (Measure.map (coordinateProjection S)
(interventionalLaw θ (W.targetPerm i))).prod
(decoderInterventionCoordinateMeasure W hpos i) := by rw [hpredMap]
Mapping a product measure through a measurable fiber map gives a compositional product whenever the supplied kernel is the fiberwise pushforward of the second marginal. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma map_prod_compProd_of_fiber
{α β γ δ : Type*} [MeasurableSpace α] [MeasurableSpace β]
[MeasurableSpace γ] [MeasurableSpace δ]
(μ : Measure α) (ν : Measure β) [SFinite μ] [SFinite ν]
(g : α → δ) (hg : Measurable g) (f : α × β → γ) (hf : Measurable f)
(κ : Kernel δ γ) [IsSFiniteKernel κ]
(hκ : ∀ a, κ (g a) = Measure.map (fun b => f (a, b)) ν) :
Measure.map (fun z : α × β => (g z.1, f z)) (μ.prod ν) =
Measure.map g μ ⊗ₘ κ := by
have hpair : Measurable (fun z : α × β => (g z.1, f z)) := by
fun_prop
ext s hs
rw [Measure.map_apply hpair hs,
Measure.prod_apply (hs.preimage hpair),
Measure.compProd_apply hs]
rw [MeasureTheory.lintegral_map (Kernel.measurable_kernel_prodMk_left hs) hg]
apply lintegral_congr
intro a
rw [hκ a, Measure.map_apply (by fun_prop) (hs.preimage (by fun_prop))]
rfl
def retainedProjectionToPredecessorCube
Reindex and clamp retained latent predecessor coordinates into the compact decoder cube.
Definition (Lean source)
parameters
W :
ObservedWorld G θ
order :
Fin n → ℕ
i :
Fin n
y :
(k : (decoderRetainedLatentSet W order i).erase (W.targetPerm i)) → ℝ
def
given by
fun j
=> ⟨max 0 (min 1 (y ⟨W.targetPerm j, by apply Finset.mem_erase.mpr constructor · intro h have hji : j.1 = i := W.targetPerm.injective h have hjlt : order j < order i := by simpa [predecessorSet] using j.2 exact (hji ▸ hjlt).false · rw [decoderRetainedLatentSet] exact mem_insert_of_mem (Finset.mem_map.mpr ⟨j, j.2, rfl⟩)⟩)), ⟨le_max_left _ _, max_le zero_le_one (min_le_left _ _)⟩⟩
theorem measurable_retainedProjectionToPredecessorCube
Reindexing and clamping the retained predecessor projection is measurable.
Formal statement
hypotheses
conclusion
Measurable (retainedProjectionToPredecessorCube W order i)
Proof (Lean source)
@[fun_prop] lemma measurable_retainedProjectionToPredecessorCube
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) (order : Fin n → ℕ) (i : Fin n) :
Measurable (retainedProjectionToPredecessorCube W order i) := by
apply measurable_pi_lambda
intro j
apply Measurable.subtype_mk
fun_prop
theorem retainedProjectionToPredecessorCube_coordinateProjection
On the latent cube, the clamped retained projection is the actual predecessor restriction. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
retainedProjectionToPredecessorCube W order i (coordinateProjection ((decoderRetainedLatentSet W order i).erase (W.targetPerm i)) v)
= predecessorLatentCubeOfState W order i v hv
Proof (Lean source)
lemma retainedProjectionToPredecessorCube_coordinateProjection
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) (order : Fin n → ℕ) (i : Fin n)
(v : LatentState n) (hv : v ∈ latentCube n) :
retainedProjectionToPredecessorCube W order i
(coordinateProjection
((decoderRetainedLatentSet W order i).erase (W.targetPerm i)) v) =
predecessorLatentCubeOfState W order i v hv := by
funext j
apply Subtype.ext
simp only [retainedProjectionToPredecessorCube,
coordinateProjection,
predecessorLatentCubeOfState]
have hj := hv (W.targetPerm j) (Set.mem_univ _)
rw [min_eq_right hj.2, max_eq_right hj.1]
def clampedPredecessorCoordinates
The globally measurable clamped predecessor restriction of an ambient latent state.
Definition (Lean source)
parameters
def
given by
retainedProjectionToPredecessorCube W order i (coordinateProjection ((decoderRetainedLatentSet W order i).erase (W.targetPerm i)) v)
theorem measurable_clampedPredecessorCoordinates
The clamped predecessor-coordinate restriction is measurable.
Formal statement
hypotheses
conclusion
Measurable (clampedPredecessorCoordinates W order i)
Proof (Lean source)
@[fun_prop] lemma measurable_clampedPredecessorCoordinates
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) (order : Fin n → ℕ) (i : Fin n) :
Measurable (clampedPredecessorCoordinates W order i) := by
exact (measurable_retainedProjectionToPredecessorCube W order i).comp
(measurable_coordinateProjection _)
theorem clampedPredecessorCoordinates_of_mem
On the latent cube, clamped predecessor coordinates equal the genuine restriction. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
clampedPredecessorCoordinates W order i v = predecessorLatentCubeOfState W order i v hv
Proof (Lean source)
lemma clampedPredecessorCoordinates_of_mem
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) (order : Fin n → ℕ) (i : Fin n)
(v : LatentState n) (hv : v ∈ latentCube n) :
clampedPredecessorCoordinates W order i v =
predecessorLatentCubeOfState W order i v hv :=
retainedProjectionToPredecessorCube_coordinateProjection W order i v hv
theorem predecessorCoordinates_independent_target
Equation (10) transported to the compact predecessor cube: under intervention i, clamped predecessor coordinates and the target coordinate have a product law. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
W :
ObservedWorld G θ
hpos :
order :
Fin n → ℕ
horder :
hgraphOrder :
PermutedGraphOrdered W order
i :
Fin n
conclusion
Measure.map (fun v => (clampedPredecessorCoordinates W order i v, v (W.targetPerm i))) (interventionalLaw θ (W.targetPerm i))
= (Measure.map (clampedPredecessorCoordinates W order i) (interventionalLaw θ (W.targetPerm i))).prod (decoderInterventionCoordinateMeasure W hpos i)
Proof (Lean source)
lemma predecessorCoordinates_independent_target
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) (hpos : PositiveNormalizedSmoothMechanisms G θ)
{order : Fin n → ℕ}
(horder : IsTopologicalOrdering (observedLawRatioGraph gaussianFeatureMap W.law) order)
(hgraphOrder : PermutedGraphOrdered W order) (i : Fin n) :
Measure.map (fun v => (clampedPredecessorCoordinates W order i v,
v (W.targetPerm i))) (interventionalLaw θ (W.targetPerm i)) =
(Measure.map (clampedPredecessorCoordinates W order i)
(interventionalLaw θ (W.targetPerm i))).prod
(decoderInterventionCoordinateMeasure W hpos i) := by
let A := decoderRetainedLatentSet W order i
let S := A.erase (W.targetPerm i)
let pred := coordinateProjection
(X := fun _ : Fin n => ℝ) S
let F := retainedProjectionToPredecessorCube W order i
let μ := interventionalLaw θ (W.targetPerm i)
let ν := decoderInterventionCoordinateMeasure W hpos i
letI : IsProbabilityMeasure μ := interventionalLaw_isProbabilityMeasure W hpos i
have hprod := equationTen_predecessorCoordinates_independent_target
W hpos horder hgraphOrder i
change Measure.map (fun v => (pred v, v (W.targetPerm i))) μ =
(Measure.map pred μ).prod ν at hprod
have hF : Measurable F := measurable_retainedProjectionToPredecessorCube W order i
have hpair : Measurable (fun v : LatentState n => (pred v, v (W.targetPerm i))) := by
fun_prop
have hmap := congrArg (Measure.map (map F id)) hprod
calc
Measure.map (fun v => (clampedPredecessorCoordinates W order i v,
v (W.targetPerm i))) μ =
Measure.map (map F id)
(Measure.map (fun v => (pred v, v (W.targetPerm i))) μ) := by
rw [Measure.map_map (hF.prodMap measurable_id) hpair]
rfl
_ = Measure.map (map F id) ((Measure.map pred μ).prod ν) := hmap
_ = (Measure.map F (Measure.map pred μ)).prod ν := by
rw [← Measure.map_prod_map _ _ hF measurable_id]
simp
_ = (Measure.map (clampedPredecessorCoordinates W order i) μ).prod ν := by
rw [Measure.map_map hF (by fun_prop)]
rfl
theorem equationElevenAmbientKernel_apply_scoreMap
On a triangular predecessor score, the ambient equation-(11) kernel is precisely the pushforward of the replacement coordinate law along the corresponding own-score fiber. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
s :
θ :
Mechanism n G
W :
ObservedWorld G θ
hpos :
hmix :
hone :
hsign :
FixedOwnDerivativeSign G s θ
order :
Fin n → ℕ
horder :
hgraphOrder :
PermutedGraphOrdered W order
i :
Fin n
z :
PredecessorLatentCube order i
conclusion
(equationElevenAmbientKernel s W hpos hmix hone hsign horder hgraphOrder i) (predecessorScoreMap W order i z)
= Measure.map (fun w => equationElevenClampedScore W order i (z, w)) (decoderInterventionCoordinateMeasure W hpos i)
Proof (Lean source)
lemma equationElevenAmbientKernel_apply_scoreMap
{n : ℕ} {G : DAG (Fin n)} (s : SignVector n)
{θ : Mechanism n G} (W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W)
(hsign : FixedOwnDerivativeSign G s θ)
{order : Fin n → ℕ}
(horder : IsTopologicalOrdering (observedLawRatioGraph gaussianFeatureMap W.law) order)
(hgraphOrder : PermutedGraphOrdered W order) (i : Fin n)
(z : PredecessorLatentCube order i) :
(equationElevenAmbientKernel s W hpos hmix hone hsign horder hgraphOrder i)
(predecessorScoreMap W order i z) =
Measure.map (fun w => equationElevenClampedScore W order i (z, w))
(decoderInterventionCoordinateMeasure W hpos i) := by
let e := predecessorScoreHomeomorph s W hpos hmix hone hsign horder hgraphOrder i
have hrange : predecessorScoreMap W order i z ∈
range (predecessorScoreMap W order i) := ⟨z, rfl⟩
have hretract : predecessorScoreRangeRetraction W order i
(predecessorScoreMap W order i z) =
⟨predecessorScoreMap W order i z, hrange⟩ := by
simp [predecessorScoreRangeRetraction, hrange]
have hinv : e.symm ⟨predecessorScoreMap W order i z, hrange⟩ = z := by
apply e.injective
rw [e.apply_symm_apply]
rfl
rw [equationElevenAmbientKernel, Kernel.comap_apply,
equationElevenScoreImageKernel, Kernel.comap_apply, hretract, hinv,
equationElevenPredecessorKernel]
ext u hu
rw [Kernel.map_apply' _ (measurable_equationElevenClampedScore W hpos order i) _ hu,
Kernel.comap_apply, Kernel.parallelComp_apply, Kernel.id_apply, Kernel.const_apply,
Measure.dirac_prod]
rw [Measure.map_apply (by fun_prop)
(hu.preimage (measurable_equationElevenClampedScore W hpos order i))]
rw [Measure.map_apply (by fun_prop) hu]
rfl
theorem equationElevenClampedScore_of_latentState
On a latent cube point, the clamped equation-(11) fiber score equals the observed target log-ratio, because every target parent occurs among the reconstructed predecessors. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
W :
ObservedWorld G θ
hpos :
hmix :
hone :
order :
Fin n → ℕ
horder :
hgraphOrder :
PermutedGraphOrdered W order
i :
Fin n
v :
hv :
v ∈ latentCube n
conclusion
equationElevenClampedScore W order i (clampedPredecessorCoordinates W order i v, v (W.targetPerm i))
= observedLawLogRatio W.law i (W.mix v)
Proof (Lean source)
lemma equationElevenClampedScore_of_latentState
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W)
{order : Fin n → ℕ}
(horder : IsTopologicalOrdering (observedLawRatioGraph gaussianFeatureMap W.law) order)
(hgraphOrder : PermutedGraphOrdered W order) (i : Fin n)
(v : LatentState n) (hv : v ∈ latentCube n) :
equationElevenClampedScore W order i
(clampedPredecessorCoordinates W order i v, v (W.targetPerm i)) =
observedLawLogRatio W.law i (W.mix v) := by
rw [clampedPredecessorCoordinates_of_mem W order i v hv]
rw [observedLawLogRatio_comp_mix_eq W hpos hmix hone i v hv]
unfold equationElevenClampedScore
have hvi := hv (W.targetPerm i) (Set.mem_univ _)
rw [min_eq_right hvi.2, max_eq_right hvi.1]
dsimp only
have hp : θ.p (W.targetPerm i)
(update (predecessorCubeLatentState W order i
(predecessorLatentCubeOfState W order i v hv))
(W.targetPerm i) (v (W.targetPerm i))) = θ.p (W.targetPerm i) v := by
apply θ.parent_local (W.targetPerm i)
· simp
· intro a ha
let j := W.targetPerm.symm a
have hjparent : j ∈ environmentParentSet W i := by
simpa [j, environmentParentSet, parents] using ha
have hjpred := environmentParentSet_subset_predecessorSet_of_transitiveClosure
W horder hgraphOrder i hjparent
have hbase := predecessorCubeLatentState_ofState_apply W order i v hv j hjpred
have hat : a ≠ W.targetPerm i := by
intro h
have hji : j = i := W.targetPerm.injective (by simpa [j] using h)
have hjlt : order j < order i := by simpa [predecessorSet] using hjpred
exact (hji ▸ hjlt).false
simpa [update, hat, j] using hbase
rw [hp]
Under intervention i, the observed predecessor-score and target-score law is the predecessor marginal composed with the ambient equation-(11) Markov kernel. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
s :
θ :
Mechanism n G
W :
ObservedWorld G θ
hpos :
hmix :
hone :
hsign :
FixedOwnDerivativeSign G s θ
order :
Fin n → ℕ
horder :
hgraphOrder :
PermutedGraphOrdered W order
i :
Fin n
conclusion
Measure.map (fun x ↦ (familyProjection (observedLawLogRatio W.law) (predecessorSet order i) x, observedLawLogRatio W.law i x)) (W.law i.succ)
= Measure.map (familyProjection (observedLawLogRatio W.law) (predecessorSet order i)) (W.law i.succ) ⊗ₘ equationElevenAmbientKernel s W hpos hmix hone hsign horder hgraphOrder i
Proof (Lean source)
lemma observedRatioPredecessor_compProd
{n : ℕ} {G : DAG (Fin n)} (s : SignVector n)
{θ : Mechanism n G} (W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W)
(hsign : FixedOwnDerivativeSign G s θ)
{order : Fin n → ℕ}
(horder : IsTopologicalOrdering (observedLawRatioGraph gaussianFeatureMap W.law) order)
(hgraphOrder : PermutedGraphOrdered W order) (i : Fin n) :
Measure.map
(fun x ↦
(familyProjection (observedLawLogRatio W.law) (predecessorSet order i) x,
observedLawLogRatio W.law i x)) (W.law i.succ) =
Measure.map
(familyProjection (observedLawLogRatio W.law) (predecessorSet order i))
(W.law i.succ) ⊗ₘ
equationElevenAmbientKernel s W hpos hmix hone hsign horder hgraphOrder i := by
let μ := interventionalLaw θ (W.targetPerm i)
let ν := decoderInterventionCoordinateMeasure W hpos i
let z := clampedPredecessorCoordinates W order i
let g := predecessorScoreMap W order i
let f := equationElevenClampedScore W order i
let κ := equationElevenAmbientKernel s W hpos hmix hone hsign horder hgraphOrder i
let latentJoint : LatentState n → PredecessorLogRatios order i × ℝ := fun v =>
(g (z v), f (z v, v (W.targetPerm i)))
let observedJoint : LatentState n → PredecessorLogRatios order i × ℝ := fun x =>
(familyProjection (observedLawLogRatio W.law) (predecessorSet order i) x,
observedLawLogRatio W.law i x)
have hμ : IsProbabilityMeasure μ := interventionalLaw_isProbabilityMeasure W hpos i
letI : IsProbabilityMeasure μ := hμ
have hν : IsProbabilityMeasure ν := decoderInterventionCoordinateMeasure_isProbability W hpos i
letI : IsProbabilityMeasure ν := hν
have hz : Measurable z := measurable_clampedPredecessorCoordinates W order i
have hg : Measurable g := (continuous_predecessorScoreMap W hpos order i).measurable
have hf : Measurable f := measurable_equationElevenClampedScore W hpos order i
have hκ : ∀ a, κ (g a) = Measure.map (fun b => f (a, b)) ν :=
equationElevenAmbientKernel_apply_scoreMap
s W hpos hmix hone hsign horder hgraphOrder i
have hbase := predecessorCoordinates_independent_target W hpos horder hgraphOrder i
change Measure.map (fun v => (z v, v (W.targetPerm i))) μ =
(Measure.map z μ).prod ν at hbase
have hpair : Measurable (fun v : LatentState n => (z v, v (W.targetPerm i))) := by
fun_prop
have hgf : Measurable (fun p : PredecessorLatentCube order i × ℝ =>
(g p.1, f p)) := by fun_prop
have hmap := congrArg (Measure.map (fun p : PredecessorLatentCube order i × ℝ =>
(g p.1, f p))) hbase
have hlatent : Measure.map latentJoint μ = Measure.map g (Measure.map z μ) ⊗ₘ κ := by
calc
Measure.map latentJoint μ = Measure.map (fun p : PredecessorLatentCube order i × ℝ =>
(g p.1, f p)) (Measure.map (fun v => (z v, v (W.targetPerm i))) μ) := by
rw [Measure.map_map hgf hpair]
rfl
_ = Measure.map (fun p : PredecessorLatentCube order i × ℝ =>
(g p.1, f p)) ((Measure.map z μ).prod ν) := hmap
_ = Measure.map g (Measure.map z μ) ⊗ₘ κ :=
map_prod_compProd_of_fiber (Measure.map z μ) ν g hg f hf κ hκ
have hcube : ∀ᵐ v ∂μ, v ∈ latentCube n := by
filter_upwards [Measure.support_mem_ae (μ := μ)] with v hv
simpa [μ, interventionalLaw_support_eq_latentCube W hpos i] using hv
have hlatent_obs : latentJoint =ᵐ[μ] observedJoint ∘ W.mix := by
filter_upwards [hcube] with v hv
apply Prod.ext
· dsimp only [latentJoint, observedJoint, g, z]
rw [clampedPredecessorCoordinates_of_mem W order i v hv]
exact predecessorScoreMap_of_latentState W hpos hmix hone horder hgraphOrder i v hv
· exact equationElevenClampedScore_of_latentState
W hpos hmix hone horder hgraphOrder i v hv
have hmixAE : AEMeasurable W.mix μ := by
apply aemeasurable_of_supportMeasurableOn (by
rw [latentCube]
exact MeasurableSet.pi Set.countable_univ (fun _ _ ↦ measurableSet_Icc))
hcube hmix.1.continuousOn.domRestrict.measurable
have hobsJoint : Measurable observedJoint := by
dsimp only [observedJoint]
apply Measurable.prodMk
· apply measurable_pi_lambda
intro j
exact measurable_observedLawLogRatio W.law j
· exact measurable_observedLawLogRatio W.law i
have hpushed : Measure.map observedJoint (W.law i.succ) = Measure.map latentJoint μ := by
rw [hone.2.1 i]
rw [AEMeasurable.map_map_of_aemeasurable hobsJoint.aemeasurable hmixAE]
exact Measure.map_congr hlatent_obs.symm
have hpredAE : (g ∘ z) =ᵐ[μ]
(familyProjection (observedLawLogRatio W.law) (predecessorSet order i)) ∘ W.mix := by
filter_upwards [hcube] with v hv
dsimp only [g, z, Function.comp_apply]
rw [clampedPredecessorCoordinates_of_mem W order i v hv]
exact predecessorScoreMap_of_latentState W hpos hmix hone horder hgraphOrder i v hv
have hpredMeas : Measurable
(familyProjection (observedLawLogRatio W.law) (predecessorSet order i)) := by
apply measurable_pi_lambda
intro j
exact measurable_observedLawLogRatio W.law j
have hpredPushed : Measure.map
(familyProjection (observedLawLogRatio W.law) (predecessorSet order i))
(W.law i.succ) = Measure.map g (Measure.map z μ) := by
rw [hone.2.1 i]
rw [AEMeasurable.map_map_of_aemeasurable hpredMeas.aemeasurable hmixAE]
rw [Measure.map_map hg hz]
exact Measure.map_congr hpredAE.symm
rw [hpushed, hlatent, hpredPushed]
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.DecoderConditionalKernel 26 declarations This file supplies the factor-elimination identity underlying equation (10): when a parent-closed retained set contains the intervention target, its marginal density is the replacement density times the product of the re
Intervention marginal density for the decoder
This file supplies the factor-elimination identity underlying equation (10): when a parent-closed retained set contains the intervention target, its marginal density is the replacement density times the product of the remaining retained observational factors. It also constructs the equation-(11) Markov kernel on the compact predecessor cube and transports it through the triangular predecessor-score homeomorphism.
theorem interventionMarginalDensity_eq_replacement_mul_partialDensity Lemma interventionMarginalDensity_eq_replacement_mul_partialDensity in the paper ↗
Eliminating all coordinates outside a parent-closed set containing the intervention target leaves the intervention density times the observational factors at the other retained nodes. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
V :
∀ i, MeasurableSpace (X i)
μ :
∀ i, Measure (X i)
∀ i, SigmaFinite (μ i)
G :
DAG V
B :
Factorization G X μ
A :
Finset V
hA :
ParentClosed G A
j :
V
hj :
j ∈ A
q :
InterventionDensity j X μ
conclusion
(∫⋯∫⁻_(univ \ A), B.interventionDensity j q ∂μ)
= fun v => q.density (v j) * B.partialDensity (A.erase j) v
Proof (Lean source)
lemma interventionMarginalDensity_eq_replacement_mul_partialDensity
{V : Type*} [DecidableEq V] [Fintype V]
{X : V → Type*} [∀ i, MeasurableSpace (X i)]
{μ : ∀ i, Measure (X i)} [∀ i, SigmaFinite (μ i)]
{G : DAG V} (B : Factorization G X μ)
{A : Finset V} (hA : ParentClosed G A) {j : V} (hj : j ∈ A)
(q : InterventionDensity j X μ) :
(∫⋯∫⁻_(univ \ A), B.interventionDensity j q ∂μ) =
fun v => q.density (v j) * B.partialDensity (A.erase j) v := by
rw [← B.observationalDensity_intervene]
rw [(B.intervene j q).lmarginal_compl_observationalDensity_eq hA]
funext v
unfold Factorization.partialDensity Factorization.intervene
rw [← Finset.prod_erase_mul _ _ hj]
have hprod :
(∏ i ∈ A.erase j,
if i = j then q.density (v j) else B.factor i v) =
∏ i ∈ A.erase j, B.factor i v := by
apply Finset.prod_congr rfl
intro i hi
have hij : i ≠ j := (Finset.mem_erase.mp hi).1
simp [hij]
rw [hprod]
simp only [if_pos]
exact mul_comm _ _
def decoderRetainedLatentSet
The latent-coordinate image of a decoder predecessor set, together with the current intervention target.
theorem decoderRetainedLatentSet_parentClosed
If the selected order respects the recovered ancestral order, the target and its decoder predecessors form a parent-closed latent set. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
W :
ObservedWorld G θ
order :
Fin n → ℕ
horder :
hgraphOrder :
PermutedGraphOrdered W order
i :
Fin n
conclusion
ParentClosed G (decoderRetainedLatentSet W order i)
Proof (Lean source)
lemma decoderRetainedLatentSet_parentClosed
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) {order : Fin n → ℕ}
(horder : IsTopologicalOrdering (observedLawRatioGraph gaussianFeatureMap W.law) order)
(hgraphOrder : PermutedGraphOrdered W order) (i : Fin n) :
ParentClosed G (decoderRetainedLatentSet W order i) := by
intro k hk a ha
rw [decoderRetainedLatentSet, Finset.mem_insert] at hk ⊢
rcases hk with rfl | hk
· right
refine Finset.mem_map.mpr ⟨W.targetPerm.symm a, ?_, ?_⟩
· apply environmentParentSet_subset_predecessorSet_of_transitiveClosure W horder hgraphOrder i
simpa only [environmentParentSet, mem_filter, Finset.mem_univ, true_and,
Equiv.apply_symm_apply] using G.mem_parents.mp ha
· simp
· right
rcases Finset.mem_map.mp hk with ⟨b, hb, rfl⟩
refine Finset.mem_map.mpr ⟨W.targetPerm.symm a, ?_, ?_⟩
· have hab : W.targetPerm.symm a ∈ predecessorSet order b := by
apply environmentParentSet_subset_predecessorSet_of_transitiveClosure W horder hgraphOrder b
have hedge : G.edge a (W.targetPerm b) := by
simpa using G.mem_parents.mp ha
simpa only [environmentParentSet, mem_filter, Finset.mem_univ, true_and,
Equiv.apply_symm_apply] using hedge
simp only [predecessorSet, mem_filter, Finset.mem_univ, true_and] at hab hb ⊢
exact hab.trans hb
· simp
theorem equationTen_predecessorDensity_update_target
The predecessor-factor part of equation (10) is constant along the current target's own-coordinate fiber. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
W :
ObservedWorld G θ
hpos :
order :
Fin n → ℕ
horder :
hgraphOrder :
PermutedGraphOrdered W order
i :
Fin n
v :
w :
ℝ
conclusion
B :
A :
decoderRetainedLatentSet W order i
B.partialDensity (A.erase (W.targetPerm i)) (update v (W.targetPerm i) w)
= B.partialDensity (A.erase (W.targetPerm i)) v
Proof (Lean source)
lemma equationTen_predecessorDensity_update_target
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) (hpos : PositiveNormalizedSmoothMechanisms G θ)
{order : Fin n → ℕ}
(horder : IsTopologicalOrdering (observedLawRatioGraph gaussianFeatureMap W.law) order)
(hgraphOrder : PermutedGraphOrdered W order) (i : Fin n)
(v : LatentState n) (w : ℝ) :
let B := mechanismUnitCubeFactorization hpos
let A := decoderRetainedLatentSet W order i
B.partialDensity (A.erase (W.targetPerm i))
(update v (W.targetPerm i) w) =
B.partialDensity (A.erase (W.targetPerm i)) v := by
dsimp only
unfold partialDensity
apply Finset.prod_congr rfl
intro k hk
apply (mechanismUnitCubeFactorization hpos).local_factor k
intro a ha
have hatarget : a ≠ W.targetPerm i := by
intro hEq
subst a
rcases Finset.mem_insert.mp ha with hki | hparent
· exact (Finset.mem_erase.mp hk).1 hki.symm
· have hkA := (Finset.mem_erase.mp hk).2
rw [decoderRetainedLatentSet, Finset.mem_insert] at hkA
rcases hkA with hkTarget | hkPred
· exact (Finset.mem_erase.mp hk).1 hkTarget
· rcases Finset.mem_map.mp hkPred with ⟨b, hb, hkb⟩
subst k
exact target_not_parent_of_mem_predecessorSet W horder hgraphOrder hb hparent
simp [update, hatarget]
Equation (10) at the density-elimination level: under the current intervention, retaining the target and all predecessor coordinates leaves the replacement density times the remaining retained observational factors. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
W :
ObservedWorld G θ
hpos :
order :
Fin n → ℕ
horder :
hgraphOrder :
PermutedGraphOrdered W order
i :
Fin n
conclusion
= fun v => q.density (v (W.targetPerm i)) * B.partialDensity (A.erase (W.targetPerm i)) v
Proof (Lean source)
lemma equationTen_interventionMarginalDensity
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) (hpos : PositiveNormalizedSmoothMechanisms G θ)
{order : Fin n → ℕ}
(horder : IsTopologicalOrdering (observedLawRatioGraph gaussianFeatureMap W.law) order)
(hgraphOrder : PermutedGraphOrdered W order) (i : Fin n) :
let B := mechanismUnitCubeFactorization hpos
let q := mechanismInterventionDensity W hpos i
let A := decoderRetainedLatentSet W order i
(∫⋯∫⁻_(univ \ A), B.interventionDensity (W.targetPerm i) q
∂fun _ : Fin n => unitIntervalReference) =
fun v => q.density (v (W.targetPerm i)) * B.partialDensity (A.erase (W.targetPerm i)) v := by
dsimp only
have hσ : ∀ _ : Fin n,
SigmaFinite unitIntervalReference := fun _ => by
unfold unitIntervalReference
infer_instance
exact @interventionMarginalDensity_eq_replacement_mul_partialDensity
(Fin n) _ _ (fun _ : Fin n => ℝ) (fun _ : Fin n => inferInstance)
(fun _ : Fin n => unitIntervalReference) hσ G
(mechanismUnitCubeFactorization hpos) (decoderRetainedLatentSet W order i)
(decoderRetainedLatentSet_parentClosed W horder hgraphOrder i) (W.targetPerm i)
(by simp [decoderRetainedLatentSet]) (mechanismInterventionDensity W hpos i)
Equation (10) in explicit fiber-product form: after fixing predecessor coordinates, the retained intervention density is the own-coordinate replacement density times a factor that is constant along that coordinate's fiber. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
W :
ObservedWorld G θ
hpos :
order :
Fin n → ℕ
horder :
hgraphOrder :
PermutedGraphOrdered W order
i :
Fin n
v :
w :
ℝ
conclusion
(∫⋯∫⁻_(univ \ A), B.interventionDensity (W.targetPerm i) q ∂fun _ : Fin n => unitIntervalReference) (update v (W.targetPerm i) w)
= q.density w * B.partialDensity (A.erase (W.targetPerm i)) v
Proof (Lean source)
lemma equationTen_interventionMarginalDensity_fiber_product
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) (hpos : PositiveNormalizedSmoothMechanisms G θ)
{order : Fin n → ℕ}
(horder : IsTopologicalOrdering (observedLawRatioGraph gaussianFeatureMap W.law) order)
(hgraphOrder : PermutedGraphOrdered W order) (i : Fin n)
(v : LatentState n) (w : ℝ) :
let B := mechanismUnitCubeFactorization hpos
let q := mechanismInterventionDensity W hpos i
let A := decoderRetainedLatentSet W order i
(∫⋯∫⁻_(univ \ A),
B.interventionDensity (W.targetPerm i) q
∂fun _ : Fin n => unitIntervalReference)
(update v (W.targetPerm i) w) =
q.density w * B.partialDensity (A.erase (W.targetPerm i)) v := by
dsimp only
have hten := congrFun
(equationTen_interventionMarginalDensity W hpos horder hgraphOrder i)
(update v (W.targetPerm i) w)
rw [equationTen_predecessorDensity_update_target W hpos horder hgraphOrder i v w] at hten
simpa using hten
def decoderInterventionCoordinateMeasure
The normalized replacement density, viewed as a probability measure on the target's scalar coordinate.
Definition (Lean source)
parameters
def
given by
Causalean.Graph.FiniteDensity.unitIntervalReference.withDensity (mechanismInterventionDensity W hpos i).density
instance decoderInterventionCoordinateMeasure_isProbability
Normalization of the replacement density makes its coordinate measure a probability measure.
Definition (Lean source)
instance decoderInterventionCoordinateMeasure_isProbability
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) (hpos : PositiveNormalizedSmoothMechanisms G θ)
(i : Fin n) : IsProbabilityMeasure (decoderInterventionCoordinateMeasure W hpos i) := by
constructor
rw [decoderInterventionCoordinateMeasure, withDensity_apply _ MeasurableSet.univ]
simpa only [Measure.restrict_univ] using
(mechanismInterventionDensity W hpos i).normalized_density
def equationElevenClampedScore
A globally measurable clamped extension of the target log-ratio score along a predecessor fiber. On the unit interval it is the score appearing in equation (11).
Definition (Lean source)
parameters
def
equationElevenClampedScore W order i zw :
ℝ
given by
v :
predecessorCubeLatentState W order i zw.1
theorem measurable_equationElevenClampedScore
Smoothness on the compact cube makes the clamped equation-(11) score jointly measurable in predecessor coordinates and the target coordinate. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
Measurable (equationElevenClampedScore W order i)
Proof (Lean source)
@[fun_prop]
lemma measurable_equationElevenClampedScore
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) (hpos : PositiveNormalizedSmoothMechanisms G θ)
(order : Fin n → ℕ) (i : Fin n) : Measurable (equationElevenClampedScore W order i) := by
have hw : Continuous (fun zw : PredecessorLatentCube order i × ℝ =>
max 0 (min 1 zw.2)) := by fun_prop
have hwmem : ∀ zw : PredecessorLatentCube order i × ℝ,
max 0 (min 1 zw.2) ∈ Icc (0 : ℝ) 1 := by
intro zw
exact ⟨le_max_left _ _, max_le zero_le_one (min_le_left _ _)⟩
have hbase : Continuous (predecessorCubeLatentState W order i) := by
apply continuous_pi
intro a
simp only [predecessorCubeLatentState]
split <;> fun_prop
have hv : Continuous (fun zw : PredecessorLatentCube order i × ℝ =>
update (predecessorCubeLatentState W order i zw.1)
(W.targetPerm i) (max 0 (min 1 zw.2))) := by
apply continuous_pi
intro k
by_cases hki : k = W.targetPerm i
· subst k
simpa using hw
· have heq : (fun a : PredecessorLatentCube order i × ℝ =>
update (predecessorCubeLatentState W order i a.1)
(W.targetPerm i) (max 0 (min 1 a.2)) k) =
fun a => predecessorCubeLatentState W order i a.1 k := by
funext a
simp [update, hki]
rw [heq]
exact (continuous_apply k).comp (hbase.comp continuous_fst)
have hvmem : ∀ zw : PredecessorLatentCube order i × ℝ,
update (predecessorCubeLatentState W order i zw.1)
(W.targetPerm i) (max 0 (min 1 zw.2)) ∈ latentCube n := by
intro zw k hk
by_cases hki : k = W.targetPerm i
· subst k
simpa using hwmem zw
· simp [update, hki]
exact predecessorCubeLatentState_mem_latentCube W order i zw.1 k (Set.mem_univ k)
have hq := (hpos.2.2.2.1 (W.targetPerm i)).continuousOn.comp_continuous hw hwmem
have hp := (hpos.2.2.1 (W.targetPerm i)).continuousOn.comp_continuous hv hvmem
have hpne : ∀ zw : PredecessorLatentCube order i × ℝ, θ.p (W.targetPerm i)
(update (predecessorCubeLatentState W order i zw.1)
(W.targetPerm i) (max 0 (min 1 zw.2))) ≠ 0 := fun zw =>
ne_of_gt (hpos.1 _ _ (hvmem zw))
have hqne : ∀ zw : PredecessorLatentCube order i × ℝ,
θ.q (W.targetPerm i) (max 0 (min 1 zw.2)) ≠ 0 := fun zw =>
ne_of_gt (hpos.2.1 _ _ (hwmem zw))
exact (hq.div hp hpne |>.log (fun zw => div_ne_zero (hqne zw) (hpne zw))).measurable
def equationElevenPredecessorKernel
The equation-(11) Markov kernel on predecessor latent coordinates: draw the target from its replacement density and push it through the target log-ratio score.
Definition (Lean source)
parameters
def
given by
(((Kernel.id : Kernel (PredecessorLatentCube order i) _) ∥ₖ Kernel.const Unit (decoderInterventionCoordinateMeasure W hpos i)).comap (fun z => (z, ())) (by fun_prop)).map (equationElevenClampedScore W order i)
instance equationElevenPredecessorKernel_isMarkov
The equation-(11) predecessor kernel has unit mass on every predecessor fiber.
Definition (Lean source)
instance equationElevenPredecessorKernel_isMarkov
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) (hpos : PositiveNormalizedSmoothMechanisms G θ)
(order : Fin n → ℕ) (i : Fin n) :
IsMarkovKernel (equationElevenPredecessorKernel W hpos order i) := by
unfold equationElevenPredecessorKernel
let base : Kernel (PredecessorLatentCube order i × Unit)
(PredecessorLatentCube order i × ℝ) :=
Kernel.id ∥ₖ Kernel.const Unit (decoderInterventionCoordinateMeasure W hpos i)
haveI : IsMarkovKernel base := by dsimp only [base]; infer_instance
haveI : IsMarkovKernel (base.comap (fun z => (z, ())) (by fun_prop)) :=
Kernel.IsMarkovKernel.comap base (by fun_prop)
exact Kernel.IsMarkovKernel.map _ (measurable_equationElevenClampedScore W hpos order i)
theorem equationElevenPredecessorKernel_apply_Iic
Every lower-interval probability of the predecessor kernel is exactly the explicit equation-(11) conditional-CDF integral. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
= equationElevenConditionalRatioCDF θ (W.targetPerm i) t (predecessorCubeLatentState W order i z)
Proof (Lean source)
lemma equationElevenPredecessorKernel_apply_Iic
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) (hpos : PositiveNormalizedSmoothMechanisms G θ)
(order : Fin n → ℕ) (i : Fin n) (z : PredecessorLatentCube order i) (t : ℝ) :
((equationElevenPredecessorKernel W hpos order i) z (Iic t)).toReal =
equationElevenConditionalRatioCDF θ (W.targetPerm i) t
(predecessorCubeLatentState W order i z) := by
rw [equationElevenPredecessorKernel,
Kernel.map_apply' _ (measurable_equationElevenClampedScore W hpos order i) _
measurableSet_Iic,
Kernel.comap_apply, Kernel.parallelComp_apply, Kernel.id_apply, Kernel.const_apply]
rw [Measure.dirac_prod]
rw [Measure.map_apply (by fun_prop)
(measurableSet_Iic.preimage (measurable_equationElevenClampedScore W hpos order i))]
change ((decoderInterventionCoordinateMeasure W hpos i)
{w | equationElevenClampedScore W order i (z, w) ≤ t}).toReal = _
rw [decoderInterventionCoordinateMeasure]
have hset : MeasurableSet {w | equationElevenClampedScore W order i (z, w) ≤ t} :=
measurableSet_Iic.preimage ((measurable_equationElevenClampedScore W hpos order i).comp
(measurable_const.prodMk measurable_id))
rw [withDensity_apply _ hset]
let v := predecessorCubeLatentState W order i z
let score : ℝ → ℝ := fun w => log
(θ.q (W.targetPerm i) w /
θ.p (W.targetPerm i) (update v (W.targetPerm i) w))
let f : ℝ → ℝ := fun w => if score w ≤ t then θ.q (W.targetPerm i) w else 0
have hf_nonneg : 0 ≤ᵐ[volume.restrict (Icc (0 : ℝ) 1)] f := by
filter_upwards [ae_restrict_mem measurableSet_Icc] with w hw
simp only [f]
split
· exact le_of_lt (hpos.2.1 _ w hw)
· exact le_rfl
have hf_meas : AEStronglyMeasurable f (volume.restrict (Icc (0 : ℝ) 1)) := by
have hscore : ContinuousOn score (Icc (0 : ℝ) 1) := by
dsimp only [score, v]
have hq := (hpos.2.2.2.1 (W.targetPerm i)).continuousOn
have hbase := predecessorCubeLatentState_mem_latentCube W order i z
have hp : ContinuousOn (fun w => θ.p (W.targetPerm i)
(update (predecessorCubeLatentState W order i z)
(W.targetPerm i) w)) (Icc (0 : ℝ) 1) := by
apply (hpos.2.2.1 (W.targetPerm i)).continuousOn.comp
· fun_prop
· intro w hw k hk
by_cases hki : k = W.targetPerm i
· subst k
simpa using hw
· simp [update, hki]
exact hbase k (Set.mem_univ k)
have hpne : ∀ w ∈ Icc (0 : ℝ) 1, θ.p (W.targetPerm i)
(update (predecessorCubeLatentState W order i z)
(W.targetPerm i) w) ≠ 0 := fun w hw => ne_of_gt (hpos.1 _ _ (by
intro k hk
by_cases hki : k = W.targetPerm i
· subst k
simpa using hw
· simp [update, hki]
exact hbase k (Set.mem_univ k)))
exact (hq.div hp hpne).log (fun w hw => div_ne_zero
(ne_of_gt (hpos.2.1 _ w hw)) (hpne w hw))
have hqae : AEMeasurable (θ.q (W.targetPerm i))
(volume.restrict (Icc (0 : ℝ) 1)) :=
(hpos.2.2.2.1 (W.targetPerm i)).continuousOn.aemeasurable measurableSet_Icc
have hscoreae : AEMeasurable score (volume.restrict (Icc (0 : ℝ) 1)) :=
hscore.aemeasurable measurableSet_Icc
let qm := hqae.mk
let sm := hscoreae.mk
have hfm : Measurable (fun w => if sm w ≤ t then qm w else 0) := by
exact Measurable.ite (measurableSet_le hscoreae.measurable_mk measurable_const)
hqae.measurable_mk measurable_const
apply hfm.aestronglyMeasurable.congr
filter_upwards [hqae.ae_eq_mk, hscoreae.ae_eq_mk] with w hqw hsw
simp only [qm, sm, f] at hqw hsw ⊢
rw [← hqw, ← hsw]
rw [equationElevenConditionalRatioCDF]
change _ = ∫ w in Icc (0 : ℝ) 1, f w
rw [integral_eq_lintegral_of_nonneg_ae hf_nonneg hf_meas]
congr 1
rw [← lintegral_indicator hset]
apply lintegral_congr_ae
filter_upwards [ae_restrict_mem measurableSet_Icc] with w hw
have hclamp : max 0 (min 1 w) = w := by
rw [min_eq_right hw.2, max_eq_right hw.1]
have hscoreeq : equationElevenClampedScore W order i (z, w) = score w := by
simp only [equationElevenClampedScore, score, v, hclamp]
simp only [indicator, Set.mem_ofPred_eq, hscoreeq]
simp only [mechanismInterventionDensity,
interventionDensityOfRealCube,
interventionDensityOfCube, hclamp]
by_cases hs : score w ≤ t
· simp [f, hs]
· simp [f, hs]
def equationElevenScoreImageKernel
Transporting the predecessor kernel through the compact triangular inverse gives the equation-(11) kernel on the realized predecessor-score image.
Definition (Lean source)
parameters
s :
θ :
Mechanism n G
W :
ObservedWorld G θ
hpos :
hmix :
hone :
hsign :
FixedOwnDerivativeSign G s θ
order :
Fin n → ℕ
horder :
hgraphOrder :
PermutedGraphOrdered W order
i :
Fin n
def
equationElevenScoreImageKernel s W hpos hmix hone hsign horder hgraphOrder i :
given by
(equationElevenPredecessorKernel W hpos order i).comap (predecessorScoreHomeomorph s W hpos hmix hone hsign horder hgraphOrder i).symm (continuous_predecessorScoreInverse s W hpos hmix hone hsign horder hgraphOrder i).measurable
instance equationElevenScoreImageKernel_isMarkov
The score-image transport of the equation-(11) kernel remains Markov.
Definition (Lean source)
instance equationElevenScoreImageKernel_isMarkov
{n : ℕ} {G : DAG (Fin n)} (s : SignVector n)
{θ : Mechanism n G} (W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W)
(hsign : FixedOwnDerivativeSign G s θ)
{order : Fin n → ℕ}
(horder : IsTopologicalOrdering (observedLawRatioGraph gaussianFeatureMap W.law) order)
(hgraphOrder : PermutedGraphOrdered W order) (i : Fin n) :
IsMarkovKernel
(equationElevenScoreImageKernel s W hpos hmix hone hsign horder hgraphOrder i) := by
exact Kernel.IsMarkovKernel.comap _
(continuous_predecessorScoreInverse s W hpos hmix hone hsign horder hgraphOrder i).measurable
theorem equationElevenScoreImageKernel_apply_Iic
On every realized predecessor score, the transported kernel's lower-interval probability is the explicit equation-(11) integral at the reconstructed predecessor coordinates. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
s :
θ :
Mechanism n G
W :
ObservedWorld G θ
hpos :
hmix :
hone :
hsign :
FixedOwnDerivativeSign G s θ
order :
Fin n → ℕ
horder :
hgraphOrder :
PermutedGraphOrdered W order
i :
Fin n
ell :
range (predecessorScoreMap W order i)
t :
ℝ
conclusion
= equationElevenConditionalRatioCDF θ (W.targetPerm i) t (predecessorCubeLatentState W order i ((predecessorScoreHomeomorph s W hpos hmix hone hsign horder hgraphOrder i).symm ell))
Proof (Lean source)
lemma equationElevenScoreImageKernel_apply_Iic
{n : ℕ} {G : DAG (Fin n)} (s : SignVector n)
{θ : Mechanism n G} (W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W)
(hsign : FixedOwnDerivativeSign G s θ)
{order : Fin n → ℕ}
(horder : IsTopologicalOrdering (observedLawRatioGraph gaussianFeatureMap W.law) order)
(hgraphOrder : PermutedGraphOrdered W order) (i : Fin n)
(ell : range (predecessorScoreMap W order i)) (t : ℝ) :
((equationElevenScoreImageKernel s W hpos hmix hone hsign horder hgraphOrder i)
ell (Iic t)).toReal =
equationElevenConditionalRatioCDF θ (W.targetPerm i) t
(predecessorCubeLatentState W order i
((predecessorScoreHomeomorph s W hpos hmix hone hsign horder hgraphOrder i).symm ell)) := by
rw [equationElevenScoreImageKernel, Kernel.comap_apply]
exact equationElevenPredecessorKernel_apply_Iic W hpos order i _ t
theorem equationElevenScoreImageKernel_apply_Iic_of_latentState
At every predecessor score realized by a latent cube point, the transported Markov kernel has exactly the equation-(11) lower-interval probability at that latent state. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
s :
θ :
Mechanism n G
W :
ObservedWorld G θ
hpos :
hmix :
hone :
hsign :
FixedOwnDerivativeSign G s θ
order :
Fin n → ℕ
horder :
hgraphOrder :
PermutedGraphOrdered W order
i :
Fin n
v :
hv :
v ∈ latentCube n
t :
ℝ
conclusion
ell :
⟨familyProjection (observedLawLogRatio W.law) (predecessorSet order i) (W.mix v), observedPredecessorLogRatio_mem_scoreRange W hpos hmix hone horder hgraphOrder i v hv⟩ (: range (predecessorScoreMap W order i))
= equationElevenConditionalRatioCDF θ (W.targetPerm i) t v
Proof (Lean source)
lemma equationElevenScoreImageKernel_apply_Iic_of_latentState
{n : ℕ} {G : DAG (Fin n)} (s : SignVector n)
{θ : Mechanism n G} (W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W)
(hsign : FixedOwnDerivativeSign G s θ)
{order : Fin n → ℕ}
(horder : IsTopologicalOrdering (observedLawRatioGraph gaussianFeatureMap W.law) order)
(hgraphOrder : PermutedGraphOrdered W order)
(i : Fin n) (v : LatentState n) (hv : v ∈ latentCube n) (t : ℝ) :
let ell : range (predecessorScoreMap W order i) :=
⟨familyProjection (observedLawLogRatio W.law) (predecessorSet order i) (W.mix v),
observedPredecessorLogRatio_mem_scoreRange W hpos hmix hone horder hgraphOrder i v hv⟩
((equationElevenScoreImageKernel s W hpos hmix hone hsign horder hgraphOrder i)
ell (Iic t)).toReal =
equationElevenConditionalRatioCDF θ (W.targetPerm i) t v := by
dsimp only
let z := predecessorLatentCubeOfState W order i v hv
let e := predecessorScoreHomeomorph s W hpos hmix hone hsign horder hgraphOrder i
let ell : range (predecessorScoreMap W order i) :=
⟨familyProjection (observedLawLogRatio W.law) (predecessorSet order i) (W.mix v),
observedPredecessorLogRatio_mem_scoreRange W hpos hmix hone horder hgraphOrder i v hv⟩
have hinv : e.symm ell = z := by
apply e.injective
rw [e.apply_symm_apply]
apply Subtype.ext
exact (predecessorScoreMap_of_latentState W hpos hmix hone horder hgraphOrder i v hv).symm
rw [equationElevenScoreImageKernel_apply_Iic s W hpos hmix hone hsign horder hgraphOrder i ell t]
apply equationElevenConditionalRatioCDF_eq_of_parents_eq θ (W.targetPerm i) t
intro a ha
let k := W.targetPerm.symm a
have hkParent : k ∈ environmentParentSet W i := by
simpa [k, environmentParentSet, parents] using ha
have hkPred := environmentParentSet_subset_predecessorSet_of_transitiveClosure
W horder hgraphOrder i hkParent
rw [hinv]
simpa [z, k] using predecessorCubeLatentState_ofState_apply W order i v hv k hkPred
def predecessorScoreRangeRetraction
A measurable retraction of the ambient predecessor-score space onto the compact realized score image. Off the image it uses the score of the zero predecessor vector.
Definition (Lean source)
parameters
def
given by
by classical exact if h : ell ∈ range (predecessorScoreMap W order i) then ⟨ell, h⟩ else ⟨predecessorScoreMap W order i (fun _ => ⟨0, by norm_num⟩), ⟨fun _
=> ⟨0, by norm_num⟩, rfl⟩⟩
theorem measurable_predecessorScoreRangeRetraction
The compact-score retraction is measurable. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
Measurable (predecessorScoreRangeRetraction W order i)
Proof (Lean source)
@[fun_prop]
lemma measurable_predecessorScoreRangeRetraction
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) (hpos : PositiveNormalizedSmoothMechanisms G θ)
(order : Fin n → ℕ) (i : Fin n) :
Measurable (predecessorScoreRangeRetraction W order i) := by
classical
have hclosed : IsClosed (range (predecessorScoreMap W order i)) :=
by simpa only [Set.image_univ] using
(isCompact_univ.image (continuous_predecessorScoreMap W hpos order i)).isClosed
unfold predecessorScoreRangeRetraction
apply Measurable.dite measurable_id measurable_const hclosed.measurableSet
def equationElevenAmbientKernel
The equation-(11) kernel on the full predecessor-score space, obtained by a measurable retraction to the compact realized score image. Its off-support values are immaterial.
Definition (Lean source)
parameters
s :
θ :
Mechanism n G
W :
ObservedWorld G θ
hpos :
hmix :
hone :
hsign :
FixedOwnDerivativeSign G s θ
order :
Fin n → ℕ
horder :
hgraphOrder :
PermutedGraphOrdered W order
i :
Fin n
def
equationElevenAmbientKernel s W hpos hmix hone hsign horder hgraphOrder i :
Kernel (PredecessorLogRatios order i) ℝ
given by
(equationElevenScoreImageKernel s W hpos hmix hone hsign horder hgraphOrder i).comap (predecessorScoreRangeRetraction W order i) (measurable_predecessorScoreRangeRetraction W hpos order i)
instance equationElevenAmbientKernel_isMarkov
The ambient extension of the equation-(11) kernel remains Markov.
Definition (Lean source)
instance equationElevenAmbientKernel_isMarkov
{n : ℕ} {G : DAG (Fin n)} (s : SignVector n)
{θ : Mechanism n G} (W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W)
(hsign : FixedOwnDerivativeSign G s θ)
{order : Fin n → ℕ}
(horder : IsTopologicalOrdering (observedLawRatioGraph gaussianFeatureMap W.law) order)
(hgraphOrder : PermutedGraphOrdered W order) (i : Fin n) :
IsMarkovKernel
(equationElevenAmbientKernel s W hpos hmix hone hsign horder hgraphOrder i) := by
exact Kernel.IsMarkovKernel.comap _
(measurable_predecessorScoreRangeRetraction W hpos order i)
theorem equationElevenAmbientKernel_apply_Iic_of_latentState
On every predecessor score realized by a latent cube point, the ambient kernel evaluates to the explicit equation-(11) conditional CDF. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
s :
θ :
Mechanism n G
W :
ObservedWorld G θ
hpos :
hmix :
hone :
hsign :
FixedOwnDerivativeSign G s θ
order :
Fin n → ℕ
horder :
hgraphOrder :
PermutedGraphOrdered W order
i :
Fin n
v :
hv :
v ∈ latentCube n
t :
ℝ
conclusion
((equationElevenAmbientKernel s W hpos hmix hone hsign horder hgraphOrder i) (familyProjection (observedLawLogRatio W.law) (predecessorSet order i) (W.mix v)) (Iic t)).toReal
= equationElevenConditionalRatioCDF θ (W.targetPerm i) t v
Proof (Lean source)
lemma equationElevenAmbientKernel_apply_Iic_of_latentState
{n : ℕ} {G : DAG (Fin n)} (s : SignVector n)
{θ : Mechanism n G} (W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W)
(hsign : FixedOwnDerivativeSign G s θ)
{order : Fin n → ℕ}
(horder : IsTopologicalOrdering (observedLawRatioGraph gaussianFeatureMap W.law) order)
(hgraphOrder : PermutedGraphOrdered W order)
(i : Fin n) (v : LatentState n) (hv : v ∈ latentCube n) (t : ℝ) :
((equationElevenAmbientKernel s W hpos hmix hone hsign horder hgraphOrder i)
(familyProjection (observedLawLogRatio W.law) (predecessorSet order i) (W.mix v))
(Iic t)).toReal =
equationElevenConditionalRatioCDF θ (W.targetPerm i) t v := by
let ell := familyProjection (observedLawLogRatio W.law) (predecessorSet order i) (W.mix v)
have hell : ell ∈ range (predecessorScoreMap W order i) :=
observedPredecessorLogRatio_mem_scoreRange W hpos hmix hone horder hgraphOrder i v hv
rw [equationElevenAmbientKernel, Kernel.comap_apply]
have hretract : predecessorScoreRangeRetraction W order i ell = ⟨ell, hell⟩ := by
simp [predecessorScoreRangeRetraction, hell]
rw [hretract]
exact equationElevenScoreImageKernel_apply_Iic_of_latentState
s W hpos hmix hone hsign horder hgraphOrder i v hv t
Integrating out coordinates on which a measurable function is invariant leaves the function unchanged when every coordinate reference measure has unit mass. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
V :
∀ i, MeasurableSpace (X i)
μ :
∀ i, Measure (X i)
∀ i, IsProbabilityMeasure (μ i)
T :
Finset V
f :
(∀ i, X i) → ℝ≥0∞
hf :
hinv :
∀ k ∈ T, ∀ x y, f (update x k y) = f x
conclusion
(∫⋯∫⁻_T, f ∂μ) = f
Proof (Lean source)
lemma lmarginal_eq_self_of_update_eq
{V : Type*} [DecidableEq V]
{X : V → Type*} [∀ i, MeasurableSpace (X i)]
(μ : ∀ i, Measure (X i)) [∀ i, IsProbabilityMeasure (μ i)]
(T : Finset V) (f : (∀ i, X i) → ℝ≥0∞) (hf : Measurable f)
(hinv : ∀ k ∈ T, ∀ x y, f (update x k y) = f x) :
(∫⋯∫⁻_T, f ∂μ) = f := by
induction T using Finset.induction with
| empty => simp
| @insert k T hk ih =>
rw [MeasureTheory.lmarginal_insert' _ hf hk]
have hinner : (fun x => ∫⁻ y, f (update x k y) ∂μ k) = f := by
funext x
simp_rw [hinv k (mem_insert_self k T) x]
simp
rw [hinner]
exact ih (fun j hj => hinv j (mem_insert_of_mem hj))
theorem equationTen_retainedDensity_update_outside
The factorized retained density from equation (10) is invariant under every coordinate outside the target-plus-predecessor set. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
W :
ObservedWorld G θ
hpos :
order :
Fin n → ℕ
horder :
hgraphOrder :
PermutedGraphOrdered W order
i k :
Fin n
hk :
k ∉ decoderRetainedLatentSet W order i
v :
x :
ℝ
conclusion
Proof (Lean source)
lemma equationTen_retainedDensity_update_outside
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) (hpos : PositiveNormalizedSmoothMechanisms G θ)
{order : Fin n → ℕ}
(horder : IsTopologicalOrdering (observedLawRatioGraph gaussianFeatureMap W.law) order)
(hgraphOrder : PermutedGraphOrdered W order) (i k : Fin n)
(hk : k ∉ decoderRetainedLatentSet W order i)
(v : LatentState n) (x : ℝ) :
let B := mechanismUnitCubeFactorization hpos
let q := mechanismInterventionDensity W hpos i
let A := decoderRetainedLatentSet W order i
q.density ((update v k x) (W.targetPerm i)) *
B.partialDensity (A.erase (W.targetPerm i)) (update v k x) =
q.density (v (W.targetPerm i)) *
B.partialDensity (A.erase (W.targetPerm i)) v := by
dsimp only
have hkt : k ≠ W.targetPerm i := by
intro h
subst k
exact hk (by simp [decoderRetainedLatentSet])
congr 1
· simp [update, hkt.symm]
· unfold partialDensity
apply Finset.prod_congr rfl
intro l hl
apply (mechanismUnitCubeFactorization hpos).local_factor l
intro a ha
have haA : a ∈ decoderRetainedLatentSet W order i := by
rcases Finset.mem_insert.mp ha with rfl | ha
· exact (Finset.mem_erase.mp hl).2
· exact decoderRetainedLatentSet_parentClosed W horder hgraphOrder i
(Finset.mem_erase.mp hl).2 ha
have hka : k ≠ a := fun h => hk (h ▸ haA)
simp [update, hka.symm]
theorem equationTen_retainedMarginalMeasure
Equation (10) as an equality of retained-coordinate measures: the actual intervention law has the same target-plus-predecessor marginal as the explicit replacement-density product. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
W :
ObservedWorld G θ
hpos :
order :
Fin n → ℕ
horder :
hgraphOrder :
PermutedGraphOrdered W order
i :
Fin n
conclusion
Measure.map (coordinateProjection (X := fun _ : Fin n => ℝ) A) (interventionalLaw θ (W.targetPerm i))
= Measure.map (coordinateProjection (X := fun _ : Fin n => ℝ) A) ((Measure.pi (fun _ : Fin n => unitIntervalReference)).withDensity (fun v => q.density (v (W.targetPerm i)) * B.partialDensity (A.erase (W.targetPerm i)) v))
Proof (Lean source)
lemma equationTen_retainedMarginalMeasure
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) (hpos : PositiveNormalizedSmoothMechanisms G θ)
{order : Fin n → ℕ}
(horder : IsTopologicalOrdering (observedLawRatioGraph gaussianFeatureMap W.law) order)
(hgraphOrder : PermutedGraphOrdered W order) (i : Fin n) :
let B := mechanismUnitCubeFactorization hpos
let q := mechanismInterventionDensity W hpos i
let A := decoderRetainedLatentSet W order i
Measure.map (coordinateProjection
(X := fun _ : Fin n => ℝ) A)
(interventionalLaw θ (W.targetPerm i)) =
Measure.map (coordinateProjection
(X := fun _ : Fin n => ℝ) A)
((Measure.pi (fun _ : Fin n =>
Causalean.Graph.FiniteDensity.unitIntervalReference)).withDensity
(fun v => q.density (v (W.targetPerm i)) *
B.partialDensity (A.erase (W.targetPerm i)) v)) := by
dsimp only
let B := mechanismUnitCubeFactorization hpos
let q := mechanismInterventionDensity W hpos i
let A := decoderRetainedLatentSet W order i
have hσ : ∀ _ : Fin n,
SigmaFinite Causalean.Graph.FiniteDensity.unitIntervalReference := fun _ => by
unfold Causalean.Graph.FiniteDensity.unitIntervalReference
infer_instance
have hprob : ∀ _ : Fin n,
IsProbabilityMeasure Causalean.Graph.FiniteDensity.unitIntervalReference := fun _ => by
constructor
unfold Causalean.Graph.FiniteDensity.unitIntervalReference
simp [Real.volume_Icc]
have hBint : Measurable (B.interventionDensity (W.targetPerm i) q) :=
@measurable_interventionDensity
(Fin n) _ _ (fun _ : Fin n => ℝ) _
(fun _ : Fin n => Causalean.Graph.FiniteDensity.unitIntervalReference) hσ G B
(W.targetPerm i) q
have hBpartial : Measurable (B.partialDensity (A.erase (W.targetPerm i))) :=
@measurable_partialDensity
(Fin n) _ _ (fun _ : Fin n => ℝ) _
(fun _ : Fin n => Causalean.Graph.FiniteDensity.unitIntervalReference) hσ G B
(A.erase (W.targetPerm i))
rw [← mechanismUnitCubeFactorization_interventionMeasure W hpos i]
unfold interventionMeasure
apply @map_coordinateProjection_withDensity_eq_of_lmarginal_eq
(Fin n) _ (fun _ : Fin n => ℝ) _ _
(fun _ : Fin n => Causalean.Graph.FiniteDensity.unitIntervalReference) hσ A
· exact hBint
· exact q.measurable_density.comp (measurable_pi_apply _) |>.mul hBpartial
· rw [equationTen_interventionMarginalDensity W hpos horder hgraphOrder i]
symm
apply @lmarginal_eq_self_of_update_eq (Fin n) _ (fun _ : Fin n => ℝ) _
(fun _ : Fin n => Causalean.Graph.FiniteDensity.unitIntervalReference) hprob
(univ \ A)
· exact (q.measurable_density.comp (measurable_pi_apply _)).mul hBpartial
· intro k hk v x
exact equationTen_retainedDensity_update_outside W hpos horder hgraphOrder i k
(show k ∉ decoderRetainedLatentSet W order i from (Finset.mem_sdiff.mp hk).2) v x
theorem rawObservedConditionalRatioCDF_ae_eq_kernel_of_compProd Lemma rawObservedConditionalRatioCDF_ae_eq_kernel_of_compProd in the paper ↗
A candidate finite kernel whose compositional product is the observed predecessor/score law is the raw conditional-ratio CDF at every fixed threshold. This is the conditional-distribution uniqueness step used after equation (10). Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
n :
ℕ
laws :
order :
Fin n → ℕ
i :
Fin n
κ :
Kernel (PredecessorLogRatios order i) ℝ
hjoint :
Measure.map (fun x ↦ (familyProjection (observedLawLogRatio laws.1) (predecessorSet order i) x, observedLawLogRatio laws.1 i x)) (laws.1 i.succ)
= Measure.map (familyProjection (observedLawLogRatio laws.1) (predecessorSet order i)) (laws.1 i.succ) ⊗ₘ κ
t :
conclusion
(fun ell ↦ rawObservedConditionalRatioCDF laws order i t ell)
=ᵐ[ Measure.map (familyProjection (observedLawLogRatio laws.1) (predecessorSet order i)) (laws.1 i.succ)] fun ell ↦ (κ ell (Iic t)).toReal
Proof (Lean source)
lemma rawObservedConditionalRatioCDF_ae_eq_kernel_of_compProd
{n : ℕ}
(laws : ObservedProbabilityLawFamily n) (order : Fin n → ℕ) (i : Fin n)
(κ : Kernel (PredecessorLogRatios order i) ℝ) [IsFiniteKernel κ]
(hjoint : Measure.map
(fun x ↦
(familyProjection (observedLawLogRatio laws.1) (predecessorSet order i) x,
observedLawLogRatio laws.1 i x)) (laws.1 i.succ) =
Measure.map
(familyProjection (observedLawLogRatio laws.1) (predecessorSet order i))
(laws.1 i.succ) ⊗ₘ κ) :
∀ t, (fun ell ↦ rawObservedConditionalRatioCDF laws order i t ell) =ᵐ[
Measure.map
(familyProjection (observedLawLogRatio laws.1) (predecessorSet order i))
(laws.1 i.succ)] fun ell ↦ (κ ell (Iic t)).toReal := by
letI : IsProbabilityMeasure (laws.1 i.succ) := laws.2 i.succ
have hcond := ProbabilityTheory.condDistrib_ae_eq_of_measure_eq_compProd
(familyProjection (observedLawLogRatio laws.1) (predecessorSet order i))
(measurable_observedLawLogRatio laws.1 i).aemeasurable hjoint
intro t
filter_upwards [hcond] with ell hell
have hfinite : ∀ e, IsFiniteMeasure (laws.1 e) := by
intro e
letI : IsProbabilityMeasure (laws.1 e) := laws.2 e
infer_instance
rw [rawObservedConditionalRatioCDF, dif_pos hfinite, hell]
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.DecoderContinuousVersion 11 declarations This file proves continuity of the equation-(11) fiber integral and packages the ambient Markov kernel as the continuous conditional-CDF version selected by the decoder.
Continuous equation-(11) conditional-CDF version
This file proves continuity of the equation-(11) fiber integral and packages the ambient Markov kernel as the continuous conditional-CDF version selected by the decoder.
theorem continuous_equationElevenClampedScore
The clamped equation-(11) score is jointly continuous in predecessor and target coordinates. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
Continuous (equationElevenClampedScore W order i)
Proof (Lean source)
lemma continuous_equationElevenClampedScore
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) (hpos : PositiveNormalizedSmoothMechanisms G θ)
(order : Fin n → ℕ) (i : Fin n) : Continuous (equationElevenClampedScore W order i) := by
have hw : Continuous (fun zw : PredecessorLatentCube order i × ℝ =>
max 0 (min 1 zw.2)) := by fun_prop
have hwmem : ∀ zw : PredecessorLatentCube order i × ℝ,
max 0 (min 1 zw.2) ∈ Icc (0 : ℝ) 1 := by
intro zw
exact ⟨le_max_left _ _, max_le zero_le_one (min_le_left _ _)⟩
have hbase : Continuous (predecessorCubeLatentState W order i) := by
apply continuous_pi
intro a
simp only [predecessorCubeLatentState]
split <;> fun_prop
have hv : Continuous (fun zw : PredecessorLatentCube order i × ℝ =>
update (predecessorCubeLatentState W order i zw.1)
(W.targetPerm i) (max 0 (min 1 zw.2))) := by
apply continuous_pi
intro k
by_cases hki : k = W.targetPerm i
· subst k
simpa using hw
· have heq : (fun a : PredecessorLatentCube order i × ℝ =>
update (predecessorCubeLatentState W order i a.1)
(W.targetPerm i) (max 0 (min 1 a.2)) k) =
fun a => predecessorCubeLatentState W order i a.1 k := by
funext a
simp [update, hki]
rw [heq]
exact (continuous_apply k).comp (hbase.comp continuous_fst)
have hvmem : ∀ zw : PredecessorLatentCube order i × ℝ,
update (predecessorCubeLatentState W order i zw.1)
(W.targetPerm i) (max 0 (min 1 zw.2)) ∈ latentCube n := by
intro zw k hk
by_cases hki : k = W.targetPerm i
· subst k
simpa using hwmem zw
· simp [update, hki]
exact predecessorCubeLatentState_mem_latentCube W order i zw.1 k (Set.mem_univ k)
have hq := (hpos.2.2.2.1 (W.targetPerm i)).continuousOn.comp_continuous hw hwmem
have hp := (hpos.2.2.1 (W.targetPerm i)).continuousOn.comp_continuous hv hvmem
have hpne : ∀ zw : PredecessorLatentCube order i × ℝ, θ.p (W.targetPerm i)
(update (predecessorCubeLatentState W order i zw.1)
(W.targetPerm i) (max 0 (min 1 zw.2))) ≠ 0 := fun zw =>
ne_of_gt (hpos.1 _ _ (hvmem zw))
have hqne : ∀ zw : PredecessorLatentCube order i × ℝ,
θ.q (W.targetPerm i) (max 0 (min 1 zw.2)) ≠ 0 := fun zw =>
ne_of_gt (hpos.2.1 _ _ (hwmem zw))
exact (hq.div hp hpne).log (fun zw => div_ne_zero (hqne zw) (hpne zw))
def equationElevenPredecessorCDF
The equation-(11) fiber integral on the compact predecessor cube.
Definition (Lean source)
parameters
def
equationElevenPredecessorCDF W order i z :
ℝ
given by
∫ w in Icc (0 : ℝ) 1, if equationElevenClampedScore W order i (z.2, w)
≤ z.1 then θ.q (W.targetPerm i) w else 0
theorem continuous_equationElevenPredecessorCDF
The equation-(11) fiber integral varies continuously with both its threshold and predecessor coordinates. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
s :
θ :
Mechanism n G
W :
ObservedWorld G θ
hpos :
hsign :
FixedOwnDerivativeSign G s θ
order :
Fin n → ℕ
i :
Fin n
conclusion
Continuous (equationElevenPredecessorCDF W order i)
Proof (Lean source)
lemma continuous_equationElevenPredecessorCDF
{n : ℕ} {G : DAG (Fin n)} (s : SignVector n)
{θ : Mechanism n G} (W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hsign : FixedOwnDerivativeSign G s θ)
(order : Fin n → ℕ) (i : Fin n) :
Continuous (equationElevenPredecessorCDF W order i) := by
let μ := volume.restrict (Icc (0 : ℝ) 1)
let F : (ℝ × PredecessorLatentCube order i) → ℝ → ℝ := fun z w =>
if equationElevenClampedScore W order i (z.2, w) ≤ z.1
then θ.q (W.targetPerm i) w else 0
have hqcont : ContinuousOn (θ.q (W.targetPerm i)) (Icc (0 : ℝ) 1) :=
(hpos.2.2.2.1 (W.targetPerm i)).continuousOn
have hqint : Integrable (fun w => ‖θ.q (W.targetPerm i) w‖) μ := by
exact (hqcont.norm.integrableOn_compact isCompact_Icc)
rw [continuous_iff_continuousAt]
intro z
have hmono : StrictMonoOn
(fun w => equationElevenClampedScore W order i (z.2, w))
(Icc (0 : ℝ) 1) ∨
StrictAntiOn (fun w => equationElevenClampedScore W order i (z.2, w))
(Icc (0 : ℝ) 1) := by
let v := predecessorCubeLatentState W order i z.2
have hv : v ∈ latentCube n := predecessorCubeLatentState_mem_latentCube W order i z.2
rcases s.signed (W.targetPerm i) with hneg | hposi
· right
have h := fixedOwnDerivativeSign_strictAntiOn s θ hpos hsign
(W.targetPerm i) v hv hneg
apply h.congr
intro w hw
simp only [equationElevenClampedScore]
rw [min_eq_right hw.2, max_eq_right hw.1]
· left
have h := fixedOwnDerivativeSign_strictMonoOn s θ hpos hsign
(W.targetPerm i) v hv hposi
apply h.congr
intro w hw
simp only [equationElevenClampedScore]
rw [min_eq_right hw.2, max_eq_right hw.1]
have hlevel : Subsingleton {w : ℝ | w ∈ Icc (0 : ℝ) 1 ∧
equationElevenClampedScore W order i (z.2, w) = z.1} := by
intro a ha b hb
rcases hmono with hm | hm
· apply hm.injOn ha.1 hb.1
calc
equationElevenClampedScore W order i (z.2, a) = z.1 := ha.2
_ = equationElevenClampedScore W order i (z.2, b) := hb.2.symm
· apply hm.injOn ha.1 hb.1
calc
equationElevenClampedScore W order i (z.2, a) = z.1 := ha.2
_ = equationElevenClampedScore W order i (z.2, b) := hb.2.symm
have hne : ∀ᵐ w ∂μ,
equationElevenClampedScore W order i (z.2, w) ≠ z.1 := by
rw [MeasureTheory.ae_restrict_iff' measurableSet_Icc, MeasureTheory.ae_iff]
have hset : {w | ¬(w ∈ Icc (0 : ℝ) 1 →
equationElevenClampedScore W order i (z.2, w) ≠ z.1)} =
{w | w ∈ Icc (0 : ℝ) 1 ∧
equationElevenClampedScore W order i (z.2, w) = z.1} := by
ext w
by_cases hwmem : w ∈ Icc (0 : ℝ) 1 <;>
by_cases heq : equationElevenClampedScore W order i (z.2, w) = z.1 <;>
simp [hwmem, heq]
rw [hset]
exact hlevel.measure_zero volume
apply MeasureTheory.continuousAt_of_dominated
(F := F) (bound := fun w => ‖θ.q (W.targetPerm i) w‖)
· filter_upwards with z'
have hscore : Measurable (fun w => equationElevenClampedScore W order i (z'.2, w)) :=
(continuous_equationElevenClampedScore W hpos order i).measurable.comp
(measurable_const.prodMk measurable_id)
have hq : AEMeasurable (θ.q (W.targetPerm i)) μ :=
hqcont.aemeasurable measurableSet_Icc
let qm := hq.mk (θ.q (W.targetPerm i))
have hm : Measurable (fun w =>
if equationElevenClampedScore W order i (z'.2, w) ≤ z'.1
then qm w else 0) :=
Measurable.ite (measurableSet_le hscore measurable_const)
hq.measurable_mk measurable_const
apply hm.aestronglyMeasurable.congr
filter_upwards [hq.ae_eq_mk] with w hw
simp only [F, qm]
rw [← hw]
· filter_upwards with z'
filter_upwards with w
by_cases h : equationElevenClampedScore W order i (z'.2, w) ≤ z'.1
· simp [F, h]
· simp [F, h]
· exact hqint
· filter_upwards [hne] with w hw
have hscore : ContinuousAt
(fun z' : ℝ × PredecessorLatentCube order i =>
equationElevenClampedScore W order i (z'.2, w)) z :=
(continuous_equationElevenClampedScore W hpos order i).continuousAt.comp
(continuousAt_snd.prodMk continuousAt_const)
rcases lt_or_gt_of_ne hw with hlt | hgt
· have hev := hscore.eventually_lt continuousAt_fst hlt
have hc : ContinuousAt
(fun _ : ℝ × PredecessorLatentCube order i => θ.q (W.targetPerm i) w) z :=
continuousAt_const
apply hc.congr_of_eventuallyEq
filter_upwards [hev] with z' hz'
simp [F, hz'.le]
· have hev := continuousAt_fst.eventually_lt hscore hgt
have hc : ContinuousAt (fun _ : ℝ × PredecessorLatentCube order i => (0 : ℝ)) z :=
continuousAt_const
apply hc.congr_of_eventuallyEq
filter_upwards [hev] with z' hz'
simp [F, not_le_of_gt hz']
theorem equationElevenPredecessorCDF_eq
The compact fiber CDF is the explicit equation-(11) integral. the stated conclusion follows.
Formal statement
hypotheses
conclusion
equationElevenPredecessorCDF W order i (t, z)
= equationElevenConditionalRatioCDF θ (W.targetPerm i) t (predecessorCubeLatentState W order i z)
Proof (Lean source)
lemma equationElevenPredecessorCDF_eq
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) (order : Fin n → ℕ) (i : Fin n)
(t : ℝ) (z : PredecessorLatentCube order i) :
equationElevenPredecessorCDF W order i (t, z) =
equationElevenConditionalRatioCDF θ (W.targetPerm i) t
(predecessorCubeLatentState W order i z) := by
unfold equationElevenPredecessorCDF equationElevenConditionalRatioCDF
apply MeasureTheory.integral_congr_ae
filter_upwards [ae_restrict_mem measurableSet_Icc] with w hw
simp only [equationElevenClampedScore]
have hclamp : max 0 (min 1 w) = w := by
simp [min_eq_right hw.2, max_eq_right hw.1]
simp [hclamp]
def equationElevenAmbientCDF
The ambient equation-(11) Markov kernel evaluated on lower intervals, packaged as a unit-interval-valued conditional CDF.
Definition (Lean source)
parameters
s :
θ :
Mechanism n G
W :
ObservedWorld G θ
hpos :
hmix :
hone :
hsign :
FixedOwnDerivativeSign G s θ
order :
Fin n → ℕ
horder :
hgraphOrder :
PermutedGraphOrdered W order
i :
Fin n
t :
ℝ
ell :
PredecessorLogRatios order i
def
given by
by let κ := equationElevenAmbientKernel s W hpos hmix hone hsign horder hgraphOrder i let y := (κ ell (Iic t)).toReal have hle : κ ell (Iic t)
theorem measurable_equationElevenAmbientCDF
The ambient equation-(11) conditional CDF is jointly measurable. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
s :
θ :
Mechanism n G
W :
ObservedWorld G θ
hpos :
hmix :
hone :
hsign :
FixedOwnDerivativeSign G s θ
order :
Fin n → ℕ
horder :
hgraphOrder :
PermutedGraphOrdered W order
i :
Fin n
conclusion
Measurable (fun z : ℝ × PredecessorLogRatios order i => equationElevenAmbientCDF s W hpos hmix hone hsign horder hgraphOrder i z.1 z.2)
Proof (Lean source)
lemma measurable_equationElevenAmbientCDF
{n : ℕ} {G : DAG (Fin n)} (s : SignVector n)
{θ : Mechanism n G} (W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W)
(hsign : FixedOwnDerivativeSign G s θ)
{order : Fin n → ℕ}
(horder : IsTopologicalOrdering (observedLawRatioGraph gaussianFeatureMap W.law) order)
(hgraphOrder : PermutedGraphOrdered W order) (i : Fin n) :
Measurable (fun z : ℝ × PredecessorLogRatios order i =>
equationElevenAmbientCDF s W hpos hmix hone hsign horder hgraphOrder i z.1 z.2) := by
apply Measurable.subtype_mk
exact ENNReal.measurable_toReal.comp
(measurable_kernel_Iic_uncurry
(equationElevenAmbientKernel s W hpos hmix hone hsign horder hgraphOrder i))
theorem continuousOn_equationElevenAmbientCDF_scoreRange
On the realized predecessor-score range, the ambient equation-(11) CDF is continuous. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
s :
θ :
Mechanism n G
W :
ObservedWorld G θ
hpos :
hmix :
hone :
hsign :
FixedOwnDerivativeSign G s θ
order :
Fin n → ℕ
horder :
hgraphOrder :
PermutedGraphOrdered W order
i :
Fin n
conclusion
ContinuousOn (fun z : ℝ × PredecessorLogRatios order i => (equationElevenAmbientCDF s W hpos hmix hone hsign horder hgraphOrder i z.1 z.2 : ℝ)) (univ ×ˢ range (predecessorScoreMap W order i))
Proof (Lean source)
lemma continuousOn_equationElevenAmbientCDF_scoreRange
{n : ℕ} {G : DAG (Fin n)} (s : SignVector n)
{θ : Mechanism n G} (W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W)
(hsign : FixedOwnDerivativeSign G s θ)
{order : Fin n → ℕ}
(horder : IsTopologicalOrdering (observedLawRatioGraph gaussianFeatureMap W.law) order)
(hgraphOrder : PermutedGraphOrdered W order) (i : Fin n) :
ContinuousOn (fun z : ℝ × PredecessorLogRatios order i =>
(equationElevenAmbientCDF s W hpos hmix hone hsign horder hgraphOrder i z.1 z.2 : ℝ))
(univ ×ˢ range (predecessorScoreMap W order i)) := by
classical
let R := range (predecessorScoreMap W order i)
let r := predecessorScoreRangeRetraction W order i
let e := predecessorScoreHomeomorph s W hpos hmix hone hsign horder hgraphOrder i
have hr : ContinuousOn r R := by
rw [continuousOn_iff_continuous_restrict]
have hre : R.domRestrict r = id := by
funext ell
apply Subtype.ext
simp [R, r, predecessorScoreRangeRetraction, ell.property]
rw [hre]
exact continuous_id
let H : ℝ × PredecessorLogRatios order i →
ℝ × PredecessorLatentCube order i := fun z => (z.1, e.symm (r z.2))
have hH : ContinuousOn H (univ ×ˢ R) := by
apply ContinuousOn.prodMk continuousOn_fst
apply e.symm.continuous.comp_continuousOn
apply hr.comp continuousOn_snd
intro z hz
exact hz.2
have hcont : ContinuousOn (equationElevenPredecessorCDF W order i ∘ H)
(univ ×ˢ R) :=
(continuous_equationElevenPredecessorCDF s W hpos hsign order i).comp_continuousOn hH
apply hcont.congr
intro z hz
have hzrange : z.2 ∈ range (predecessorScoreMap W order i) := hz.2
let ell : R := ⟨z.2, hzrange⟩
have hretract : r z.2 = ell := by
apply Subtype.ext
change (predecessorScoreRangeRetraction W order i z.2).1 = z.2
simp [predecessorScoreRangeRetraction, hzrange]
have hinv : e.symm ell = e.symm (r z.2) := by rw [hretract]
change (equationElevenAmbientKernel s W hpos hmix hone hsign horder hgraphOrder i
z.2 (Iic z.1)).toReal = equationElevenPredecessorCDF W order i (H z)
rw [equationElevenAmbientKernel, Kernel.comap_apply,
show predecessorScoreRangeRetraction W order i z.2 = ell from hretract,
equationElevenScoreImageKernel, Kernel.comap_apply]
change ((equationElevenPredecessorKernel W hpos order i)
(e.symm ell) (Iic z.1)).toReal = _
rw [equationElevenPredecessorKernel_apply_Iic]
rw [equationElevenPredecessorCDF_eq]
exact equationElevenConditionalRatioCDF_eq_of_parents_eq θ (W.targetPerm i) z.1
(fun _ _ => congrFun
(congrArg (fun z0 => predecessorCubeLatentState W order i z0) hinv) _)
theorem observedConditionalRatioSupport_subset_scoreRange
The observed joint ratio/predecessor support has predecessor component in the compact triangular score range. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
W :
ObservedWorld G θ
hpos :
hmix :
hone :
order :
Fin n → ℕ
horder :
hgraphOrder :
PermutedGraphOrdered W order
i :
Fin n
conclusion
Proof (Lean source)
lemma observedConditionalRatioSupport_subset_scoreRange
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W)
{order : Fin n → ℕ}
(horder : IsTopologicalOrdering (observedLawRatioGraph gaussianFeatureMap W.law) order)
(hgraphOrder : PermutedGraphOrdered W order) (i : Fin n) :
observedConditionalRatioSupport W.law order i ⊆
univ ×ˢ range (predecessorScoreMap W order i) := by
let μ := interventionalLaw θ (W.targetPerm i)
let pred := familyProjection (observedLawLogRatio W.law) (predecessorSet order i)
let R := range (predecessorScoreMap W order i)
have hclosedRange : IsClosed (range (predecessorScoreMap W order i)) := by
simpa only [Set.image_univ] using
(isCompact_univ.image (continuous_predecessorScoreMap W hpos order i)).isClosed
have hcube : ∀ᵐ v ∂μ, v ∈ latentCube n := by
filter_upwards [Measure.support_mem_ae (μ := μ)] with v hv
simpa [μ, interventionalLaw_support_eq_latentCube W hpos i] using hv
have hmixAE : AEMeasurable W.mix μ := by
apply aemeasurable_of_supportMeasurableOn (by
rw [latentCube]
exact MeasurableSet.pi Set.countable_univ (fun _ _ => measurableSet_Icc))
hcube hmix.1.continuousOn.domRestrict.measurable
have hpred : Measurable pred := by
dsimp only [pred]
apply measurable_pi_lambda
intro j
exact measurable_observedLawLogRatio W.law j
have hcomp : AEMeasurable (pred ∘ W.mix) μ :=
hpred.aemeasurable.comp_aemeasurable hmixAE
have hRae : R ∈ ae (Measure.map (pred ∘ W.mix) μ) := by
refine (MeasureTheory.ae_map_iff hcomp (p := fun y => y ∈ R)
hclosedRange.measurableSet).2 ?_
filter_upwards [hcube] with v hv
exact observedPredecessorLogRatio_mem_scoreRange
W hpos hmix hone horder hgraphOrder i v hv
have hsupp : Measure.support (Measure.map pred (W.law i.succ)) ⊆ R := by
apply Measure.support_subset_of_isClosed hclosedRange
rw [hone.2.1 i]
rw [AEMeasurable.map_map_of_aemeasurable hpred.aemeasurable hmixAE]
exact hRae
rw [observedConditionalRatioSupport]
intro z hz
exact ⟨Set.mem_univ _, hsupp hz.2⟩
theorem rawObservedConditionalRatioCDF_joint_ae_eq_kernel_of_compProd Lemma rawObservedConditionalRatioCDF_joint_ae_eq_kernel_of_compProd in the paper ↗
Conditional-kernel uniqueness holds jointly under the observed threshold/predecessor law, not merely separately at each fixed threshold. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
n :
ℕ
laws :
order :
Fin n → ℕ
i :
Fin n
κ :
Kernel (PredecessorLogRatios order i) ℝ
hjoint :
Measure.map (fun x ↦ (familyProjection (observedLawLogRatio laws.1) (predecessorSet order i) x, observedLawLogRatio laws.1 i x)) (laws.1 i.succ)
= Measure.map (familyProjection (observedLawLogRatio laws.1) (predecessorSet order i)) (laws.1 i.succ) ⊗ₘ κ
conclusion
=ᵐ[ Measure.map (conditionalRatioArgument laws.1 order i) (laws.1 i.succ)] (fun z => rawObservedConditionalRatioCDF laws order i z.1 z.2)
Proof (Lean source)
lemma rawObservedConditionalRatioCDF_joint_ae_eq_kernel_of_compProd
{n : ℕ}
(laws : ObservedProbabilityLawFamily n) (order : Fin n → ℕ) (i : Fin n)
(κ : Kernel (PredecessorLogRatios order i) ℝ) [IsFiniteKernel κ]
(hjoint : Measure.map
(fun x ↦
(familyProjection (observedLawLogRatio laws.1) (predecessorSet order i) x,
observedLawLogRatio laws.1 i x)) (laws.1 i.succ) =
Measure.map
(familyProjection (observedLawLogRatio laws.1) (predecessorSet order i))
(laws.1 i.succ) ⊗ₘ κ) :
(fun z : ℝ × PredecessorLogRatios order i => (κ z.2 (Iic z.1)).toReal) =ᵐ[
Measure.map (conditionalRatioArgument laws.1 order i) (laws.1 i.succ)]
(fun z => rawObservedConditionalRatioCDF laws order i z.1 z.2) := by
letI : IsProbabilityMeasure (laws.1 i.succ) := laws.2 i.succ
let pred := familyProjection (observedLawLogRatio laws.1) (predecessorSet order i)
let score := observedLawLogRatio laws.1 i
let predLaw := Measure.map pred (laws.1 i.succ)
have hcond := ProbabilityTheory.condDistrib_ae_eq_of_measure_eq_compProd
pred (measurable_observedLawLogRatio laws.1 i).aemeasurable hjoint
have hall : ∀ᵐ ell ∂predLaw, ∀ t,
(κ ell (Iic t)).toReal = rawObservedConditionalRatioCDF laws order i t ell := by
filter_upwards [hcond] with ell hell
intro t
have hfinite : ∀ e, IsFiniteMeasure (laws.1 e) := by
intro e
letI : IsProbabilityMeasure (laws.1 e) := laws.2 e
infer_instance
rw [rawObservedConditionalRatioCDF, dif_pos hfinite, hell]
let joint := Measure.map (conditionalRatioArgument laws.1 order i) (laws.1 i.succ)
have hmap : Measure.map snd joint = predLaw := by
dsimp only [joint, predLaw, pred]
rw [Measure.map_map measurable_snd
(measurable_conditionalRatioArgument laws.1 order i)]
rfl
rw [← hmap] at hall
have hpull := MeasureTheory.mem_ae_of_mem_ae_map measurable_snd.aemeasurable hall
filter_upwards [hpull] with z hz
exact hz z.1
theorem equationElevenAmbientCDF_isContinuousVersion
The ambient equation-(11) CDF is a continuous conditional-distribution version for the observed intervention law. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
s :
θ :
Mechanism n G
W :
ObservedWorld G θ
hpos :
hmix :
hone :
hsign :
FixedOwnDerivativeSign G s θ
order :
Fin n → ℕ
horder :
hgraphOrder :
PermutedGraphOrdered W order
i :
Fin n
conclusion
IsContinuousConditionalRatioCDFVersion (observedProbabilityLawFamily W.law) order i (equationElevenAmbientCDF s W hpos hmix hone hsign horder hgraphOrder i)
Proof (Lean source)
lemma equationElevenAmbientCDF_isContinuousVersion
{n : ℕ} {G : DAG (Fin n)} (s : SignVector n)
{θ : Mechanism n G} (W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W)
(hsign : FixedOwnDerivativeSign G s θ)
{order : Fin n → ℕ}
(horder : IsTopologicalOrdering (observedLawRatioGraph gaussianFeatureMap W.law) order)
(hgraphOrder : PermutedGraphOrdered W order) (i : Fin n) :
IsContinuousConditionalRatioCDFVersion (observedProbabilityLawFamily W.law) order i
(equationElevenAmbientCDF s W hpos hmix hone hsign horder hgraphOrder i) := by
let laws := observedProbabilityLawFamily W.law
let κ := equationElevenAmbientKernel s W hpos hmix hone hsign horder hgraphOrder i
have hprob : ∀ e, IsProbabilityMeasure (W.law e) :=
observedWorld_laws_isProbabilityMeasure W hpos hmix hone
have hlaws : laws.1 = W.law := by
simp [laws, observedProbabilityLawFamily, hprob]
have hjoint := observedRatioPredecessor_compProd
s W hpos hmix hone hsign horder hgraphOrder i
have hjoint' : Measure.map
(fun x ↦
(familyProjection (observedLawLogRatio laws.1) (predecessorSet order i) x,
observedLawLogRatio laws.1 i x)) (laws.1 i.succ) =
Measure.map
(familyProjection (observedLawLogRatio laws.1) (predecessorSet order i))
(laws.1 i.succ) ⊗ₘ κ := by
simpa only [hlaws] using hjoint
have hfixed := rawObservedConditionalRatioCDF_ae_eq_kernel_of_compProd
laws order i κ hjoint'
have hjointAE := rawObservedConditionalRatioCDF_joint_ae_eq_kernel_of_compProd
laws order i κ hjoint'
have hsupp : observedConditionalRatioSupport laws.1 order i ⊆
univ ×ˢ range (predecessorScoreMap W order i) := by
rw [hlaws]
exact observedConditionalRatioSupport_subset_scoreRange
W hpos hmix hone horder hgraphOrder i
refine ⟨?_, ?_, ?_, ?_⟩
· exact measurable_subtype_coe.comp
(measurable_equationElevenAmbientCDF
s W hpos hmix hone hsign horder hgraphOrder i)
· intro t
simpa [κ, equationElevenAmbientCDF] using (hfixed t).symm
· simpa [κ, equationElevenAmbientCDF] using hjointAE
· exact (continuousOn_equationElevenAmbientCDF_scoreRange
s W hpos hmix hone hsign horder hgraphOrder i).mono
hsupp
theorem equationElevenAmbientCDF_apply_of_latentState
On every realized predecessor score, the continuous version evaluates to the explicit equation-(11) integral. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
s :
θ :
Mechanism n G
W :
ObservedWorld G θ
hpos :
hmix :
hone :
hsign :
FixedOwnDerivativeSign G s θ
order :
Fin n → ℕ
horder :
hgraphOrder :
PermutedGraphOrdered W order
i :
Fin n
t :
ℝ
v :
hv :
v ∈ latentCube n
conclusion
(equationElevenAmbientCDF s W hpos hmix hone hsign horder hgraphOrder i t (familyProjection (observedLawLogRatio W.law) (predecessorSet order i) (W.mix v)) : ℝ)
= equationElevenConditionalRatioCDF θ (W.targetPerm i) t v
Proof (Lean source)
lemma equationElevenAmbientCDF_apply_of_latentState
{n : ℕ} {G : DAG (Fin n)} (s : SignVector n)
{θ : Mechanism n G} (W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W)
(hsign : FixedOwnDerivativeSign G s θ)
{order : Fin n → ℕ}
(horder : IsTopologicalOrdering (observedLawRatioGraph gaussianFeatureMap W.law) order)
(hgraphOrder : PermutedGraphOrdered W order)
(i : Fin n) (t : ℝ) (v : LatentState n) (hv : v ∈ latentCube n) :
(equationElevenAmbientCDF s W hpos hmix hone hsign horder hgraphOrder i t
(familyProjection (observedLawLogRatio W.law) (predecessorSet order i) (W.mix v)) : ℝ) =
equationElevenConditionalRatioCDF θ (W.targetPerm i) t v := by
exact equationElevenAmbientKernel_apply_Iic_of_latentState
s W hpos hmix hone hsign horder hgraphOrder i v hv t
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.DecoderCore 14 declarations This file collects paper-local consequences of the primitive observed-world assumptions that feed directly into the exact population decoder.
Structural pieces of the exact population decoder
This file collects paper-local consequences of the primitive observed-world assumptions that feed directly into the exact population decoder.
theorem interventionalLaw_support_eq_latentCube
Strict positivity of every retained factor and the replacement density gives each latent single-target intervention law the full closed cube as its support. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
Measure.support (interventionalLaw θ (W.targetPerm e)) = latentCube n
Proof (Lean source)
lemma interventionalLaw_support_eq_latentCube
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) (hpos : PositiveNormalizedSmoothMechanisms G θ) (e : Fin n) :
Measure.support (interventionalLaw θ (W.targetPerm e)) = latentCube n := by
letI : SigmaFinite unitIntervalReference := by
unfold unitIntervalReference
infer_instance
let B := mechanismUnitCubeFactorization hpos
let q := mechanismInterventionDensity W hpos e
let μ := unitCubeReference (Fin n)
have hint_μ : interventionalLaw θ (W.targetPerm e) ≪ μ := by
rw [← mechanismUnitCubeFactorization_interventionMeasure W hpos e]
exact withDensity_absolutelyContinuous _ _
have hdens_ne : ∀ v, B.interventionDensity (W.targetPerm e) q v ≠ 0 := by
intro v
apply mul_ne_zero
· change ofReal
(θ.q (W.targetPerm e) (max 0 (min 1 (v (W.targetPerm e))))) ≠ 0
apply (ENNReal.ofReal_pos.mpr (hpos.2.1 (W.targetPerm e) _ ?_)).ne'
exact ⟨le_max_left _ _, max_le zero_le_one (min_le_left _ _)⟩
· unfold partialDensity
apply Finset.prod_ne_zero_iff.mpr
intro i _
change ofReal
(θ.p i (clampCube (Fin n) v)) ≠ 0
exact (ENNReal.ofReal_pos.mpr (hpos.1 i _ (by
simpa only [latentCube, unitCube] using
clampCube_mem (Fin n) v))).ne'
have hμ_int : μ ≪ interventionalLaw θ (W.targetPerm e) := by
rw [← mechanismUnitCubeFactorization_interventionMeasure W hpos e]
exact withDensity_absolutelyContinuous'
(B.measurable_interventionDensity (W.targetPerm e) q).aemeasurable
(Filter.Eventually.of_forall hdens_ne)
apply Set.Subset.antisymm
· exact hint_μ.support_mono.trans (by
rw [unitCubeReference_support_eq_latentCube n])
· rw [← unitCubeReference_support_eq_latentCube n]
exact hμ_int.support_mono
theorem observedLaw_support_eq_observedSupport
The observational observed law has exactly the image of the latent cube as its support. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
Measure.support (W.law 0) = observedSupport G W
Proof (Lean source)
lemma observedLaw_support_eq_observedSupport
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W) :
Measure.support (W.law 0) = observedSupport G W := by
have hcube_compact : IsCompact (latentCube n) := by
rw [latentCube]
exact isCompact_univ_pi fun _ => isCompact_Icc
have hsupport_closed : IsClosed (observedSupport G W) := by
exact (hcube_compact.image_of_continuousOn hmix.1.continuousOn).isClosed
have hobs_cube : ∀ᵐ v ∂observationalLaw θ, v ∈ latentCube n := by
filter_upwards [Measure.support_mem_ae (μ := observationalLaw θ)] with v hv
rw [observationalLaw_support_eq_latentCube hpos] at hv
exact hv
have hcube_meas : MeasurableSet (latentCube n) := by
rw [latentCube]
exact MeasurableSet.pi Set.countable_univ (fun _ _ ↦ measurableSet_Icc)
have hmixAE : AEMeasurable W.mix (observationalLaw θ) :=
aemeasurable_of_supportMeasurableOn hcube_meas
(ae_iff.mp hobs_cube) hmix.1.continuousOn.domRestrict.measurable
have hfull : W.law 0 (observedSupport G W)ᶜ = 0 := by
rw [hone.1, Measure.map_apply_of_aemeasurable hmixAE hsupport_closed.isOpen_compl.measurableSet]
apply measure_mono_null (t := (latentCube n)ᶜ)
· intro v hv hvcube
exact hv ⟨v, hvcube, rfl⟩
· exact ae_iff.mp hobs_cube
apply Set.Subset.antisymm
· exact Measure.support_subset_of_isClosed hsupport_closed (ae_iff.mpr hfull)
· rintro x ⟨v, hv, rfl⟩
rw [Measure.mem_support_iff_forall]
intro U hU
rcases mem_nhds_iff.mp hU with ⟨O, hOU, hOopen, hmixO⟩
rw [hone.1]
apply lt_of_lt_of_le _ (measure_mono hOU)
rw [Measure.map_apply_of_aemeasurable hmixAE hOopen.measurableSet]
have hpre : W.mix ⁻¹' O ∈ nhdsWithin v (latentCube n) :=
(hmix.1.continuousOn v hv) (hOopen.mem_nhds hmixO)
rcases mem_nhdsWithin_iff_exists_mem_nhds_inter.mp hpre with ⟨V, hV, hVsub⟩
have hVpos : 0 < observationalLaw θ V :=
(Measure.mem_support_iff_forall v).mp
((observationalLaw_support_eq_latentCube hpos).symm.subset hv) V hV
rw [← Measure.measure_inter_eq_of_ae hobs_cube] at hVpos
exact hVpos.trans_le (measure_mono (by simpa [inter_comm] using hVsub))
theorem observationalLaw_isProbabilityMeasure
Normalized local factors make the observational latent law a probability measure. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
hpos :
conclusion
Proof (Lean source)
lemma observationalLaw_isProbabilityMeasure
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(hpos : PositiveNormalizedSmoothMechanisms G θ) :
IsProbabilityMeasure (observationalLaw θ) := by
constructor
rw [← mechanismUnitCubeFactorization_observationalMeasure hpos]
exact factorization_observationalMeasure_univ (mechanismUnitCubeFactorization hpos)
theorem interventionalLaw_isProbabilityMeasure
Normalized local and replacement factors make every interventional latent law a probability measure. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
IsProbabilityMeasure (interventionalLaw θ (W.targetPerm e))
Proof (Lean source)
lemma interventionalLaw_isProbabilityMeasure
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) (hpos : PositiveNormalizedSmoothMechanisms G θ) (e : Fin n) :
IsProbabilityMeasure (interventionalLaw θ (W.targetPerm e)) := by
constructor
rw [← mechanismUnitCubeFactorization_interventionMeasure W hpos e]
exact factorization_interventionMeasure_univ (mechanismUnitCubeFactorization hpos)
(W.targetPerm e) (mechanismInterventionDensity W hpos e)
theorem observedWorld_laws_isProbabilityMeasure
The pushforward assumptions and normalized latent factors make every observed environment law a probability measure. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma observedWorld_laws_isProbabilityMeasure
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W) :
∀ e, IsProbabilityMeasure (W.law e) := by
intro e
let μ := unitCubeReference (Fin n)
have hcube : MeasurableSet (latentCube n) := by
rw [latentCube]
exact MeasurableSet.pi Set.countable_univ (fun _ _ ↦ measurableSet_Icc)
have hμ_cube : μ (latentCube n)ᶜ = 0 := by
simpa only [μ, latentCube, unitCube] using
unitCubeReference_compl (Fin n)
refine Fin.cases ?_ (fun i => ?_) e
· letI : IsProbabilityMeasure (observationalLaw θ) :=
observationalLaw_isProbabilityMeasure hpos
have hobs_μ : observationalLaw θ ≪ μ := by
rw [← mechanismUnitCubeFactorization_observationalMeasure hpos]
exact withDensity_absolutelyContinuous _ _
have hmixAE : AEMeasurable W.mix (observationalLaw θ) :=
aemeasurable_of_supportMeasurableOn hcube (hobs_μ hμ_cube)
hmix.1.continuousOn.domRestrict.measurable
rw [hone.1]
exact Measure.isProbabilityMeasure_map hmixAE
· letI : IsProbabilityMeasure (interventionalLaw θ (W.targetPerm i)) :=
interventionalLaw_isProbabilityMeasure W hpos i
let B := mechanismUnitCubeFactorization hpos
let q := mechanismInterventionDensity W hpos i
have hint_μ : B.interventionMeasure (W.targetPerm i) q ≪ μ := by
rw [interventionMeasure]
exact withDensity_absolutelyContinuous _ _
have hint_cube : interventionalLaw θ (W.targetPerm i) (latentCube n)ᶜ = 0 := by
rw [← mechanismUnitCubeFactorization_interventionMeasure W hpos i]
exact hint_μ hμ_cube
have hmixAE : AEMeasurable W.mix (interventionalLaw θ (W.targetPerm i)) :=
aemeasurable_of_supportMeasurableOn hcube hint_cube
hmix.1.continuousOn.domRestrict.measurable
rw [hone.2.1 i]
exact Measure.isProbabilityMeasure_map hmixAE
theorem canonicalRatio_map_eq_observedLawRatio_map_mix
Transporting the latent canonical Radon--Nikodym ratio through the support diffeomorphism does not change its law under any latent base measure dominated by the observational law. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
W :
ObservedWorld G θ
hpos :
hmix :
hone :
i :
Fin n
ρ :
Measure (LatentState n)
hρ :
ρ ≪ observationalLaw θ
conclusion
Measure.map (fun v ↦ ((interventionalLaw θ (W.targetPerm i)).rnDeriv (observationalLaw θ) v).toReal) ρ
= Measure.map (observedLawRatio W.law i) (Measure.map W.mix ρ)
Proof (Lean source)
lemma canonicalRatio_map_eq_observedLawRatio_map_mix
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W)
(i : Fin n) (ρ : Measure (LatentState n))
(hρ : ρ ≪ observationalLaw θ) :
Measure.map (fun v ↦ ((interventionalLaw θ (W.targetPerm i)).rnDeriv
(observationalLaw θ) v).toReal) ρ =
Measure.map (observedLawRatio W.law i) (Measure.map W.mix ρ) := by
let m := interventionalLaw θ (W.targetPerm i)
let ν := observationalLaw θ
let S := latentCube n
let T := observedSupport G W
letI : IsProbabilityMeasure m := interventionalLaw_isProbabilityMeasure W hpos i
letI : IsProbabilityMeasure ν := observationalLaw_isProbabilityMeasure hpos
have hS : MeasurableSet S := by
change MeasurableSet (latentCube n)
rw [latentCube]
exact MeasurableSet.pi Set.countable_univ (fun _ _ ↦ measurableSet_Icc)
have hcompact : IsCompact S := by
change IsCompact (latentCube n)
rw [latentCube]
exact isCompact_univ_pi fun _ => isCompact_Icc
have hT : MeasurableSet T :=
(hcompact.image_of_continuousOn hmix.1.continuousOn).isClosed.measurableSet
have hmS : m Sᶜ = 0 := by
change interventionalLaw θ (W.targetPerm i) (latentCube n)ᶜ = 0
rw [← interventionalLaw_support_eq_latentCube W hpos i]
exact Measure.measure_compl_support
have hνS : ν Sᶜ = 0 := by
change observationalLaw θ (latentCube n)ᶜ = 0
rw [← observationalLaw_support_eq_latentCube hpos]
exact Measure.measure_compl_support
have hmn : m ≪ ν := interventionalLaw_absolutelyContinuous_observational W hpos i
have hrn := rnDeriv_map_of_support_equiv m ν hmn S T hS hT hmS hνS
W.mix W.unmix hmix.1.continuousOn.domRestrict.measurable
hmix.2.1.continuousOn.domRestrict.measurable
(fun x hx ↦ ⟨x, hx, rfl⟩)
(fun y hy ↦ by
rcases hy with ⟨x, hx, rfl⟩
simpa only [hmix.2.2.1 x hx] using hx)
hmix.2.2.1 hmix.2.2.2
have hrnW :
(fun v ↦ ((W.law i.succ).rnDeriv (W.law 0) (W.mix v)).toReal) =ᵐ[ν]
(fun v ↦ (m.rnDeriv ν v).toReal) := by
filter_upwards [hrn] with v hv
change ((W.law i.succ).rnDeriv (W.law 0) (W.mix v)).toReal =
((interventionalLaw θ (W.targetPerm i)).rnDeriv (observationalLaw θ) v).toReal
rw [hone.2.1 i, hone.1]
exact congrArg toReal hv
have hrnρ := hρ.ae_eq hrnW
have hρS : ρ Sᶜ = 0 := hρ hνS
have hmixρ : AEMeasurable W.mix ρ :=
aemeasurable_of_supportMeasurableOn hS hρS
hmix.1.continuousOn.domRestrict.measurable
calc
Measure.map (fun v ↦ (m.rnDeriv ν v).toReal) ρ =
Measure.map (observedLawRatio W.law i ∘ W.mix) ρ :=
Measure.map_congr hrnρ.symm
_ = Measure.map (observedLawRatio W.law i) (Measure.map W.mix ρ) := by
symm
exact AEMeasurable.map_map_of_aemeasurable
(measurable_observedLawRatio W.law i).aemeasurable hmixρ
theorem observationalLaw_ae_mem_latentCube
The observational latent law is concentrated on the closed latent cube. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
Proof (Lean source)
-- @node: observationalLaw_ae_mem_latentCube
lemma observationalLaw_ae_mem_latentCube
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(hpos : PositiveNormalizedSmoothMechanisms G θ) :
∀ᵐ v ∂observationalLaw θ, v ∈ latentCube n := by
let μ := unitCubeReference (Fin n)
have hobs_μ : observationalLaw θ ≪ μ := by
rw [← mechanismUnitCubeFactorization_observationalMeasure hpos]
exact withDensity_absolutelyContinuous _ _
apply ae_iff.mpr
change observationalLaw θ (latentCube n)ᶜ = 0
exact hobs_μ (by
simpa only [μ, latentCube, unitCube] using
unitCubeReference_compl (Fin n))
theorem observedObservational_absolutelyContinuous_interventional
Strict positivity of all latent factors makes the observational and target-intervention observed laws equivalent; this is the reverse direction not needed by the ratio-law bridge. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma observedObservational_absolutelyContinuous_interventional
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W) (e : Fin n) :
W.law 0 ≪ W.law e.succ := by
let B := mechanismUnitCubeFactorization hpos
let q := mechanismInterventionDensity W hpos e
let μ := unitCubeReference (Fin n)
have hmeas_int : Measurable (B.interventionDensity (W.targetPerm e) q) := by
unfold interventionDensity
exact (q.measurable_density.comp (measurable_pi_apply _)).mul
(Finset.measurable_prod _ fun i _ ↦ B.measurable_factor i)
have hint_pos : ∀ v, B.interventionDensity (W.targetPerm e) q v ≠ 0 := by
intro v
unfold interventionDensity
apply mul_ne_zero
· change ofReal
(θ.q (W.targetPerm e) (max 0 (min 1 (v (W.targetPerm e))))) ≠ 0
exact (ENNReal.ofReal_pos.mpr (hpos.2.1 (W.targetPerm e) _ ⟨le_max_left _ _,
max_le zero_le_one (min_le_left _ _)⟩)).ne'
· exact Finset.prod_ne_zero_iff.mpr fun i _ ↦ by
change ofReal
(θ.p i (clampCube (Fin n) v)) ≠ 0
exact (ENNReal.ofReal_pos.mpr (hpos.1 i _ (by
simpa only [latentCube, unitCube] using
clampCube_mem (Fin n) v))).ne'
have hμ_int : μ ≪ B.interventionMeasure (W.targetPerm e) q := by
rw [interventionMeasure]
exact withDensity_absolutelyContinuous' hmeas_int.aemeasurable
(Filter.Eventually.of_forall hint_pos)
have hobs_μ : B.observationalMeasure ≪ μ := by
rw [observationalMeasure]
exact withDensity_absolutelyContinuous _ _
have hlatentB : B.observationalMeasure ≪
B.interventionMeasure (W.targetPerm e) q := hobs_μ.trans hμ_int
have hlatent : observationalLaw θ ≪ interventionalLaw θ (W.targetPerm e) := by
simpa only [B, q, mechanismUnitCubeFactorization_observationalMeasure hpos,
mechanismUnitCubeFactorization_interventionMeasure W hpos e] using hlatentB
let ν := interventionalLaw θ (W.targetPerm e)
have hcube : MeasurableSet (latentCube n) := by
rw [latentCube]
exact MeasurableSet.pi Set.countable_univ (fun _ _ ↦ measurableSet_Icc)
have hobs_cube : observationalLaw θ (latentCube n)ᶜ = 0 := by
rw [← mechanismUnitCubeFactorization_observationalMeasure hpos]
exact hobs_μ (by
simpa only [μ, latentCube, unitCube] using
unitCubeReference_compl (Fin n))
have hint_μ : B.interventionMeasure (W.targetPerm e) q ≪ μ := by
rw [interventionMeasure]
exact withDensity_absolutelyContinuous _ _
have hint_cube : ν (latentCube n)ᶜ = 0 := by
change interventionalLaw θ (W.targetPerm e) (latentCube n)ᶜ = 0
rw [← mechanismUnitCubeFactorization_interventionMeasure W hpos e]
exact hint_μ (by
simpa only [μ, latentCube, unitCube] using
unitCubeReference_compl (Fin n))
have hmixObs : AEMeasurable W.mix (observationalLaw θ) :=
aemeasurable_of_supportMeasurableOn hcube hobs_cube
hmix.1.continuousOn.domRestrict.measurable
have hmixInt : AEMeasurable W.mix ν :=
aemeasurable_of_supportMeasurableOn hcube hint_cube
hmix.1.continuousOn.domRestrict.measurable
let mix' := hmixInt.mk W.mix
have heqInt : W.mix =ᵐ[ν] mix' := hmixInt.ae_eq_mk
have heqObs : W.mix =ᵐ[observationalLaw θ] mix' := hlatent.ae_eq heqInt
rw [hone.1, hone.2.1 e]
calc
Measure.map W.mix (observationalLaw θ) =
Measure.map mix' (observationalLaw θ) := Measure.map_congr heqObs
_ ≪ Measure.map mix' ν :=
@Measure.AbsolutelyContinuous.map _ _ _ _ _ _ hlatent mix' hmixInt.measurable_mk
_ = Measure.map W.mix ν := (Measure.map_congr heqInt).symm
theorem observedWorldLawCoherent_of_assumptions
The supplied observed-world ratio agrees almost everywhere with the canonical law ratio, and the law-selected rank construction is unchanged by a compatible representation. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma observedWorldLawCoherent_of_assumptions
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W) :
ObservedWorldLawCoherent W := by
classical
constructor
· intro i
let μ := unitCubeReference (Fin n)
have hcube : MeasurableSet (latentCube n) := by
rw [latentCube]
exact MeasurableSet.pi Set.countable_univ (fun _ _ ↦ measurableSet_Icc)
have hμ_cube : μ (latentCube n)ᶜ = 0 := by
simpa only [μ, latentCube, unitCube] using
unitCubeReference_compl (Fin n)
have hobs_μ : observationalLaw θ ≪ μ := by
rw [← mechanismUnitCubeFactorization_observationalMeasure hpos]
exact withDensity_absolutelyContinuous _ _
have hobs_cube : observationalLaw θ (latentCube n)ᶜ = 0 := hobs_μ hμ_cube
have hmixAE : AEMeasurable W.mix (observationalLaw θ) :=
aemeasurable_of_supportMeasurableOn hcube hobs_cube
hmix.1.continuousOn.domRestrict.measurable
have hint_cube : interventionalLaw θ (W.targetPerm i) (latentCube n)ᶜ = 0 := by
let B := mechanismUnitCubeFactorization hpos
let q := mechanismInterventionDensity W hpos i
have hint_μ : B.interventionMeasure (W.targetPerm i) q ≪ μ := by
rw [interventionMeasure]
exact withDensity_absolutelyContinuous _ _
rw [← mechanismUnitCubeFactorization_interventionMeasure W hpos i]
exact hint_μ hμ_cube
have hmixAEInt : AEMeasurable W.mix
(interventionalLaw θ (W.targetPerm i)) :=
aemeasurable_of_supportMeasurableOn hcube hint_cube
hmix.1.continuousOn.domRestrict.measurable
have hrn_pos : ∀ᵐ x ∂W.law 0,
0 < (W.law i.succ).rnDeriv (W.law 0) x :=
by
letI : IsProbabilityMeasure (observationalLaw θ) :=
observationalLaw_isProbabilityMeasure hpos
letI : IsProbabilityMeasure (interventionalLaw θ (W.targetPerm i)) :=
interventionalLaw_isProbabilityMeasure W hpos i
letI : IsProbabilityMeasure (W.law 0) := ⟨by
rw [hone.1, Measure.map_apply_of_aemeasurable hmixAE MeasurableSet.univ,
preimage_univ, measure_univ]⟩
letI : IsProbabilityMeasure (W.law i.succ) := ⟨by
rw [hone.2.1 i, Measure.map_apply_of_aemeasurable hmixAEInt MeasurableSet.univ,
preimage_univ, measure_univ]⟩
exact Measure.rnDeriv_pos'
(observedObservational_absolutelyContinuous_interventional W hpos hmix hone i)
filter_upwards [hone.2.2.1 i, hrn_pos] with x hx hrnpos
have hposx : 0 < W.ratio i x := by
by_contra hnot
have hle : W.ratio i x ≤ 0 := le_of_not_gt hnot
have : ofReal (W.ratio i x) = 0 := ENNReal.ofReal_eq_zero.mpr hle
rw [hx] at this
exact hrnpos.ne' this
calc
W.ratio i x = (ofReal (W.ratio i x)).toReal :=
(ENNReal.toReal_ofReal hposx.le).symm
_ = ((W.law i.succ).rnDeriv (W.law 0) x).toReal := congrArg toReal hx
_ = observedLawRatio W.law i x := rfl
· intro G' θ' W' hcompat order i x hx
rcases hcompat with ⟨hlaw, _hsupport⟩
rw [← hlaw]
def smoothObservedRatio
The smooth mechanism formula for an observed ratio, extended through the unmixing map.
Definition (Lean source)
parameters
def
smoothObservedRatio W i x :
ℝ
given by
θ.q (W.targetPerm i) (W.unmix x (W.targetPerm i)) / θ.p (W.targetPerm i) (W.unmix x)
theorem smoothObservedRatio_isVersion
Under the positive smooth mechanism assumptions, shared mixing assumptions, and perfect-intervention assumptions, the smooth mechanism ratio is a continuous observed-ratio version for world W and environment i.
Formal statement
hypotheses
conclusion
IsContinuousObservedRatioVersion W.law i (smoothObservedRatio W i)
Proof (Lean source)
lemma smoothObservedRatio_isVersion
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W) (i : Fin n) :
IsContinuousObservedRatioVersion W.law i (smoothObservedRatio W i) := by
constructor
· have hratio := (observedWorldLawCoherent_of_assumptions W hpos hmix hone).1 i
apply Filter.EventuallyEq.trans _ hratio
filter_upwards [Measure.support_mem_ae (μ := W.law 0)] with x hx
rw [observedLaw_support_eq_observedSupport W hpos hmix hone] at hx
rcases hx with ⟨v, hv, rfl⟩
rw [smoothObservedRatio, show W.unmix (W.mix v) = v from hmix.2.2.1 v hv]
exact (hone.2.2.2 i v hv).symm
· rw [observedLaw_support_eq_observedSupport W hpos hmix hone]
have hunmix_mem : Set.MapsTo W.unmix (observedSupport G W) (latentCube n) := by
rintro x ⟨v, hv, rfl⟩
simpa only [hmix.2.2.1 v hv] using hv
have hunmix_cont : ContinuousOn W.unmix (observedSupport G W) :=
hmix.2.1.continuousOn
have hcoord_cont : ContinuousOn
(fun x ↦ W.unmix x (W.targetPerm i)) (observedSupport G W) :=
(continuous_apply (W.targetPerm i)).comp_continuousOn hunmix_cont
have hcoord_mem : Set.MapsTo
(fun x ↦ W.unmix x (W.targetPerm i)) (observedSupport G W) (Icc 0 1) := by
intro x hx
exact hunmix_mem hx (W.targetPerm i) (Set.mem_univ _)
apply ((hpos.2.2.2.1 (W.targetPerm i)).continuousOn.comp
hcoord_cont hcoord_mem).div
((hpos.2.2.1 (W.targetPerm i)).continuousOn.comp hunmix_cont hunmix_mem)
intro x hx
exact ne_of_gt (hpos.1 (W.targetPerm i) _ (hunmix_mem hx))
theorem exists_continuousObservedRatioVersion
Under positive smooth mechanisms, shared diffeomorphic mixing, and one perfect intervention per node, each observed ratio law has a continuous version.
Formal statement
hypotheses
conclusion
∃ R, IsContinuousObservedRatioVersion W.law i R
Proof (Lean source)
lemma exists_continuousObservedRatioVersion
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W) (i : Fin n) :
∃ R, IsContinuousObservedRatioVersion W.law i R := by
exact ⟨smoothObservedRatio W i,
smoothObservedRatio_isVersion W hpos hmix hone i⟩
theorem observedLawLogRatio_comp_mix_eq
The law-selected logarithmic ratio pulled back through the mixing map agrees pointwise on the latent cube with the smooth mechanism ratio, under the positive smooth mechanism assumptions, shared mixing assumptions, and perfect-intervention assumptions, for world W and environment i.
Formal statement
hypotheses
conclusion
observedLawLogRatio W.law i (W.mix v)
= log (θ.q (W.targetPerm i) (v (W.targetPerm i)) / θ.p (W.targetPerm i) v)
Proof (Lean source)
-- @node: observedLawLogRatio_comp_mix_eq
lemma observedLawLogRatio_comp_mix_eq
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W) (i : Fin n) :
∀ v, v ∈ latentCube n →
observedLawLogRatio W.law i (W.mix v) =
log (θ.q (W.targetPerm i) (v (W.targetPerm i)) /
θ.p (W.targetPerm i) v) := by
intro v hv
have hselected := observedContinuousRatio_isVersion W.law i
(exists_continuousObservedRatioVersion W hpos hmix hone i)
have hR := smoothObservedRatio_isVersion W hpos hmix hone i
have heq := hselected.eqOn hR
have hmix_support : W.mix v ∈ Measure.support (W.law 0) := by
rw [observedLaw_support_eq_observedSupport W hpos hmix hone]
exact ⟨v, hv, rfl⟩
rw [observedLawLogRatio, heq hmix_support, smoothObservedRatio,
hmix.2.2.1 v hv]
theorem observedLawLogRatio_comp_mix_ae_eq
The observed-law logarithmic ratio pulled back through the mixing map agrees almost everywhere with the mechanism's scalar log ratio. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
(fun v => observedLawLogRatio W.law i (W.mix v))
=ᵐ[observationalLaw θ] fun v => log (θ.q (W.targetPerm i) (v (W.targetPerm i)) / θ.p (W.targetPerm i) v)
Proof (Lean source)
-- @node: observedLawLogRatio_comp_mix_ae_eq
lemma observedLawLogRatio_comp_mix_ae_eq
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W) (i : Fin n) :
(fun v => observedLawLogRatio W.law i (W.mix v)) =ᵐ[observationalLaw θ]
fun v => log
(θ.q (W.targetPerm i) (v (W.targetPerm i)) / θ.p (W.targetPerm i) v) := by
filter_upwards [observationalLaw_ae_mem_latentCube hpos] with v hv
exact observedLawLogRatio_comp_mix_eq W hpos hmix hone i v hv
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.DecoderGraph 19 declarations This file proves equations (4)--(6): nonancestor interventions cannot create ratio-law edges, ancestral covers generate the latent transitive closure, and every ratio-graph topological ordering places latent parents befo
Ratio-graph reconstruction for the exact decoder
This file proves equations (4)--(6): nonancestor interventions cannot create ratio-law edges, ancestral covers generate the latent transitive closure, and every ratio-graph topological ordering places latent parents before their children.
def environmentParentSet
Environment-label parents obtained by pulling back the latent parent set.
A numeric order respects every latent edge after intervention-label relabeling.
Definition (Lean source)
parameters
def
PermutedGraphOrdered W order :
Prop
given by
order j < order i
theorem permutedGraphOrdered_of_transitiveClosure
Transitive-closure recovery makes every ratio-graph topological order respect latent edges. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
PermutedGraphOrdered W order
Proof (Lean source)
lemma permutedGraphOrdered_of_transitiveClosure
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) (order : Fin n → ℕ)
(horder : IsTopologicalOrdering
(observedLawRatioGraph gaussianFeatureMap W.law) order)
(htc : TransGen (observedLawRatioGraph gaussianFeatureMap W.law) =
TransGen (permutedGraph G W)) :
PermutedGraphOrdered W order := by
intro j i hji
have hpath : TransGen
(observedLawRatioGraph gaussianFeatureMap W.law) j i := by
rw [htc]
exact Relation.TransGen.single hji
have path_lt : ∀ {a b : Fin n},
TransGen (observedLawRatioGraph gaussianFeatureMap W.law) a b →
order a < order b := by
intro a b hab
induction hab with
| single h => exact horder.2 h
| tail hab h ih => exact lt_trans ih (horder.2 h)
exact path_lt hpath
theorem canonical_observationalRatioLaw_eq_supplied
The observational canonical-ratio law is invariant under the supplied support diffeomorphism. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
observationalRatioLaw (canonicalObservedWorld G θ W.targetPerm) i
= observationalRatioLaw W i
Proof (Lean source)
lemma canonical_observationalRatioLaw_eq_supplied
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) (hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W) (i : Fin n) :
observationalRatioLaw (canonicalObservedWorld G θ W.targetPerm) i =
observationalRatioLaw W i := by
have hcanonical : observedLawRatio (canonicalObservedWorld G θ W.targetPerm).law i =
fun v => ((interventionalLaw θ (W.targetPerm i)).rnDeriv
(observationalLaw θ) v).toReal := by
rfl
rw [observationalRatioLaw, observationalRatioLaw, hcanonical, hone.1]
exact canonicalRatio_map_eq_observedLawRatio_map_mix W hpos hmix hone i
(observationalLaw θ) Measure.AbsolutelyContinuous.rfl
theorem canonical_interventionalRatioLaw_eq_supplied
Every intervention-base canonical-ratio law is likewise invariant under the supplied support diffeomorphism. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
interventionalRatioLaw (canonicalObservedWorld G θ W.targetPerm) j i
= interventionalRatioLaw W j i
Proof (Lean source)
lemma canonical_interventionalRatioLaw_eq_supplied
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) (hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W) (j i : Fin n) :
interventionalRatioLaw (canonicalObservedWorld G θ W.targetPerm) j i =
interventionalRatioLaw W j i := by
have hcanonical : observedLawRatio (canonicalObservedWorld G θ W.targetPerm).law i =
fun v => ((interventionalLaw θ (W.targetPerm i)).rnDeriv
(observationalLaw θ) v).toReal := by
rfl
rw [interventionalRatioLaw, interventionalRatioLaw, hcanonical, hone.2.1 j]
exact canonicalRatio_map_eq_observedLawRatio_map_mix W hpos hmix hone i
(interventionalLaw θ (W.targetPerm j))
(interventionalLaw_absolutelyContinuous_observational W hpos j)
theorem canonical_populationDiscrepancy_eq_observedLawDiscrepancy
Hence the canonical-world discrepancy used by cover separation is exactly the observable discrepancy of the supplied law family. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
populationDiscrepancy U (canonicalObservedWorld G θ W.targetPerm) j i
= observedLawDiscrepancy U W.law j i
Proof (Lean source)
lemma canonical_populationDiscrepancy_eq_observedLawDiscrepancy
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
{H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℝ H] [CompleteSpace H]
(U : UnitNormFeatureMap H) (W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W) (j i : Fin n) :
populationDiscrepancy U (canonicalObservedWorld G θ W.targetPerm) j i =
observedLawDiscrepancy U W.law j i := by
unfold populationDiscrepancy observedLawDiscrepancy
rw [canonical_observationalRatioLaw_eq_supplied W hpos hmix hone i,
canonical_interventionalRatioLaw_eq_supplied W hpos hmix hone j i]
rfl
theorem canonical_coverDiscrepancy_to_supplied
Cover separation stated for the identity-mixing canonical world supplies exactly the cover edges required for the observed world. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
H :
U :
W :
ObservedWorld G θ
hpos :
hmix :
hone :
hcover :
∀ ⦃a b : Fin n⦄
if
ancestralCover G a b
then
0 < populationDiscrepancy U (canonicalObservedWorld G θ W.targetPerm) (W.targetPerm.symm a) (W.targetPerm.symm b)
a b :
Fin n
ancestralCover G a b
conclusion
0 < populationDiscrepancy U W (W.targetPerm.symm a) (W.targetPerm.symm b)
Proof (Lean source)
lemma canonical_coverDiscrepancy_to_supplied
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
{H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℝ H] [CompleteSpace H]
(U : UnitNormFeatureMap H) (W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W)
(hcover : ∀ ⦃a b : Fin n⦄, ancestralCover G a b →
0 < populationDiscrepancy U (canonicalObservedWorld G θ W.targetPerm)
(W.targetPerm.symm a) (W.targetPerm.symm b)) :
∀ ⦃a b : Fin n⦄, ancestralCover G a b →
0 < populationDiscrepancy U W
(W.targetPerm.symm a) (W.targetPerm.symm b) := by
intro a b hab
have hc := hcover hab
rw [canonical_populationDiscrepancy_eq_observedLawDiscrepancy
U W hpos hmix hone] at hc
exact hc
A non-ancestor intervention leaves the corresponding ratio law unchanged, hence has zero MMD. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
n :
W :
ObservedWorld G θ
hpos :
hmix :
hone :
j i :
Fin n
hne :
j ≠ i
hna :
¬ G.isAncestor (W.targetPerm j) (W.targetPerm i)
conclusion
populationDiscrepancy gaussianFeatureMap W j i = 0
Proof (Lean source)
lemma ratio_nonancestor_zero
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
{θ : Mechanism n G} (W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W)
{j i : Fin n} (hne : j ≠ i)
(hna : ¬ G.isAncestor (W.targetPerm j) (W.targetPerm i)) :
populationDiscrepancy gaussianFeatureMap W j i = 0 := by
apply populationDiscrepancy_eq_zero_of_ratioLaw_eq
exact FiniteDensityObservedWorldBridge.ratioLaw_eq_of_nonancestor
(finiteDensityObservedWorldBridge_of_assumptions W hpos hmix hone) hne hna
theorem isAncestor_transGen_ancestralCover
Every strict ancestor relation in a finite DAG factors through ancestral covers. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
TransGen (ancestralCover G) a b
Proof (Lean source)
lemma isAncestor_transGen_ancestralCover
{n : ℕ} (G : DAG (Fin n)) {a b : Fin n}
(h : G.isAncestor a b) : TransGen (ancestralCover G) a b := by
letI : LE (Fin n) := ⟨fun x y => x = y ∨ G.isAncestor x y⟩
letI : LT (Fin n) := ⟨G.isAncestor⟩
letI : PartialOrder (Fin n) := {
le_refl x := inl rfl
le_trans x y z hxy hyz := by
rcases hxy with rfl | hxy
· exact hyz
rcases hyz with rfl | hyz
· exact inr hxy
· exact inr (G.isAncestor_trans hxy hyz)
le_antisymm x y hxy hyx := by
rcases hxy with rfl | hxy
· rfl
rcases hyx with rfl | hyx
· rfl
exact elim (G.isAncestor_irrefl x (G.isAncestor_trans hxy hyx))
lt_iff_le_not_ge x y := by
constructor
· intro hxy
exact ⟨Or.inr hxy, fun hyx => by
rcases hyx with rfl | hyx
· exact G.isAncestor_irrefl _ hxy
· exact G.isAncestor_irrefl _ (G.isAncestor_trans hxy hyx)⟩
· rintro ⟨hxy, hnxy⟩
rcases hxy with rfl | hxy
· exact elim (hnxy (inl rfl))
· exact hxy }
letI : DecidableLE (Fin n) := Classical.decRel _
letI : DecidableLT (Fin n) := Classical.decRel _
letI : LocallyFiniteOrder (Fin n) := Fintype.toLocallyFiniteOrder
exact transGen_covBy_of_lt h
theorem observedLawRatioGraph_edge_isAncestor
Every observable ratio-graph edge points along the latent ancestral order. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
n :
W :
ObservedWorld G θ
hpos :
hmix :
hone :
j i :
Fin n
hji :
observedLawRatioGraph gaussianFeatureMap W.law j i
conclusion
G.isAncestor (W.targetPerm j) (W.targetPerm i)
Proof (Lean source)
lemma observedLawRatioGraph_edge_isAncestor
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
{θ : Mechanism n G} (W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W)
{j i : Fin n}
(hji : observedLawRatioGraph gaussianFeatureMap W.law j i) :
G.isAncestor (W.targetPerm j) (W.targetPerm i) := by
by_contra hna
have hz := ratio_nonancestor_zero (s := s) W hpos hmix hone hji.1 hna
have hz' : observedLawDiscrepancy gaussianFeatureMap W.law j i = 0 := by
exact hz
exact (not_lt_of_ge hz'.le) hji.2
theorem ratioGraph_transitiveClosure_eq_of_cover
Sound ratio edges plus all ancestral covers recover exactly the permuted latent transitive closure. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
n :
W :
ObservedWorld G θ
hpos :
hmix :
hone :
hcover :
∀ ⦃a b : Fin n⦄
if
ancestralCover G a b
then
0 < populationDiscrepancy gaussianFeatureMap W (W.targetPerm.symm a) (W.targetPerm.symm b)
conclusion
= TransGen (permutedGraph G W)
Proof (Lean source)
lemma ratioGraph_transitiveClosure_eq_of_cover
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
{θ : Mechanism n G} (W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W)
(hcover : ∀ ⦃a b : Fin n⦄, ancestralCover G a b →
0 < populationDiscrepancy gaussianFeatureMap W
(W.targetPerm.symm a) (W.targetPerm.symm b)) :
TransGen (observedLawRatioGraph gaussianFeatureMap W.law) =
TransGen (permutedGraph G W) := by
have hliftAncestor : ∀ {a b}, G.isAncestor a b →
TransGen (permutedGraph G W) (W.targetPerm.symm a) (W.targetPerm.symm b) := by
intro a b hji
induction hji with
| edge h => exact Relation.TransGen.single (by simpa [permutedGraph] using h)
| trans hab hbc ih =>
exact Relation.TransGen.tail ih (by simpa [permutedGraph] using hbc)
have hancestorPath : ∀ {j i}, G.isAncestor (W.targetPerm j) (W.targetPerm i) →
TransGen (permutedGraph G W) j i := by
intro j i hji
simpa using hliftAncestor hji
have hforward : TransGen (observedLawRatioGraph gaussianFeatureMap W.law) ≤
TransGen (permutedGraph G W) := by
apply Relation.TransGen.closed
intro j i hji
exact hancestorPath
(observedLawRatioGraph_edge_isAncestor (s := s) W hpos hmix hone hji)
have hcoverEdge : ∀ {a b : Fin n}, ancestralCover G a b →
observedLawRatioGraph gaussianFeatureMap W.law
(W.targetPerm.symm a) (W.targetPerm.symm b) := by
intro a b hab
have hne : W.targetPerm.symm a ≠ W.targetPerm.symm b := by
intro heq
have : a = b := W.targetPerm.symm.injective heq
subst b
exact G.isAncestor_irrefl a (@CovBy.lt (Fin n) ⟨G.isAncestor⟩ a a hab)
refine ⟨hne, ?_⟩
exact hcover hab
have hmapCoverPath : ∀ {a b}, TransGen (ancestralCover G) a b →
TransGen (observedLawRatioGraph gaussianFeatureMap W.law)
(W.targetPerm.symm a) (W.targetPerm.symm b) := by
intro a b hc
exact hc.lift W.targetPerm.symm (fun _ _ h => hcoverEdge h)
have hpermEdge : ∀ {j i}, permutedGraph G W j i →
TransGen (observedLawRatioGraph gaussianFeatureMap W.law) j i := by
intro j i hji
have hc := isAncestor_transGen_ancestralCover G (Causalean.DAG.isAncestor.edge hji)
simpa using hmapCoverPath hc
exact le_antisymm hforward (Relation.TransGen.closed (fun _ _ h => hpermEdge h))
theorem environmentParentSet_subset_predecessorSet_of_transitiveClosure
Once the ratio graph has the latent transitive closure, every topological ordering puts all environment-label parents before their child. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
W :
ObservedWorld G θ
order :
Fin n → ℕ
horder :
hgraphOrder :
PermutedGraphOrdered W order
i :
Fin n
conclusion
environmentParentSet W i ⊆ predecessorSet order i
Proof (Lean source)
lemma environmentParentSet_subset_predecessorSet_of_transitiveClosure
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ)
{order : Fin n → ℕ}
(horder : IsTopologicalOrdering (observedLawRatioGraph gaussianFeatureMap W.law) order)
(hgraphOrder : PermutedGraphOrdered W order) (i : Fin n) :
environmentParentSet W i ⊆ predecessorSet order i := by
intro j hj
have hedge : permutedGraph G W j i := by
simpa [environmentParentSet, permutedGraph] using hj
have hlt : order j < order i := hgraphOrder hedge
simpa [predecessorSet] using hlt
theorem finTwo_edge_ancestralCover
In a two-vertex DAG every directed edge is automatically a cover of the ancestral order, since there is no third vertex that can lie strictly between its endpoints. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma finTwo_edge_ancestralCover (G : DAG (Fin 2)) {j i : Fin 2}
(hji : G.edge j i) : ancestralCover G j i := by
refine ⟨Causalean.DAG.isAncestor.edge hji, ?_⟩
intro c hjc hci
have hj_ne_i : j ≠ i := by
intro h
subst i
exact G.acyclic j (Relation.TransGen.single hji)
have hj_ne_c : j ≠ c := by
intro h
subst c
exact G.acyclic j ((G.isAncestor_iff_transGen).mp hjc)
have hc_ne_i : c ≠ i := by
intro h
subst c
exact G.acyclic i ((G.isAncestor_iff_transGen).mp hci)
fin_cases j <;> fin_cases i <;> fin_cases c <;> simp_all
theorem bivariate_edge_discrepancy_pos_of_coverSeparated
Cover separation supplies the bivariate Gaussian-MMD edge witness because every edge of a two-vertex DAG is an ancestral cover. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
W :
hcover :
∀ ⦃a b : Fin 2⦄
if
ancestralCover G a b
then
0 < populationDiscrepancy gaussianFeatureMap W (W.targetPerm.symm a) (W.targetPerm.symm b)
j i :
G.edge j i
conclusion
0 < observedLawDiscrepancy gaussianFeatureMap W.law (W.targetPerm.symm j) (W.targetPerm.symm i)
Proof (Lean source)
lemma bivariate_edge_discrepancy_pos_of_coverSeparated
{G : DAG (Fin 2)} {theta : Mechanism 2 G}
(W : ObservedWorld G theta)
(hcover : ∀ ⦃a b : Fin 2⦄, ancestralCover G a b →
0 < populationDiscrepancy gaussianFeatureMap W
(W.targetPerm.symm a) (W.targetPerm.symm b)) :
∀ ⦃j i⦄, G.edge j i →
0 < observedLawDiscrepancy gaussianFeatureMap W.law
(W.targetPerm.symm j) (W.targetPerm.symm i) := by
intro j i hji
simpa [observedLawDiscrepancy, populationDiscrepancy,
observationalRatioLaw, interventionalRatioLaw] using
hcover (finTwo_edge_ancestralCover G hji)
theorem target_not_parent_of_mem_predecessorSet
The intervention target of a node cannot be a parent of any earlier node in a valid ratio-graph topological ordering. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
W :
ObservedWorld G θ
order :
Fin n → ℕ
horder :
hgraphOrder :
PermutedGraphOrdered W order
j i :
Fin n
hj :
j ∈ predecessorSet order i
conclusion
W.targetPerm i ∉ G.parents (W.targetPerm j)
Proof (Lean source)
lemma target_not_parent_of_mem_predecessorSet
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) {order : Fin n → ℕ}
(horder : IsTopologicalOrdering (observedLawRatioGraph gaussianFeatureMap W.law) order)
(hgraphOrder : PermutedGraphOrdered W order)
{j i : Fin n} (hj : j ∈ predecessorSet order i) :
W.targetPerm i ∉ G.parents (W.targetPerm j) := by
intro hparent
have hi_env : i ∈ environmentParentSet W j := by
simpa [environmentParentSet, parents] using hparent
have hi_pred := environmentParentSet_subset_predecessorSet_of_transitiveClosure
W horder hgraphOrder j hi_env
have hji : order j < order i := by simpa [predecessorSet] using hj
have hij : order i < order j := by simpa [predecessorSet] using hi_pred
exact lt_asymm hji hij
theorem predecessorLogRatioProjection_injective_on_predecessors
The triangular predecessor-score map recovers every predecessor's latent coordinate. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
s :
θ :
Mechanism n G
W :
ObservedWorld G θ
hpos :
hmix :
hone :
hsign :
FixedOwnDerivativeSign G s θ
order :
Fin n → ℕ
horder :
hgraphOrder :
PermutedGraphOrdered W order
i :
Fin n
v w :
hv :
v ∈ latentCube n
hw :
w ∈ latentCube n
heq :
familyProjection (observedLawLogRatio W.law) (predecessorSet order i) (W.mix v)
= familyProjection (observedLawLogRatio W.law) (predecessorSet order i) (W.mix w)
j ∈ predecessorSet order i :
conclusion
v (W.targetPerm j) = w (W.targetPerm j)
Proof (Lean source)
lemma predecessorLogRatioProjection_injective_on_predecessors
{n : ℕ} {G : DAG (Fin n)} (s : SignVector n)
{θ : Mechanism n G} (W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W)
(hsign : FixedOwnDerivativeSign G s θ)
{order : Fin n → ℕ}
(horder : IsTopologicalOrdering (observedLawRatioGraph gaussianFeatureMap W.law) order)
(hgraphOrder : PermutedGraphOrdered W order)
(i : Fin n) {v w : LatentState n} (hv : v ∈ latentCube n) (hw : w ∈ latentCube n)
(heq : familyProjection (observedLawLogRatio W.law) (predecessorSet order i) (W.mix v) =
familyProjection (observedLawLogRatio W.law) (predecessorSet order i) (W.mix w)) :
∀ j ∈ predecessorSet order i, v (W.targetPerm j) = w (W.targetPerm j) := by
have hind : ∀ m : ℕ, ∀ j : Fin n, order j = m → j ∈ predecessorSet order i →
v (W.targetPerm j) = w (W.targetPerm j) := by
intro m
induction m using Nat.strong_induction_on with
| h m ih =>
intro j hjorder hjpred
have hparents : ∀ a ∈ G.parents (W.targetPerm j), v a = w a := by
intro a ha
let k := W.targetPerm.symm a
have hk_env : k ∈ environmentParentSet W j := by
simpa [k, environmentParentSet, parents] using ha
have hk_pred_j := environmentParentSet_subset_predecessorSet_of_transitiveClosure
W horder hgraphOrder j hk_env
have hkj : order k < order j := by
simpa [predecessorSet] using hk_pred_j
have hji : order j < order i := by simpa [predecessorSet] using hjpred
have hk_pred_i : k ∈ predecessorSet order i := by
simpa [predecessorSet] using hkj.trans hji
have hk := ih (order k) (by simpa [hjorder] using hkj) k rfl hk_pred_i
simpa [k] using hk
have hscore :
log (θ.q (W.targetPerm j) (v (W.targetPerm j)) /
θ.p (W.targetPerm j) v) =
log (θ.q (W.targetPerm j) (w (W.targetPerm j)) /
θ.p (W.targetPerm j) w) := by
have hjscore := congrFun heq ⟨j, hjpred⟩
simpa only [familyProjection,
observedLawLogRatio_comp_mix_eq W hpos hmix hone j v hv,
observedLawLogRatio_comp_mix_eq W hpos hmix hone j w hw] using hjscore
exact (mechanismLogRatio_eq_iff_own_eq_of_parents_eq s θ hpos hsign
(W.targetPerm j) hv hw hparents).mp hscore
intro j hj
exact hind (order j) j rfl hj
theorem observedLawRatioGraph_hasTopologicalOrdering
Positivity and the shared perfect-intervention representation make the observed ratio graph acyclic, witnessed by the latent DAG's topological order transported to environment labels. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma observedLawRatioGraph_hasTopologicalOrdering
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
{θ : Mechanism n G} (W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W) :
∃ order, IsTopologicalOrdering
(observedLawRatioGraph gaussianFeatureMap W.law) order := by
let order : Fin n → ℕ := fun i => G.topoOrder (W.targetPerm i)
refine ⟨order, ?_, ?_⟩
· exact G.topoOrder_injective.comp W.targetPerm.injective
· intro j i hji
exact G.isAncestor_topoOrder_lt
(observedLawRatioGraph_edge_isAncestor (s := s) W hpos hmix hone hji)
theorem selectedTopologicalOrder_valid_of_assumptions
Under the model assumptions, the decoder's internally selected ordering is a valid topological ordering of the observed ratio graph. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma selectedTopologicalOrder_valid_of_assumptions
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
{θ : Mechanism n G} (W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W) :
IsTopologicalOrdering (observedLawRatioGraph gaussianFeatureMap W.law)
(selectedTopologicalOrder (observedProbabilityLawFamily W.law)) := by
classical
let laws := observedProbabilityLawFamily W.law
have hprob : ∀ e, IsProbabilityMeasure (W.law e) :=
observedWorld_laws_isProbabilityMeasure W hpos hmix hone
have hlaws : laws.1 = W.law := by
simp [laws, observedProbabilityLawFamily, hprob]
have hex : ∃ order, IsTopologicalOrdering
(observedLawRatioGraph gaussianFeatureMap laws.1) order := by
rw [hlaws]
exact observedLawRatioGraph_hasTopologicalOrdering (s := s) W hpos hmix hone
simpa only [hlaws] using selectedTopologicalOrder_isTopologicalOrdering laws hex
theorem ratioGraph_reconstruction_order_and_predecessors
Equations (4)--(6) assemble into transitive-closure recovery, validity of the decoder's selected order, and containment of every true parent among every valid order's predecessors. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
n :
W :
ObservedWorld G θ
hpos :
hmix :
hone :
hcover :
∀ ⦃a b : Fin n⦄
if
ancestralCover G a b
then
0 < populationDiscrepancy gaussianFeatureMap W (W.targetPerm.symm a) (W.targetPerm.symm b)
conclusion
conclusion 1
= TransGen (permutedGraph G W)
conclusion 2
conclusion 3
environmentParentSet W i ⊆ predecessorSet order i
Proof (Lean source)
lemma ratioGraph_reconstruction_order_and_predecessors
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
{θ : Mechanism n G} (W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W)
(hcover : ∀ ⦃a b : Fin n⦄, ancestralCover G a b →
0 < populationDiscrepancy gaussianFeatureMap W
(W.targetPerm.symm a) (W.targetPerm.symm b)) :
TransGen (observedLawRatioGraph gaussianFeatureMap W.law) =
TransGen (permutedGraph G W) ∧
IsTopologicalOrdering (observedLawRatioGraph gaussianFeatureMap W.law)
(selectedTopologicalOrder (observedProbabilityLawFamily W.law)) ∧
∀ order, IsTopologicalOrdering (observedLawRatioGraph gaussianFeatureMap W.law) order →
∀ i, environmentParentSet W i ⊆ predecessorSet order i := by
have htc := ratioGraph_transitiveClosure_eq_of_cover (s := s) W hpos hmix hone hcover
refine ⟨htc, selectedTopologicalOrder_valid_of_assumptions (s := s) W hpos hmix hone, ?_⟩
intro order horder i
exact environmentParentSet_subset_predecessorSet_of_transitiveClosure W horder
(permutedGraphOrdered_of_transitiveClosure W order horder htc) i
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.DecoderOrderedLocalMarkov 7 declarations This module packages the finite-density ordered local Markov theorem in the paper's CondIndepGiven interface under its observational law.
Ordered local Markov bridge for decoder pruning
This module packages the finite-density ordered local Markov theorem in the
paper's CondIndepGiven interface under its observational law.
theorem permutedGraph_edge_lt_of_transitiveClosure
A numeric topological order for the ratio graph also orders every latent edge once their transitive closures agree. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
W :
ObservedWorld G θ
order :
Fin n → ℕ
horder :
htc :
= TransGen (permutedGraph G W)
j i :
permutedGraph G W j i
conclusion
order j < order i
Proof (Lean source)
lemma permutedGraph_edge_lt_of_transitiveClosure
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) (order : Fin n → ℕ)
(horder : IsTopologicalOrdering
(observedLawRatioGraph gaussianFeatureMap W.law) order)
(htc : TransGen (observedLawRatioGraph gaussianFeatureMap W.law) =
TransGen (permutedGraph G W)) :
∀ ⦃j i⦄, permutedGraph G W j i → order j < order i := by
intro j i hji
have hpath : TransGen
(observedLawRatioGraph gaussianFeatureMap W.law) j i := by
rw [htc]
exact Relation.TransGen.single hji
have path_lt : ∀ {a b : Fin n},
TransGen (observedLawRatioGraph gaussianFeatureMap W.law) a b →
order a < order b := by
intro a b hab
induction hab with
| single h => exact horder.2 h
| tail hab h ih => exact lt_trans ih (horder.2 h)
exact path_lt hpath
def latentTopologicalRankingOfPermutedOrder
A topological order of the environment-label graph pulls back along the target permutation to a topological ranking of the latent graph.
Definition (Lean source)
parameters
W :
ObservedWorld G θ
order :
Fin n → ℕ
hinj :
Injective order
hedge :
∀ ⦃j i⦄
if
permutedGraph G W j i
then
order j < order i
def
given by
clause 1
rank := fun k ↦ order (W.targetPerm.symm k)
clause 2
injective_rank := hinj.comp W.targetPerm.symm.injective
clause 3
edge_lt := by intro j i hji apply hedge simpa [permutedGraph] using hji
theorem latentTopologicalRankingOfPermutedOrder_predecessors
Predecessors in the pulled-back latent ranking are exactly target-permutation images of the environment-label predecessors. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
W :
ObservedWorld G θ
order :
Fin n → ℕ
hinj :
Injective order
hedge :
∀ ⦃j i⦄
if
permutedGraph G W j i
then
order j < order i
k i :
Fin n
conclusion
k ∈ predecessors (latentTopologicalRankingOfPermutedOrder W order hinj hedge) (W.targetPerm i)
↔ W.targetPerm.symm k ∈ predecessorSet order i
Proof (Lean source)
lemma latentTopologicalRankingOfPermutedOrder_predecessors
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) (order : Fin n → ℕ)
(hinj : Injective order)
(hedge : ∀ ⦃j i⦄, permutedGraph G W j i → order j < order i)
(k i : Fin n) :
k ∈ predecessors
(latentTopologicalRankingOfPermutedOrder W order hinj hedge) (W.targetPerm i) ↔
W.targetPerm.symm k ∈ predecessorSet order i := by
simp [predecessors, predecessorSet,
latentTopologicalRankingOfPermutedOrder]
theorem mechanism_condIndepGiven_orderedLocalMarkov
A positive normalized paper mechanism makes a latent coordinate conditionally independent of the nonconditioned predecessors whenever the conditioning set contains all latent parents. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
CondIndepGiven (observationalLaw θ) (fun v : LatentState n ↦ v i) (coordinateProjection (X := fun _ : Fin n ↦ ℝ) (predecessors τ i \ A)) (coordinateProjection (X := fun _ : Fin n ↦ ℝ) A)
Proof (Lean source)
lemma mechanism_condIndepGiven_orderedLocalMarkov
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(τ : TopologicalRanking G)
(i : Fin n) (A : Finset (Fin n))
(hA : A ⊆ predecessors τ i)
(hpa : G.parents i ⊆ A) :
CondIndepGiven (observationalLaw θ) (fun v : LatentState n ↦ v i)
(coordinateProjection
(X := fun _ : Fin n ↦ ℝ)
(predecessors τ i \ A))
(coordinateProjection
(X := fun _ : Fin n ↦ ℝ) A) := by
rw [← mechanismUnitCubeFactorization_observationalMeasure hpos]
rcases mechanism_orderedLocalMarkov hpos τ i A hA hpa with ⟨hfinite, hci⟩
exact ⟨hfinite, measurable_pi_apply i,
measurable_coordinateProjection _,
measurable_coordinateProjection _, hci⟩
theorem mechanism_condIndepGiven_permutedOrderedLocalMarkov
The ordered local-Markov property pulled back to environment labels: whenever an environment conditioning set contains the target's environment-label parents, the target latent coordinate is independent of all remaining predecessors. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
W :
ObservedWorld G θ
hpos :
order :
Fin n → ℕ
horder :
hgraphOrder :
PermutedGraphOrdered W order
i :
Fin n
hA :
A ⊆ predecessorSet order i
hpa :
environmentParentSet W i ⊆ A
conclusion
CondIndepGiven (observationalLaw θ) (fun v : LatentState n ↦ v (W.targetPerm i)) (coordinateProjection (X := fun _ : Fin n ↦ ℝ) ((predecessorSet order i).map W.targetPerm.toEmbedding \ A.map W.targetPerm.toEmbedding)) (coordinateProjection (X := fun _ : Fin n ↦ ℝ) (A.map W.targetPerm.toEmbedding))
Proof (Lean source)
lemma mechanism_condIndepGiven_permutedOrderedLocalMarkov
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(order : Fin n → ℕ)
(horder : IsTopologicalOrdering
(observedLawRatioGraph gaussianFeatureMap W.law) order)
(hgraphOrder : PermutedGraphOrdered W order)
(i : Fin n) (A : Finset (Fin n))
(hA : A ⊆ predecessorSet order i)
(hpa : environmentParentSet W i ⊆ A) :
CondIndepGiven (observationalLaw θ)
(fun v : LatentState n ↦ v (W.targetPerm i))
(coordinateProjection
(X := fun _ : Fin n ↦ ℝ)
((predecessorSet order i).map W.targetPerm.toEmbedding \
A.map W.targetPerm.toEmbedding))
(coordinateProjection
(X := fun _ : Fin n ↦ ℝ) (A.map W.targetPerm.toEmbedding)) := by
let τ := latentTopologicalRankingOfPermutedOrder W order horder.1 hgraphOrder
have hpred : predecessors τ (W.targetPerm i) =
(predecessorSet order i).map W.targetPerm.toEmbedding := by
ext k
rw [latentTopologicalRankingOfPermutedOrder_predecessors
W order horder.1 hgraphOrder k i]
simp
rw [← hpred]
apply mechanism_condIndepGiven_orderedLocalMarkov hpos τ (W.targetPerm i)
(A.map W.targetPerm.toEmbedding)
· rw [hpred]
intro k hk
rcases Finset.mem_map.mp hk with ⟨e, he, rfl⟩
exact Finset.mem_map.mpr ⟨e, hA he, rfl⟩
· intro k hk
have hkEnv : W.targetPerm.symm k ∈ environmentParentSet W i := by
simpa [environmentParentSet, parents] using hk
exact Finset.mem_map.mpr ⟨W.targetPerm.symm k, hpa hkEnv,
W.targetPerm.apply_symm_apply k⟩
theorem condIndepCoordinates_singletons_iff_condIndepGiven
Singleton-block conditional independence is equivalent to its scalar-coordinate presentation, with the same finite conditioning projection. the stated conclusion follows.
Formal statement
hypotheses
conclusion
CondIndepCoordinates θ {i} {b} Z
↔ CondIndepGiven (observationalLaw θ) (fun v : LatentState n ↦ v i) (fun v : LatentState n ↦ v b) (coordinateProjection Z)
Proof (Lean source)
lemma condIndepCoordinates_singletons_iff_condIndepGiven
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(i b : Fin n) (Z : Finset (Fin n)) :
CondIndepCoordinates θ {i} {b} Z ↔
CondIndepGiven (observationalLaw θ)
(fun v : LatentState n ↦ v i) (fun v : LatentState n ↦ v b)
(coordinateProjection Z) := by
constructor
· rintro ⟨hfinite, hXi, hXb, hZ, hCI⟩
letI := hfinite
let evalI : ((j : {j // j ∈ ({i} : Finset (Fin n))}) → ℝ) → ℝ :=
fun x ↦ x ⟨i, Finset.mem_singleton_self i⟩
let evalB : ((j : {j // j ∈ ({b} : Finset (Fin n))}) → ℝ) → ℝ :=
fun x ↦ x ⟨b, Finset.mem_singleton_self b⟩
have hevalI : Measurable evalI := measurable_pi_apply _
have hevalB : Measurable evalB := measurable_pi_apply _
refine ⟨hfinite, measurable_pi_apply i, measurable_pi_apply b, hZ, ?_⟩
simpa [evalI, evalB, coordinateProjection, Function.comp_def] using
hCI.comp hevalI hevalB
· rintro ⟨hfinite, hi, hb, hZ, hCI⟩
letI := hfinite
let singletonI : ℝ → ((j : {j // j ∈ ({i} : Finset (Fin n))}) → ℝ) :=
fun r _ ↦ r
let singletonB : ℝ → ((j : {j // j ∈ ({b} : Finset (Fin n))}) → ℝ) :=
fun r _ ↦ r
have hsingletonI : Measurable singletonI :=
measurable_pi_lambda _ fun _ ↦ measurable_id
have hsingletonB : Measurable singletonB :=
measurable_pi_lambda _ fun _ ↦ measurable_id
refine ⟨hfinite,
measurable_coordinateProjection _,
measurable_coordinateProjection _, hZ, ?_⟩
have hraw := hCI.comp hsingletonI hsingletonB
convert hraw using 1
· funext v j
have hj : (j : Fin n) = i := Finset.mem_singleton.mp j.2
change v j = v i
exact congrArg v hj
· funext v j
have hj : (j : Fin n) = b := Finset.mem_singleton.mp j.2
change v j = v b
exact congrArg v hj
theorem parentOmission_intersection_core
The decoder's weak-union independence and the ordered local-Markov independence combine, by strict-positive-density intersection, to remove every nonparent predecessor from the conditioning set of a putatively omitted parent. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
W :
ObservedWorld G θ
hpos :
hmix :
hone :
i b :
Fin n
hU :
∀ c, Measurable (observedLatentCoordinate W c)
hPS :
P ⊆ S
hAS :
A ⊆ S
hiS :
i ∉ S
hbP :
b ∈ P
hbA :
b ∉ A
hCI :
CondIndepGiven (W.law 0) (observedLatentCoordinate W i) (familyProjection (observedLatentCoordinate W) (S \ A)) (familyProjection (observedLatentCoordinate W) A)
hMarkov :
CondIndepGiven (W.law 0) (observedLatentCoordinate W i) (familyProjection (observedLatentCoordinate W) (S \ P)) (familyProjection (observedLatentCoordinate W) P)
conclusion
CondIndepGiven (W.law 0) (observedLatentCoordinate W i) (observedLatentCoordinate W b) (familyProjection (observedLatentCoordinate W) (P.erase b))
Proof (Lean source)
lemma parentOmission_intersection_core
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W)
(i b : Fin n) (S P A : Finset (Fin n))
(hU : ∀ c, Measurable (observedLatentCoordinate W c))
(hPS : P ⊆ S) (hAS : A ⊆ S) (hiS : i ∉ S)
(hbP : b ∈ P) (hbA : b ∉ A)
(hCI : CondIndepGiven (W.law 0) (observedLatentCoordinate W i)
(familyProjection (observedLatentCoordinate W) (S \ A))
(familyProjection (observedLatentCoordinate W) A))
(hMarkov : CondIndepGiven (W.law 0) (observedLatentCoordinate W i)
(familyProjection (observedLatentCoordinate W) (S \ P))
(familyProjection (observedLatentCoordinate W) P)) :
CondIndepGiven (W.law 0) (observedLatentCoordinate W i)
(observedLatentCoordinate W b)
(familyProjection (observedLatentCoordinate W) (P.erase b)) := by
classical
let U := observedLatentCoordinate W
have hU' : ∀ c, Measurable (U c) := hU
letI : IsFiniteMeasure (W.law 0) := hCI.choose
have hbS : b ∈ S := hPS hbP
have hweak : CondIndepGiven (W.law 0) (U i) (U b)
(familyProjection U (S.erase b)) :=
(condIndep_coordSplit_prodMk U i b A S hU' hAS hbS hbA hCI).2
let Z := P.erase b
let C := S \ P
have hib : i ≠ b := by
intro h
subst b
exact hiS (hPS hbP)
have hiZ : i ∉ Z := by
intro hi
exact hiS (hPS (mem_of_mem_erase hi))
have hiC : i ∉ C := by
intro hi
exact hiS (Finset.mem_sdiff.mp hi).1
have hbZ : b ∉ Z := by simp [Z]
have hbC : b ∉ C := by
intro hb
exact (Finset.mem_sdiff.mp hb).2 hbP
have hZC : Disjoint Z C := by
refine Finset.disjoint_left.mpr ?_
intro x hxZ hxC
exact (Finset.mem_sdiff.mp hxC).2 (mem_of_mem_erase hxZ)
have hZCeq : Z ∪ C = S.erase b := by
ext x
simp only [Z, C, mem_union, Finset.mem_erase, mem_sdiff]
constructor
· rintro (⟨hxb, hxP⟩ | ⟨hxS, hxnotP⟩)
· exact ⟨hxb, hPS hxP⟩
· exact ⟨by intro h; subst x; exact hxnotP hbP, hxS⟩
· rintro ⟨hxb, hxS⟩
by_cases hxP : x ∈ P
· exact inl ⟨hxb, hxP⟩
· exact inr ⟨hxS, hxP⟩
let eZC : ((j : {j // j ∈ S.erase b}) → ℝ) ≃ᵐ
(((j : {j // j ∈ Z}) → ℝ) × ((j : {j // j ∈ C}) → ℝ)) :=
(valuesEquivOfEq (Ω := fun _ : Fin n ↦ ℝ) hZCeq.symm).trans
(MeasurableEquiv.piFinsetUnion (fun _ : Fin n ↦ ℝ) hZC).symm
have heZC : eZC ∘ familyProjection U (S.erase b) =
fun x => (familyProjection U Z x, familyProjection U C x) := by
funext x
ext j <;> rfl
have h₁ : CondIndepGiven (W.law 0) (U i) (U b)
(fun x => (familyProjection U Z x, familyProjection U C x)) := by
rw [← heZC]
exact (condIndepGiven_measurableEquiv_comp (U i) (U b)
(familyProjection U (S.erase b)) (MeasurableEquiv.refl ℝ)
(MeasurableEquiv.refl ℝ) eZC).2 hweak
let singletonB : ((j : {j // j ∈ ({b} : Finset (Fin n))}) → ℝ) ≃ᵐ ℝ := {
toFun x := x ⟨b, mem_singleton_self b⟩
invFun r _ := r
left_inv x := by
funext j
have hj : j = ⟨b, mem_singleton_self b⟩ :=
Subtype.ext (Finset.mem_singleton.mp j.2)
subst j
rfl
right_inv _ := rfl
measurable_toFun := measurable_pi_apply _
measurable_invFun := measurable_pi_lambda _ fun _ ↦ measurable_id }
have hZb : Disjoint Z ({b} : Finset (Fin n)) := by
simpa [Finset.disjoint_singleton_right]
have hZbeq : Z ∪ {b} = P := by
rw [union_comm]
exact Finset.insert_erase hbP
let eZb : ((j : {j // j ∈ P}) → ℝ) ≃ᵐ
(((j : {j // j ∈ Z}) → ℝ) × ℝ) :=
(valuesEquivOfEq (Ω := fun _ : Fin n ↦ ℝ) hZbeq.symm).trans
((MeasurableEquiv.piFinsetUnion (fun _ : Fin n ↦ ℝ) hZb).symm.trans
(MeasurableEquiv.prodCongr (MeasurableEquiv.refl _) singletonB))
have heZb : eZb ∘ familyProjection U P =
fun x => (familyProjection U Z x, U b x) := by
funext x
ext j <;> rfl
have h₂ : CondIndepGiven (W.law 0) (U i) (familyProjection U C)
(fun x => (familyProjection U Z x, U b x)) := by
rw [← heZb]
exact (condIndepGiven_measurableEquiv_comp (U i) (familyProjection U C)
(familyProjection U P) (MeasurableEquiv.refl ℝ)
(MeasurableEquiv.refl _) eZb).2 hMarkov
exact (condIndep_intersection_of_pos W hpos hmix hone i b C Z hib hiC hiZ
hbC hbZ hZC.symm h₁ h₂).1
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.DecoderPruning 3 declarations This file packages the set-theoretic conclusion of equation (19): once the conditional-independence test characterizes exactly the supersets of the true parent set, that set is the unique inclusion-minimal admissible set
Exact parent pruning
This file packages the set-theoretic conclusion of equation (19): once the conditional-independence test characterizes exactly the supersets of the true parent set, that set is the unique inclusion-minimal admissible set and the decoder's selected graph is the permuted latent graph.
theorem exactParentPruning_at_order
At one common order, the conditional-independence characterization makes the true parent set the unique inclusion-minimal admissible set. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
W :
ObservedWorld G θ
laws :
order :
Fin n → ℕ
hparents :
∀ i, environmentParentSet W i ⊆ predecessorSet order i
hiff :
∀ i A
if
A ⊆ predecessorSet order i
then
(CondIndepGiven (laws.1 0) (observedLawRankCoordinate laws order i) (familyProjection (observedLawRankCoordinate laws order) (predecessorSet order i \ A)) (familyProjection (observedLawRankCoordinate laws order) A) ↔ environmentParentSet W i ⊆ A)
i :
conclusion
conclusion 1
MinimalAdmissibleParentSet laws order i (environmentParentSet W i)
Proof (Lean source)
lemma exactParentPruning_at_order
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) (laws : ObservedProbabilityLawFamily n)
(order : Fin n → ℕ)
(hparents : ∀ i, environmentParentSet W i ⊆ predecessorSet order i)
(hiff : ∀ i A, A ⊆ predecessorSet order i →
(CondIndepGiven (laws.1 0) (observedLawRankCoordinate laws order i)
(familyProjection (observedLawRankCoordinate laws order)
(predecessorSet order i \ A))
(familyProjection (observedLawRankCoordinate laws order) A) ↔
environmentParentSet W i ⊆ A)) :
∀ i, MinimalAdmissibleParentSet laws order i (environmentParentSet W i) ∧
∀ A, MinimalAdmissibleParentSet laws order i A →
A = environmentParentSet W i := by
intro i
have hp := hparents i
have hpa : AdmissibleParentSet laws order i (environmentParentSet W i) :=
⟨hp, (hiff i _ hp).2 Finset.Subset.rfl⟩
have hpmin : MinimalAdmissibleParentSet laws order i (environmentParentSet W i) := by
refine ⟨hpa, ?_⟩
intro B hB _
exact (hiff i B hB.1).1 hB.2
refine ⟨hpmin, ?_⟩
intro A hA
apply Finset.Subset.antisymm
· exact hA.2 _ hpa ((hiff i A hA.1.1).1 hA.1.2)
· exact (hiff i A hA.1.1).1 hA.1.2
theorem exactParentPruning_of_condIndepCharacterization
If admissibility is equivalent to containing the true environment-label parents for every valid ordering, parent pruning has that parent set as its unique minimum. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
W :
ObservedWorld G θ
laws :
hpred :
∀ order
if
then
∀ i, environmentParentSet W i ⊆ predecessorSet order i
hiff :
∀ order,
IsTopologicalOrdering (observedLawRatioGraph gaussianFeatureMap laws.1) order → ∀ i A, A ⊆ predecessorSet order i → (CondIndepGiven (laws.1 0) (observedLawRankCoordinate laws order i) (familyProjection (observedLawRankCoordinate laws order) (predecessorSet order i \ A)) (familyProjection (observedLawRankCoordinate laws order) A) ↔ environmentParentSet W i ⊆ A)
order :
i :
conclusion
conclusion 1
MinimalAdmissibleParentSet laws order i (environmentParentSet W i)
Proof (Lean source)
lemma exactParentPruning_of_condIndepCharacterization
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ)
(laws : ObservedProbabilityLawFamily n)
(hpred : ∀ order, IsTopologicalOrdering
(observedLawRatioGraph gaussianFeatureMap laws.1) order →
∀ i, environmentParentSet W i ⊆ predecessorSet order i)
(hiff : ∀ order, IsTopologicalOrdering
(observedLawRatioGraph gaussianFeatureMap laws.1) order →
∀ i A, A ⊆ predecessorSet order i →
(CondIndepGiven (laws.1 0) (observedLawRankCoordinate laws order i)
(familyProjection (observedLawRankCoordinate laws order)
(predecessorSet order i \ A))
(familyProjection (observedLawRankCoordinate laws order) A) ↔
environmentParentSet W i ⊆ A)) :
∀ order, IsTopologicalOrdering
(observedLawRatioGraph gaussianFeatureMap laws.1) order →
∀ i, MinimalAdmissibleParentSet laws order i (environmentParentSet W i) ∧
∀ A, MinimalAdmissibleParentSet laws order i A →
A = environmentParentSet W i := by
intro order horder i
have hparents : environmentParentSet W i ⊆ predecessorSet order i :=
hpred order horder i
have hparentsAdmissible :
AdmissibleParentSet laws order i (environmentParentSet W i) := by
refine ⟨hparents, ?_⟩
exact (hiff order horder i (environmentParentSet W i) hparents).2
(Finset.Subset.rfl)
have hparentsMinimal :
MinimalAdmissibleParentSet laws order i (environmentParentSet W i) := by
refine ⟨hparentsAdmissible, ?_⟩
intro B hB _hBparents
exact (hiff order horder i B hB.1).1 hB.2
refine ⟨hparentsMinimal, ?_⟩
intro A hA
apply Finset.Subset.antisymm
· exact hA.2 (environmentParentSet W i) hparentsAdmissible
((hiff order horder i A hA.1.1).1 hA.1.2)
· exact (hiff order horder i A hA.1.1).1 hA.1.2
theorem selectedParentDAG_eq_permutedGraph_of_exactPruning
Exact unique parent pruning at the selected topological order makes the decoder's returned DAG equal to the environment-label pullback of the latent DAG. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
W :
ObservedWorld G θ
laws :
hprune :
∀ i
if
∀ A, MinimalAdmissibleParentSet laws (selectedTopologicalOrder laws) i A
then
A = environmentParentSet W i
conclusion
Proof (Lean source)
lemma selectedParentDAG_eq_permutedGraph_of_exactPruning
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ)
(laws : ObservedProbabilityLawFamily n)
(hprune : ∀ i,
MinimalAdmissibleParentSet laws (selectedTopologicalOrder laws) i
(environmentParentSet W i) ∧
∀ A, MinimalAdmissibleParentSet laws (selectedTopologicalOrder laws) i A →
A = environmentParentSet W i) :
(populationDecoder laws).2.2.edge = permutedGraph G W := by
funext j i
apply propext
change selectedParentRelation laws (selectedTopologicalOrder laws) j i ↔
permutedGraph G W j i
have hunique : ∃! A : Finset (Fin n),
MinimalAdmissibleParentSet laws (selectedTopologicalOrder laws) i A := by
refine ⟨environmentParentSet W i, (hprune i).1, ?_⟩
intro A hA
exact (hprune i).2 A hA
have hselected : selectedParentSet laws (selectedTopologicalOrder laws) i =
some (environmentParentSet W i) := by
rw [selectedParentSet, dif_pos hunique]
congr
exact (hprune i).2 _ (Classical.choose_spec hunique.exists)
constructor
· rintro ⟨A, hA, hjA⟩
rw [hselected] at hA
injection hA with hA
subst A
simpa [environmentParentSet, permutedGraph] using hjA
· intro hji
refine ⟨environmentParentSet W i, hselected, ?_⟩
simpa [environmentParentSet, permutedGraph] using hji
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.DecoderRank 16 declarations This file proves the one-dimensional integration step in equation (12): evaluating a conditional CDF at a strictly monotone score returns the intervention CDF, reflected when the score is decreasing.
Scalar conditional-rank identities
This file proves the one-dimensional integration step in equation (12): evaluating a conditional CDF at a strictly monotone score returns the intervention CDF, reflected when the score is decreasing.
def equationElevenConditionalRatioCDF
Equation (11): conditional on predecessor ranks (hence on the parents), the target log-ratio CDF integrates the intervention density over own-coordinate values below the log-ratio threshold.
theorem interventionCDF_mem_Icc
The normalized positive intervention density has a distribution function valued in the unit interval at every point of the unit interval. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
interventionCDF θ i z ∈ Icc (0 : ℝ) 1
Proof (Lean source)
lemma interventionCDF_mem_Icc
{n : ℕ} {G : DAG (Fin n)} (θ : Mechanism n G)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(i : Fin n) (z : ℝ) (hz : z ∈ Icc (0 : ℝ) 1) :
interventionCDF θ i z ∈ Icc (0 : ℝ) 1 := by
have hq : IntervalIntegrable (θ.q i) volume 0 1 := by
have hcont : ContinuousOn (θ.q i) (uIcc (0 : ℝ) 1) := by
simpa [uIcc_of_le (show (0 : ℝ) ≤ 1 by norm_num)] using
(hpos.2.2.2.1 i).continuousOn
exact hcont.intervalIntegrable
have hq_nonneg : ∀ᵐ w ∂volume.restrict (Ioc (0 : ℝ) 1), 0 ≤ θ.q i w := by
filter_upwards [ae_restrict_mem measurableSet_Ioc] with w hw
exact le_of_lt (hpos.2.1 i w ⟨le_of_lt hw.1, hw.2⟩)
have hnorm := hpos.2.2.2.2.2 i
rw [MeasureTheory.integral_Icc_eq_integral_Ioc,
← intervalIntegral.integral_of_le zero_le_one] at hnorm
rw [interventionCDF]
constructor
· exact intervalIntegral.integral_nonneg hz.1 fun w hw ↦
le_of_lt (hpos.2.1 i w ⟨hw.1, hw.2.trans hz.2⟩)
· rw [← hnorm]
exact intervalIntegral.integral_mono_interval le_rfl hz.1 hz.2 hq_nonneg hq
theorem equationElevenConditionalRatioCDF_mem_Icc
The equation-(11) conditional kernel is a genuine unit-interval-valued CDF at every threshold and every latent state in the cube. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
equationElevenConditionalRatioCDF θ i t v ∈ Icc (0 : ℝ) 1
Proof (Lean source)
lemma equationElevenConditionalRatioCDF_mem_Icc
{n : ℕ} {G : DAG (Fin n)} (θ : Mechanism n G)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(i : Fin n) (t : ℝ) (v : LatentState n) (hv : v ∈ latentCube n) :
equationElevenConditionalRatioCDF θ i t v ∈ Icc (0 : ℝ) 1 := by
let f : ℝ → ℝ := fun w ↦
if log (θ.q i w / θ.p i (update v i w)) ≤ t then θ.q i w else 0
have hq : IntegrableOn (θ.q i) (Icc (0 : ℝ) 1) := by
exact (hpos.2.2.2.1 i).continuousOn.integrableOn_compact isCompact_Icc
have hscore : ContinuousOn
(fun w ↦ log (θ.q i w / θ.p i (update v i w)))
(Icc (0 : ℝ) 1) := by
have hqcont := (hpos.2.2.2.1 i).continuousOn
have hpcont : ContinuousOn (fun w ↦ θ.p i (update v i w))
(Icc (0 : ℝ) 1) := by
apply (hpos.2.2.1 i).continuousOn.comp
· fun_prop
· intro w hw j hj
by_cases hji : j = i
· subst j
simpa using hw
· simp [hji]
exact hv j (Set.mem_univ j)
have hpne : ∀ w ∈ Icc (0 : ℝ) 1, θ.p i (update v i w) ≠ 0 :=
fun w hw ↦ ne_of_gt (hpos.1 i _ (by
intro j hj
by_cases hji : j = i
· subst j
simpa using hw
· simp [hji]
exact hv j (Set.mem_univ j)))
have hqne : ∀ w ∈ Icc (0 : ℝ) 1, θ.q i w ≠ 0 :=
fun w hw ↦ ne_of_gt (hpos.2.1 i w hw)
exact (hqcont.div hpcont hpne).log
(fun w hw ↦ div_ne_zero (hqne w hw) (hpne w hw))
let μ := volume.restrict (Icc (0 : ℝ) 1)
have hqae : AEMeasurable (θ.q i) μ :=
(hpos.2.2.2.1 i).continuousOn.aemeasurable measurableSet_Icc
have hscoreae : AEMeasurable
(fun w ↦ log (θ.q i w / θ.p i (update v i w))) μ :=
hscore.aemeasurable measurableSet_Icc
let qm := hqae.mk
let sm := hscoreae.mk
have hfm : Measurable (fun w ↦ if sm w ≤ t then qm w else 0) := by
exact Measurable.ite (measurableSet_le hscoreae.measurable_mk measurable_const)
hqae.measurable_mk measurable_const
have hfeq : (fun w ↦ if sm w ≤ t then qm w else 0) =ᵐ[μ] f := by
filter_upwards [hqae.ae_eq_mk, hscoreae.ae_eq_mk] with w hqw hsw
simp only [qm, sm] at hqw hsw ⊢
rw [← hqw, ← hsw]
have hfstrong : AEStronglyMeasurable f μ :=
hfm.aestronglyMeasurable.congr hfeq
have hf : IntegrableOn f (Icc (0 : ℝ) 1) := by
apply hq.mono' hfstrong
· filter_upwards [ae_restrict_mem measurableSet_Icc] with w hw
simp only [f]
split
· rw [Real.norm_eq_abs, abs_of_pos (hpos.2.1 i w hw)]
· simpa only [norm_zero] using le_of_lt (hpos.2.1 i w hw)
have hnonneg : 0 ≤ ∫ w in Icc (0 : ℝ) 1, f w := by
apply MeasureTheory.integral_nonneg_of_ae
filter_upwards [ae_restrict_mem measurableSet_Icc] with w hw
simp only [f]
split
· exact le_of_lt (hpos.2.1 i w hw)
· exact le_rfl
have hle : (∫ w in Icc (0 : ℝ) 1, f w) ≤
∫ w in Icc (0 : ℝ) 1, θ.q i w := by
apply MeasureTheory.integral_mono_ae hf hq
filter_upwards [ae_restrict_mem measurableSet_Icc] with w hw
simp only [f]
split
· exact le_rfl
· exact le_of_lt (hpos.2.1 i w hw)
rw [hpos.2.2.2.2.2 i] at hle
change 0 ≤ ∫ w in Icc (0 : ℝ) 1, f w ∧
(∫ w in Icc (0 : ℝ) 1, f w) ≤ 1
exact ⟨hnonneg, hle⟩
theorem equationElevenConditionalRatioCDF_eq_of_parents_eq
The equation-(11) kernel depends on the conditioning state only through the target's parent coordinates. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
equationElevenConditionalRatioCDF θ i t v = equationElevenConditionalRatioCDF θ i t v'
Proof (Lean source)
lemma equationElevenConditionalRatioCDF_eq_of_parents_eq
{n : ℕ} {G : DAG (Fin n)} (θ : Mechanism n G)
(i : Fin n) (t : ℝ) {v v' : LatentState n}
(hparents : ∀ j ∈ G.parents i, v j = v' j) :
equationElevenConditionalRatioCDF θ i t v =
equationElevenConditionalRatioCDF θ i t v' := by
apply MeasureTheory.integral_congr_ae
filter_upwards with w
have hp : θ.p i (update v i w) =
θ.p i (update v' i w) := by
apply θ.parent_local i
· simp
· intro j hj
have hji : j ≠ i := by
intro hEq
subst j
exact G.acyclic i (Relation.TransGen.single (G.mem_parents.mp hj))
simpa [update, hji] using hparents j hj
rw [hp]
On the unit interval, a score whose lower level set at z is [0,z] has conditional CDF equal to the integral of its density from zero to z. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma conditionalCDF_at_strictMono_score
(q score : ℝ → ℝ) (z : ℝ) (hz : z ∈ Icc (0 : ℝ) 1)
(hlevel : ∀ w ∈ Icc (0 : ℝ) 1, score w ≤ score z ↔ w ≤ z) :
(∫ w in Icc (0 : ℝ) 1, if score w ≤ score z then q w else 0) =
∫ w in (0 : ℝ)..z, q w := by
rw [intervalIntegral.integral_of_le hz.1,
← MeasureTheory.integral_Icc_eq_integral_Ioc]
rw [← MeasureTheory.integral_indicator measurableSet_Icc,
← MeasureTheory.integral_indicator measurableSet_Icc]
apply MeasureTheory.integral_congr_ae
filter_upwards with w
by_cases hw : w ∈ Icc (0 : ℝ) 1
· simp only [Set.indicator_of_mem hw]
have hm := hlevel w hw
by_cases hwz : w ≤ z
· have hwmem : w ∈ Icc (0 : ℝ) z := ⟨hw.1, hwz⟩
simp [hm, hwz, hwmem]
· have hwnmem : w ∉ Icc (0 : ℝ) z := fun h ↦ hwz h.2
simp [hm, hwz, hwnmem]
· have hw' : w ∉ Icc (0 : ℝ) z := by
intro hwz
exact hw ⟨hwz.1, hwz.2.trans hz.2⟩
simp [hw, hw']
On the unit interval, a decreasing score has conditional CDF equal to one minus the integral of its normalized density from zero to z. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma conditionalCDF_at_strictAnti_score
(q score : ℝ → ℝ) (z : ℝ) (hz : z ∈ Icc (0 : ℝ) 1)
(hq : IntervalIntegrable q volume 0 1)
(hnorm : ∫ w in Icc (0 : ℝ) 1, q w = 1)
(hlevel : ∀ w ∈ Icc (0 : ℝ) 1, score w ≤ score z ↔ z ≤ w) :
(∫ w in Icc (0 : ℝ) 1, if score w ≤ score z then q w else 0) =
1 - ∫ w in (0 : ℝ)..z, q w := by
have htail :
(∫ w in Icc (0 : ℝ) 1, if score w ≤ score z then q w else 0) =
∫ w in z..(1 : ℝ), q w := by
rw [intervalIntegral.integral_of_le hz.2,
← MeasureTheory.integral_Icc_eq_integral_Ioc]
rw [← MeasureTheory.integral_indicator measurableSet_Icc,
← MeasureTheory.integral_indicator measurableSet_Icc]
apply MeasureTheory.integral_congr_ae
filter_upwards with w
by_cases hw : w ∈ Icc (0 : ℝ) 1
· simp only [Set.indicator_of_mem hw]
have hm := hlevel w hw
by_cases hzw : z ≤ w
· have hwmem : w ∈ Icc z (1 : ℝ) := ⟨hzw, hw.2⟩
simp [hm, hzw, hwmem]
· have hwnmem : w ∉ Icc z (1 : ℝ) := fun h ↦ hzw h.1
simp [hm, hzw, hwnmem]
· have hw' : w ∉ Icc z (1 : ℝ) := by
intro hwz
exact hw ⟨hz.1.trans hwz.1, hwz.2⟩
simp [hw, hw']
rw [htail]
have hq0z : IntervalIntegrable q volume 0 z := by
apply hq.mono_set
rw [uIcc_of_le hz.1, uIcc_of_le zero_le_one]
exact Set.Icc_subset_Icc_right hz.2
have hqz1 : IntervalIntegrable q volume z 1 := by
apply hq.mono_set
rw [uIcc_of_le hz.2, uIcc_of_le zero_le_one]
exact Set.Icc_subset_Icc_left hz.1
have hadd := intervalIntegral.integral_add_adjacent_intervals hq0z hqz1
rw [MeasureTheory.integral_Icc_eq_integral_Ioc,
← intervalIntegral.integral_of_le zero_le_one] at hnorm
linarith
theorem conditionalCDF_at_strictMonoOn_score Lemma conditionalCDF_at_strictMonoOn_score in the paper ↗
A normalized density evaluated through a strictly increasing score produces its ordinary distribution function. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
(∫ w in Icc (0 : ℝ) 1, if score w ≤ score z then q w else 0) = ∫ w in (0 : ℝ)..z, q w
Proof (Lean source)
lemma conditionalCDF_at_strictMonoOn_score
(q score : ℝ → ℝ) (z : ℝ) (hz : z ∈ Icc (0 : ℝ) 1)
(hmono : StrictMonoOn score (Icc (0 : ℝ) 1)) :
(∫ w in Icc (0 : ℝ) 1, if score w ≤ score z then q w else 0) =
∫ w in (0 : ℝ)..z, q w := by
apply conditionalCDF_at_strictMono_score q score z hz
intro w hw
exact hmono.le_iff_le hw hz
theorem conditionalCDF_at_strictAntiOn_score Lemma conditionalCDF_at_strictAntiOn_score in the paper ↗
A normalized density evaluated through a strictly decreasing score produces its reflected distribution function. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
q score :
ℝ → ℝ
z :
ℝ
hz :
z ∈ Icc (0 : ℝ) 1
hq :
IntervalIntegrable q volume 0 1
hnorm :
∫ w in Icc (0 : ℝ) 1, q w = 1
hanti :
StrictAntiOn score (Icc (0 : ℝ) 1)
conclusion
(∫ w in Icc (0 : ℝ) 1, if score w ≤ score z then q w else 0)
= 1 - ∫ w in (0 : ℝ)..z, q w
Proof (Lean source)
lemma conditionalCDF_at_strictAntiOn_score
(q score : ℝ → ℝ) (z : ℝ) (hz : z ∈ Icc (0 : ℝ) 1)
(hq : IntervalIntegrable q volume 0 1)
(hnorm : ∫ w in Icc (0 : ℝ) 1, q w = 1)
(hanti : StrictAntiOn score (Icc (0 : ℝ) 1)) :
(∫ w in Icc (0 : ℝ) 1, if score w ≤ score z then q w else 0) =
1 - ∫ w in (0 : ℝ)..z, q w := by
apply conditionalCDF_at_strictAnti_score q score z hz hq hnorm
intro w hw
exact hanti.le_iff_ge hw hz
theorem equationEleven_at_strictMonoOwnScore
Equation (11), evaluated at the realized own-coordinate score, is the intervention CDF when that score is strictly increasing on the unit interval. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
equationElevenConditionalRatioCDF θ i (log (θ.q i (v i) / θ.p i v)) v
= interventionCDF θ i (v i)
Proof (Lean source)
lemma equationEleven_at_strictMonoOwnScore
{n : ℕ} {G : DAG (Fin n)} (θ : Mechanism n G)
(i : Fin n) (v : LatentState n) (hv : v ∈ latentCube n)
(hmono : StrictMonoOn
(fun w => log (θ.q i w / θ.p i (update v i w)))
(Icc (0 : ℝ) 1)) :
equationElevenConditionalRatioCDF θ i
(log (θ.q i (v i) / θ.p i v)) v =
interventionCDF θ i (v i) := by
have hvi : v i ∈ Icc (0 : ℝ) 1 := hv i (Set.mem_univ i)
have hvupdate : update v i (v i) = v := Function.update_eq_self i v
simpa only [equationElevenConditionalRatioCDF, interventionCDF, hvupdate] using
conditionalCDF_at_strictMonoOn_score (θ.q i)
(fun w => log (θ.q i w / θ.p i (update v i w))) (v i) hvi hmono
theorem equationEleven_at_strictAntiOwnScore
Equation (11), evaluated at the realized own-coordinate score, is the reflected intervention CDF when that score is strictly decreasing on the unit interval. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
equationElevenConditionalRatioCDF θ i (log (θ.q i (v i) / θ.p i v)) v
= 1 - interventionCDF θ i (v i)
Proof (Lean source)
lemma equationEleven_at_strictAntiOwnScore
{n : ℕ} {G : DAG (Fin n)} (θ : Mechanism n G)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(i : Fin n) (v : LatentState n) (hv : v ∈ latentCube n)
(hanti : StrictAntiOn
(fun w => log (θ.q i w / θ.p i (update v i w)))
(Icc (0 : ℝ) 1)) :
equationElevenConditionalRatioCDF θ i
(log (θ.q i (v i) / θ.p i v)) v =
1 - interventionCDF θ i (v i) := by
have hvi : v i ∈ Icc (0 : ℝ) 1 := hv i (Set.mem_univ i)
have hvupdate : update v i (v i) = v := Function.update_eq_self i v
have hq : IntervalIntegrable (θ.q i) volume 0 1 := by
have hcont : ContinuousOn (θ.q i) (uIcc (0 : ℝ) 1) := by
simpa [uIcc_of_le (show (0 : ℝ) ≤ 1 by norm_num)] using
(hpos.2.2.2.1 i).continuousOn
exact hcont.intervalIntegrable
simpa only [equationElevenConditionalRatioCDF, interventionCDF, hvupdate] using
conditionalCDF_at_strictAntiOn_score (θ.q i)
(fun w => log (θ.q i w / θ.p i (update v i w)))
(v i) hvi hq (hpos.2.2.2.2.2 i) hanti
theorem fixedOwnDerivativeSign_strictMonoOn
A positive prescribed own-score derivative makes the own-coordinate score strictly increasing on the closed unit interval. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
s :
θ :
Mechanism n G
hpos :
hsign :
FixedOwnDerivativeSign G s θ
i :
Fin n
v :
hv :
v ∈ latentCube n
hsi :
s.value i = 1
conclusion
Proof (Lean source)
lemma fixedOwnDerivativeSign_strictMonoOn
{n : ℕ} {G : DAG (Fin n)} (s : SignVector n)
(θ : Mechanism n G) (hpos : PositiveNormalizedSmoothMechanisms G θ)
(hsign : FixedOwnDerivativeSign G s θ)
(i : Fin n) (v : LatentState n) (hv : v ∈ latentCube n)
(hsi : s.value i = 1) :
StrictMonoOn
(fun w => log (θ.q i w / θ.p i (Function.update v i w)))
(Icc (0 : ℝ) 1) := by
let score : ℝ → ℝ := fun w => log (θ.q i w / θ.p i (Function.update v i w))
apply strictMonoOn_of_deriv_pos (convex_Icc (0 : ℝ) 1)
· have hq : ContinuousOn (θ.q i) (Icc (0 : ℝ) 1) := (hpos.2.2.2.1 i).continuousOn
have hp : ContinuousOn (fun w => θ.p i (Function.update v i w)) (Icc (0 : ℝ) 1) := by
apply (hpos.2.2.1 i).continuousOn.comp
· fun_prop
· intro w hw j hj
by_cases hji : j = i
· subst j
simpa using hw
· simp [hji]
exact hv j (Set.mem_univ j)
have hpne : ∀ w ∈ Icc (0 : ℝ) 1, θ.p i (Function.update v i w) ≠ 0 := fun w hw =>
ne_of_gt (hpos.1 i _ (by
intro j hj
by_cases hji : j = i
· subst j
simpa using hw
· simp [hji]
exact hv j (Set.mem_univ j)))
have hqne : ∀ w ∈ Icc (0 : ℝ) 1, θ.q i w ≠ 0 := fun w hw =>
ne_of_gt (hpos.2.1 i w hw)
exact (hq.div hp hpne).log (fun w hw => div_ne_zero (hqne w hw) (hpne w hw))
· intro w hw
rw [interior_Icc] at hw
have hwIcc : w ∈ Icc (0 : ℝ) 1 := ⟨le_of_lt hw.1, le_of_lt hw.2⟩
let vw := Function.update v i w
have hvw : vw ∈ latentCube n := by
intro j hj
by_cases hji : j = i
· subst j
simpa [vw] using hwIcc
· simp [vw, hji]
exact hv j (Set.mem_univ j)
have hs := hsign i vw hvw
rw [hsi, one_mul] at hs
have hscore : (fun z => log (θ.q i z / θ.p i (Function.update vw i z))) = score := by
funext z
simp [vw, score, Function.update_idem]
have hderivWithin : derivWithin score (Icc (0 : ℝ) 1) w =
ownLogRatioDerivative θ i vw := by
simp only [ownLogRatioDerivative, hscore, vw, Function.update_self]
rw [← derivWithin_of_mem_nhds
(Filter.mem_of_superset (Ioo_mem_nhds hw.1 hw.2) Set.Ioo_subset_Icc_self)]
rw [hderivWithin]
exact hs
theorem fixedOwnDerivativeSign_strictAntiOn
A negative prescribed own-score derivative makes the own-coordinate score strictly decreasing on the closed unit interval. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
s :
θ :
Mechanism n G
hpos :
hsign :
FixedOwnDerivativeSign G s θ
i :
Fin n
v :
hv :
v ∈ latentCube n
hsi :
s.value i = -1
conclusion
Proof (Lean source)
lemma fixedOwnDerivativeSign_strictAntiOn
{n : ℕ} {G : DAG (Fin n)} (s : SignVector n)
(θ : Mechanism n G) (hpos : PositiveNormalizedSmoothMechanisms G θ)
(hsign : FixedOwnDerivativeSign G s θ)
(i : Fin n) (v : LatentState n) (hv : v ∈ latentCube n)
(hsi : s.value i = -1) :
StrictAntiOn
(fun w => log (θ.q i w / θ.p i (Function.update v i w)))
(Icc (0 : ℝ) 1) := by
let score : ℝ → ℝ := fun w => log (θ.q i w / θ.p i (Function.update v i w))
apply strictAntiOn_of_deriv_neg (convex_Icc (0 : ℝ) 1)
· have hq : ContinuousOn (θ.q i) (Icc (0 : ℝ) 1) := (hpos.2.2.2.1 i).continuousOn
have hp : ContinuousOn (fun w => θ.p i (Function.update v i w)) (Icc (0 : ℝ) 1) := by
apply (hpos.2.2.1 i).continuousOn.comp
· fun_prop
· intro w hw j hj
by_cases hji : j = i
· subst j
simpa using hw
· simp [hji]
exact hv j (Set.mem_univ j)
have hpne : ∀ w ∈ Icc (0 : ℝ) 1, θ.p i (Function.update v i w) ≠ 0 := fun w hw =>
ne_of_gt (hpos.1 i _ (by
intro j hj
by_cases hji : j = i
· subst j
simpa using hw
· simp [hji]
exact hv j (Set.mem_univ j)))
have hqne : ∀ w ∈ Icc (0 : ℝ) 1, θ.q i w ≠ 0 := fun w hw =>
ne_of_gt (hpos.2.1 i w hw)
exact (hq.div hp hpne).log (fun w hw => div_ne_zero (hqne w hw) (hpne w hw))
· intro w hw
rw [interior_Icc] at hw
have hwIcc : w ∈ Icc (0 : ℝ) 1 := ⟨le_of_lt hw.1, le_of_lt hw.2⟩
let vw := Function.update v i w
have hvw : vw ∈ latentCube n := by
intro j hj
by_cases hji : j = i
· subst j
simpa [vw] using hwIcc
· simp [vw, hji]
exact hv j (Set.mem_univ j)
have hs := hsign i vw hvw
rw [hsi] at hs
have hscore : (fun z => log (θ.q i z / θ.p i (Function.update vw i z))) = score := by
funext z
simp [vw, score, Function.update_idem]
have hderivWithin : derivWithin score (Icc (0 : ℝ) 1) w =
ownLogRatioDerivative θ i vw := by
simp only [ownLogRatioDerivative, hscore, vw, Function.update_self]
rw [← derivWithin_of_mem_nhds
(Filter.mem_of_superset (Ioo_mem_nhds hw.1 hw.2) Set.Ioo_subset_Icc_self)]
rw [hderivWithin]
linarith
theorem mechanismLogRatio_eq_iff_own_eq_of_parents_eq
With parent coordinates fixed, equality of a node's log-ratio score is equivalent to equality of its own coordinate. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
s :
θ :
Mechanism n G
hpos :
hsign :
FixedOwnDerivativeSign G s θ
i :
Fin n
v w :
hv :
v ∈ latentCube n
hw :
w ∈ latentCube n
hparents :
∀ j ∈ G.parents i, v j = w j
Proof (Lean source)
lemma mechanismLogRatio_eq_iff_own_eq_of_parents_eq
{n : ℕ} {G : DAG (Fin n)} (s : SignVector n)
(θ : Mechanism n G) (hpos : PositiveNormalizedSmoothMechanisms G θ)
(hsign : FixedOwnDerivativeSign G s θ) (i : Fin n)
{v w : LatentState n} (hv : v ∈ latentCube n) (hw : w ∈ latentCube n)
(hparents : ∀ j ∈ G.parents i, v j = w j) :
log (θ.q i (v i) / θ.p i v) = log (θ.q i (w i) / θ.p i w) ↔
v i = w i := by
let score : ℝ → ℝ := fun z ↦
log (θ.q i z / θ.p i (update v i z))
have hvi : v i ∈ Icc (0 : ℝ) 1 := hv i (Set.mem_univ i)
have hwi : w i ∈ Icc (0 : ℝ) 1 := hw i (Set.mem_univ i)
have hscore_v : score (v i) = log (θ.q i (v i) / θ.p i v) := by
simp [score, Function.update_eq_self]
have hp_update : θ.p i (update v i (w i)) = θ.p i w := by
apply θ.parent_local i
· simp
· intro j hj
have hji : j ≠ i := by
intro hEq
subst j
exact G.acyclic i (Relation.TransGen.single (G.mem_parents.mp hj))
simpa [update, hji] using hparents j hj
have hscore_w : score (w i) = log (θ.q i (w i) / θ.p i w) := by
simp only [score, hp_update]
constructor
· intro heq
have hscore_eq : score (v i) = score (w i) := by
rw [hscore_v, hscore_w]
exact heq
rcases s.signed i with hneg | hpossign
· exact (fixedOwnDerivativeSign_strictAntiOn s θ hpos hsign i v hv hneg).injOn
hvi hwi hscore_eq
· exact (fixedOwnDerivativeSign_strictMonoOn s θ hpos hsign i v hv hpossign).injOn
hvi hwi hscore_eq
· intro hi
have hp : θ.p i v = θ.p i w := θ.parent_local i v w hi hparents
rw [hi, hp]
theorem equationEleven_at_fixedOwnDerivativeSign
Equation (12) follows from equation (11) and the prescribed own-coordinate derivative sign: the recovered rank is the intervention CDF, reflected exactly for negative sign. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
s :
θ :
Mechanism n G
hpos :
hsign :
FixedOwnDerivativeSign G s θ
i :
Fin n
v :
hv :
v ∈ latentCube n
conclusion
equationElevenConditionalRatioCDF θ i (log (θ.q i (v i) / θ.p i v)) v
= if s.value i = 1 then interventionCDF θ i (v i) else 1 - interventionCDF θ i (v i)
Proof (Lean source)
lemma equationEleven_at_fixedOwnDerivativeSign
{n : ℕ} {G : DAG (Fin n)} (s : SignVector n)
(θ : Mechanism n G) (hpos : PositiveNormalizedSmoothMechanisms G θ)
(hsign : FixedOwnDerivativeSign G s θ)
(i : Fin n) (v : LatentState n) (hv : v ∈ latentCube n) :
equationElevenConditionalRatioCDF θ i
(log (θ.q i (v i) / θ.p i v)) v =
if s.value i = 1 then interventionCDF θ i (v i)
else 1 - interventionCDF θ i (v i) := by
rcases s.signed i with hneg | hposi
· rw [if_neg (by linarith : s.value i ≠ 1)]
exact equationEleven_at_strictAntiOwnScore θ hpos i v hv
(fixedOwnDerivativeSign_strictAntiOn s θ hpos hsign i v hv hneg)
· rw [if_pos hposi]
exact equationEleven_at_strictMonoOwnScore θ i v hv
(fixedOwnDerivativeSign_strictMonoOn s θ hpos hsign i v hv hposi)
theorem equationEleven_at_fixedOwnDerivativeSign_mem_Icc
At the realized score, the signed equation-(11) formula has the unit-interval range required by the decoder's conditional-CDF codomain. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
s :
θ :
Mechanism n G
hpos :
hsign :
FixedOwnDerivativeSign G s θ
i :
Fin n
v :
hv :
v ∈ latentCube n
conclusion
Proof (Lean source)
lemma equationEleven_at_fixedOwnDerivativeSign_mem_Icc
{n : ℕ} {G : DAG (Fin n)} (s : SignVector n)
(θ : Mechanism n G) (hpos : PositiveNormalizedSmoothMechanisms G θ)
(hsign : FixedOwnDerivativeSign G s θ)
(i : Fin n) (v : LatentState n) (hv : v ∈ latentCube n) :
equationElevenConditionalRatioCDF θ i
(log (θ.q i (v i) / θ.p i v)) v ∈ Icc (0 : ℝ) 1 := by
rw [equationEleven_at_fixedOwnDerivativeSign s θ hpos hsign i v hv]
have hvi : v i ∈ Icc (0 : ℝ) 1 := hv i (Set.mem_univ i)
have hQ := interventionCDF_mem_Icc θ hpos i (v i) hvi
split
· exact hQ
· constructor <;> linarith [hQ.1, hQ.2]
theorem observedLawRankCoordinate_eq_of_equationEleven
Once the law-selected conditional CDF is identified with equation (11), evaluating it at the pointwise identified log ratio gives the signed intervention-CDF rank of equation (12). Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
s :
θ :
Mechanism n G
W :
ObservedWorld G θ
laws :
order :
Fin n → ℕ
hpos :
hsign :
FixedOwnDerivativeSign G s θ
i :
Fin n
hEleven :
∀ t v
if
v ∈ latentCube n
then
(observedConditionalRatioCDF laws order i t (familyProjection (observedLawLogRatio laws.1) (predecessorSet order i) (W.mix v)) : ℝ)
= equationElevenConditionalRatioCDF θ (W.targetPerm i) t v
hRatio :
∀ v
if
v ∈ latentCube n
then
observedLawLogRatio laws.1 i (W.mix v)
= log (θ.q (W.targetPerm i) (v (W.targetPerm i)) / θ.p (W.targetPerm i) v)
v :
v ∈ latentCube n
conclusion
(observedLawRankCoordinate laws order i (W.mix v) : ℝ)
= if s.value (W.targetPerm i) = 1 then interventionCDF θ (W.targetPerm i) (v (W.targetPerm i)) else 1
- interventionCDF θ (W.targetPerm i) (v (W.targetPerm i))
Proof (Lean source)
lemma observedLawRankCoordinate_eq_of_equationEleven
{n : ℕ} {G : DAG (Fin n)} (s : SignVector n)
(θ : Mechanism n G) (W : ObservedWorld G θ)
(laws : ObservedProbabilityLawFamily n) (order : Fin n → ℕ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hsign : FixedOwnDerivativeSign G s θ)
(i : Fin n)
(hEleven : ∀ t v, v ∈ latentCube n →
(observedConditionalRatioCDF laws order i t
(familyProjection (observedLawLogRatio laws.1)
(predecessorSet order i) (W.mix v)) : ℝ) =
equationElevenConditionalRatioCDF θ (W.targetPerm i) t v)
(hRatio : ∀ v, v ∈ latentCube n →
observedLawLogRatio laws.1 i (W.mix v) =
log (θ.q (W.targetPerm i) (v (W.targetPerm i)) /
θ.p (W.targetPerm i) v)) :
∀ v, v ∈ latentCube n →
(observedLawRankCoordinate laws order i (W.mix v) : ℝ) =
if s.value (W.targetPerm i) = 1
then interventionCDF θ (W.targetPerm i) (v (W.targetPerm i))
else 1 - interventionCDF θ (W.targetPerm i) (v (W.targetPerm i)) := by
intro v hv
rw [observedLawRankCoordinate, hEleven _ v hv, hRatio v hv]
exact equationEleven_at_fixedOwnDerivativeSign s θ hpos hsign
(W.targetPerm i) v hv
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.DecoderRankAssembly 2 declarations This file assembles the support, continuous-version, explicit equation-(11), and signed-rank facts into the complete conditional-rank clause consumed by the exact population decoder theorem.
Continuous conditional-rank assembly
This file assembles the support, continuous-version, explicit equation-(11), and signed-rank facts into the complete conditional-rank clause consumed by the exact population decoder theorem.
theorem exactRatioDecoder_continuousRankClauses_of_order
A ratio-graph order that also respects the latent edges has the law-selected continuous conditional CDF, its equation-(11) realization, and the signed intervention-CDF rank. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
s :
θ :
Mechanism n G
W :
ObservedWorld G θ
hpos :
hmix :
hone :
hsign :
FixedOwnDerivativeSign G s θ
order :
Fin n → ℕ
horder :
hgraphOrder :
PermutedGraphOrdered W order
i :
conclusion
laws :
IsContinuousConditionalRatioCDFVersion laws order i (observedConditionalRatioCDF laws order i)
∃ C,
IsContinuousConditionalRatioCDFVersion laws order i C
t v :
v ∈ latentCube n
= equationElevenConditionalRatioCDF θ (W.targetPerm i) t v
C :
IsContinuousConditionalRatioCDFVersion laws order i C
z ∈ observedConditionalRatioSupport W.law order i :
C z.1 z.2 = observedConditionalRatioCDF laws order i z.1 z.2
t v :
v ∈ latentCube n
(observedConditionalRatioCDF laws order i t (familyProjection (observedLawLogRatio W.law) (predecessorSet order i) (W.mix v)) : ℝ)
= equationElevenConditionalRatioCDF θ (W.targetPerm i) t v
v :
v ∈ latentCube n
(observedLawRankCoordinate laws order i (W.mix v) : ℝ)
= if s.value (W.targetPerm i) = 1 then interventionCDF θ (W.targetPerm i) (v (W.targetPerm i)) else 1
- interventionCDF θ (W.targetPerm i) (v (W.targetPerm i))
Proof (Lean source)
lemma exactRatioDecoder_continuousRankClauses_of_order
{n : ℕ} {G : DAG (Fin n)} (s : SignVector n)
{θ : Mechanism n G} (W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W)
(hsign : FixedOwnDerivativeSign G s θ)
(order : Fin n → ℕ)
(horder : IsTopologicalOrdering
(observedLawRatioGraph gaussianFeatureMap W.law) order)
(hgraphOrder : PermutedGraphOrdered W order) :
∀ i,
let laws := observedProbabilityLawFamily W.law
IsContinuousConditionalRatioCDFVersion laws order i
(observedConditionalRatioCDF laws order i) ∧
(∃ C, IsContinuousConditionalRatioCDFVersion laws order i C ∧
∀ t v, v ∈ latentCube n →
(C t (familyProjection (observedLawLogRatio W.law)
(predecessorSet order i) (W.mix v)) : ℝ) =
equationElevenConditionalRatioCDF θ (W.targetPerm i) t v) ∧
(∀ C, IsContinuousConditionalRatioCDFVersion laws order i C →
∀ z ∈ observedConditionalRatioSupport W.law order i,
C z.1 z.2 = observedConditionalRatioCDF laws order i z.1 z.2) ∧
(∀ t v, v ∈ latentCube n →
(observedConditionalRatioCDF laws order i t
(familyProjection (observedLawLogRatio W.law)
(predecessorSet order i) (W.mix v)) : ℝ) =
equationElevenConditionalRatioCDF θ (W.targetPerm i) t v) ∧
∀ v, v ∈ latentCube n →
(observedLawRankCoordinate laws order i (W.mix v) : ℝ) =
if s.value (W.targetPerm i) = 1
then interventionCDF θ (W.targetPerm i) (v (W.targetPerm i))
else 1 - interventionCDF θ (W.targetPerm i) (v (W.targetPerm i)) := by
intro i
let laws := observedProbabilityLawFamily W.law
have hprob : ∀ e, IsProbabilityMeasure (W.law e) :=
observedWorld_laws_isProbabilityMeasure W hpos hmix hone
have hlaws : laws.1 = W.law := by
simp [laws, observedProbabilityLawFamily, hprob]
let C := equationElevenAmbientCDF s W hpos hmix hone hsign horder hgraphOrder i
have hC : IsContinuousConditionalRatioCDFVersion laws order i C := by
exact equationElevenAmbientCDF_isContinuousVersion
s W hpos hmix hone hsign horder hgraphOrder i
have hex : ∃ D, IsContinuousConditionalRatioCDFVersion laws order i D := ⟨C, hC⟩
have hselected : IsContinuousConditionalRatioCDFVersion laws order i
(observedConditionalRatioCDF laws order i) :=
observedConditionalRatioCDF_isContinuousVersion laws order i hex
have hC_apply : ∀ t v, v ∈ latentCube n →
(C t (familyProjection (observedLawLogRatio W.law)
(predecessorSet order i) (W.mix v)) : ℝ) =
equationElevenConditionalRatioCDF θ (W.targetPerm i) t v := by
intro t v hv
exact equationElevenAmbientCDF_apply_of_latentState
s W hpos hmix hone hsign horder hgraphOrder i t v hv
have hselected_apply : ∀ t v, v ∈ latentCube n →
(observedConditionalRatioCDF laws order i t
(familyProjection (observedLawLogRatio W.law)
(predecessorSet order i) (W.mix v)) : ℝ) =
equationElevenConditionalRatioCDF θ (W.targetPerm i) t v := by
intro t v hv
let ell := familyProjection (observedLawLogRatio W.law)
(predecessorSet order i) (W.mix v)
have hz : (t, ell) ∈ observedConditionalRatioSupport laws.1 order i := by
rw [hlaws]
exact ⟨Set.mem_univ t,
observedPredecessorLogRatio_mem_support W hpos hmix hone i v hv⟩
have heq := observedConditionalRatioCDF_eq_on_support laws order i hex C hC
(t, ell) hz
calc
(observedConditionalRatioCDF laws order i t ell : ℝ) = (C t ell : ℝ) :=
congrArg val heq.symm
_ = equationElevenConditionalRatioCDF θ (W.targetPerm i) t v := hC_apply t v hv
refine ⟨hselected, ⟨C, hC, hC_apply⟩, ?_, hselected_apply, ?_⟩
· intro D hD
simpa only [hlaws] using
(observedConditionalRatioCDF_eq_on_support laws order i hex D hD)
· have hselected_apply' : ∀ t v, v ∈ latentCube n →
(observedConditionalRatioCDF laws order i t
(familyProjection (observedLawLogRatio laws.1)
(predecessorSet order i) (W.mix v)) : ℝ) =
equationElevenConditionalRatioCDF θ (W.targetPerm i) t v := by
simpa only [hlaws] using hselected_apply
have hratio' : ∀ v, v ∈ latentCube n →
observedLawLogRatio laws.1 i (W.mix v) =
log (θ.q (W.targetPerm i) (v (W.targetPerm i)) /
θ.p (W.targetPerm i) v) := by
simpa only [hlaws] using
(observedLawLogRatio_comp_mix_eq W hpos hmix hone i)
exact observedLawRankCoordinate_eq_of_equationEleven
s θ W laws order hpos hsign i hselected_apply' hratio'
Transitive-closure recovery supplies latent-edge compatibility for every valid ratio order. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
s :
θ :
Mechanism n G
W :
ObservedWorld G θ
hpos :
hmix :
hone :
hsign :
FixedOwnDerivativeSign G s θ
htc :
= TransGen (permutedGraph G W)
order :
i :
conclusion
laws :
IsContinuousConditionalRatioCDFVersion laws order i (observedConditionalRatioCDF laws order i)
∃ C,
IsContinuousConditionalRatioCDFVersion laws order i C
t v :
v ∈ latentCube n
= equationElevenConditionalRatioCDF θ (W.targetPerm i) t v
C :
IsContinuousConditionalRatioCDFVersion laws order i C
z ∈ observedConditionalRatioSupport W.law order i :
C z.1 z.2 = observedConditionalRatioCDF laws order i z.1 z.2
t v :
v ∈ latentCube n
(observedConditionalRatioCDF laws order i t (familyProjection (observedLawLogRatio W.law) (predecessorSet order i) (W.mix v)) : ℝ)
= equationElevenConditionalRatioCDF θ (W.targetPerm i) t v
v :
v ∈ latentCube n
(observedLawRankCoordinate laws order i (W.mix v) : ℝ)
= if s.value (W.targetPerm i) = 1 then interventionCDF θ (W.targetPerm i) (v (W.targetPerm i)) else 1
- interventionCDF θ (W.targetPerm i) (v (W.targetPerm i))
Proof (Lean source)
lemma exactRatioDecoder_continuousRankClauses
{n : ℕ} {G : DAG (Fin n)} (s : SignVector n)
{θ : Mechanism n G} (W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W)
(hsign : FixedOwnDerivativeSign G s θ)
(htc : TransGen (observedLawRatioGraph gaussianFeatureMap W.law) =
TransGen (permutedGraph G W)) :
∀ order, IsTopologicalOrdering
(observedLawRatioGraph gaussianFeatureMap W.law) order →
∀ i,
let laws := observedProbabilityLawFamily W.law
IsContinuousConditionalRatioCDFVersion laws order i
(observedConditionalRatioCDF laws order i) ∧
(∃ C, IsContinuousConditionalRatioCDFVersion laws order i C ∧
∀ t v, v ∈ latentCube n →
(C t (familyProjection (observedLawLogRatio W.law)
(predecessorSet order i) (W.mix v)) : ℝ) =
equationElevenConditionalRatioCDF θ (W.targetPerm i) t v) ∧
(∀ C, IsContinuousConditionalRatioCDFVersion laws order i C →
∀ z ∈ observedConditionalRatioSupport W.law order i,
C z.1 z.2 = observedConditionalRatioCDF laws order i z.1 z.2) ∧
(∀ t v, v ∈ latentCube n →
(observedConditionalRatioCDF laws order i t
(familyProjection (observedLawLogRatio W.law)
(predecessorSet order i) (W.mix v)) : ℝ) =
equationElevenConditionalRatioCDF θ (W.targetPerm i) t v) ∧
∀ v, v ∈ latentCube n →
(observedLawRankCoordinate laws order i (W.mix v) : ℝ) =
if s.value (W.targetPerm i) = 1
then interventionCDF θ (W.targetPerm i) (v (W.targetPerm i))
else 1 - interventionCDF θ (W.targetPerm i) (v (W.targetPerm i)) := by
intro order horder
exact exactRatioDecoder_continuousRankClauses_of_order s W hpos hmix hone hsign
order horder (permutedGraphOrdered_of_transitiveClosure W order horder htc)
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.DecoderRankCondIndep 4 declarations This module records the measurable-embedding invariance used to pass between the decoder's signed CDF ranks and the corresponding latent coordinates.
Rank-coordinate conditional-independence transport
This module records the measurable-embedding invariance used to pass between the decoder's signed CDF ranks and the corresponding latent coordinates.
theorem condIndepGiven_measurableEmbedding_comp
Applying measurable embeddings separately to both variables and the conditioning variable preserves and reflects conditional independence. the stated conclusion follows.
Formal statement
hypotheses
Ω X Y Z X' Y' Z' :
UX :
Ω → X
UY :
Ω → Y
UZ :
Ω → Z
fX :
X → X'
fY :
Y → Y'
fZ :
Z → Z'
eX :
eY :
eZ :
conclusion
CondIndepGiven μ (fX ∘ UX) (fY ∘ UY) (fZ ∘ UZ) ↔ CondIndepGiven μ UX UY UZ
Proof (Lean source)
lemma condIndepGiven_measurableEmbedding_comp
{Ω X Y Z X' Y' Z' : Type*}
[MeasurableSpace Ω] [StandardBorelSpace Ω]
[MeasurableSpace X] [MeasurableSpace Y] [MeasurableSpace Z]
[MeasurableSpace X'] [MeasurableSpace Y'] [MeasurableSpace Z']
[Nonempty X] [Nonempty Y] [Nonempty Z]
{μ : Measure Ω} (UX : Ω → X) (UY : Ω → Y) (UZ : Ω → Z)
{fX : X → X'} {fY : Y → Y'} {fZ : Z → Z'}
(eX : MeasurableEmbedding fX) (eY : MeasurableEmbedding fY)
(eZ : MeasurableEmbedding fZ) :
CondIndepGiven μ (fX ∘ UX) (fY ∘ UY) (fZ ∘ UZ) ↔
CondIndepGiven μ UX UY UZ := by
constructor
· rintro ⟨hμ, hX, hY, hZ, hCI⟩
letI := hμ
have hUX : Measurable UX := by
convert eX.measurable_invFun.comp hX using 1
funext ω
exact (eX.leftInverse_invFun (UX ω)).symm
have hUY : Measurable UY := by
convert eY.measurable_invFun.comp hY using 1
funext ω
exact (eY.leftInverse_invFun (UY ω)).symm
have hUZ : Measurable UZ := by
convert eZ.measurable_invFun.comp hZ using 1
funext ω
exact (eZ.leftInverse_invFun (UZ ω)).symm
refine ⟨hμ, hUX, hUY, hUZ, ?_⟩
have hraw := hCI.comp eX.measurable_invFun eY.measurable_invFun
have hs : CondIndepFun (comap (fZ ∘ UZ) inferInstance)
hZ.comap_le UX UY μ := by
convert hraw using 1 <;> funext ω
· exact (eX.leftInverse_invFun (UX ω)).symm
· exact (eY.leftInverse_invFun (UY ω)).symm
simpa only [← MeasurableSpace.comap_comp, eZ.comap_eq] using hs
· rintro ⟨hμ, hX, hY, hZ, hCI⟩
letI := hμ
refine ⟨hμ, eX.measurable.comp hX, eY.measurable.comp hY,
eZ.measurable.comp hZ, ?_⟩
have hraw := hCI.comp eX.measurable eY.measurable
simpa only [← MeasurableSpace.comap_comp, eZ.comap_eq] using hraw
def signedInterventionCDFChart
The signed intervention CDF restricted to the unit interval, with its range proof.
Definition (Lean source)
parameters
given by
fun z
=> ⟨signedInterventionCDF s θ i z, by unfold signedInterventionCDF have hQ := interventionCDF_mem_Icc θ hpos i z z.2 split · exact hQ · constructor <;> linarith [hQ.1, hQ.2]⟩
theorem continuous_signedInterventionCDFChart
The signed intervention CDF chart is continuous on the closed unit interval. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
Continuous (signedInterventionCDFChart s θ hpos i)
Proof (Lean source)
lemma continuous_signedInterventionCDFChart
{n : ℕ} {G : DAG (Fin n)} (s : SignVector n)
(θ : Mechanism n G) (hpos : PositiveNormalizedSmoothMechanisms G θ) (i : Fin n) :
Continuous (signedInterventionCDFChart s θ hpos i) := by
have hqint : IntegrableOn (θ.q i) (Icc (0 : ℝ) 1) :=
(hpos.2.2.2.1 i).continuousOn.integrableOn_compact isCompact_Icc
have hcset := intervalIntegral.continuousOn_primitive hqint
have hcdf : Continuous (fun z : Icc (0 : ℝ) 1 => interventionCDF θ i z) := by
have h := hcset.restrict
convert h using 1
funext z
change (∫ u in (0 : ℝ)..(z : ℝ), θ.q i u) =
∫ u in Ioc (0 : ℝ) (z : ℝ), θ.q i u
exact intervalIntegral.integral_of_le z.2.1
have hsigned : Continuous (fun z : Icc (0 : ℝ) 1 =>
signedInterventionCDF s θ i z) := by
unfold signedInterventionCDF
split
· exact hcdf
· fun_prop
exact hsigned.subtype_mk _
theorem signedInterventionCDFChart_measurableEmbedding
The positive-density signed CDF chart is a measurable embedding of the unit interval. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
MeasurableEmbedding (signedInterventionCDFChart s θ hpos i)
Proof (Lean source)
lemma signedInterventionCDFChart_measurableEmbedding
{n : ℕ} {G : DAG (Fin n)} (s : SignVector n)
(θ : Mechanism n G) (hpos : PositiveNormalizedSmoothMechanisms G θ) (i : Fin n) :
MeasurableEmbedding (signedInterventionCDFChart s θ hpos i) := by
apply (continuous_signedInterventionCDFChart s θ hpos i).measurableEmbedding
intro a b hab
apply Subtype.ext
exact signedInterventionCDF_injOn s θ hpos i a.2 b.2 (congrArg val hab)
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.DecoderRankPruning 15 declarations This module transports conditional independence between law-selected signed-CDF ranks and measurable support-restricted latent coordinates, then combines ordered local Markov and positive-density intersection to characte
Rank-coordinate parent-pruning assembly
This module transports conditional independence between law-selected signed-CDF ranks and measurable support-restricted latent coordinates, then combines ordered local Markov and positive-density intersection to characterize the admissible parent sets exactly.
def observedLatentUnitCoordinate
The support-restricted observed latent coordinate, bundled with its unit-interval range.
Definition (Lean source)
parameters
def
given by
fun x
=> ⟨observedLatentCoordinate W e x, by classical by_cases hx : x ∈ observedSupport G W · rcases hx with ⟨v, hv, rfl⟩ rw [observedLatentCoordinate_mix W hmix e v hv] exact hv _ (Set.mem_univ _) · simp [observedLatentCoordinate, hx]⟩
theorem measurable_observedLatentUnitCoordinate
The unit-interval-valued observed latent coordinate is measurable. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
Measurable (observedLatentUnitCoordinate W hmix e)
Proof (Lean source)
lemma measurable_observedLatentUnitCoordinate
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) (hmix : SharedDiffeomorphicMixing G θ W) (e : Fin n) :
Measurable (observedLatentUnitCoordinate W hmix e) :=
(measurable_observedLatentCoordinate W hmix e).subtype_mk
def signedInterventionCDFFamilyChart
Coordinatewise signed-CDF chart on a finite family of environment labels.
Definition (Lean source)
parameters
def
given by
fun z e => signedInterventionCDFChart s θ hpos (W.targetPerm e) (z e)
theorem signedInterventionCDFFamilyChart_measurableEmbedding
Applying the signed intervention CDF separately in finitely many coordinates is a measurable embedding. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
MeasurableEmbedding (signedInterventionCDFFamilyChart s θ hpos W A)
Proof (Lean source)
lemma signedInterventionCDFFamilyChart_measurableEmbedding
{n : ℕ} {G : DAG (Fin n)} (s : SignVector n)
(θ : Mechanism n G) (hpos : PositiveNormalizedSmoothMechanisms G θ)
(W : ObservedWorld G θ) (A : Finset (Fin n)) :
MeasurableEmbedding (signedInterventionCDFFamilyChart s θ hpos W A) := by
apply Continuous.measurableEmbedding
· apply continuous_pi
intro e
exact (continuous_signedInterventionCDFChart s θ hpos (W.targetPerm e)).comp
(continuous_apply e)
· intro x y hxy
funext e
exact (signedInterventionCDFChart_measurableEmbedding s θ hpos (W.targetPerm e)).injective
(congrFun hxy e)
Coordinatewise coercion from unit-interval-valued families to real-valued families.
theorem unitFamilyVal_measurableEmbedding
Coordinatewise coercion from a finite product of unit intervals is a measurable embedding. the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma unitFamilyVal_measurableEmbedding (A : Finset (Fin n)) :
MeasurableEmbedding (unitFamilyVal A) := by
apply Continuous.measurableEmbedding
· exact continuous_pi fun e => continuous_subtype_val.comp (continuous_apply e)
· intro x y hxy
funext e
exact Subtype.ext (congrFun hxy e)
theorem rankCoordinate_condIndep_iff_observedLatent
Pointwise identification of every selected rank with its signed intervention CDF transports the decoder's conditional-independence test exactly to the support-restricted latent coordinates. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
s :
θ :
Mechanism n G
W :
ObservedWorld G θ
hpos :
hmix :
hone :
laws :
order :
Fin n → ℕ
hrank :
∀ e v
if
v ∈ latentCube n
then
(observedLawRankCoordinate laws order e (W.mix v) : ℝ)
= signedInterventionCDF s θ (W.targetPerm e) (v (W.targetPerm e))
i :
Fin n
conclusion
CondIndepGiven (W.law 0) (observedLawRankCoordinate laws order i) (familyProjection (observedLawRankCoordinate laws order) Y) (familyProjection (observedLawRankCoordinate laws order) Z)
↔ CondIndepGiven (W.law 0) (observedLatentCoordinate W i) (familyProjection (observedLatentCoordinate W) Y) (familyProjection (observedLatentCoordinate W) Z)
Proof (Lean source)
lemma rankCoordinate_condIndep_iff_observedLatent
{n : ℕ} {G : DAG (Fin n)} (s : SignVector n)
{θ : Mechanism n G} (W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W)
(laws : ObservedProbabilityLawFamily n) (order : Fin n → ℕ)
(hrank : ∀ e v, v ∈ latentCube n →
(observedLawRankCoordinate laws order e (W.mix v) : ℝ) =
signedInterventionCDF s θ (W.targetPerm e) (v (W.targetPerm e)))
(i : Fin n) (Y Z : Finset (Fin n)) :
CondIndepGiven (W.law 0) (observedLawRankCoordinate laws order i)
(familyProjection (observedLawRankCoordinate laws order) Y)
(familyProjection (observedLawRankCoordinate laws order) Z) ↔
CondIndepGiven (W.law 0) (observedLatentCoordinate W i)
(familyProjection (observedLatentCoordinate W) Y)
(familyProjection (observedLatentCoordinate W) Z) := by
classical
let U := fun e => observedLatentUnitCoordinate W hmix e
let phi := fun e => signedInterventionCDFChart s θ hpos (W.targetPerm e)
have hsupp : ∀ᵐ x ∂W.law 0, x ∈ observedSupport G W := by
filter_upwards [Measure.support_mem_ae (μ := W.law 0)] with x hx
rwa [observedLaw_support_eq_observedSupport W hpos hmix hone] at hx
have hi : observedLawRankCoordinate laws order i =ᵐ[W.law 0] phi i ∘ U i := by
filter_upwards [hsupp] with x hx
rcases hx with ⟨v, hv, rfl⟩
apply Subtype.ext
simpa [phi, U, signedInterventionCDFChart, observedLatentUnitCoordinate,
observedLatentCoordinate_mix W hmix i v hv] using hrank i v hv
have hY : familyProjection (observedLawRankCoordinate laws order) Y =ᵐ[W.law 0]
signedInterventionCDFFamilyChart s θ hpos W Y ∘ familyProjection U Y := by
filter_upwards [hsupp] with x hx
rcases hx with ⟨v, hv, rfl⟩
funext e
apply Subtype.ext
simpa [signedInterventionCDFFamilyChart, familyProjection, U,
signedInterventionCDFChart, observedLatentUnitCoordinate,
observedLatentCoordinate_mix W hmix e v hv] using hrank e v hv
have hZ : familyProjection (observedLawRankCoordinate laws order) Z =ᵐ[W.law 0]
signedInterventionCDFFamilyChart s θ hpos W Z ∘ familyProjection U Z := by
filter_upwards [hsupp] with x hx
rcases hx with ⟨v, hv, rfl⟩
funext e
apply Subtype.ext
simpa [signedInterventionCDFFamilyChart, familyProjection, U,
signedInterventionCDFChart, observedLatentUnitCoordinate,
observedLatentCoordinate_mix W hmix e v hv] using hrank e v hv
have hU (e : Fin n) : Measurable (U e) :=
measurable_observedLatentUnitCoordinate W hmix e
have hUY (A : Finset (Fin n)) : Measurable (familyProjection U A) :=
measurable_pi_lambda _ fun e => hU e
have hcongr := condIndepGiven_congr_ae
(measurable_observedLawRankCoordinate laws order i)
((signedInterventionCDFChart_measurableEmbedding s θ hpos (W.targetPerm i)).measurable.comp
(hU i))
(measurable_rankFamilyProjection laws order Y)
((signedInterventionCDFFamilyChart_measurableEmbedding s θ hpos W Y).measurable.comp (hUY Y))
(measurable_rankFamilyProjection laws order Z)
((signedInterventionCDFFamilyChart_measurableEmbedding s θ hpos W Z).measurable.comp (hUY Z))
hi hY hZ
have hchart := condIndepGiven_measurableEmbedding_comp
(μ := W.law 0)
(U i) (familyProjection U Y) (familyProjection U Z)
(signedInterventionCDFChart_measurableEmbedding s θ hpos (W.targetPerm i))
(signedInterventionCDFFamilyChart_measurableEmbedding s θ hpos W Y)
(signedInterventionCDFFamilyChart_measurableEmbedding s θ hpos W Z)
have hval := condIndepGiven_measurableEmbedding_comp
(μ := W.law 0)
(U i) (familyProjection U Y) (familyProjection U Z)
(MeasurableEmbedding.subtype_coe measurableSet_Icc)
(unitFamilyVal_measurableEmbedding Y) (unitFamilyVal_measurableEmbedding Z)
have hvalI : ((fun z : Icc (0 : ℝ) 1 => (z : ℝ)) ∘ U i) =
observedLatentCoordinate W i := by
funext x
rfl
have hvalY : unitFamilyVal Y ∘ familyProjection U Y =
familyProjection (observedLatentCoordinate W) Y := by
funext x e
rfl
have hvalZ : unitFamilyVal Z ∘ familyProjection U Z =
familyProjection (observedLatentCoordinate W) Z := by
funext x e
rfl
have hc : CondIndepGiven (W.law 0) (phi i ∘ U i)
(signedInterventionCDFFamilyChart s θ hpos W Y ∘ familyProjection U Y)
(signedInterventionCDFFamilyChart s θ hpos W Z ∘ familyProjection U Z) ↔
CondIndepGiven (W.law 0) (U i) (familyProjection U Y) (familyProjection U Z) := by
simpa [phi] using hchart
have hv : CondIndepGiven (W.law 0) (observedLatentCoordinate W i)
(familyProjection (observedLatentCoordinate W) Y)
(familyProjection (observedLatentCoordinate W) Z) ↔
CondIndepGiven (W.law 0) (U i) (familyProjection U Y) (familyProjection U Z) := by
rw [hvalI, hvalY, hvalZ] at hval
exact hval
exact hcongr.trans (hc.trans hv.symm)
def measurableMixVersion
A globally measurable version of the mixing map, equal to it on the latent cube.
Definition (Lean source)
parameters
W :
ObservedWorld G θ
def
given by
by classical exact (latentCube n).piecewise W.mix (fun _ => 0)
theorem measurable_measurableMixVersion
The support-restricted mixing-map version is measurable. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma measurable_measurableMixVersion
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) (hmix : SharedDiffeomorphicMixing G θ W) :
Measurable (measurableMixVersion W) := by
classical
have hcube : MeasurableSet (latentCube n) := by
rw [latentCube]
measurability
unfold measurableMixVersion
exact hmix.1.continuousOn.measurable_piecewise continuous_const.continuousOn hcube
theorem observedLatent_condIndep_iff_latent
Conditional independence of support-restricted observed latent coordinates is equivalent to conditional independence of the target-permuted coordinates under the latent observational law. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
CondIndepGiven (W.law 0) (observedLatentCoordinate W i) (familyProjection (observedLatentCoordinate W) Y) (familyProjection (observedLatentCoordinate W) Z)
↔ CondIndepGiven (observationalLaw θ) (fun v => v (W.targetPerm i)) (familyProjection (fun e v => v (W.targetPerm e)) Y) (familyProjection (fun e v => v (W.targetPerm e)) Z)
Proof (Lean source)
lemma observedLatent_condIndep_iff_latent
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W)
(i : Fin n) (Y Z : Finset (Fin n)) :
CondIndepGiven (W.law 0) (observedLatentCoordinate W i)
(familyProjection (observedLatentCoordinate W) Y)
(familyProjection (observedLatentCoordinate W) Z) ↔
CondIndepGiven (observationalLaw θ) (fun v => v (W.targetPerm i))
(familyProjection (fun e v => v (W.targetPerm e)) Y)
(familyProjection (fun e v => v (W.targetPerm e)) Z) := by
classical
let mix' := measurableMixVersion W
let V := fun e : Fin n => observedLatentCoordinate W e
let X := fun e : Fin n => fun v : LatentState n => v (W.targetPerm e)
have hmix' : Measurable mix' := measurable_measurableMixVersion W hmix
have hV (e : Fin n) : Measurable (V e) := measurable_observedLatentCoordinate W hmix e
have hVproj (A : Finset (Fin n)) : Measurable (familyProjection V A) :=
measurable_pi_lambda _ fun e => hV e
have hcube : ∀ᵐ v ∂observationalLaw θ, v ∈ latentCube n :=
observationalLaw_ae_mem_latentCube hpos
have hmix_eq : W.mix =ᵐ[observationalLaw θ] mix' := by
filter_upwards [hcube] with v hv
simp [mix', measurableMixVersion, hv]
have hlaw : W.law 0 = Measure.map mix' (observationalLaw θ) := by
rw [hone.1]
exact Measure.map_congr hmix_eq
have hi : V i ∘ mix' =ᵐ[observationalLaw θ] X i := by
filter_upwards [hcube, hmix_eq] with v hv hmv
change V i (mix' v) = X i v
rw [← hmv]
exact observedLatentCoordinate_mix W hmix i v hv
have hY : familyProjection V Y ∘ mix' =ᵐ[observationalLaw θ]
familyProjection X Y := by
filter_upwards [hcube, hmix_eq] with v hv hmv
change familyProjection V Y (mix' v) = familyProjection X Y v
rw [← hmv]
funext e
exact observedLatentCoordinate_mix W hmix e v hv
have hZ : familyProjection V Z ∘ mix' =ᵐ[observationalLaw θ]
familyProjection X Z := by
filter_upwards [hcube, hmix_eq] with v hv hmv
change familyProjection V Z (mix' v) = familyProjection X Z v
rw [← hmv]
funext e
exact observedLatentCoordinate_mix W hmix e v hv
letI : IsProbabilityMeasure (observationalLaw θ) :=
observationalLaw_isProbabilityMeasure hpos
have hmap := condIndepGiven_map_iff (mu := observationalLaw θ)
hmix' (hV i) (hVproj Y) (hVproj Z)
rw [← hlaw] at hmap
have hcongr := condIndepGiven_congr_ae
((hV i).comp hmix') (measurable_pi_apply (W.targetPerm i))
((hVproj Y).comp hmix')
(measurable_pi_lambda _ fun e => measurable_pi_apply (W.targetPerm e))
((hVproj Z).comp hmix')
(measurable_pi_lambda _ fun e => measurable_pi_apply (W.targetPerm e))
hi hY hZ
exact hmap.symm.trans hcongr
def permutedValuesEquiv
Reindexing a finite coordinate family along the world's target permutation.
Definition (Lean source)
parameters
def
permutedValuesEquiv W A :
((k : {k // k ∈ A.map W.targetPerm.toEmbedding}) → ℝ) ≃ᵐ ((e : {e // e ∈ A}) → ℝ)
given by
clause 1
toFun x e := x ⟨W.targetPerm e, Finset.mem_map.mpr ⟨e, e.2, rfl⟩⟩
clause 2
invFun y k := y ⟨W.targetPerm.symm k, by rcases Finset.mem_map.mp k.2 with ⟨e, he, hek⟩ have heq : e
= W.targetPerm.symm k := by apply W.targetPerm.injective simpa using hek simpa only [← heq] using he⟩
clause 5
measurable_toFun := measurable_pi_lambda _ fun e => measurable_pi_apply _
clause 6
measurable_invFun := measurable_pi_lambda _ fun k => measurable_pi_apply _
A one-coordinate finite family is measurably equivalent to its scalar coordinate.
Definition (Lean source)
parameters
n :
ℕ
b :
Fin n
given by
clause 1
toFun x := x ⟨b, mem_singleton_self b⟩
clause 2
invFun r _ := r
clause 3
clause 4
right_inv _ := rfl
clause 5
measurable_toFun := measurable_pi_apply _
clause 6
measurable_invFun := measurable_pi_lambda _ fun _ => measurable_id
theorem mechanism_condIndepGiven_permutedOrderedLocalMarkov_env
Ordered local Markov in environment-label coordinates, reindexing the latent projection along the intervention-target permutation. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
W :
ObservedWorld G θ
hpos :
order :
Fin n → ℕ
horder :
hgraphOrder :
PermutedGraphOrdered W order
i :
Fin n
hA :
A ⊆ predecessorSet order i
hpa :
environmentParentSet W i ⊆ A
conclusion
CondIndepGiven (observationalLaw θ) (fun v => v (W.targetPerm i)) (familyProjection (fun e v => v (W.targetPerm e)) (predecessorSet order i \ A)) (familyProjection (fun e v => v (W.targetPerm e)) A)
Proof (Lean source)
lemma mechanism_condIndepGiven_permutedOrderedLocalMarkov_env
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(order : Fin n → ℕ)
(horder : IsTopologicalOrdering
(observedLawRatioGraph gaussianFeatureMap W.law) order)
(hgraphOrder : PermutedGraphOrdered W order)
(i : Fin n) (A : Finset (Fin n))
(hA : A ⊆ predecessorSet order i)
(hpa : environmentParentSet W i ⊆ A) :
CondIndepGiven (observationalLaw θ) (fun v => v (W.targetPerm i))
(familyProjection (fun e v => v (W.targetPerm e))
(predecessorSet order i \ A))
(familyProjection (fun e v => v (W.targetPerm e)) A) := by
classical
let S := predecessorSet order i
have hraw := mechanism_condIndepGiven_permutedOrderedLocalMarkov
W hpos order horder hgraphOrder i A hA hpa
have hmapdiff : S.map W.targetPerm.toEmbedding \ A.map W.targetPerm.toEmbedding =
(S \ A).map W.targetPerm.toEmbedding := by
ext k
simp [S]
rw [hmapdiff] at hraw
have htransport := (condIndepGiven_measurableEquiv_comp
(μ := observationalLaw θ)
(fun v : LatentState n => v (W.targetPerm i))
(coordinateProjection
(X := fun _ : Fin n => ℝ) ((S \ A).map W.targetPerm.toEmbedding))
(coordinateProjection
(X := fun _ : Fin n => ℝ) (A.map W.targetPerm.toEmbedding))
(MeasurableEquiv.refl ℝ) (permutedValuesEquiv W (S \ A))
(permutedValuesEquiv W A)).2 hraw
convert htransport using 1
· rfl
· funext x e
rfl
· funext x e
rfl
theorem exactRankCondIndepCharacterization_of_order
For every valid ratio-graph ordering, rank-coordinate conditional independence holds exactly when the conditioning set contains all environment-label parents. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
s :
θ :
Mechanism n G
W :
ObservedWorld G θ
hpos :
hminimal :
CausalMinimality G θ
hmix :
hone :
laws :
order :
Fin n → ℕ
horder :
hgraphOrder :
PermutedGraphOrdered W order
hrank :
∀ e v
if
v ∈ latentCube n
then
(observedLawRankCoordinate laws order e (W.mix v) : ℝ)
= signedInterventionCDF s θ (W.targetPerm e) (v (W.targetPerm e))
i A :
A ⊆ predecessorSet order i
conclusion
CondIndepGiven (W.law 0) (observedLawRankCoordinate laws order i) (familyProjection (observedLawRankCoordinate laws order) (predecessorSet order i \ A)) (familyProjection (observedLawRankCoordinate laws order) A)
↔ environmentParentSet W i ⊆ A
Proof (Lean source)
lemma exactRankCondIndepCharacterization_of_order
{n : ℕ} {G : DAG (Fin n)} (s : SignVector n)
{θ : Mechanism n G} (W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hminimal : CausalMinimality G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W)
(laws : ObservedProbabilityLawFamily n)
(order : Fin n → ℕ)
(horder : IsTopologicalOrdering
(observedLawRatioGraph gaussianFeatureMap W.law) order)
(hgraphOrder : PermutedGraphOrdered W order)
(hrank : ∀ e v, v ∈ latentCube n →
(observedLawRankCoordinate laws order e (W.mix v) : ℝ) =
signedInterventionCDF s θ (W.targetPerm e) (v (W.targetPerm e))) :
∀ i A, A ⊆ predecessorSet order i →
(CondIndepGiven (W.law 0) (observedLawRankCoordinate laws order i)
(familyProjection (observedLawRankCoordinate laws order)
(predecessorSet order i \ A))
(familyProjection (observedLawRankCoordinate laws order) A) ↔
environmentParentSet W i ⊆ A) := by
classical
intro i A hAS
let S := predecessorSet order i
let P := environmentParentSet W i
have hPS : P ⊆ S :=
environmentParentSet_subset_predecessorSet_of_transitiveClosure W horder
hgraphOrder i
have hrankObs := rankCoordinate_condIndep_iff_observedLatent
s W hpos hmix hone laws order hrank i (S \ A) A
constructor
· intro hCIrank
have hCIobs : CondIndepGiven (W.law 0) (observedLatentCoordinate W i)
(familyProjection (observedLatentCoordinate W) (S \ A))
(familyProjection (observedLatentCoordinate W) A) := hrankObs.1 hCIrank
by_contra hnsub
rw [Finset.not_subset] at hnsub
rcases hnsub with ⟨b, hbP, hbA⟩
have hmarkLat := mechanism_condIndepGiven_permutedOrderedLocalMarkov_env
W hpos order horder hgraphOrder i P hPS (Finset.Subset.rfl)
have hmarkObs := (observedLatent_condIndep_iff_latent
W hpos hmix hone i (S \ P) P).2 hmarkLat
have hiS : i ∉ S := by simp [S, predecessorSet]
have hscalarObs := parentOmission_intersection_core W hpos hmix hone
i b S P A (fun c => measurable_observedLatentCoordinate W hmix c)
hPS hAS hiS hbP hbA hCIobs hmarkObs
have hsingleObs : CondIndepGiven (W.law 0) (observedLatentCoordinate W i)
(familyProjection (observedLatentCoordinate W) {b})
(familyProjection (observedLatentCoordinate W) (P.erase b)) := by
have h := (condIndepGiven_measurableEquiv_comp
(μ := W.law 0) (observedLatentCoordinate W i)
(familyProjection (observedLatentCoordinate W) {b})
(familyProjection (observedLatentCoordinate W) (P.erase b))
(MeasurableEquiv.refl ℝ) (singletonFamilyEquiv b)
(MeasurableEquiv.refl _)).1
apply h
simpa [singletonFamilyEquiv, familyProjection, Function.comp_def] using hscalarObs
have hsingleLat := (observedLatent_condIndep_iff_latent
W hpos hmix hone i {b} (P.erase b)).1 hsingleObs
have hscalarLat : CondIndepGiven (observationalLaw θ)
(fun v => v (W.targetPerm i)) (fun v => v (W.targetPerm b))
(familyProjection (fun e v => v (W.targetPerm e)) (P.erase b)) := by
have h := (condIndepGiven_measurableEquiv_comp
(μ := observationalLaw θ) (fun v : LatentState n => v (W.targetPerm i))
(familyProjection (fun (e : Fin n) (v : LatentState n) => v (W.targetPerm e)) {b})
(familyProjection (fun (e : Fin n) (v : LatentState n) => v (W.targetPerm e)) (P.erase b))
(MeasurableEquiv.refl ℝ) (singletonFamilyEquiv b)
(MeasurableEquiv.refl _)).2 hsingleLat
simpa [singletonFamilyEquiv, familyProjection, Function.comp_def] using h
have hmapP : P.map W.targetPerm.toEmbedding = G.parents (W.targetPerm i) := by
ext k
simp [P, environmentParentSet, parents]
have hmapErase : (P.erase b).map W.targetPerm.toEmbedding =
(G.parents (W.targetPerm i)).erase (W.targetPerm b) := by
rw [Finset.map_erase, hmapP]
rfl
have hactual : CondIndepGiven (observationalLaw θ)
(fun v => v (W.targetPerm i)) (fun v => v (W.targetPerm b))
(coordinateProjection
(X := fun _ : Fin n => ℝ) ((G.parents (W.targetPerm i)).erase (W.targetPerm b))) := by
have h := (condIndepGiven_measurableEquiv_comp
(μ := observationalLaw θ) (fun v : LatentState n => v (W.targetPerm i))
(fun v : LatentState n => v (W.targetPerm b))
(coordinateProjection
(X := fun _ : Fin n => ℝ) ((P.erase b).map W.targetPerm.toEmbedding))
(MeasurableEquiv.refl ℝ) (MeasurableEquiv.refl ℝ)
(permutedValuesEquiv W (P.erase b))).1 hscalarLat
rw [← hmapErase]
exact h
have hedge : G.edge (W.targetPerm b) (W.targetPerm i) := by
simpa [P, environmentParentSet, parents] using hbP
exact hminimal hedge
((condIndepCoordinates_singletons_iff_condIndepGiven
(W.targetPerm i) (W.targetPerm b)
((G.parents (W.targetPerm i)).erase (W.targetPerm b))).2 hactual)
· intro hpa
have hlat := mechanism_condIndepGiven_permutedOrderedLocalMarkov_env
W hpos order horder hgraphOrder i A hAS hpa
have hobs := (observedLatent_condIndep_iff_latent
W hpos hmix hone i (S \ A) A).2 hlat
exact hrankObs.2 hobs
theorem exactRankCondIndepCharacterization
Transitive-closure recovery specializes the ordered characterization to every ratio order. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
s :
θ :
Mechanism n G
W :
ObservedWorld G θ
hpos :
hminimal :
CausalMinimality G θ
hmix :
hone :
laws :
htc :
= TransGen (permutedGraph G W)
order :
Fin n → ℕ
horder :
hrank :
∀ e v
if
v ∈ latentCube n
then
(observedLawRankCoordinate laws order e (W.mix v) : ℝ)
= signedInterventionCDF s θ (W.targetPerm e) (v (W.targetPerm e))
i A :
A ⊆ predecessorSet order i
conclusion
CondIndepGiven (W.law 0) (observedLawRankCoordinate laws order i) (familyProjection (observedLawRankCoordinate laws order) (predecessorSet order i \ A)) (familyProjection (observedLawRankCoordinate laws order) A)
↔ environmentParentSet W i ⊆ A
Proof (Lean source)
lemma exactRankCondIndepCharacterization
{n : ℕ} {G : DAG (Fin n)} (s : SignVector n)
{θ : Mechanism n G} (W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hminimal : CausalMinimality G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W)
(laws : ObservedProbabilityLawFamily n)
(htc : TransGen (observedLawRatioGraph gaussianFeatureMap W.law) =
TransGen (permutedGraph G W))
(order : Fin n → ℕ)
(horder : IsTopologicalOrdering
(observedLawRatioGraph gaussianFeatureMap W.law) order)
(hrank : ∀ e v, v ∈ latentCube n →
(observedLawRankCoordinate laws order e (W.mix v) : ℝ) =
signedInterventionCDF s θ (W.targetPerm e) (v (W.targetPerm e))) :
∀ i A, A ⊆ predecessorSet order i →
(CondIndepGiven (W.law 0) (observedLawRankCoordinate laws order i)
(familyProjection (observedLawRankCoordinate laws order)
(predecessorSet order i \ A))
(familyProjection (observedLawRankCoordinate laws order) A) ↔
environmentParentSet W i ⊆ A) :=
exactRankCondIndepCharacterization_of_order s W hpos hminimal hmix hone laws
order horder (permutedGraphOrdered_of_transitiveClosure W order horder htc) hrank
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.DecoderRepresentation 11 declarations These lemmas isolate the law-support and graph-relabeling parts of the compatible- representation argument.
Compatible-representation bookkeeping
These lemmas isolate the law-support and graph-relabeling parts of the compatible- representation argument. The remaining analytic step is the componentwise coordinate identification from the common law-selected ranks.
def signedInterventionCDF
The scalar rank chart selected by an intervention CDF and the prescribed ratio-score sign.
Definition (Lean source)
parameters
def
signedInterventionCDF s θ i z :
ℝ
given by
if s.value i = 1 then interventionCDF θ i z else 1 - interventionCDF θ i z
theorem interventionCDF_strictMonoOn
Strict positivity of the intervention density makes its CDF strictly increasing on the latent interval. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma interventionCDF_strictMonoOn
{n : ℕ} {G : DAG (Fin n)} (θ : Mechanism n G)
(hpos : PositiveNormalizedSmoothMechanisms G θ) (i : Fin n) :
StrictMonoOn (interventionCDF θ i) (Icc (0 : ℝ) 1) := by
intro a ha b hb hab
have hcont : ContinuousOn (θ.q i) (Icc (0 : ℝ) 1) :=
(hpos.2.2.2.1 i).continuousOn
have hcont₀a : ContinuousOn (θ.q i) (uIcc (0 : ℝ) a) := by
apply hcont.mono
rw [Set.uIcc_of_le ha.1]
exact Set.Icc_subset_Icc le_rfl ha.2
have hcontab : ContinuousOn (θ.q i) (uIcc a b) := by
apply hcont.mono
rw [Set.uIcc_of_le hab.le]
exact Set.Icc_subset_Icc ha.1 hb.2
have hintpos : 0 < ∫ z in a..b, θ.q i z := by
apply intervalIntegral.integral_pos hab
· simpa only [Set.uIcc_of_le hab.le] using hcontab
· intro z hz
exact (hpos.2.1 i z ⟨ha.1.trans hz.1.le, hz.2.trans hb.2⟩).le
· exact ⟨a, ⟨le_rfl, hab.le⟩, hpos.2.1 i a ha⟩
have hadd : (∫ z in (0 : ℝ)..a, θ.q i z) + ∫ z in a..b, θ.q i z =
∫ z in (0 : ℝ)..b, θ.q i z :=
intervalIntegral.integral_add_adjacent_intervals
hcont₀a.intervalIntegrable hcontab.intervalIntegrable
unfold interventionCDF
linarith
theorem signedInterventionCDF_injOn
Reflection for a negative score sign preserves injectivity of the scalar intervention-CDF chart. Given the stated inputs and conditions, the stated conclusion follows.
Proof (Lean source)
lemma signedInterventionCDF_injOn
{n : ℕ} {G : DAG (Fin n)} (s : SignVector n)
(θ : Mechanism n G) (hpos : PositiveNormalizedSmoothMechanisms G θ) (i : Fin n) :
InjOn (signedInterventionCDF s θ i) (Icc (0 : ℝ) 1) := by
intro a ha b hb heq
have hinj := (interventionCDF_strictMonoOn θ hpos i).injOn
unfold signedInterventionCDF at heq
split at heq
· exact hinj ha hb heq
· apply hinj ha hb
linarith
theorem observedProbabilityLawFamily_eq_of_law_eq
The probability-family packaging, selected order, and rank coordinates are literally shared by representations whose supplied law families are equal. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
observedProbabilityLawFamily W₂.law = observedProbabilityLawFamily W₁.law
Proof (Lean source)
lemma observedProbabilityLawFamily_eq_of_law_eq
{n : ℕ} {G₁ G₂ : DAG (Fin n)}
{θ₁ : Mechanism n G₁} {θ₂ : Mechanism n G₂}
(W₁ : ObservedWorld G₁ θ₁) (W₂ : ObservedWorld G₂ θ₂)
(hlaw : W₂.law = W₁.law) :
observedProbabilityLawFamily W₂.law = observedProbabilityLawFamily W₁.law := by
rw [hlaw]
theorem observedSupport_eq_of_observationalLaw_eq
Compatible smooth observed worlds with the same observational law have the same observed support. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
W₁ :
ObservedWorld G₁ θ₁
W₂ :
ObservedWorld G₂ θ₂
hpos₁ :
hmix₁ :
SharedDiffeomorphicMixing G₁ θ₁ W₁
hone₁ :
OnePerfectInterventionPerNode G₁ θ₁ W₁
hpos₂ :
hmix₂ :
SharedDiffeomorphicMixing G₂ θ₂ W₂
hone₂ :
OnePerfectInterventionPerNode G₂ θ₂ W₂
hlaw :
W₂.law = W₁.law
conclusion
observedSupport G₁ W₁ = observedSupport G₂ W₂
Proof (Lean source)
lemma observedSupport_eq_of_observationalLaw_eq
{n : ℕ} {G₁ G₂ : DAG (Fin n)}
{θ₁ : Mechanism n G₁} {θ₂ : Mechanism n G₂}
(W₁ : ObservedWorld G₁ θ₁) (W₂ : ObservedWorld G₂ θ₂)
(hpos₁ : PositiveNormalizedSmoothMechanisms G₁ θ₁)
(hmix₁ : SharedDiffeomorphicMixing G₁ θ₁ W₁)
(hone₁ : OnePerfectInterventionPerNode G₁ θ₁ W₁)
(hpos₂ : PositiveNormalizedSmoothMechanisms G₂ θ₂)
(hmix₂ : SharedDiffeomorphicMixing G₂ θ₂ W₂)
(hone₂ : OnePerfectInterventionPerNode G₂ θ₂ W₂)
(hlaw : W₂.law = W₁.law) :
observedSupport G₁ W₁ = observedSupport G₂ W₂ := by
rw [← observedLaw_support_eq_observedSupport W₁ hpos₁ hmix₁ hone₁,
← observedLaw_support_eq_observedSupport W₂ hpos₂ hmix₂ hone₂,
hlaw]
theorem aligned_edge_iff_of_permutedGraph_eq
Equality of environment-label graphs gives graph isomorphism under the intervention-target alignment permutation. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
W₁ :
ObservedWorld G₁ θ₁
W₂ :
ObservedWorld G₂ θ₂
hgraph :
permutedGraph G₁ W₁ = permutedGraph G₂ W₂
j i :
conclusion
G₂.edge ((W₁.targetPerm.symm.trans W₂.targetPerm) j) ((W₁.targetPerm.symm.trans W₂.targetPerm) i)
↔ G₁.edge j i
Proof (Lean source)
lemma aligned_edge_iff_of_permutedGraph_eq
{n : ℕ} {G₁ G₂ : DAG (Fin n)}
{θ₁ : Mechanism n G₁} {θ₂ : Mechanism n G₂}
(W₁ : ObservedWorld G₁ θ₁) (W₂ : ObservedWorld G₂ θ₂)
(hgraph : permutedGraph G₁ W₁ = permutedGraph G₂ W₂) :
∀ j i,
G₂.edge ((W₁.targetPerm.symm.trans W₂.targetPerm) j)
((W₁.targetPerm.symm.trans W₂.targetPerm) i) ↔
G₁.edge j i := by
intro j i
have h := congrFun (congrFun hgraph (W₁.targetPerm.symm j))
(W₁.targetPerm.symm i)
simpa [permutedGraph] using h.symm
theorem aligned_edge_iff_of_common_exactPruning
If law-only exact pruning identifies both representations' parent relations, their latent graphs align by the target permutation. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
W₁ :
ObservedWorld G₁ θ₁
W₂ :
ObservedWorld G₂ θ₂
laws :
hprune₁ :
∀ i
if
∀ A, MinimalAdmissibleParentSet laws (selectedTopologicalOrder laws) i A
then
A = environmentParentSet W₁ i
hprune₂ :
∀ i
if
∀ A, MinimalAdmissibleParentSet laws (selectedTopologicalOrder laws) i A
then
A = environmentParentSet W₂ i
j i :
conclusion
G₂.edge ((W₁.targetPerm.symm.trans W₂.targetPerm) j) ((W₁.targetPerm.symm.trans W₂.targetPerm) i)
↔ G₁.edge j i
Proof (Lean source)
lemma aligned_edge_iff_of_common_exactPruning
{n : ℕ} {G₁ G₂ : DAG (Fin n)}
{θ₁ : Mechanism n G₁} {θ₂ : Mechanism n G₂}
(W₁ : ObservedWorld G₁ θ₁) (W₂ : ObservedWorld G₂ θ₂)
(laws : ObservedProbabilityLawFamily n)
(hprune₁ : ∀ i,
MinimalAdmissibleParentSet laws (selectedTopologicalOrder laws) i
(environmentParentSet W₁ i) ∧
∀ A, MinimalAdmissibleParentSet laws (selectedTopologicalOrder laws) i A →
A = environmentParentSet W₁ i)
(hprune₂ : ∀ i,
MinimalAdmissibleParentSet laws (selectedTopologicalOrder laws) i
(environmentParentSet W₂ i) ∧
∀ A, MinimalAdmissibleParentSet laws (selectedTopologicalOrder laws) i A →
A = environmentParentSet W₂ i) :
∀ j i,
G₂.edge ((W₁.targetPerm.symm.trans W₂.targetPerm) j)
((W₁.targetPerm.symm.trans W₂.targetPerm) i) ↔
G₁.edge j i := by
apply aligned_edge_iff_of_permutedGraph_eq W₁ W₂
rw [← selectedParentDAG_eq_permutedGraph_of_exactPruning W₁ laws hprune₁,
← selectedParentDAG_eq_permutedGraph_of_exactPruning W₂ laws hprune₂]
theorem competitorTopologicalOrder_is_common_of_law_eq
A topological order pulled back from a compatible competitor orders the common observable ratio graph and, once the reference transitive closure is identified, the reference graph too. No cover-separation premise is required for the competitor. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
n :
W₁ :
ObservedWorld G₁ θ₁
W₂ :
ObservedWorld G₂ θ₂
hpos₂ :
hmix₂ :
SharedDiffeomorphicMixing G₂ θ₂ W₂
hone₂ :
OnePerfectInterventionPerNode G₂ θ₂ W₂
hlaw :
W₂.law = W₁.law
htc₁ :
= TransGen (permutedGraph G₁ W₁)
conclusion
order :
fun e ↦ G₂.topoOrder (W₂.targetPerm e) (: Fin n → ℕ)
j i :
permutedGraph G₁ W₁ j i
order j < order i
j i :
permutedGraph G₂ W₂ j i
order j < order i
i :
environmentParentSet W₁ i ⊆ predecessorSet order i
i :
environmentParentSet W₂ i ⊆ predecessorSet order i
Proof (Lean source)
lemma competitorTopologicalOrder_is_common_of_law_eq
{n : ℕ} {G₁ G₂ : DAG (Fin n)}
{s : SignVector n} {θ₁ : Mechanism n G₁} {θ₂ : Mechanism n G₂}
(W₁ : ObservedWorld G₁ θ₁) (W₂ : ObservedWorld G₂ θ₂)
(hpos₂ : PositiveNormalizedSmoothMechanisms G₂ θ₂)
(hmix₂ : SharedDiffeomorphicMixing G₂ θ₂ W₂)
(hone₂ : OnePerfectInterventionPerNode G₂ θ₂ W₂)
(hlaw : W₂.law = W₁.law)
(htc₁ : TransGen (observedLawRatioGraph gaussianFeatureMap W₁.law) =
TransGen (permutedGraph G₁ W₁)) :
let order : Fin n → ℕ := fun e ↦ G₂.topoOrder (W₂.targetPerm e)
IsTopologicalOrdering (observedLawRatioGraph gaussianFeatureMap W₁.law) order ∧
(∀ ⦃j i⦄, permutedGraph G₁ W₁ j i → order j < order i) ∧
(∀ ⦃j i⦄, permutedGraph G₂ W₂ j i → order j < order i) ∧
(∀ i, environmentParentSet W₁ i ⊆ predecessorSet order i) ∧
(∀ i, environmentParentSet W₂ i ⊆ predecessorSet order i) := by
let order : Fin n → ℕ := fun e ↦ G₂.topoOrder (W₂.targetPerm e)
have hinj : Injective order :=
G₂.topoOrder_injective.comp W₂.targetPerm.injective
have hratio : ∀ ⦃j i⦄,
observedLawRatioGraph gaussianFeatureMap W₁.law j i → order j < order i := by
intro j i hji
have hji₂ : observedLawRatioGraph gaussianFeatureMap W₂.law j i := by
simpa only [hlaw] using hji
exact G₂.isAncestor_topoOrder_lt
(observedLawRatioGraph_edge_isAncestor (s := s) W₂ hpos₂ hmix₂ hone₂ hji₂)
have hpath : ∀ ⦃j i⦄,
TransGen (observedLawRatioGraph gaussianFeatureMap W₁.law) j i →
order j < order i := by
intro j i hji
induction hji with
| single h => exact hratio h
| tail _ h ih => exact lt_trans ih (hratio h)
have hedge₁ : ∀ ⦃j i⦄, permutedGraph G₁ W₁ j i → order j < order i := by
intro j i hji
apply hpath
rw [htc₁]
exact Relation.TransGen.single hji
have hedge₂ : ∀ ⦃j i⦄, permutedGraph G₂ W₂ j i → order j < order i := by
intro j i hji
exact G₂.isAncestor_topoOrder_lt (Causalean.DAG.isAncestor.edge hji)
refine ⟨⟨hinj, hratio⟩, hedge₁, hedge₂, ?_, ?_⟩
· intro j i hji
simpa [predecessorSet, order] using hedge₁ (by
simpa [environmentParentSet, permutedGraph] using hji)
· intro j i hji
simpa [predecessorSet, order] using hedge₂ (by
simpa [environmentParentSet, permutedGraph] using hji)
theorem signedInterventionCDF_eq_of_common_rank
Two representations realizing the same law-only rank coordinate have equal signed scalar intervention-CDF coordinates at every common support point. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
n :
W₁ :
ObservedWorld G₁ θ₁
W₂ :
ObservedWorld G₂ θ₂
hmix₁ :
SharedDiffeomorphicMixing G₁ θ₁ W₁
hmix₂ :
SharedDiffeomorphicMixing G₂ θ₂ W₂
laws :
order :
Fin n → ℕ
hrank₁ :
∀ e v
if
v ∈ latentCube n
then
(observedLawRankCoordinate laws order e (W₁.mix v) : ℝ)
= signedInterventionCDF s θ₁ (W₁.targetPerm e) (v (W₁.targetPerm e))
hrank₂ :
∀ e v
if
v ∈ latentCube n
then
(observedLawRankCoordinate laws order e (W₂.mix v) : ℝ)
= signedInterventionCDF s θ₂ (W₂.targetPerm e) (v (W₂.targetPerm e))
x :
hx₁ :
x ∈ observedSupport G₁ W₁
hx₂ :
x ∈ observedSupport G₂ W₂
e :
Fin n
conclusion
signedInterventionCDF s θ₁ (W₁.targetPerm e) (W₁.unmix x (W₁.targetPerm e))
= signedInterventionCDF s θ₂ (W₂.targetPerm e) (W₂.unmix x (W₂.targetPerm e))
Proof (Lean source)
lemma signedInterventionCDF_eq_of_common_rank
{n : ℕ} {G₁ G₂ : DAG (Fin n)}
{s : SignVector n} {θ₁ : Mechanism n G₁} {θ₂ : Mechanism n G₂}
(W₁ : ObservedWorld G₁ θ₁) (W₂ : ObservedWorld G₂ θ₂)
(hmix₁ : SharedDiffeomorphicMixing G₁ θ₁ W₁)
(hmix₂ : SharedDiffeomorphicMixing G₂ θ₂ W₂)
(laws : ObservedProbabilityLawFamily n) (order : Fin n → ℕ)
(hrank₁ : ∀ e v, v ∈ latentCube n →
(observedLawRankCoordinate laws order e (W₁.mix v) : ℝ) =
signedInterventionCDF s θ₁ (W₁.targetPerm e) (v (W₁.targetPerm e)))
(hrank₂ : ∀ e v, v ∈ latentCube n →
(observedLawRankCoordinate laws order e (W₂.mix v) : ℝ) =
signedInterventionCDF s θ₂ (W₂.targetPerm e) (v (W₂.targetPerm e)))
(x : LatentState n) (hx₁ : x ∈ observedSupport G₁ W₁)
(hx₂ : x ∈ observedSupport G₂ W₂) (e : Fin n) :
signedInterventionCDF s θ₁ (W₁.targetPerm e)
(W₁.unmix x (W₁.targetPerm e)) =
signedInterventionCDF s θ₂ (W₂.targetPerm e)
(W₂.unmix x (W₂.targetPerm e)) := by
have hv₁ : W₁.unmix x ∈ latentCube n := by
rcases hx₁ with ⟨v, hv, rfl⟩
simpa only [hmix₁.2.2.1 v hv] using hv
have hv₂ : W₂.unmix x ∈ latentCube n := by
rcases hx₂ with ⟨v, hv, rfl⟩
simpa only [hmix₂.2.2.1 v hv] using hv
have h₁ := hrank₁ e (W₁.unmix x) hv₁
have h₂ := hrank₂ e (W₂.unmix x) hv₂
rw [hmix₁.2.2.2 x hx₁] at h₁
rw [hmix₂.2.2.2 x hx₂] at h₂
exact h₁.symm.trans h₂
theorem alignedCoordinate_eq_of_common_rank
Equality of an aligned source coordinate forces equality of the competitor coordinate once both worlds realize the common law-only rank. This is the componentwise-dependence leaf of the representation argument. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
n :
W₁ :
ObservedWorld G₁ θ₁
W₂ :
ObservedWorld G₂ θ₂
hmix₁ :
SharedDiffeomorphicMixing G₁ θ₁ W₁
hpos₂ :
hmix₂ :
SharedDiffeomorphicMixing G₂ θ₂ W₂
laws :
order :
Fin n → ℕ
hrank₁ :
∀ e v
if
v ∈ latentCube n
then
(observedLawRankCoordinate laws order e (W₁.mix v) : ℝ)
= signedInterventionCDF s θ₁ (W₁.targetPerm e) (v (W₁.targetPerm e))
hrank₂ :
∀ e v
if
v ∈ latentCube n
then
(observedLawRankCoordinate laws order e (W₂.mix v) : ℝ)
= signedInterventionCDF s θ₂ (W₂.targetPerm e) (v (W₂.targetPerm e))
x y :
hx₁ :
x ∈ observedSupport G₁ W₁
hx₂ :
x ∈ observedSupport G₂ W₂
hy₁ :
y ∈ observedSupport G₁ W₁
hy₂ :
y ∈ observedSupport G₂ W₂
e :
Fin n
hcoord :
W₁.unmix x (W₁.targetPerm e) = W₁.unmix y (W₁.targetPerm e)
conclusion
W₂.unmix x (W₂.targetPerm e) = W₂.unmix y (W₂.targetPerm e)
Proof (Lean source)
lemma alignedCoordinate_eq_of_common_rank
{n : ℕ} {G₁ G₂ : DAG (Fin n)}
{s : SignVector n} {θ₁ : Mechanism n G₁} {θ₂ : Mechanism n G₂}
(W₁ : ObservedWorld G₁ θ₁) (W₂ : ObservedWorld G₂ θ₂)
(hmix₁ : SharedDiffeomorphicMixing G₁ θ₁ W₁)
(hpos₂ : PositiveNormalizedSmoothMechanisms G₂ θ₂)
(hmix₂ : SharedDiffeomorphicMixing G₂ θ₂ W₂)
(laws : ObservedProbabilityLawFamily n) (order : Fin n → ℕ)
(hrank₁ : ∀ e v, v ∈ latentCube n →
(observedLawRankCoordinate laws order e (W₁.mix v) : ℝ) =
signedInterventionCDF s θ₁ (W₁.targetPerm e) (v (W₁.targetPerm e)))
(hrank₂ : ∀ e v, v ∈ latentCube n →
(observedLawRankCoordinate laws order e (W₂.mix v) : ℝ) =
signedInterventionCDF s θ₂ (W₂.targetPerm e) (v (W₂.targetPerm e)))
(x y : LatentState n)
(hx₁ : x ∈ observedSupport G₁ W₁) (hx₂ : x ∈ observedSupport G₂ W₂)
(hy₁ : y ∈ observedSupport G₁ W₁) (hy₂ : y ∈ observedSupport G₂ W₂)
(e : Fin n)
(hcoord : W₁.unmix x (W₁.targetPerm e) =
W₁.unmix y (W₁.targetPerm e)) :
W₂.unmix x (W₂.targetPerm e) = W₂.unmix y (W₂.targetPerm e) := by
have hsx := signedInterventionCDF_eq_of_common_rank W₁ W₂ hmix₁ hmix₂
laws order hrank₁ hrank₂ x hx₁ hx₂ e
have hsy := signedInterventionCDF_eq_of_common_rank W₁ W₂ hmix₁ hmix₂
laws order hrank₁ hrank₂ y hy₁ hy₂ e
apply signedInterventionCDF_injOn s θ₂ hpos₂ (W₂.targetPerm e)
· rcases hx₂ with ⟨v, hv, rfl⟩
simpa only [hmix₂.2.2.1 v hv] using hv (W₂.targetPerm e) (Set.mem_univ _)
· rcases hy₂ with ⟨v, hv, rfl⟩
simpa only [hmix₂.2.2.1 v hv] using hv (W₂.targetPerm e) (Set.mem_univ _)
· calc
signedInterventionCDF s θ₂ (W₂.targetPerm e)
(W₂.unmix x (W₂.targetPerm e)) =
signedInterventionCDF s θ₁ (W₁.targetPerm e)
(W₁.unmix x (W₁.targetPerm e)) := hsx.symm
_ = signedInterventionCDF s θ₁ (W₁.targetPerm e)
(W₁.unmix y (W₁.targetPerm e)) := congrArg _ hcoord
_ = signedInterventionCDF s θ₂ (W₂.targetPerm e)
(W₂.unmix y (W₂.targetPerm e)) := hsy
theorem signedInterventionCDF_eq_of_common_law_and_transitiveClosures
Once the common law-selected order is known to be topological for both latent transitive closures, the assembled equation-(11)--(12) rank formula identifies their signed scalar CDF coordinates pointwise on the common support. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
s :
θ₁ :
Mechanism n G₁
θ₂ :
Mechanism n G₂
W₁ :
ObservedWorld G₁ θ₁
W₂ :
ObservedWorld G₂ θ₂
hpos₁ :
hmix₁ :
SharedDiffeomorphicMixing G₁ θ₁ W₁
hone₁ :
OnePerfectInterventionPerNode G₁ θ₁ W₁
hsign₁ :
FixedOwnDerivativeSign G₁ s θ₁
hpos₂ :
hmix₂ :
SharedDiffeomorphicMixing G₂ θ₂ W₂
hone₂ :
OnePerfectInterventionPerNode G₂ θ₂ W₂
hsign₂ :
FixedOwnDerivativeSign G₂ s θ₂
hlaw :
W₂.law = W₁.law
htc₁ :
= TransGen (permutedGraph G₁ W₁)
htc₂ :
= TransGen (permutedGraph G₂ W₂)
order :
Fin n → ℕ
horder :
x :
hx₁ :
x ∈ observedSupport G₁ W₁
hx₂ :
x ∈ observedSupport G₂ W₂
e :
Fin n
conclusion
signedInterventionCDF s θ₁ (W₁.targetPerm e) (W₁.unmix x (W₁.targetPerm e))
= signedInterventionCDF s θ₂ (W₂.targetPerm e) (W₂.unmix x (W₂.targetPerm e))
Proof (Lean source)
lemma signedInterventionCDF_eq_of_common_law_and_transitiveClosures
{n : ℕ} {G₁ G₂ : DAG (Fin n)}
(s : SignVector n) {θ₁ : Mechanism n G₁} {θ₂ : Mechanism n G₂}
(W₁ : ObservedWorld G₁ θ₁) (W₂ : ObservedWorld G₂ θ₂)
(hpos₁ : PositiveNormalizedSmoothMechanisms G₁ θ₁)
(hmix₁ : SharedDiffeomorphicMixing G₁ θ₁ W₁)
(hone₁ : OnePerfectInterventionPerNode G₁ θ₁ W₁)
(hsign₁ : FixedOwnDerivativeSign G₁ s θ₁)
(hpos₂ : PositiveNormalizedSmoothMechanisms G₂ θ₂)
(hmix₂ : SharedDiffeomorphicMixing G₂ θ₂ W₂)
(hone₂ : OnePerfectInterventionPerNode G₂ θ₂ W₂)
(hsign₂ : FixedOwnDerivativeSign G₂ s θ₂)
(hlaw : W₂.law = W₁.law)
(htc₁ : TransGen (observedLawRatioGraph gaussianFeatureMap W₁.law) =
TransGen (permutedGraph G₁ W₁))
(htc₂ : TransGen (observedLawRatioGraph gaussianFeatureMap W₂.law) =
TransGen (permutedGraph G₂ W₂))
(order : Fin n → ℕ)
(horder : IsTopologicalOrdering
(observedLawRatioGraph gaussianFeatureMap W₁.law) order)
(x : LatentState n) (hx₁ : x ∈ observedSupport G₁ W₁)
(hx₂ : x ∈ observedSupport G₂ W₂) (e : Fin n) :
signedInterventionCDF s θ₁ (W₁.targetPerm e)
(W₁.unmix x (W₁.targetPerm e)) =
signedInterventionCDF s θ₂ (W₂.targetPerm e)
(W₂.unmix x (W₂.targetPerm e)) := by
let laws := observedProbabilityLawFamily W₁.law
have horder₂ : IsTopologicalOrdering
(observedLawRatioGraph gaussianFeatureMap W₂.law) order := by
simpa only [hlaw] using horder
apply signedInterventionCDF_eq_of_common_rank W₁ W₂ hmix₁ hmix₂ laws order
(fun e v hv ↦ ?_) (fun e v hv ↦ ?_) x hx₁ hx₂ e
· simpa only [laws, signedInterventionCDF] using
(exactRatioDecoder_continuousRankClauses s W₁ hpos₁ hmix₁ hone₁ hsign₁
htc₁ order horder e).2.2.2.2 v hv
· simpa only [laws, hlaw, signedInterventionCDF] using
(exactRatioDecoder_continuousRankClauses s W₂ hpos₂ hmix₂ hone₂ hsign₂
htc₂ order horder₂ e).2.2.2.2 v hv
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.DecoderSupport 5 declarations This file proves that every predecessor-score vector produced by a latent cube point lies in the support of its observed interventional predecessor law.
Realized predecessor-score support
This file proves that every predecessor-score vector produced by a latent cube point lies in the support of its observed interventional predecessor law. It is the support bridge needed to turn continuous-version uniqueness into the pointwise equation-(11) identity.
theorem observedInterventionalLaw_support_eq_observedSupport
Every supplied observed single-target intervention law has the full observed model support, because its latent density is strictly positive on the full cube. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
Measure.support (W.law e.succ) = observedSupport G W
Proof (Lean source)
lemma observedInterventionalLaw_support_eq_observedSupport
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W) (e : Fin n) :
Measure.support (W.law e.succ) = observedSupport G W := by
let μ := interventionalLaw θ (W.targetPerm e)
have hcube_compact : IsCompact (latentCube n) := by
rw [latentCube]
exact isCompact_univ_pi fun _ => isCompact_Icc
have hsupport_closed : IsClosed (observedSupport G W) :=
(hcube_compact.image_of_continuousOn hmix.1.continuousOn).isClosed
have hμ_cube : ∀ᵐ v ∂μ, v ∈ latentCube n := by
filter_upwards [Measure.support_mem_ae (μ := μ)] with v hv
simpa [μ, interventionalLaw_support_eq_latentCube W hpos e] using hv
have hcube_meas : MeasurableSet (latentCube n) := by
rw [latentCube]
exact MeasurableSet.pi Set.countable_univ (fun _ _ => measurableSet_Icc)
have hmixAE : AEMeasurable W.mix μ :=
aemeasurable_of_supportMeasurableOn hcube_meas
(ae_iff.mp hμ_cube) hmix.1.continuousOn.domRestrict.measurable
have hfull : W.law e.succ (observedSupport G W)ᶜ = 0 := by
rw [hone.2.1 e,
Measure.map_apply_of_aemeasurable hmixAE hsupport_closed.isOpen_compl.measurableSet]
apply measure_mono_null (t := (latentCube n)ᶜ)
· intro v hv hvcube
exact hv ⟨v, hvcube, rfl⟩
· exact ae_iff.mp hμ_cube
apply Set.Subset.antisymm
· exact Measure.support_subset_of_isClosed hsupport_closed (ae_iff.mpr hfull)
· rintro x ⟨v, hv, rfl⟩
rw [Measure.mem_support_iff_forall]
intro U hU
rcases mem_nhds_iff.mp hU with ⟨O, hOU, hOopen, hmixO⟩
rw [hone.2.1 e]
apply lt_of_lt_of_le _ (measure_mono hOU)
rw [Measure.map_apply_of_aemeasurable hmixAE hOopen.measurableSet]
have hpre : W.mix ⁻¹' O ∈ nhdsWithin v (latentCube n) :=
(hmix.1.continuousOn v hv) (hOopen.mem_nhds hmixO)
rcases mem_nhdsWithin_iff_exists_mem_nhds_inter.mp hpre with ⟨V, hV, hVsub⟩
have hVpos : 0 < μ V :=
(Measure.mem_support_iff_forall v).mp
((interventionalLaw_support_eq_latentCube W hpos e).symm.subset hv) V hV
rw [← Measure.measure_inter_eq_of_ae hμ_cube] at hVpos
exact hVpos.trans_le (measure_mono (by simpa [inter_comm] using hVsub))
theorem continuousOn_observedLawLogRatio_observedSupport
The law-selected log ratio is continuous on the common observed support. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma continuousOn_observedLawLogRatio_observedSupport
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W) (i : Fin n) :
ContinuousOn (observedLawLogRatio W.law i) (observedSupport G W) := by
have hex := exists_continuousObservedRatioVersion W hpos hmix hone i
have hselected := observedContinuousRatio_isVersion W.law i hex
have hsmooth := smoothObservedRatio_isVersion W hpos hmix hone i
have heq := hselected.eqOn hsmooth
rw [← observedLaw_support_eq_observedSupport W hpos hmix hone]
apply ContinuousOn.log hselected.2
intro x hx
rw [heq hx]
rw [observedLaw_support_eq_observedSupport W hpos hmix hone] at hx
rcases hx with ⟨v, hv, rfl⟩
rw [smoothObservedRatio, hmix.2.2.1 v hv]
exact div_ne_zero (ne_of_gt (hpos.2.1 _ _ (hv _ (Set.mem_univ _))))
(ne_of_gt (hpos.1 _ _ hv))
theorem continuousOn_predecessorLogRatioProjection_observedSupport
The vector of predecessor log ratios is continuous on the common observed support. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
ContinuousOn (familyProjection (observedLawLogRatio W.law) (predecessorSet order i)) (observedSupport G W)
Proof (Lean source)
lemma continuousOn_predecessorLogRatioProjection_observedSupport
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W)
(order : Fin n → ℕ) (i : Fin n) :
ContinuousOn
(familyProjection (observedLawLogRatio W.law) (predecessorSet order i))
(observedSupport G W) := by
rw [continuousOn_pi]
intro j
exact continuousOn_observedLawLogRatio_observedSupport W hpos hmix hone j
A function continuous on the support of a measure sends support points into the support of its pushforward. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
X Y :
hf :
AEMeasurable f μ
hcont :
ContinuousOn f (Measure.support μ)
x :
X
hx :
x ∈ Measure.support μ
conclusion
f x ∈ Measure.support (Measure.map f μ)
Proof (Lean source)
lemma continuousOn_map_mem_support
{X Y : Type*} [TopologicalSpace X] [MeasurableSpace X]
[HereditarilyLindelofSpace X]
[TopologicalSpace Y] [MeasurableSpace Y] [BorelSpace Y]
{μ : Measure X} {f : X → Y} (hf : AEMeasurable f μ)
(hcont : ContinuousOn f (Measure.support μ)) {x : X}
(hx : x ∈ Measure.support μ) : f x ∈ Measure.support (Measure.map f μ) := by
rw [Measure.mem_support_iff_forall]
intro U hU
rcases mem_nhds_iff.mp hU with ⟨O, hOU, hOopen, hfxO⟩
apply lt_of_lt_of_le _ (measure_mono hOU)
rw [Measure.map_apply_of_aemeasurable hf hOopen.measurableSet]
have hpre : f ⁻¹' O ∈ nhdsWithin x (Measure.support μ) :=
(hcont x hx) (hOopen.mem_nhds hfxO)
rcases mem_nhdsWithin_iff_exists_mem_nhds_inter.mp hpre with ⟨V, hV, hVsub⟩
have hVpos : 0 < μ V := (Measure.mem_support_iff_forall x).mp hx V hV
rw [← Measure.measure_inter_eq_of_ae (Measure.support_mem_ae (μ := μ))] at hVpos
exact hVpos.trans_le (measure_mono (fun y hy => hVsub ⟨hy.2, hy.1⟩))
theorem observedPredecessorLogRatio_mem_support
Every predecessor-score vector realized by a latent cube point belongs to the support of the observed predecessor law under the current intervention. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
familyProjection (observedLawLogRatio W.law) (predecessorSet order i) (W.mix v) ∈ Measure.support (Measure.map (familyProjection (observedLawLogRatio W.law) (predecessorSet order i)) (W.law i.succ))
Proof (Lean source)
lemma observedPredecessorLogRatio_mem_support
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W)
{order : Fin n → ℕ} (i : Fin n) (v : LatentState n)
(hv : v ∈ latentCube n) :
familyProjection (observedLawLogRatio W.law) (predecessorSet order i) (W.mix v) ∈
Measure.support (Measure.map
(familyProjection (observedLawLogRatio W.law) (predecessorSet order i))
(W.law i.succ)) := by
let pred := familyProjection (observedLawLogRatio W.law) (predecessorSet order i)
have hpred : Measurable pred := by
dsimp only [pred, familyProjection]
apply measurable_pi_lambda
intro j
exact measurable_observedLawLogRatio W.law j
have hx : W.mix v ∈ Measure.support (W.law i.succ) := by
rw [observedInterventionalLaw_support_eq_observedSupport W hpos hmix hone i]
exact ⟨v, hv, rfl⟩
apply continuousOn_map_mem_support (μ := W.law i.succ) (f := pred)
hpred.aemeasurable
(show ContinuousOn pred (Measure.support (W.law i.succ)) by
rw [observedInterventionalLaw_support_eq_observedSupport W hpos hmix hone i]
exact continuousOn_predecessorLogRatioProjection_observedSupport
W hpos hmix hone order i)
hx
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.DecoderTriangularInverse 12 declarations This file packages equations (7)--(9): predecessor latent coordinates form a compact cube, their triangular log-ratio score map is a continuous injection, and hence it has a continuous inverse on its image.
Compact triangular predecessor-score inverse
This file packages equations (7)--(9): predecessor latent coordinates form a compact cube, their triangular log-ratio score map is a continuous injection, and hence it has a continuous inverse on its image.
The compact product cube of latent coordinates indexed by predecessors of i.
Definition (Lean source)
parameters
def
PredecessorLatentCube order i :
Type
given by
(j : {j // j ∈ predecessorSet order i}) → Icc (0 : ℝ) 1
def predecessorCubeLatentState
Embed predecessor coordinates into the full latent cube, filling nonpredecessors with zero.
Definition (Lean source)
parameters
def
given by
fun k
=> if h : W.targetPerm.symm k ∈ predecessorSet order i then z ⟨W.targetPerm.symm k, h⟩ else 0
def predecessorLatentCubeOfState
Restrict a latent cube point to the coordinates indexed by decoder predecessors.
Definition (Lean source)
parameters
def
given by
fun j => ⟨v (W.targetPerm j), hv _ (Set.mem_univ _)⟩
theorem predecessorCubeLatentState_ofState_apply
Embedding the predecessor restriction back into the ambient cube preserves every predecessor coordinate. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
W :
ObservedWorld G θ
order :
Fin n → ℕ
i :
Fin n
v :
hv :
v ∈ latentCube n
j :
Fin n
hj :
j ∈ predecessorSet order i
conclusion
predecessorCubeLatentState W order i (predecessorLatentCubeOfState W order i v hv) (W.targetPerm j)
= v (W.targetPerm j)
Proof (Lean source)
lemma predecessorCubeLatentState_ofState_apply
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) (order : Fin n → ℕ) (i : Fin n)
(v : LatentState n) (hv : v ∈ latentCube n)
(j : Fin n) (hj : j ∈ predecessorSet order i) :
predecessorCubeLatentState W order i
(predecessorLatentCubeOfState W order i v hv) (W.targetPerm j) =
v (W.targetPerm j) := by
simp [predecessorCubeLatentState, predecessorLatentCubeOfState, hj]
def predecessorScoreMap
The triangular predecessor log-ratio map, written directly in latent coordinates.
Definition (Lean source)
parameters
def
given by
fun j
=> log (θ.q (W.targetPerm j) ((predecessorCubeLatentState W order i z) (W.targetPerm j)) / θ.p (W.targetPerm j) (predecessorCubeLatentState W order i z))
theorem predecessorCubeLatentState_mem_latentCube
The zero-filled predecessor-coordinate embedding always lies in the full latent cube. the stated conclusion follows.
Formal statement
hypotheses
conclusion
predecessorCubeLatentState W order i z ∈ latentCube n
Proof (Lean source)
lemma predecessorCubeLatentState_mem_latentCube
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) (order : Fin n → ℕ) (i : Fin n)
(z : PredecessorLatentCube order i) :
predecessorCubeLatentState W order i z ∈ latentCube n := by
intro k hk
simp only [predecessorCubeLatentState]
split
· exact (z _).property
· exact ⟨by norm_num, by norm_num⟩
theorem predecessorScoreMap_of_latentState
On a latent cube point, the compact triangular score map is exactly the observed predecessor log-ratio projection pulled back through the mixing map. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
W :
ObservedWorld G θ
hpos :
hmix :
hone :
order :
Fin n → ℕ
horder :
hgraphOrder :
PermutedGraphOrdered W order
i :
Fin n
v :
hv :
v ∈ latentCube n
conclusion
predecessorScoreMap W order i (predecessorLatentCubeOfState W order i v hv)
Proof (Lean source)
lemma predecessorScoreMap_of_latentState
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W)
{order : Fin n → ℕ}
(horder : IsTopologicalOrdering (observedLawRatioGraph gaussianFeatureMap W.law) order)
(hgraphOrder : PermutedGraphOrdered W order)
(i : Fin n) (v : LatentState n) (hv : v ∈ latentCube n) :
predecessorScoreMap W order i (predecessorLatentCubeOfState W order i v hv) =
familyProjection (observedLawLogRatio W.law) (predecessorSet order i) (W.mix v) := by
funext j
unfold familyProjection
rw [observedLawLogRatio_comp_mix_eq W hpos hmix hone j v hv]
unfold predecessorScoreMap
congr 2
· exact congrArg (θ.q (W.targetPerm j))
(predecessorCubeLatentState_ofState_apply W order i v hv j j.property)
· apply θ.parent_local (W.targetPerm j)
· exact predecessorCubeLatentState_ofState_apply W order i v hv j j.property
· intro a ha
let k := W.targetPerm.symm a
have hkParent : k ∈ environmentParentSet W j := by
simpa [k, environmentParentSet, parents] using ha
have hkBeforeJ := environmentParentSet_subset_predecessorSet_of_transitiveClosure
W horder hgraphOrder j hkParent
have hjBeforeI : order j < order i := by
simpa [predecessorSet] using j.property
have hkBeforeI : k ∈ predecessorSet order i := by
simpa [predecessorSet] using
(show order k < order i from (by
simpa [predecessorSet] using hkBeforeJ : order k < order j).trans hjBeforeI)
simpa [k] using predecessorCubeLatentState_ofState_apply
W order i v hv k hkBeforeI
theorem observedPredecessorLogRatio_mem_scoreRange
Every predecessor log-ratio vector realized by a latent cube point belongs to the image of the compact triangular predecessor-score map. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
W :
ObservedWorld G θ
hpos :
hmix :
hone :
order :
Fin n → ℕ
horder :
hgraphOrder :
PermutedGraphOrdered W order
i :
Fin n
v :
hv :
v ∈ latentCube n
conclusion
familyProjection (observedLawLogRatio W.law) (predecessorSet order i) (W.mix v) ∈ range (predecessorScoreMap W order i)
Proof (Lean source)
lemma observedPredecessorLogRatio_mem_scoreRange
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W)
{order : Fin n → ℕ}
(horder : IsTopologicalOrdering (observedLawRatioGraph gaussianFeatureMap W.law) order)
(hgraphOrder : PermutedGraphOrdered W order)
(i : Fin n) (v : LatentState n) (hv : v ∈ latentCube n) :
familyProjection (observedLawLogRatio W.law) (predecessorSet order i) (W.mix v) ∈
range (predecessorScoreMap W order i) := by
refine ⟨predecessorLatentCubeOfState W order i v hv, ?_⟩
exact predecessorScoreMap_of_latentState W hpos hmix hone horder hgraphOrder i v hv
theorem continuous_predecessorScoreMap
Smooth positive mechanisms make the triangular predecessor score map continuous. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
Continuous (predecessorScoreMap W order i)
Proof (Lean source)
lemma continuous_predecessorScoreMap
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) (hpos : PositiveNormalizedSmoothMechanisms G θ)
(order : Fin n → ℕ) (i : Fin n) : Continuous (predecessorScoreMap W order i) := by
have hlatent : Continuous (predecessorCubeLatentState W order i) := by
apply continuous_pi
intro k
simp only [predecessorCubeLatentState]
split <;> fun_prop
apply continuous_pi
intro j
have hcoord : ∀ z, predecessorCubeLatentState W order i z (W.targetPerm j) ∈
Icc (0 : ℝ) 1 := by
intro z
simpa only [predecessorCubeLatentState, Equiv.symm_apply_apply,
dif_pos j.property] using (z j).property
have hpne : ∀ z, θ.p (W.targetPerm j)
(predecessorCubeLatentState W order i z) ≠ 0 := fun z =>
ne_of_gt (hpos.1 _ _ (predecessorCubeLatentState_mem_latentCube W order i z))
have hqne : ∀ z, θ.q (W.targetPerm j)
(predecessorCubeLatentState W order i z (W.targetPerm j)) ≠ 0 := fun z =>
ne_of_gt (hpos.2.1 _ _ (hcoord z))
apply Continuous.log _ (fun z => div_ne_zero (hqne z) (hpne z))
apply Continuous.div _ _ hpne
· exact (hpos.2.2.2.1 (W.targetPerm j)).continuousOn.comp_continuous
(continuous_apply (W.targetPerm j) |>.comp hlatent) hcoord
· exact (hpos.2.2.1 (W.targetPerm j)).continuousOn.comp_continuous hlatent
(predecessorCubeLatentState_mem_latentCube W order i)
theorem predecessorScoreMap_injective
The prescribed own-score signs make the triangular predecessor score map injective. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
s :
θ :
Mechanism n G
W :
ObservedWorld G θ
hpos :
hmix :
hone :
hsign :
FixedOwnDerivativeSign G s θ
order :
Fin n → ℕ
horder :
hgraphOrder :
PermutedGraphOrdered W order
i :
Fin n
conclusion
Injective (predecessorScoreMap W order i)
Proof (Lean source)
lemma predecessorScoreMap_injective
{n : ℕ} {G : DAG (Fin n)} (s : SignVector n)
{θ : Mechanism n G} (W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W)
(hsign : FixedOwnDerivativeSign G s θ)
{order : Fin n → ℕ}
(horder : IsTopologicalOrdering (observedLawRatioGraph gaussianFeatureMap W.law) order)
(hgraphOrder : PermutedGraphOrdered W order)
(i : Fin n) : Injective (predecessorScoreMap W order i) := by
intro z w heq
let vz := predecessorCubeLatentState W order i z
let vw := predecessorCubeLatentState W order i w
have hvz : vz ∈ latentCube n := predecessorCubeLatentState_mem_latentCube W order i z
have hvw : vw ∈ latentCube n := predecessorCubeLatentState_mem_latentCube W order i w
have heq' : familyProjection (observedLawLogRatio W.law) (predecessorSet order i)
(W.mix vz) = familyProjection (observedLawLogRatio W.law)
(predecessorSet order i) (W.mix vw) := by
funext j
simp only [familyProjection]
rw [observedLawLogRatio_comp_mix_eq W hpos hmix hone j vz hvz,
observedLawLogRatio_comp_mix_eq W hpos hmix hone j vw hvw]
exact congrFun heq j
have hcoords := predecessorLogRatioProjection_injective_on_predecessors
s W hpos hmix hone hsign horder hgraphOrder i hvz hvw heq'
funext j
apply Subtype.ext
simpa only [vz, vw, predecessorCubeLatentState, Equiv.symm_apply_apply,
dif_pos j.property] using hcoords j j.property
The compact triangular predecessor score map is a homeomorphism onto its image.
Definition (Lean source)
parameters
s :
θ :
Mechanism n G
W :
ObservedWorld G θ
hpos :
hmix :
hone :
hsign :
FixedOwnDerivativeSign G s θ
order :
Fin n → ℕ
horder :
hgraphOrder :
PermutedGraphOrdered W order
i :
Fin n
def
predecessorScoreHomeomorph s W hpos hmix hone hsign horder hgraphOrder i :
given by
((continuous_predecessorScoreMap W hpos order i).isClosedEmbedding (predecessorScoreMap_injective s W hpos hmix hone hsign horder hgraphOrder i)).isEmbedding |>.toHomeomorph
theorem continuous_predecessorScoreInverse
The inverse predecessor-coordinate reconstruction is continuous on the score image. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
s :
θ :
Mechanism n G
W :
ObservedWorld G θ
hpos :
hmix :
hone :
hsign :
FixedOwnDerivativeSign G s θ
order :
Fin n → ℕ
horder :
hgraphOrder :
PermutedGraphOrdered W order
i :
Fin n
conclusion
Continuous (predecessorScoreHomeomorph s W hpos hmix hone hsign horder hgraphOrder i).symm
Proof (Lean source)
lemma continuous_predecessorScoreInverse
{n : ℕ} {G : DAG (Fin n)} (s : SignVector n)
{θ : Mechanism n G} (W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W)
(hsign : FixedOwnDerivativeSign G s θ)
{order : Fin n → ℕ}
(horder : IsTopologicalOrdering (observedLawRatioGraph gaussianFeatureMap W.law) order)
(hgraphOrder : PermutedGraphOrdered W order)
(i : Fin n) :
Continuous (predecessorScoreHomeomorph s W hpos hmix hone hsign horder hgraphOrder i).symm :=
(predecessorScoreHomeomorph s W hpos hmix hone hsign horder hgraphOrder i).symm.continuous
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.FiniteDensityBridge 13 declarations This file records the model-specific identification needed to instantiate the general finite-DAG nonancestor marginal theorem with the paper's observed laws.
Finite-density bridge for observed ratio laws
This file records the model-specific identification needed to instantiate the general finite-DAG nonancestor marginal theorem with the paper's observed laws.
structure FiniteDensityObservedWorldBridge
A paper mechanism and observed world are represented by the reusable unit-cube factorization, including identification of the canonical observed ratio pushforwards.
Definition (Lean source)
parameters
W :
ObservedWorld G θ
fields
factorization :
UnitCubeFactorization (Fin n) G
intervention :
∀ e : Fin n,
numerator :
Fin n → ℝ → ℝ≥0∞
ratio_measurable :
∀ i,
Measurable (fun v : LatentState n => numerator i (v (W.targetPerm i)) / factorization.factor (W.targetPerm i) v)
observational_identification :
∀ i,
= Measure.map (fun v : LatentState n => (numerator i (v (W.targetPerm i)) / factorization.factor (W.targetPerm i) v).toReal) factorization.observationalMeasure
interventional_identification :
∀ j i,
interventionalRatioLaw W j i
= Measure.map (fun v : LatentState n => (numerator i (v (W.targetPerm i)) / factorization.factor (W.targetPerm i) v).toReal) (factorization.interventionMeasure (W.targetPerm j) (intervention j))
def mechanismUnitCubeFactorization
A positive normalized smooth latent mechanism determines the reusable unit-cube factorization of its observational conditional densities.
Definition (Lean source)
parameters
hpos :
def
given by
by apply unitCubeFactorizationOfRealCubeFactors θ.p · apply measurableOnSet_of_continuousOn_cube intro i exact (hpos.2.2.1 i).continuousOn · intro i v hv exact (hpos.1 i v hv).le · intro i v w hv hw hvw apply θ.parent_local i v w · exact hvw i (mem_insert_self i (G.parents i)) · intro k hk exact hvw k (mem_insert_of_mem hk) · exact hpos.2.2.2.2.1
def mechanismInterventionDensity
A positive normalized smooth latent mechanism and observed target permutation determine the reusable unit-cube intervention density for an environment.
Definition (Lean source)
parameters
def
mechanismInterventionDensity W hpos e :
given by
by apply interventionDensityOfRealCube (W.targetPerm e) (θ.q (W.targetPerm e)) · exact (hpos.2.2.2.1 (W.targetPerm e)).continuousOn.domRestrict.measurable · intro x hx exact (hpos.2.1 (W.targetPerm e) x hx).le · exact hpos.2.2.2.2.2 (W.targetPerm e)
theorem mechanismUnitCubeFactorization_observationalMeasure
A positive normalized smooth mechanism has the same observational law as the observational measure of its clamped unit-cube factorization.
Formal statement
hypotheses
hpos :
conclusion
(mechanismUnitCubeFactorization hpos).observationalMeasure = observationalLaw θ
Proof (Lean source)
theorem mechanismUnitCubeFactorization_observationalMeasure
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(hpos : PositiveNormalizedSmoothMechanisms G θ) :
(mechanismUnitCubeFactorization hpos).observationalMeasure = observationalLaw θ := by
-- Unfold the paper law after applying the substrate's real cube-product identification.
change (mechanismUnitCubeFactorization hpos).observationalMeasure =
realCubeProductDensityMeasure θ.p
simpa only [mechanismUnitCubeFactorization, latentCube] using
observationalMeasure_unitCubeFactorizationOfRealCubeFactors θ.p
(measurableOnSet_of_continuousOn_cube θ.p
(fun i => (hpos.2.2.1 i).continuousOn))
(fun i v hv => (hpos.1 i v hv).le)
(fun i v w _ _ hvw => θ.parent_local i v w
(hvw i (mem_insert_self _ _))
(fun k hk => hvw k (mem_insert_of_mem hk)))
hpos.2.2.2.2.1
theorem mechanismUnitCubeFactorization_interventionMeasure
A positive normalized smooth mechanism has the same target-intervention law as the intervention measure of its clamped unit-cube factorization.
Formal statement
hypotheses
conclusion
(mechanismUnitCubeFactorization hpos).interventionMeasure (W.targetPerm e) (mechanismInterventionDensity W hpos e)
= interventionalLaw θ (W.targetPerm e)
Proof (Lean source)
theorem mechanismUnitCubeFactorization_interventionMeasure
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) (hpos : PositiveNormalizedSmoothMechanisms G θ) (e : Fin n) :
(mechanismUnitCubeFactorization hpos).interventionMeasure (W.targetPerm e)
(mechanismInterventionDensity W hpos e) =
interventionalLaw θ (W.targetPerm e) := by
-- Unfold the paper law after applying the substrate's real cube-intervention identification.
change (mechanismUnitCubeFactorization hpos).interventionMeasure (W.targetPerm e)
(mechanismInterventionDensity W hpos e) =
realCubeInterventionDensityMeasure θ.p (W.targetPerm e) (θ.q (W.targetPerm e))
simpa only [mechanismUnitCubeFactorization, mechanismInterventionDensity, latentCube] using
interventionMeasure_unitCubeFactorizationOfRealCubeFactors θ.p
(measurableOnSet_of_continuousOn_cube θ.p
(fun i => (hpos.2.2.1 i).continuousOn))
(fun i v hv => (hpos.1 i v hv).le)
(fun i v w _ _ hvw => θ.parent_local i v w
(hvw i (mem_insert_self _ _))
(fun k hk => hvw k (mem_insert_of_mem hk)))
hpos.2.2.2.2.1 (W.targetPerm e) (θ.q (W.targetPerm e))
((hpos.2.2.2.1 (W.targetPerm e)).continuousOn.domRestrict.measurable)
(fun x hx => (hpos.2.1 (W.targetPerm e) x hx).le)
(hpos.2.2.2.2.2 (W.targetPerm e))
def mechanismRatioNumerator
The positive normalized smooth mechanism and environment target determine the globally measurable clamped numerator used in the canonical target ratio.
Definition (Lean source)
parameters
def
mechanismRatioNumerator W hpos e :
ℝ → ℝ≥0∞
given by
(mechanismInterventionDensity W hpos e).density
theorem mechanismTargetRatio_toReal_eq
On the latent unit cube, the real value of the clamped factor ratio is the paper's ordinary replacement-to-observational density ratio. With the stated inputs and conditions, the documented conclusion follows.
Formal statement
hypotheses
conclusion
(mechanismRatioNumerator W hpos e (v (W.targetPerm e)) / (mechanismUnitCubeFactorization hpos).factor (W.targetPerm e) v).toReal
= θ.q (W.targetPerm e) (v (W.targetPerm e)) / θ.p (W.targetPerm e) v
Proof (Lean source)
theorem mechanismTargetRatio_toReal_eq
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) (hpos : PositiveNormalizedSmoothMechanisms G θ)
(e : Fin n) {v : LatentState n} (hv : v ∈ latentCube n) :
(mechanismRatioNumerator W hpos e (v (W.targetPerm e)) /
(mechanismUnitCubeFactorization hpos).factor (W.targetPerm e) v).toReal =
θ.q (W.targetPerm e) (v (W.targetPerm e)) / θ.p (W.targetPerm e) v := by
-- Both clamped extensions are unchanged on the cube; positivity removes `ofReal` truncation.
have hj := hv (W.targetPerm e) (Set.mem_univ _)
have hnum : mechanismRatioNumerator W hpos e (v (W.targetPerm e)) =
ofReal (θ.q (W.targetPerm e) (v (W.targetPerm e))) := by
simp [mechanismRatioNumerator, mechanismInterventionDensity,
interventionDensityOfRealCube, interventionDensityOfCube,
min_eq_right hj.2, max_eq_right hj.1]
have hden : (mechanismUnitCubeFactorization hpos).factor (W.targetPerm e) v =
ofReal (θ.p (W.targetPerm e) v) := by
simpa only [mechanismUnitCubeFactorization, latentCube] using
unitCubeFactorizationOfRealCubeFactors_factor_eq θ.p
(measurableOnSet_of_continuousOn_cube θ.p
(fun i => (hpos.2.2.1 i).continuousOn))
(fun i w hw => (hpos.1 i w hw).le)
(fun i x y _ _ hxy => θ.parent_local i x y
(hxy i (mem_insert_self _ _))
(fun k hk => hxy k (mem_insert_of_mem hk)))
hpos.2.2.2.2.1 (W.targetPerm e) hv
rw [hnum, hden, ENNReal.toReal_div,
ENNReal.toReal_ofReal (hpos.2.1 _ _ hj).le,
ENNReal.toReal_ofReal (hpos.1 _ _ hv).le]
theorem measurable_mechanismTargetRatio
The clamped target ratio is globally measurable.
Formal statement
hypotheses
conclusion
Measurable (fun v : LatentState n => mechanismRatioNumerator W hpos e (v (W.targetPerm e)) / (mechanismUnitCubeFactorization hpos).factor (W.targetPerm e) v)
Proof (Lean source)
theorem measurable_mechanismTargetRatio
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) (hpos : PositiveNormalizedSmoothMechanisms G θ) (e : Fin n) :
Measurable (fun v : LatentState n =>
mechanismRatioNumerator W hpos e (v (W.targetPerm e)) /
(mechanismUnitCubeFactorization hpos).factor (W.targetPerm e) v) := by
exact ((mechanismInterventionDensity W hpos e).measurable_density.comp
(measurable_pi_apply (W.targetPerm e))).div
((mechanismUnitCubeFactorization hpos).measurable_factor (W.targetPerm e))
theorem interventionalLaw_absolutelyContinuous_observational
Every positive single-target latent intervention law is absolutely continuous with respect to the corresponding observational latent law. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
interventionalLaw θ (W.targetPerm e) ≪ observationalLaw θ
Proof (Lean source)
lemma interventionalLaw_absolutelyContinuous_observational
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) (hpos : PositiveNormalizedSmoothMechanisms G θ) (e : Fin n) :
interventionalLaw θ (W.targetPerm e) ≪ observationalLaw θ := by
letI : SigmaFinite unitIntervalReference := by
unfold unitIntervalReference
infer_instance
let B := mechanismUnitCubeFactorization hpos
let q := mechanismInterventionDensity W hpos e
let μ := unitCubeReference (Fin n)
have hfactor_pos : ∀ i v, 0 < B.factor i v := by
intro i v
change 0 < ofReal (θ.p i (clampCube (Fin n) v))
rw [ENNReal.ofReal_pos]
exact hpos.1 i _ (by
simpa only [latentCube, unitCube] using
clampCube_mem (Fin n) v)
have hμ_obs : μ ≪ B.observationalMeasure := by
rw [observationalMeasure]
apply withDensity_absolutelyContinuous' B.measurable_observationalDensity.aemeasurable
filter_upwards with v
unfold observationalDensity
partialDensity
exact Finset.prod_ne_zero_iff.mpr (fun i _ => (hfactor_pos i v).ne')
have hint_μ : B.interventionMeasure (W.targetPerm e) q ≪ μ := by
rw [interventionMeasure]
exact withDensity_absolutelyContinuous _ _
simpa only [B, q, mechanismUnitCubeFactorization_interventionMeasure W hpos e,
mechanismUnitCubeFactorization_observationalMeasure hpos] using hint_μ.trans hμ_obs
theorem observedInterventional_absolutelyContinuous_observational
Under positive latent densities, support-local shared mixing, and the supplied pushforward laws, every observed single-target intervention law is absolutely continuous with respect to the observed observational law.
Formal statement
Proof (Lean source)
theorem observedInterventional_absolutelyContinuous_observational
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W) (e : Fin n) :
W.law e.succ ≪ W.law 0 := by
-- First compare both cube-restricted densities to the common restricted-volume reference,
-- then transport absolute continuity through the map using its a.e. measurability on the cube.
let B := mechanismUnitCubeFactorization hpos
let q := mechanismInterventionDensity W hpos e
let μ := unitCubeReference (Fin n)
have hfactor_pos : ∀ i v, 0 < B.factor i v := by
intro i v
change 0 < ofReal (θ.p i (clampCube (Fin n) v))
rw [ENNReal.ofReal_pos]
exact hpos.1 i _ (by
simpa only [latentCube, unitCube] using
clampCube_mem (Fin n) v)
have hmeas_obs : Measurable B.observationalDensity := by
unfold observationalDensity
partialDensity
exact Finset.measurable_prod univ (fun i _ => B.measurable_factor i)
have hμ_obs : μ ≪ B.observationalMeasure := by
rw [observationalMeasure]
apply withDensity_absolutelyContinuous' hmeas_obs.aemeasurable
filter_upwards with v
change (∏ i ∈ univ, B.factor i v) ≠ 0
exact Finset.prod_ne_zero_iff.mpr (fun i _ => (hfactor_pos i v).ne')
have hint_μ : B.interventionMeasure (W.targetPerm e) q ≪ μ := by
rw [interventionMeasure]
exact withDensity_absolutelyContinuous _ _
have hlatentB : B.interventionMeasure (W.targetPerm e) q ≪ B.observationalMeasure :=
hint_μ.trans hμ_obs
have hlatent : interventionalLaw θ (W.targetPerm e) ≪ observationalLaw θ := by
simpa only [B, q, mechanismUnitCubeFactorization_interventionMeasure W hpos e,
mechanismUnitCubeFactorization_observationalMeasure hpos] using hlatentB
have hcube : MeasurableSet (latentCube n) := by
rw [latentCube]
exact MeasurableSet.pi Set.countable_univ (fun _ _ => measurableSet_Icc)
have hμ_cube : μ (latentCube n)ᶜ = 0 := by
simpa only [μ, latentCube, unitCube] using
unitCubeReference_compl (Fin n)
have hobs_μ : observationalLaw θ ≪ μ := by
rw [← mechanismUnitCubeFactorization_observationalMeasure hpos]
exact withDensity_absolutelyContinuous _ _
have hobs_cube : observationalLaw θ (latentCube n)ᶜ = 0 := hobs_μ hμ_cube
have hmixOn : SupportMeasurableOn W.mix (latentCube n) :=
hmix.1.continuousOn.domRestrict.measurable
have hmixAE : AEMeasurable W.mix (observationalLaw θ) :=
aemeasurable_of_supportMeasurableOn hcube hobs_cube hmixOn
let mix' := hmixAE.mk W.mix
have hmix_eq_obs : W.mix =ᵐ[observationalLaw θ] mix' := hmixAE.ae_eq_mk
have hmix_eq_int : W.mix =ᵐ[interventionalLaw θ (W.targetPerm e)] mix' :=
hlatent.ae_eq hmix_eq_obs
have hmap : Measure.map mix' (interventionalLaw θ (W.targetPerm e)) ≪
Measure.map mix' (observationalLaw θ) :=
@Measure.AbsolutelyContinuous.map _ _ _ _ _ _ hlatent mix' hmixAE.measurable_mk
rw [hone.2.1 e, hone.1]
calc
Measure.map W.mix (interventionalLaw θ (W.targetPerm e)) =
Measure.map mix' (interventionalLaw θ (W.targetPerm e)) :=
Measure.map_congr hmix_eq_int
_ ≪ Measure.map mix' (observationalLaw θ) := hmap
_ = Measure.map W.mix (observationalLaw θ) := (Measure.map_congr hmix_eq_obs).symm
def finiteDensityObservedWorldBridge_of_assumptions
The [primitive smooth-mechanism, support-local mixing, and observed-law hypotheses] (hyp:hpos,hmix,hone) construct the paper's finite-density observed-world bridge.
Definition (Lean source)
parameters
def
finiteDensityObservedWorldBridge_of_assumptions W hpos hmix hone :
given by
clause 1
factorization := mechanismUnitCubeFactorization hpos
clause 2
intervention := mechanismInterventionDensity W hpos
clause 3
numerator := mechanismRatioNumerator W hpos
clause 4
ratio_measurable := measurable_mechanismTargetRatio W hpos
clause 5
observational_identification := by intro i let μ := unitCubeReference (Fin n) have hcube : MeasurableSet (latentCube n) := by rw [latentCube] exact MeasurableSet.pi Set.countable_univ (fun _ _ => measurableSet_Icc) have hμ_cube : μ (latentCube n)ᶜ
= 0 := by simpa only [μ, latentCube, unitCube] using unitCubeReference_compl (Fin n) have hobs_μ : observationalLaw θ ≪ μ := by rw [← mechanismUnitCubeFactorization_observationalMeasure hpos] exact withDensity_absolutelyContinuous _ _ have hobs_cube : observationalLaw θ (latentCube n)ᶜ = 0 := hobs_μ hμ_cube have hmixOn : SupportMeasurableOn W.mix (latentCube n) := hmix.1.continuousOn.domRestrict.measurable have hmixAE : AEMeasurable W.mix (observationalLaw θ) := aemeasurable_of_supportMeasurableOn hcube hobs_cube hmixOn let mix' := hmixAE.mk W.mix have hmix_eq : W.mix =ᵐ[observationalLaw θ] mix' := hmixAE.ae_eq_mk have htend : Tendsto W.mix (ae (observationalLaw θ)) (ae (W.law 0)) := by have hrep := (hmixAE.measurable_mk.quasiMeasurePreserving (observationalLaw θ)).tendsto_ae rw [← Measure.map_congr hmix_eq, ← hone.1] at hrep exact Filter.Tendsto.congr' hmix_eq.symm hrep have hpull := (hone.2.2.1 i).comp_tendsto htend let r := fun v : LatentState n => mechanismRatioNumerator W hpos i (v (W.targetPerm i)) / (mechanismUnitCubeFactorization hpos).factor (W.targetPerm i) v have hr : (observedLawRatio W.law i ∘ W.mix) =ᵐ[observationalLaw θ] (toReal ∘ r) := by filter_upwards [hpull, ae_iff.mpr hobs_cube] with v hvcanon hv have hratio_pos : 0 < W.ratio i (W.mix v) := by rw [hone.2.2.2 i v hv] exact div_pos (hpos.2.1 _ _ (hv (W.targetPerm i) (Set.mem_univ _))) (hpos.1 _ _ hv) change ((W.law i.succ).rnDeriv (W.law 0) (W.mix v)).toReal = (r v).toReal calc _ = (ofReal (W.ratio i (W.mix v))).toReal := (congrArg toReal hvcanon).symm _ = W.ratio i (W.mix v) := ENNReal.toReal_ofReal hratio_pos.le _ = θ.q (W.targetPerm i) (v (W.targetPerm i)) / θ.p (W.targetPerm i) v := hone.2.2.2 i v hv _ = (r v).toReal := (mechanismTargetRatio_toReal_eq W hpos i hv).symm unfold observationalRatioLaw rw [hone.1] calc Measure.map (observedLawRatio W.law i) (Measure.map W.mix (observationalLaw θ)) = Measure.map (observedLawRatio W.law i ∘ W.mix) (observationalLaw θ) := AEMeasurable.map_map_of_aemeasurable (measurable_observedLawRatio W.law i).aemeasurable hmixAE _ = Measure.map (toReal ∘ r) (observationalLaw θ) := Measure.map_congr hr _ = Measure.map (toReal ∘ r) (mechanismUnitCubeFactorization hpos).observationalMeasure := by rw [mechanismUnitCubeFactorization_observationalMeasure hpos] _ = _ := by rfl
clause 6
interventional_identification := by intro j i let B := mechanismUnitCubeFactorization hpos let qj := mechanismInterventionDensity W hpos j let μ := unitCubeReference (Fin n) let ν := interventionalLaw θ (W.targetPerm j) have hcube : MeasurableSet (latentCube n) := by rw [latentCube] exact MeasurableSet.pi Set.countable_univ (fun _ _ => measurableSet_Icc) have hμ_cube : μ (latentCube n)ᶜ
= 0 := by simpa only [μ, latentCube, unitCube] using unitCubeReference_compl (Fin n) have hν_μ : ν ≪ μ := by change interventionalLaw θ (W.targetPerm j) ≪ unitCubeReference (Fin n) rw [← mechanismUnitCubeFactorization_interventionMeasure W hpos j] rw [interventionMeasure] exact withDensity_absolutelyContinuous _ _ have hν_cube : ν (latentCube n)ᶜ = 0 := hν_μ hμ_cube have hmixOn : SupportMeasurableOn W.mix (latentCube n) := hmix.1.continuousOn.domRestrict.measurable have hmixAE : AEMeasurable W.mix ν := aemeasurable_of_supportMeasurableOn hcube hν_cube hmixOn let mix' := hmixAE.mk W.mix have hmix_eq : W.mix =ᵐ[ν] mix' := hmixAE.ae_eq_mk have htend : Tendsto W.mix (ae ν) (ae (W.law j.succ)) := by have hrep := (hmixAE.measurable_mk.quasiMeasurePreserving ν).tendsto_ae rw [← Measure.map_congr hmix_eq, ← hone.2.1 j] at hrep exact Filter.Tendsto.congr' hmix_eq.symm hrep have hcanon_j := (observedInterventional_absolutelyContinuous_observational W hpos hmix hone j).ae_eq (hone.2.2.1 i) have hpull := hcanon_j.comp_tendsto htend let r := fun v : LatentState n => mechanismRatioNumerator W hpos i (v (W.targetPerm i)) / (mechanismUnitCubeFactorization hpos).factor (W.targetPerm i) v have hr : (observedLawRatio W.law i ∘ W.mix) =ᵐ[ν] (toReal ∘ r) := by filter_upwards [hpull, ae_iff.mpr hν_cube] with v hvcanon hv have hratio_pos : 0 < W.ratio i (W.mix v) := by rw [hone.2.2.2 i v hv] exact div_pos (hpos.2.1 _ _ (hv (W.targetPerm i) (Set.mem_univ _))) (hpos.1 _ _ hv) change ((W.law i.succ).rnDeriv (W.law 0) (W.mix v)).toReal = (r v).toReal calc _ = (ofReal (W.ratio i (W.mix v))).toReal := (congrArg toReal hvcanon).symm _ = W.ratio i (W.mix v) := ENNReal.toReal_ofReal hratio_pos.le _ = θ.q (W.targetPerm i) (v (W.targetPerm i)) / θ.p (W.targetPerm i) v := hone.2.2.2 i v hv _ = (r v).toReal := (mechanismTargetRatio_toReal_eq W hpos i hv).symm unfold interventionalRatioLaw rw [hone.2.1 j] calc Measure.map (observedLawRatio W.law i) (Measure.map W.mix ν) = Measure.map (observedLawRatio W.law i ∘ W.mix) ν := AEMeasurable.map_map_of_aemeasurable (measurable_observedLawRatio W.law i).aemeasurable hmixAE _ = Measure.map (toReal ∘ r) ν := Measure.map_congr hr _ = Measure.map (toReal ∘ r) (B.interventionMeasure (W.targetPerm j) qj) := by change Measure.map (toReal ∘ r) (interventionalLaw θ (W.targetPerm j)) = Measure.map (toReal ∘ r) ((mechanismUnitCubeFactorization hpos).interventionMeasure (W.targetPerm j) (mechanismInterventionDensity W hpos j)) rw [mechanismUnitCubeFactorization_interventionMeasure W hpos j] _ = _ := by rfl
theorem ratioLaw_eq_of_nonancestor
The finite-density bridge turns nonancestry into equality of the two canonical real-valued ratio laws. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
W :
ObservedWorld G θ
j i :
Fin n
hji :
j ≠ i
hna :
¬ G.isAncestor (W.targetPerm j) (W.targetPerm i)
conclusion
observationalRatioLaw W i = interventionalRatioLaw W j i
Proof (Lean source)
-- @node: FiniteDensityObservedWorldBridge.ratioLaw_eq_of_nonancestor
lemma FiniteDensityObservedWorldBridge.ratioLaw_eq_of_nonancestor
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
{W : ObservedWorld G θ} (B : FiniteDensityObservedWorldBridge W)
{j i : Fin n} (hji : j ≠ i)
(hna : ¬ G.isAncestor (W.targetPerm j) (W.targetPerm i)) :
observationalRatioLaw W i = interventionalRatioLaw W j i := by
rw [B.observational_identification i, B.interventional_identification j i]
have hσ : ∀ _ : Fin n,
SigmaFinite Causalean.Graph.FiniteDensity.unitIntervalReference := fun _ => by
unfold Causalean.Graph.FiniteDensity.unitIntervalReference
infer_instance
have htarget : W.targetPerm j ≠ W.targetPerm i := by
exact fun h => hji (W.targetPerm.injective h)
have h := @targetRatio_map_eq
(Fin n) _ _ (fun _ : Fin n => ℝ) _
(fun _ : Fin n => Causalean.Graph.FiniteDensity.unitIntervalReference)
hσ G B.factorization (W.targetPerm i) (W.targetPerm j) htarget hna
(B.intervention j) (B.numerator i) (B.ratio_measurable i) (fun _ => 0)
let r := fun v : LatentState n =>
B.numerator i (v (W.targetPerm i)) / B.factorization.factor (W.targetPerm i) v
change Measure.map (toReal ∘ r) B.factorization.observationalMeasure =
Measure.map (toReal ∘ r)
(B.factorization.interventionMeasure (W.targetPerm j) (B.intervention j))
calc
_ = Measure.map toReal
(Measure.map r B.factorization.observationalMeasure) := by
simpa [r] using (Measure.map_map ENNReal.measurable_toReal
(B.ratio_measurable i)).symm
_ = Measure.map toReal
(Measure.map r (B.factorization.interventionMeasure
(W.targetPerm j) (B.intervention j))) := congrArg (Measure.map toReal) h
_ = _ := by
simpa [r] using Measure.map_map ENNReal.measurable_toReal
(B.ratio_measurable i)
theorem mechanism_orderedLocalMarkov
The observational law of a positive normalized mechanism satisfies the ordered local Markov property for every topological ranking of its latent DAG. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
Proof (Lean source)
lemma mechanism_orderedLocalMarkov
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(τ : TopologicalRanking G)
(i : Fin n) (A : Finset (Fin n))
(hA : A ⊆ predecessors τ i)
(hpa : G.parents i ⊆ A) :
let B := mechanismUnitCubeFactorization hpos
∃ hfinite : IsFiniteMeasure B.observationalMeasure,
letI := hfinite
CondIndepFun
(comap
(coordinateProjection
(X := fun _ : Fin n ↦ ℝ) A) inferInstance)
(coordinateConditioning_comap_le
(X := fun _ : Fin n ↦ ℝ) A)
(fun v : LatentState n ↦ v i)
(coordinateProjection
(X := fun _ : Fin n ↦ ℝ)
(predecessors τ i \ A))
B.observationalMeasure := by
let B := mechanismUnitCubeFactorization hpos
have hfinite : IsFiniteMeasure B.observationalMeasure :=
instIsFiniteMeasureUnitCubeObservational B
refine ⟨hfinite, ?_⟩
exact B.orderedLocalMarkov_unitCubeReference τ i A hA hpa
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.GaussianRecoveryBridge 4 declarations This file connects the paper's duplicated explicit Gaussian feature expansion and its canonical ratio laws to the neutral bounded-support recovery theorem.
Gaussian recovery bridge
This file connects the paper's duplicated explicit Gaussian feature expansion and its canonical ratio laws to the neutral bounded-support recovery theorem.
theorem canonicalObservedWorld_onePerfectInterventionPerNode
The identity-mixing canonical world satisfies the paper's perfect-intervention contract. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
OnePerfectInterventionPerNode G θ (canonicalObservedWorld G θ π)
Proof (Lean source)
lemma canonicalObservedWorld_onePerfectInterventionPerNode
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(hpos : PositiveNormalizedSmoothMechanisms G θ) (π : Perm (Fin n)) :
OnePerfectInterventionPerNode G θ (canonicalObservedWorld G θ π) := by
let W := canonicalObservedWorld G θ π
let B := mechanismUnitCubeFactorization hpos
constructor
· simp [canonicalObservedWorld]
constructor
· intro e
simp [canonicalObservedWorld]
constructor
· intro e
let q := mechanismInterventionDensity W hpos e
let r := fun v : LatentState n =>
q.density (v (W.targetPerm e)) / B.factor (W.targetPerm e) v
have hfactor_zero : ∀ v, B.factor (W.targetPerm e) v ≠ 0 := by
intro v
change ofReal (θ.p (W.targetPerm e) (clampCube (Fin n) v)) ≠ 0
exact (ENNReal.ofReal_pos.mpr (hpos.1 _ _ (by
simpa only [latentCube, unitCube] using clampCube_mem (Fin n) v))).ne'
have hfactor_top : ∀ v, B.factor (W.targetPerm e) v ≠ ∞ := by
intro v
change ofReal (θ.p (W.targetPerm e) (clampCube (Fin n) v)) ≠ ∞
exact ENNReal.ofReal_ne_top
have hmeasure : interventionalLaw θ (W.targetPerm e) =
(observationalLaw θ).withDensity r := by
rw [← mechanismUnitCubeFactorization_interventionMeasure W hpos e,
← mechanismUnitCubeFactorization_observationalMeasure hpos]
exact factorization_interventionMeasure_eq_withDensity_targetRatio B
(W.targetPerm e) q hfactor_zero hfactor_top
have hrn :
(interventionalLaw θ (W.targetPerm e)).rnDeriv (observationalLaw θ) =ᵐ[
observationalLaw θ] r := by
letI : IsProbabilityMeasure (observationalLaw θ) :=
observationalLaw_isProbabilityMeasure hpos
rw [hmeasure]
exact Measure.rnDeriv_withDensity _
((q.measurable_density.comp (measurable_pi_apply (W.targetPerm e))).div
(B.measurable_factor (W.targetPerm e)))
filter_upwards [hrn, observationalLaw_ae_mem_latentCube hpos] with v hrv hv
change ofReal
(θ.q (π e) (v (π e)) / θ.p (π e) v) =
(interventionalLaw θ (π e)).rnDeriv (observationalLaw θ) v
have hrtop : r v ≠ ∞ :=
ENNReal.div_ne_top (by
dsimp only [r, q, mechanismInterventionDensity, mechanismRatioNumerator]
exact ENNReal.ofReal_ne_top) (hfactor_zero v)
have hratio : (r v).toReal =
θ.q (π e) (v (π e)) / θ.p (π e) v := by
simpa only [r, q, B, W, mechanismRatioNumerator, canonicalObservedWorld] using
mechanismTargetRatio_toReal_eq W hpos e hv
calc
ofReal (θ.q (π e) (v (π e)) / θ.p (π e) v) =
ofReal (r v).toReal := by
rw [hratio]
_ = r v := ENNReal.ofReal_toReal hrtop
_ = _ := hrv.symm
· intro e v hv
rfl
theorem gaussianFeature_eq_recovery
The paper and neutral Gaussian feature vectors are the same explicit ℓ² vector. the stated conclusion follows.
Formal statement
hypotheses
r :
ℝ
conclusion
= CausalSmith.Substrate.GaussianMeanEmbeddingBoundedMomentRecovery.gaussianFeature r
Proof (Lean source)
lemma gaussianFeature_eq_recovery (r : ℝ) :
gaussianFeature r =
CausalSmith.Substrate.GaussianMeanEmbeddingBoundedMomentRecovery.gaussianFeature r := by
apply Subtype.ext
funext m
rfl
theorem meanEmbedding_gaussianFeatureMap_eq_recovery
The paper's Gaussian mean embedding agrees with the neutral recovery embedding. the stated conclusion follows.
Formal statement
hypotheses
μ :
Measure ℝ
conclusion
= CausalSmith.Substrate.GaussianMeanEmbeddingBoundedMomentRecovery.meanEmbedding CausalSmith.Substrate.GaussianMeanEmbeddingBoundedMomentRecovery.gaussianFeatureMap μ
Proof (Lean source)
lemma meanEmbedding_gaussianFeatureMap_eq_recovery (μ : Measure ℝ) :
meanEmbedding gaussianFeatureMap μ =
CausalSmith.Substrate.GaussianMeanEmbeddingBoundedMomentRecovery.meanEmbedding
CausalSmith.Substrate.GaussianMeanEmbeddingBoundedMomentRecovery.gaussianFeatureMap μ := by
unfold meanEmbedding
CausalSmith.Substrate.GaussianMeanEmbeddingBoundedMomentRecovery.meanEmbedding
apply integral_congr_ae
filter_upwards with r
exact gaussianFeature_eq_recovery r
theorem canonical_populationDiscrepancy_pos_of_secondMomentContrast_ne
A nonzero canonical raw second-moment contrast forces positive paper-local Gaussian MMD. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma canonical_populationDiscrepancy_pos_of_secondMomentContrast_ne
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(hpos : PositiveNormalizedSmoothMechanisms G θ) (π : Perm (Fin n))
(j i : Fin n)
(hmom : secondMomentContrast (canonicalObservedWorld G θ π) j i ≠ 0) :
0 < populationDiscrepancy gaussianFeatureMap
(canonicalObservedWorld G θ π) j i := by
let W := canonicalObservedWorld G θ π
let μ := observationalRatioLaw W i
let ν := interventionalRatioLaw W j i
let ratio := fun v : LatentState n =>
θ.q (π i) (v (π i)) / θ.p (π i) v
have hone := canonicalObservedWorld_onePerfectInterventionPerNode hpos π
have hobsCube : ∀ᵐ v ∂observationalLaw θ, v ∈ latentCube n :=
observationalLaw_ae_mem_latentCube hpos
have hac : interventionalLaw θ (π j) ≪ observationalLaw θ := by
simpa only [W, canonicalObservedWorld] using
interventionalLaw_absolutelyContinuous_observational W hpos j
have hintCube : ∀ᵐ v ∂interventionalLaw θ (π j), v ∈ latentCube n :=
hac.ae_le hobsCube
have hratioObs : observedLawRatio W.law i =ᵐ[observationalLaw θ] ratio := by
filter_upwards [hone.2.2.1 i, hobsCube] with v hrv hv
change ((interventionalLaw θ (π i)).rnDeriv (observationalLaw θ) v).toReal =
θ.q (π i) (v (π i)) / θ.p (π i) v
calc
_ = (ofReal (W.ratio i v)).toReal := congrArg toReal hrv.symm
_ = W.ratio i v := ENNReal.toReal_ofReal
(div_nonneg (hpos.2.1 _ _ (hv (π i) (Set.mem_univ _))).le
(hpos.1 _ _ hv).le)
_ = _ := by rfl
have hratioInt : observedLawRatio W.law i =ᵐ[interventionalLaw θ (π j)] ratio := by
exact hac.ae_eq hratioObs
have hratioCont : ContinuousOn ratio (latentCube n) := by
apply ((hpos.2.2.2.1 (π i)).continuousOn.comp
((continuous_apply (π i)).continuousOn)
(fun (v : LatentState n) (hv : v ∈ latentCube n) =>
hv (π i) (Set.mem_univ _))).div
((hpos.2.2.1 (π i)).continuousOn)
intro v hv
exact ne_of_gt (hpos.1 _ _ hv)
have hcubeCompact : IsCompact (latentCube n) := by
rw [latentCube]
exact isCompact_univ_pi fun _ => isCompact_Icc
obtain ⟨R, hR⟩ :=
(hcubeCompact.image_of_continuousOn hratioCont).isBounded.subset_closedBall (0 : ℝ)
let B := max R 0
have hB : 0 ≤ B := le_max_right _ _
have hratioBound : ∀ v ∈ latentCube n, ratio v ∈ Icc (-B) B := by
intro v hv
have hvball : ratio v ∈ closedBall (0 : ℝ) R :=
hR (Set.mem_image_of_mem ratio hv)
have habsR : |ratio v| ≤ R := by
simpa [Real.dist_eq] using hvball
exact (abs_le.mp (habsR.trans (le_max_left _ _)))
have hμsupport : μ (Icc (-B) B)ᶜ = 0 := by
dsimp only [μ]
rw [observationalRatioLaw,
Measure.map_apply (measurable_observedLawRatio W.law i) measurableSet_Icc.compl]
apply ae_iff.mp
filter_upwards [hratioObs, hobsCube] with v hratio hv
rw [hratio]
exact hratioBound v hv
have hνsupport : ν (Icc (-B) B)ᶜ = 0 := by
dsimp only [ν]
rw [interventionalRatioLaw, show W.law j.succ = interventionalLaw θ (π j) by rfl,
Measure.map_apply (measurable_observedLawRatio W.law i) measurableSet_Icc.compl]
apply ae_iff.mp
filter_upwards [hratioInt, hintCube] with v hratio hv
rw [hratio]
exact hratioBound v hv
have hμmoment : (∫ r, r ^ 2 ∂μ) = ∫ v, (ratio v) ^ 2 ∂observationalLaw θ := by
dsimp only [μ]
rw [observationalRatioLaw,
MeasureTheory.integral_map (measurable_observedLawRatio W.law i).aemeasurable
(by fun_prop)]
apply integral_congr_ae
filter_upwards [hratioObs] with v hv
rw [hv]
have hνmoment : (∫ r, r ^ 2 ∂ν) =
∫ v, (ratio v) ^ 2 ∂interventionalLaw θ (π j) := by
dsimp only [ν]
rw [interventionalRatioLaw, show W.law j.succ = interventionalLaw θ (π j) by rfl,
MeasureTheory.integral_map (measurable_observedLawRatio W.law i).aemeasurable
(by fun_prop)]
apply integral_congr_ae
filter_upwards [hratioInt] with v hv
rw [hv]
letI : IsProbabilityMeasure (observationalLaw θ) :=
observationalLaw_isProbabilityMeasure hpos
letI : IsProbabilityMeasure (interventionalLaw θ (π j)) := by
simpa only [W, canonicalObservedWorld] using
interventionalLaw_isProbabilityMeasure W hpos j
letI : IsFiniteMeasure (W.law 0) := by
change IsFiniteMeasure (observationalLaw θ)
infer_instance
letI : IsFiniteMeasure (W.law j.succ) := by
change IsFiniteMeasure (interventionalLaw θ (π j))
infer_instance
letI : IsFiniteMeasure μ := by
dsimp only [μ, observationalRatioLaw]
exact Measure.isFiniteMeasure_map (W.law 0) (observedLawRatio W.law i)
letI : IsFiniteMeasure ν := by
dsimp only [ν, interventionalRatioLaw]
exact Measure.isFiniteMeasure_map (W.law j.succ) (observedLawRatio W.law i)
have hraw : (∫ r, r ^ 2 ∂μ) ≠ ∫ r, r ^ 2 ∂ν := by
intro heq
apply hmom
unfold secondMomentContrast
change (∫ v, (ratio v) ^ 2 ∂interventionalLaw θ (π j)) -
(∫ v, (ratio v) ^ 2 ∂observationalLaw θ) = 0
rw [← hνmoment, ← hμmoment, ← heq, sub_self]
have hrecover :=
CausalSmith.Substrate.GaussianMeanEmbeddingBoundedMomentRecovery.norm_meanEmbedding_sub_pos_of_secondMoment_ne_of_boundedSupport
μ ν B hB hμsupport hνsupport hraw
rw [← meanEmbedding_gaussianFeatureMap_eq_recovery μ,
← meanEmbedding_gaussianFeatureMap_eq_recovery ν] at hrecover
simpa only [populationDiscrepancy, μ, ν, W] using hrecover
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.Inference 15 declarations This file defines the multi-environment evaluation sample, arbitrary training-fold ratio estimates, empirical MMDs, and the simultaneous threshold.
Independent sample-splitting and the confidence graph
This file defines the multi-environment evaluation sample, arbitrary training-fold ratio estimates, empirical MMDs, and the simultaneous threshold.
A sample-split experiment over a fixed observed environment family.
Definition (Lean source)
parameters
fields
probability :
Measure Ω
sampleSize :
Fin (n + 1) → ℕ
evaluation :
∀ e
trainingSize :
Fin (n + 1) → ℕ
training :
∀ e
ratioFit :
@realizes \(\widehat R_i\)(measurable function fitted from the full training fold)
ratioFit_joint_measurable :
∀ i, Measurable (uncurry (fun T x => ratioFit T i x))
@realizes \(\widehat R_i\)(joint measurability in training data and evaluation point)
ratioFit_measurable :
∀ T i, Measurable (ratioFit T i)
@realizes \(\widehat R_i\)(measurable as a function of the observation)
ratioFit_pos :
∀ T i x, 0 < ratioFit T i x
@realizes \(\widehat R_i\)(strictly positive range)
firstStageRadius :
ℝ → ℝ
firstStageRadius_nonneg :
∀ η, 0 ≤ firstStageRadius η
@realizes \(a_N\)(nonnegative range)
alpha :
ℝ
alpha_mem :
alpha ∈ Ioo (0 : ℝ) 1
eta :
ℝ
eta_mem :
eta ∈ Ioo (0 : ℝ) 1
The complete training fold, retained rather than compressed to an arbitrary summary.
Definition (Lean source)
parameters
def
given by
fun e r => S.training e r ω
def trainingSigma
The σ-algebra generated by the complete training fold.
Definition (Lean source)
def ratioEstimate
The fitted ratio, definitionally factored through the complete training fold.
Definition (Lean source)
parameters
def
given by
S.ratioFit (trainingFold S ω)
The training-fold finite L¹ ratio-error event used by the confidence theorem.
Definition (Lean source)
parameters
def
given by
{ω | ∀ i : Fin n, ∀ e : Fin (n + 1), Integrable (fun x => |S.ratioEstimate ω i x - observedLawRatio W.law i x|) (W.law e) ∧ ∫ x, |S.ratioEstimate ω i x - observedLawRatio W.law i x| ∂W.law e ≤ S.firstStageRadius S.eta}
def allEvaluationData
All evaluation observations as a single dependent array.
Definition (Lean source)
parameters
def
given by
fun e r => S.evaluation e r ω
Evaluation observations are independent within each environment, have the stated environment law, and the complete evaluation fold is independent of the training fold.
Definition (Lean source)
parameters
Ω :
Type u_1
def
IndependentEnvironmentSampling S :
Prop
given by
clause 1
IsProbabilityMeasure S.probability
clause 5
e r :
Measure.map (S.training e r) S.probability = W.law e
clause 8
e r :
Measure.map (S.evaluation e r) S.probability = W.law e
clause 9
clause 10
i x :
@Measurable Ω ℝ (comap (trainingFold S) inferInstance) inferInstance (fun ω => S.ratioEstimate ω i x)
clause 11
i :
@Measurable (Ω × LatentState n) ℝ ((comap (trainingFold S) inferInstance).prod inferInstance) inferInstance (fun p => S.ratioEstimate p.1 i p.2)
clause 12
def ConditionalEvaluationSampling
The part of the sample-splitting assumptions needed to condition an evaluation-sample concentration bound on the training fold. It deliberately omits the unused training-sample law and i.i.d. clauses and the measurability of the separate first-stage event.
Definition (Lean source)
parameters
Ω :
Type u_1
def
ConditionalEvaluationSampling S :
Prop
given by
clause 1
IsProbabilityMeasure S.probability
clause 3
clause 6
e r :
Measure.map (S.evaluation e r) S.probability = W.law e
clause 7
clause 8
i :
@Measurable (Ω × LatentState n) ℝ ((comap (trainingFold S) inferInstance).prod inferInstance) inferInstance (fun p => S.ratioEstimate p.1 i p.2)
theorem toConditionalEvaluationSampling
The full independent-environment sampling contract supplies the smaller collection of assumptions needed for conditional evaluation-fold concentration. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
Ω :
Type u_1
conclusion
Proof (Lean source)
lemma IndependentEnvironmentSampling.toConditionalEvaluationSampling
(h : IndependentEnvironmentSampling S) : ConditionalEvaluationSampling S := by
rcases h with
⟨hprob, hlaw, htrainMeas, _htrainIndep, _htrainLaw,
hevalMeas, hevalIndep, hevalLaw, hsplit, _hratioMeas,
hratioJoint, _hfirstMeas⟩
have htrain : Measurable (trainingFold S) := by
apply measurable_pi_lambda
intro e
apply measurable_pi_lambda
intro r
exact htrainMeas e r
exact ⟨hprob, hlaw, htrain, hevalMeas, hevalIndep, hevalLaw, hsplit, hratioJoint⟩
Event E has conditional probability at least p, on event A, relative to m. The setwise formulation avoids selecting a conditional-expectation version.
Definition (Lean source)
parameters
def
ConditionalProbabilityAtLeast μ m E A p :
Prop
given by
p * μ.real (A ∩ B) ≤ μ.real (E ∩ A ∩ B)
def minimumSampleSize
The minimum sample size across observational and intervention environments.
Definition (Lean source)
parameters
Ω :
Type u_1
def
minimumSampleSize S :
ℕ
given by
Finset.univ.inf' ⟨0, Finset.mem_univ 0⟩ S.sampleSize
def empiricalMeanEmbedding
Empirical mean embedding for ratio estimate i in environment e.
The sample-split empirical Gaussian-kernel MMD.
Definition (Lean source)
parameters
def
empiricalDiscrepancy S U j i ω :
ℝ
given by
‖empiricalMeanEmbedding S U j.succ i ω - empiricalMeanEmbedding S U 0 i ω‖
The paper's simultaneous confidence radius.
Definition (Lean source)
parameters
Ω :
Type u_1
def
confidenceRadius S :
ℝ
given by
2 * sqrt 2 * S.firstStageRadius S.eta
def sampleSplitConfidenceGraph
The graph selected by positive lower confidence bounds for population discrepancies.
Definition (Lean source)
parameters
Ω :
Type u_1
H :
U :
_hSampling :
ω :
Ω
given by
fun j i => j ≠ i ∧ 0 < empiricalDiscrepancy S U j i ω - confidenceRadius S
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.Kernel 20 declarations This file defines the paper's explicit square-summable Gaussian feature map, kernel mean embeddings, population discrepancies, and the two genericity sets.
Gaussian feature embeddings and cover separation
This file defines the paper's explicit square-summable Gaussian feature map, kernel mean embeddings, population discrepancies, and the two genericity sets.
def gaussianKernel
The Gaussian kernel exp (-(a-b)²).
Definition (Lean source)
A real Hilbert-space feature map of unit norm that realizes a specified kernel.
Definition (Lean source)
parameters
H :
fields
def gaussianFeatureCoefficient
The mth coordinate of the paper's explicit Gaussian feature expansion.
theorem gaussianFeature_memℓp
The explicit coefficient sequence is square-summable. the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma gaussianFeature_memℓp (r : ℝ) :
Memℓp (gaussianFeatureCoefficient r) 2 := by
apply memℓp_gen
norm_num
simp only [gaussianFeatureCoefficient]
have hs : Summable (fun m : ℕ => (2 * r ^ 2) ^ m / m.factorial) :=
Real.summable_pow_div_factorial _
apply (hs.mul_left (exp (-r ^ 2) ^ 2)).congr
intro m
have hnonneg : 0 ≤ (2 : ℝ) ^ m / m.factorial := by positivity
calc
_ = exp (-r ^ 2) ^ 2 * ((2 : ℝ) ^ m / m.factorial) *
r ^ (2 * m) := by ring
_ = (sqrt ((2 : ℝ) ^ m / m.factorial)) ^ 2 *
exp (-r ^ 2) ^ 2 * r ^ (2 * m) := by
rw [Real.sq_sqrt hnonneg]
ac_rfl
_ = _ := by ring
def gaussianFeature
The explicit Gaussian feature vector in real ℓ².
Definition (Lean source)
theorem gaussianFeature_inner
The inner product of two explicit Gaussian feature vectors equals the Gaussian kernel.
Formal statement
Proof (Lean source)
lemma gaussianFeature_inner (a b : ℝ) :
⟪gaussianFeature a, gaussianFeature b⟫_ℝ =
gaussianKernel a b := by
rw [lp.inner_eq_tsum]
simp_rw [RCLike.inner_apply, starRingEnd_apply, star_trivial]
change (∑' m : ℕ, gaussianFeatureCoefficient b m *
gaussianFeatureCoefficient a m) = _
have hterm : (fun m : ℕ => gaussianFeatureCoefficient b m *
gaussianFeatureCoefficient a m) =
fun m => (exp (-a ^ 2) * exp (-b ^ 2)) *
((2 * a * b) ^ m / m.factorial) := by
funext m
unfold gaussianFeatureCoefficient
have hnonneg : 0 ≤ (2 : ℝ) ^ m / m.factorial := by positivity
have hsqrt := Real.mul_self_sqrt hnonneg
calc
_ = (sqrt ((2 : ℝ) ^ m / m.factorial) *
sqrt ((2 : ℝ) ^ m / m.factorial)) *
(exp (-a ^ 2) * exp (-b ^ 2)) *
(a ^ m * b ^ m) := by ring
_ = _ := by rw [hsqrt]; ring
rw [hterm, tsum_mul_left,
(NormedSpace.expSeries_div_hasSum_exp (2 * a * b)).tsum_eq,
← Real.exp_eq_exp_ℝ, ← Real.exp_add, ← Real.exp_add]
unfold gaussianKernel
congr 1
ring
theorem gaussianFeature_norm
Every explicit Gaussian feature vector has unit norm.
Formal statement
hypotheses
r :
ℝ
conclusion
‖gaussianFeature r‖ = 1
Proof (Lean source)
lemma gaussianFeature_norm (r : ℝ) : ‖gaussianFeature r‖ = 1 := by
have hsq : ‖gaussianFeature r‖ ^ 2 = 1 := by
rw [← real_inner_self_eq_norm_sq, gaussianFeature_inner]
simp [gaussianKernel]
nlinarith [norm_nonneg (gaussianFeature r)]
The concrete unit-norm feature-map realization of the Gaussian kernel.
Definition (Lean source)
def
given by
clause 1
Φ := gaussianFeature
clause 2
norm_eq_one := gaussianFeature_norm
clause 3
inner_eq_kernel := gaussianFeature_inner
def meanEmbedding
The Bochner kernel mean embedding of a real-valued law in a Hilbert space.
Definition (Lean source)
def ObservedLawFamily
The observational law followed by the n interventional environment laws.
Definition (Lean source)
def observedLawRatio
The canonical Radon--Nikodym ratio determined by the observed laws.
Definition (Lean source)
theorem measurable_observedLawRatio
The canonical observed-law ratio is globally measurable. the stated conclusion follows.
Formal statement
hypotheses
conclusion
Measurable (observedLawRatio laws i)
Proof (Lean source)
lemma measurable_observedLawRatio (laws : ObservedLawFamily n) (i : Fin n) :
Measurable (observedLawRatio laws i) :=
ENNReal.measurable_toReal.comp (Measure.measurable_rnDeriv _ _)
def observationalRatioLaw
The law of ratio i under the observational environment.
Definition (Lean source)
def interventionalRatioLaw
The law of ratio i under intervention environment j.
Definition (Lean source)
Population Gaussian-kernel MMD between the observational and environment-j ratio laws.
Definition (Lean source)
parameters
def
populationDiscrepancy U W j i :
ℝ
given by
theorem populationDiscrepancy_eq_zero_of_ratioLaw_eq
Equal observational and interventional ratio laws have zero population discrepancy. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
H :
U :
θ :
Mechanism n G
W :
ObservedWorld G θ
j i :
Fin n
hLaw :
observationalRatioLaw W i = interventionalRatioLaw W j i
conclusion
populationDiscrepancy U W j i = 0
Proof (Lean source)
lemma populationDiscrepancy_eq_zero_of_ratioLaw_eq
{H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℝ H] [CompleteSpace H]
(U : UnitNormFeatureMap H) {θ : Mechanism n G} (W : ObservedWorld G θ)
(j i : Fin n) (hLaw : observationalRatioLaw W i = interventionalRatioLaw W j i) :
populationDiscrepancy U W j i = 0 := by
simp [populationDiscrepancy, hLaw]
The environment-j minus observational second-moment contrast of ratio i.
Definition (Lean source)
parameters
def
secondMomentContrast W j i :
ℝ
given by
(∫ x, (W.ratio i x) ^ 2 ∂W.law j.succ) - ∫ x, (W.ratio i x) ^ 2 ∂W.law 0
The observable directed ratio graph.
Definition (Lean source)
parameters
given by
fun j i => j ≠ i ∧ 0 < populationDiscrepancy U W j i
Mechanisms whose second-moment contrast is nonzero on every permuted direct edge.
Definition (Lean source)
parameters
def
given by
{θ | ∀ ⦃j i⦄, G.edge j i → secondMomentContrast (canonicalObservedWorld G θ.1 π) (π.symm j) (π.symm i) ≠ 0}
def coverSeparatedSet
Mechanisms whose Gaussian MMD is positive on every permuted ancestral cover.
Definition (Lean source)
parameters
H :
U :
def
given by
{θ | ∀ ⦃j i⦄, ancestralCover G j i → 0 < populationDiscrepancy U (canonicalObservedWorld G θ.1 π) (π.symm j) (π.symm i)}
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.OpenQuestions 1 declarations The second-stage procedure and sharp remainder criterion are intentionally left undefined by the paper, so the complete question is preserved as text.
Generated-rank frontier
The second-stage procedure and sharp remainder criterion are intentionally left undefined by the paper, so the complete question is preserved as text.
@realizes (open cross-fitted conditional-CDF estimator) @realizes (named nonassertive construction handle)
Definition (Lean source)
def
generatedRankFrontierQuestion :
_root_.String
given by
"Open, nonassertive generated-rank frontier. Regime: shared C2 diffeomorphic mixing; one \ distinct unknown-target parent-independent perfect intervention per latent node; fixed \ own-coordinate derivative signs; independent sampling within environments and independent \ training/evaluation folds; and the bounded Holder subclass Theta^beta_{G,s}(c,M,K,rho). \ Assume inf_theta Pr_theta{E_N^infty(b_N)} >
= 1-eta_N. Let h_N -> 0, b_N = o(h_N), \ and Nbar*h_N^dstar/log(Nbar) -> infinity. Question: using H_rank, can one construct \ cross-fitted conditional-CDF estimators Uhat_i with max_i \ ||Uhat_i-U_i||_{infinity,K} = O_Pr(r_N), where \ r_N = h_N^2
+ b_N/h_N? Under an additional \ uniform asymptotically linear first-stage log-ratio representation on K, derive an expansion \ separating generated-threshold indicator boundary crossings, generated-conditioning \ local-design perturbations, oracle conditional-CDF variance, and smoothing bias, and \ characterize the weakest sufficient first-stage remainder. This payload asserts neither \ construction nor attainment nor sharpness of the rate, and asserts no sufficient or necessary \ remainder bound; the influence representation and second-stage weight convention remain \ unspecified."
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.RkhsEmpiricalMean 5 declarations The statement is made against the local unit-norm feature-map interface and is designed for the reused scalar McDiarmid concentration engine.
Uniform Hilbert-valued empirical-mean deviation
The statement is made against the local unit-norm feature-map interface and is designed for the reused scalar McDiarmid concentration engine.
A training-measurable localization of the random-parameter product-law bound. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
Ω β :
Y :
Ω → β
hY :
@Measurable Ω β mΩ mβ Y
ν :
hY_law :
μ.map Y = ν
mA :
hmA :
mA ≤ mΩ
δ :
ℝ
Bad :
Ω → Set β
hBad :
@MeasurableSet (Ω × β) (mA.prod mβ) {p | p.2 ∈ Bad p.1}
hsec :
∀ ω, ν (Bad ω) ≤ ofReal δ
B :
Set Ω
hB :
@MeasurableSet Ω mA B
conclusion
μ ({ω | Y ω ∈ Bad ω} ∩ B) ≤ ofReal δ * μ B
Proof (Lean source)
-- @node: randomParam_event_inter_le
lemma randomParam_event_inter_le
{Ω β : Type*} [mΩ : MeasurableSpace Ω] [mβ : MeasurableSpace β]
{μ : Measure Ω} [IsProbabilityMeasure μ]
{Y : Ω → β} (hY : @Measurable Ω β mΩ mβ Y)
{ν : Measure β} [IsProbabilityMeasure ν] (hY_law : μ.map Y = ν)
(mA : MeasurableSpace Ω) (hmA : mA ≤ mΩ)
(hindep : @Indep Ω mA (comap Y mβ) mΩ μ)
{δ : ℝ} (Bad : Ω → Set β)
(hBad : @MeasurableSet (Ω × β) (mA.prod mβ) {p | p.2 ∈ Bad p.1})
(hsec : ∀ ω, ν (Bad ω) ≤ ofReal δ)
(B : Set Ω) (hB : @MeasurableSet Ω mA B) :
μ ({ω | Y ω ∈ Bad ω} ∩ B) ≤ ofReal δ * μ B := by
have hJ : @Measurable Ω (Ω × β) mΩ (mA.prod mβ) (fun ω => (ω, Y ω)) :=
(measurable_id'' hmA).prod hY
have hfst : @Measurable (Ω × β) Ω (mA.prod mβ) mA fst := measurable_fst
have hset : @MeasurableSet (Ω × β) (mA.prod mβ)
({p | p.2 ∈ Bad p.1} ∩ fst ⁻¹' B) :=
hBad.inter (hB.preimage hfst)
have hYa : @AEMeasurable Ω β mβ mΩ Y μ := hY.aemeasurable
have hmap := @Causalean.Mathlib.indep_trim_prod_map_eq Ω β mΩ mβ μ _
mA hmA Y hYa hindep
have heq : {ω | Y ω ∈ Bad ω} ∩ B =
(fun ω => (ω, Y ω)) ⁻¹' ({p | p.2 ∈ Bad p.1} ∩ fst ⁻¹' B) := by
ext ω
simp
rw [heq, ← Measure.map_apply hJ hset, hmap, hY_law, Measure.prod_apply hset]
calc
(∫⁻ ω, ν (Prod.mk ω ⁻¹' ({p | p.2 ∈ Bad p.1} ∩ fst ⁻¹' B))
∂μ.trim hmA)
≤ ∫⁻ ω, B.indicator (fun _ => ofReal δ) ω ∂μ.trim hmA := by
apply lintegral_mono
intro ω
change ν (Prod.mk ω ⁻¹'
({p | p.2 ∈ Bad p.1} ∩ fst ⁻¹' B)) ≤
B.indicator (fun _ => ofReal δ) ω
by_cases hω : ω ∈ B
· have he : Prod.mk ω ⁻¹'
({p | p.2 ∈ Bad p.1} ∩ fst ⁻¹' B) = Bad ω := by
ext x
simp [hω]
rw [he]
simpa [hω] using hsec ω
· have he : Prod.mk ω ⁻¹'
({p | p.2 ∈ Bad p.1} ∩ fst ⁻¹' B) = ∅ := by
ext x
simp [hω]
simp [he, hω]
_ = ofReal δ * μ B := by
rw [lintegral_indicator hB, lintegral_const,
Measure.restrict_apply_univ, trim_measurableSet_eq hmA hB]
theorem unitNormFeatureMap_dist_le
Every realization of the Gaussian kernel is sqrt 2-Lipschitz. the stated conclusion follows.
Formal statement
hypotheses
conclusion
‖U.Φ a - U.Φ b‖ ≤ sqrt 2 * |a - b|
Proof (Lean source)
-- @node: unitNormFeatureMap_dist_le
lemma unitNormFeatureMap_dist_le
{H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℝ H]
(U : UnitNormFeatureMap H) (a b : ℝ) :
‖U.Φ a - U.Φ b‖ ≤ sqrt 2 * |a - b| := by
have hsqrt : 0 ≤ sqrt 2 := Real.sqrt_nonneg _
have hsqrt_sq : (sqrt 2) ^ 2 = 2 := by norm_num
have hexp := Real.one_sub_le_exp_neg ((a - b) ^ 2)
have hsq : ‖U.Φ a - U.Φ b‖ ^ 2 ≤ 2 * (a - b) ^ 2 := by
rw [unitNormFeatureMap_dist_sq U]
unfold gaussianKernel
nlinarith
have habs : 0 ≤ |a - b| := abs_nonneg _
apply (sq_le_sq₀ (norm_nonneg _) (mul_nonneg hsqrt habs)).mp
calc
‖U.Φ a - U.Φ b‖ ^ 2 ≤ 2 * (a - b) ^ 2 := hsq
_ = (sqrt 2 * |a - b|) ^ 2 := by
rw [mul_pow, hsqrt_sq, sq_abs]
theorem meanEmbedding_map_sub_le_l1
Gaussian mean embeddings are controlled by sqrt 2 times the input L¹ distance. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
‖meanEmbedding U (Measure.map f μ) - meanEmbedding U (Measure.map g μ)‖
≤ sqrt 2 * ∫ x, |f x - g x| ∂μ
Proof (Lean source)
-- @node: meanEmbedding_map_sub_le_l1
lemma meanEmbedding_map_sub_le_l1
{X H : Type*} [MeasurableSpace X]
[NormedAddCommGroup H] [InnerProductSpace ℝ H] [CompleteSpace H]
(U : UnitNormFeatureMap H) (μ : Measure X) [IsProbabilityMeasure μ]
(f g : X → ℝ) (hf : Measurable f) (hg : Measurable g)
(hfg : Integrable (fun x => |f x - g x|) μ) :
‖meanEmbedding U (Measure.map f μ) - meanEmbedding U (Measure.map g μ)‖ ≤
sqrt 2 * ∫ x, |f x - g x| ∂μ := by
have hΦf : Integrable (fun x => U.Φ (f x)) μ :=
Integrable.of_bound
((unitNormFeatureMap_stronglyMeasurable U).comp_measurable hf).aestronglyMeasurable
1 (Filter.Eventually.of_forall fun x => by rw [U.norm_eq_one])
have hΦg : Integrable (fun x => U.Φ (g x)) μ :=
Integrable.of_bound
((unitNormFeatureMap_stronglyMeasurable U).comp_measurable hg).aestronglyMeasurable
1 (Filter.Eventually.of_forall fun x => by rw [U.norm_eq_one])
unfold meanEmbedding
rw [integral_map hf.aemeasurable
(unitNormFeatureMap_stronglyMeasurable U).aestronglyMeasurable,
integral_map hg.aemeasurable
(unitNormFeatureMap_stronglyMeasurable U).aestronglyMeasurable,
← integral_sub hΦf hΦg]
calc
‖∫ x, U.Φ (f x) - U.Φ (g x) ∂μ‖
≤ ∫ x, ‖U.Φ (f x) - U.Φ (g x)‖ ∂μ :=
norm_integral_le_integral_norm _
_ ≤ ∫ x, sqrt 2 * |f x - g x| ∂μ := by
exact integral_mono_ae (hΦf.sub hΦg).norm (hfg.const_mul _)
(Filter.Eventually.of_forall fun x => unitNormFeatureMap_dist_le U (f x) (g x))
_ = sqrt 2 * ∫ x, |f x - g x| ∂μ := integral_const_mul _ _
def fittedMeanEmbedding
Conditional population embedding of the fitted ratio under environment e.
Definition (Lean source)
parameters
def
fittedMeanEmbedding S U e i ω :
H
given by
meanEmbedding U (Measure.map (S.ratioEstimate ω i) (W.law e))
Conditional on training, all environment/ratio empirical feature means obey the stated unit-norm Hilbert-space deviation bound with probability at least 1-α. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
W :
ObservedWorld G θ
Ω H :
S :
SampleSplitWorld W Ω
U :
hSampling :
hn :
1 ≤ n
hN :
∀ e, 1 ≤ S.sampleSize e
hα :
S.alpha ∈ Ioo (0 : ℝ) 1
conclusion
ConditionalProbabilityAtLeast S.probability (comap (trainingFold S) inferInstance) {ω | ∀ i : Fin n, ∀ e : Fin (n + 1), ‖empiricalMeanEmbedding S U e i ω - fittedMeanEmbedding S U e i ω‖ ≤ (1 + sqrt (2 * log ((n : ℝ) * (n + 1 : ℝ) / S.alpha))) / sqrt (minimumSampleSize S)} univ (1 - S.alpha)
Proof (Lean source)
lemma bounded_rkhs_empirical_mean
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
{W : ObservedWorld G θ} {Ω H : Type*} [mΩ : MeasurableSpace Ω]
[NormedAddCommGroup H] [InnerProductSpace ℝ H]
[CompleteSpace H] [MeasurableSpace H] [BorelSpace H]
(S : SampleSplitWorld W Ω) (U : UnitNormFeatureMap H)
(hSampling : ConditionalEvaluationSampling S)
(hn : 1 ≤ n) (hN : ∀ e, 1 ≤ S.sampleSize e)
(hα : S.alpha ∈ Ioo (0 : ℝ) 1) :
ConditionalProbabilityAtLeast S.probability
(comap (trainingFold S) inferInstance)
{ω | ∀ i : Fin n, ∀ e : Fin (n + 1),
‖empiricalMeanEmbedding S U e i ω - fittedMeanEmbedding S U e i ω‖ ≤
(1 + sqrt (2 * log
((n : ℝ) * (n + 1 : ℝ) / S.alpha))) /
sqrt (minimumSampleSize S)} univ (1 - S.alpha) := by
classical
rcases hSampling with
⟨hprob, hlaw, htrain, hevalMeas, hevalIndep, hevalLaw, hsplit, hratioJoint⟩
letI : IsProbabilityMeasure S.probability := hprob
have hmA : trainingSigma S ≤ mΩ := htrain.comap_le
have hmin (e : Fin (n + 1)) : minimumSampleSize S ≤ S.sampleSize e :=
Finset.inf'_le S.sampleSize (Finset.mem_univ e)
have hmin_pos : 0 < minimumSampleSize S := by
rw [minimumSampleSize]
exact lt_of_lt_of_le Nat.zero_lt_one
(Finset.le_inf' ⟨0, Finset.mem_univ 0⟩ S.sampleSize (fun e _ => hN e))
let δ : ℝ := S.alpha / ((n : ℝ) * (n + 1 : ℝ))
have hnR : 0 < (n : ℝ) := by exact_mod_cast (lt_of_lt_of_le Nat.zero_lt_one hn)
have hn1 : (1 : ℝ) ≤ n := by exact_mod_cast hn
have hK : 1 ≤ (n : ℝ) * (n + 1 : ℝ) := by
calc
(1 : ℝ) = 1 * 1 := by ring
_ ≤ (n : ℝ) * (n + 1 : ℝ) :=
mul_le_mul hn1 (by linarith) (by norm_num) hnR.le
have hδpos : 0 < δ := by
dsimp [δ]
exact div_pos hα.1 (mul_pos hnR (by linarith))
have hδone : δ < 1 := by
dsimp [δ]
exact (div_le_self hα.1.le hK).trans_lt hα.2
have hradius (e : Fin (n + 1)) :
1 / sqrt (S.sampleSize e) +
sqrt (2 * log (1 / δ) / S.sampleSize e) ≤
(1 + sqrt (2 * log
((n : ℝ) * (n + 1 : ℝ) / S.alpha))) /
sqrt (minimumSampleSize S) := by
have hm0 : 0 < (S.sampleSize e : ℝ) := by exact_mod_cast hN e
have hmin0 : 0 < (minimumSampleSize S : ℝ) := by exact_mod_cast hmin_pos
have hle : sqrt (minimumSampleSize S) ≤ sqrt (S.sampleSize e) :=
Real.sqrt_le_sqrt (by exact_mod_cast hmin e)
have hlogarg : 1 / δ = (n : ℝ) * (n + 1 : ℝ) / S.alpha := by
dsimp [δ]
field_simp
have hratio_one : 1 ≤ (n : ℝ) * (n + 1 : ℝ) / S.alpha := by
apply (le_div_iff₀ hα.1).2
simpa only [one_mul] using hα.2.le.trans hK
have hlog_nonneg : 0 ≤ 2 * log
((n : ℝ) * (n + 1 : ℝ) / S.alpha) := by
exact mul_nonneg (by norm_num) (Real.log_nonneg hratio_one)
rw [hlogarg, Real.sqrt_div hlog_nonneg]
have hnum : 0 ≤ 1 + sqrt
(2 * log ((n : ℝ) * (n + 1 : ℝ) / S.alpha)) := by positivity
rw [← add_div]
exact div_le_div_of_nonneg_left hnum (Real.sqrt_pos.2 hmin0) hle
intro B hB
have hBfull : @MeasurableSet Ω mΩ B := hmA B hB
have hpair (i : Fin n) (e : Fin (n + 1)) :
MeasurableSet
{ω | (1 / sqrt (S.sampleSize e) +
sqrt (2 * log (1 / δ) / S.sampleSize e)) <
‖empiricalMeanEmbedding S U e i ω - fittedMeanEmbedding S U e i ω‖} ∧
S.probability
({ω | (1 / sqrt (S.sampleSize e) +
sqrt (2 * log (1 / δ) / S.sampleSize e)) <
‖empiricalMeanEmbedding S U e i ω - fittedMeanEmbedding S U e i ω‖} ∩ B)
≤ ofReal δ * S.probability B := by
let Y : Ω → (Fin (S.sampleSize e) → LatentState n) :=
fun ω r => S.evaluation e r ω
have hY : Measurable Y := by
apply measurable_pi_lambda
intro r
exact hevalMeas e r
let ν : Measure (Fin (S.sampleSize e) → LatentState n) :=
Measure.pi (fun _ => W.law e)
letI : IsProbabilityMeasure ν := by dsimp [ν]; infer_instance
have hYlaw : S.probability.map Y = ν := by
have hp := (ProbabilityTheory.iIndepFun_iff_map_fun_eq_pi_map
(fun r => (hevalMeas e r).aemeasurable)).mp (hevalIndep e)
calc
S.probability.map Y = Measure.pi
(fun r => S.probability.map (S.evaluation e r)) := hp
_ = ν := by
congr 1
funext r
exact hevalLaw e r
have hindepY : @Indep Ω (trainingSigma S) (comap Y inferInstance)
mΩ S.probability := by
change IndepFun (trainingFold S) Y S.probability
have he : Measurable (fun z : (∀ e, Fin (S.sampleSize e) → LatentState n) => z e) :=
measurable_pi_apply e
convert hsplit.comp measurable_id he using 1 <;> rfl
let f : Ω → LatentState n → H := fun ω x =>
U.Φ (@SampleSplitWorld.ratioEstimate n G θ W Ω mΩ S ω i x)
have hf_joint : @StronglyMeasurable (Ω × LatentState n) H
inferInstance ((trainingSigma S).prod inferInstance) (uncurry f) := by
apply (unitNormFeatureMap_stronglyMeasurable U).comp_measurable
exact hratioJoint i
have hf_int : @StronglyMeasurable Ω H inferInstance (trainingSigma S)
(fun ω => ∫ x, f ω x ∂W.law e) :=
@MeasureTheory.StronglyMeasurable.integral_prod_right Ω (LatentState n) H
(trainingSigma S) inferInstance (W.law e) inferInstance inferInstance
inferInstance f hf_joint
have hf_eval (r : Fin (S.sampleSize e)) : @StronglyMeasurable
(Ω × (Fin (S.sampleSize e) → LatentState n)) H
inferInstance ((trainingSigma S).prod inferInstance)
(fun p => f p.1 (p.2 r)) := by
exact hf_joint.comp_measurable (prodMk measurable_fst
((measurable_pi_apply r).comp measurable_snd))
let Bad : Ω → Set (Fin (S.sampleSize e) → LatentState n) := fun ω =>
{z | 1 / sqrt (S.sampleSize e) +
sqrt (2 * log (1 / δ) / S.sampleSize e) <
‖(S.sampleSize e : ℝ)⁻¹ • ∑ r, f ω (z r) - ∫ x, f ω x ∂W.law e‖}
have hBad : @MeasurableSet
(Ω × (Fin (S.sampleSize e) → LatentState n))
((trainingSigma S).prod inferInstance)
{p | p.2 ∈ Bad p.1} := by
change @MeasurableSet
(Ω × (Fin (S.sampleSize e) → LatentState n))
((trainingSigma S).prod inferInstance)
{(p : Ω × (Fin (S.sampleSize e) → LatentState n)) |
(1 / sqrt (S.sampleSize e) +
sqrt (2 * log (1 / δ) / S.sampleSize e)) <
‖(S.sampleSize e : ℝ)⁻¹ • ∑ r, f p.1 (p.2 r) -
∫ x, f p.1 x ∂W.law e‖}
apply measurableSet_lt measurable_const
apply StronglyMeasurable.measurable
apply StronglyMeasurable.norm
apply StronglyMeasurable.sub
· apply StronglyMeasurable.const_smul
rw [← Finset.sum_fn]
exact Finset.stronglyMeasurable_sum _ (fun r _ => hf_eval r)
· exact hf_int.comp_measurable measurable_fst
have hsec (ω : Ω) : ν (Bad ω) ≤ ofReal δ := by
have ht := centeredEmpiricalMean_norm_tail_le
(W.law e) (f ω) ((unitNormFeatureMap_stronglyMeasurable U).comp_measurable
(S.ratioFit_measurable (trainingFold S ω) i))
(fun x => by rw [U.norm_eq_one]) (hN e) hδpos hδone
have hne : ν (Bad ω) ≠ ⊤ := measure_ne_top _ _
rw [← ENNReal.ofReal_toReal hne]
exact ENNReal.ofReal_le_ofReal (by simpa [ν, Bad, f] using ht)
have hJ : @Measurable Ω
(Ω × (Fin (S.sampleSize e) → LatentState n)) mΩ
((trainingSigma S).prod inferInstance) (fun ω => (ω, Y ω)) :=
(measurable_id'' hmA).prod hY
have hevent : MeasurableSet {ω | Y ω ∈ Bad ω} := by
change MeasurableSet ((fun ω => (ω, Y ω)) ⁻¹' {p | p.2 ∈ Bad p.1})
exact hBad.preimage hJ
have hraw := randomParam_event_inter_le hY hYlaw (trainingSigma S) hmA
hindepY Bad hBad hsec B hB
have heq : {ω | Y ω ∈ Bad ω} =
{ω | (1 / sqrt (S.sampleSize e) +
sqrt (2 * log (1 / δ) / S.sampleSize e)) <
‖empiricalMeanEmbedding S U e i ω - fittedMeanEmbedding S U e i ω‖} := by
ext ω
simp only [Y, Bad, f, Set.mem_setOf_eq]
unfold empiricalMeanEmbedding fittedMeanEmbedding meanEmbedding
have hint : (∫ r, U.Φ r ∂Measure.map (S.ratioEstimate ω i) (W.law e)) =
∫ x, U.Φ (S.ratioEstimate ω i x) ∂W.law e :=
integral_map
(S.ratioFit_measurable (trainingFold S ω) i).aemeasurable
(unitNormFeatureMap_stronglyMeasurable U).aestronglyMeasurable
rw [hint]
rw [heq] at hevent hraw
exact ⟨hevent, hraw⟩
let BadPair : Fin n × Fin (n + 1) → Set Ω := fun ie =>
{ω | (1 / sqrt (S.sampleSize ie.2) +
sqrt (2 * log (1 / δ) / S.sampleSize ie.2)) <
‖empiricalMeanEmbedding S U ie.2 ie.1 ω -
fittedMeanEmbedding S U ie.2 ie.1 ω‖}
let BadAll : Set Ω := ⋃ ie, BadPair ie
have hBadAllMeas : MeasurableSet BadAll := by
dsimp only [BadAll]
exact MeasurableSet.iUnion fun ie => (hpair ie.1 ie.2).1
have hUnion : S.probability (BadAll ∩ B) ≤
ofReal S.alpha * S.probability B := by
calc
S.probability (BadAll ∩ B)
= S.probability (⋃ ie, BadPair ie ∩ B) := by
congr 1
dsimp only [BadAll]
ext ω
simp
_ ≤ ∑ ie : Fin n × Fin (n + 1), S.probability (BadPair ie ∩ B) :=
measure_iUnion_fintype_le _ _
_ ≤ ∑ _ie : Fin n × Fin (n + 1),
(ofReal δ * S.probability B) := by
gcongr with ie
exact (hpair ie.1 ie.2).2
_ = ofReal S.alpha * S.probability B := by
simp only [Finset.sum_const, Fintype.card_prod, Fintype.card_fin,
nsmul_eq_mul]
have hcoeff : ((Finset.univ : Finset (Fin n × Fin (n + 1))).card : ENNReal) *
ofReal δ = ofReal S.alpha := by
rw [← ENNReal.ofReal_natCast,
← ENNReal.ofReal_mul (by positivity : 0 ≤
((Finset.univ : Finset (Fin n × Fin (n + 1))).card : ℝ))]
congr 1
simp only [Finset.card_univ, Fintype.card_prod, Fintype.card_fin,
Nat.cast_mul, Nat.cast_add, cast_one]
dsimp [δ]
field_simp
calc
((Finset.univ : Finset (Fin n × Fin (n + 1))).card : ENNReal) *
(ofReal δ * S.probability B) =
(((Finset.univ : Finset (Fin n × Fin (n + 1))).card : ENNReal) *
ofReal δ) *
S.probability B := (mul_assoc _ _ _).symm
_ = _ := by rw [hcoeff]
let Good : Set Ω :=
{ω | ∀ i : Fin n, ∀ e : Fin (n + 1),
‖empiricalMeanEmbedding S U e i ω - fittedMeanEmbedding S U e i ω‖ ≤
(1 + sqrt (2 * log
((n : ℝ) * (n + 1 : ℝ) / S.alpha))) /
sqrt (minimumSampleSize S)}
have hgood : B \ BadAll ⊆ Good ∩ univ ∩ B := by
intro ω hω
dsimp only [BadAll] at hω
simp only [Set.mem_diff, Set.mem_iUnion, not_exists, BadPair, Set.mem_setOf_eq,
Set.mem_inter_iff, Set.mem_univ, true_and] at hω ⊢
refine ⟨⟨?_, trivial⟩, hω.1⟩
intro i e
exact (not_lt.mp (hω.2 (i, e))).trans (hradius e)
have hBfinite : S.probability B ≠ ⊤ := measure_ne_top _ _
have hbadfinite : S.probability (BadAll ∩ B) ≠ ⊤ := measure_ne_top _ _
have hmeasure : S.probability B - S.probability (BadAll ∩ B) ≤
S.probability (Good ∩ univ ∩ B) := by
rw [← measure_diff Set.inter_subset_right
((hBadAllMeas.inter hBfull).nullMeasurableSet)
hbadfinite]
apply measure_mono
intro ω hω
refine hgood ⟨hω.1, ?_⟩
exact fun hbad => hω.2 ⟨hbad, hω.1⟩
have hbadB : S.probability (BadAll ∩ B) ≤ S.probability B :=
measure_mono (Set.inter_subset_right)
have hUnionReal :
(S.probability (BadAll ∩ B)).toReal ≤
-- … truncated; follow the source link for the rest …
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.WitnessAssembly 14 declarations This file isolates the pointwise cancellations that turn the sparse and cancellation witness expectations into the low-dimensional integrals used by the certificate proof.
Explicit-witness density algebra
This file isolates the pointwise cancellations that turn the sparse and cancellation witness expectations into the low-dimensional integrals used by the certificate proof.
Fubini's theorem for a three-coordinate product measure, written in the explicit coordinate order used by the witness calculations. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
(∫ v, f v ∂Measure.pi fun _ : Fin 3 => μ) = ∫ x, (∫ y, (∫ z, f ![x, y, z] ∂μ) ∂μ) ∂μ
Proof (Lean source)
lemma integral_fin_three_pi_eq_iterated (μ : Measure ℝ) [SigmaFinite μ]
(f : (Fin 3 → ℝ) → ℝ) (hf : Integrable f (Measure.pi fun _ : Fin 3 => μ)) :
(∫ v, f v ∂Measure.pi fun _ : Fin 3 => μ) =
∫ x, (∫ y, (∫ z, f ![x, y, z] ∂μ) ∂μ) ∂μ := by
let e3 := MeasurableEquiv.piFinSuccAbove (fun _ : Fin 3 => ℝ) 0
have he3 := (measurePreserving_piFinSuccAbove (fun _ : Fin 3 => μ) 0).symm
have hf3 : Integrable (f ∘ e3.symm) (μ.prod (Measure.pi fun _ : Fin 2 => μ)) :=
(he3.integrable_comp_emb e3.symm.measurableEmbedding).2 hf
rw [← he3.integral_comp e3.symm.measurableEmbedding]
have hf3' := hf3
change Integrable
(fun z => f ((MeasurableEquiv.piFinSuccAbove (fun _ : Fin 3 => ℝ) 0).symm z))
(μ.prod (Measure.pi fun _ : Fin 2 => μ)) at hf3'
rw [integral_prod _ hf3']
apply integral_congr_ae
filter_upwards [hf3.prod_right_ae] with x hfx
let e2 := MeasurableEquiv.piFinSuccAbove (fun _ : Fin 2 => ℝ) 0
have he2 := (measurePreserving_piFinSuccAbove (fun _ : Fin 2 => μ) 0).symm
have hf2 : Integrable ((fun v => f (e3.symm (x, v))) ∘ e2.symm)
(μ.prod (Measure.pi fun _ : Fin 1 => μ)) :=
(he2.integrable_comp_emb e2.symm.measurableEmbedding).2 hfx
rw [← he2.integral_comp e2.symm.measurableEmbedding]
have hf2' := hf2
change Integrable (fun z => f (e3.symm (x, e2.symm z)))
(μ.prod (Measure.pi fun _ : Fin 1 => μ)) at hf2'
rw [integral_prod _ hf2']
simp only [e3, e2, MeasurableEquiv.piFinSuccAbove_symm_apply, Fin.insertNthEquiv,
Fin.insertNth_zero]
apply integral_congr_ae
filter_upwards [hf2'.prod_right_ae] with y hfy
let e1 := MeasurableEquiv.piUnique (fun _ : Fin 1 => ℝ)
have he1 := (measurePreserving_piUnique (fun _ : Fin 1 => μ)).symm
rw [← he1.integral_comp e1.symm.measurableEmbedding]
apply integral_congr_ae
filter_upwards with z
congr 1
funext i
fin_cases i <;> rfl
theorem sparse_observationalDensity_mul_ratio_sq
In the sparse observational law, one child-density factor cancels from the square of the explicit child ratio. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
observationalDensity (sparseWitness s) v * (((sparseWitness s).q 1 (v 1) / (sparseWitness s).p 1 v) ^ 2)
= ((sparseWitness s).q 1 (v 1)) ^ 2 / (sparseWitness s).p 1 v
Proof (Lean source)
lemma sparse_observationalDensity_mul_ratio_sq (s : SignVector 3)
(v : LatentState 3) (hv : v ∈ latentCube 3) :
observationalDensity (sparseWitness s) v *
(((sparseWitness s).q 1 (v 1) / (sparseWitness s).p 1 v) ^ 2) =
((sparseWitness s).q 1 (v 1)) ^ 2 / (sparseWitness s).p 1 v := by
have hp : (sparseWitness s).p 1 v ≠ 0 :=
ne_of_gt ((sparseWitness_positive_normalized_smooth s).1 1 v hv)
simp only [observationalDensity]
simp [sparseWitness, sparseP]
field_simp
theorem sparse_parentInterventionalDensity_mul_ratio_sq
Under the sparse parent intervention, the same cancellation leaves the parent tilt multiplying the reduced child integrand. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
interventionalDensity (sparseWitness s) 0 v * (((sparseWitness s).q 1 (v 1) / (sparseWitness s).p 1 v) ^ 2)
Proof (Lean source)
lemma sparse_parentInterventionalDensity_mul_ratio_sq (s : SignVector 3)
(v : LatentState 3) (hv : v ∈ latentCube 3) :
interventionalDensity (sparseWitness s) 0 v *
(((sparseWitness s).q 1 (v 1) / (sparseWitness s).p 1 v) ^ 2) =
(sparseWitness s).q 0 (v 0) *
(((sparseWitness s).q 1 (v 1)) ^ 2 / (sparseWitness s).p 1 v) := by
have hp : (sparseWitness s).p 1 v ≠ 0 :=
ne_of_gt ((sparseWitness_positive_normalized_smooth s).1 1 v hv)
simp only [interventionalDensity]
simp [sparseWitness, sparseP]
field_simp
theorem cancellation_density_difference_mul_test
The cancellation witness's parent-interventional minus observational test integrand factors into the parent tilt and the child test integrand. the stated conclusion follows.
Formal statement
hypotheses
conclusion
(interventionalDensity (cancellationWitness s) 0 v - observationalDensity (cancellationWitness s) v) * psi ((cancellationWitness s).q 1 (v 1) / (cancellationWitness s).p 1 v)
= ((cancellationWitness s).q 0 (v 0) - 1) * ((cancellationWitness s).p 1 v * psi ((cancellationWitness s).q 1 (v 1) / (cancellationWitness s).p 1 v))
Proof (Lean source)
lemma cancellation_density_difference_mul_test (s : SignVector 3)
(v : LatentState 3) (psi : ℝ → ℝ) :
(interventionalDensity (cancellationWitness s) 0 v -
observationalDensity (cancellationWitness s) v) *
psi ((cancellationWitness s).q 1 (v 1) /
(cancellationWitness s).p 1 v) =
((cancellationWitness s).q 0 (v 0) - 1) *
((cancellationWitness s).p 1 v *
psi ((cancellationWitness s).q 1 (v 1) /
(cancellationWitness s).p 1 v)) := by
simp only [interventionalDensity, observationalDensity]
simp [cancellationWitness, cancellationP]
ring
theorem cancellation_factored_test_eq_reflected_integrand
Expanding the cancellation witness identifies its factored density difference with the reflected two-coordinate integrand from the FTC argument. the stated conclusion follows.
Formal statement
hypotheses
conclusion
((cancellationWitness s).q 0 (v 0) - 1) * ((cancellationWitness s).p 1 v * psi ((cancellationWitness s).q 1 (v 1) / (cancellationWitness s).p 1 v))
= (exponentialInterventionDensity (reflectedCoordinate s 0 (v 0)) - 1) * ((1 + (1 / 10 : ℝ) * cancellationPrimitive (reflectedCoordinate s 0 (v 0)) * centeredCoordinate (reflectedCoordinate s 1 (v 1))) * psi (exponentialInterventionDensity (reflectedCoordinate s 1 (v 1)) / (1 + (1 / 10 : ℝ) * cancellationPrimitive (reflectedCoordinate s 0 (v 0)) * centeredCoordinate (reflectedCoordinate s 1 (v 1)))))
Proof (Lean source)
lemma cancellation_factored_test_eq_reflected_integrand (s : SignVector 3)
(v : LatentState 3) (psi : ℝ → ℝ) :
((cancellationWitness s).q 0 (v 0) - 1) *
((cancellationWitness s).p 1 v *
psi ((cancellationWitness s).q 1 (v 1) /
(cancellationWitness s).p 1 v)) =
(exponentialInterventionDensity (reflectedCoordinate s 0 (v 0)) - 1) *
((1 + (1 / 10 : ℝ) *
cancellationPrimitive (reflectedCoordinate s 0 (v 0)) *
centeredCoordinate (reflectedCoordinate s 1 (v 1))) *
psi (exponentialInterventionDensity (reflectedCoordinate s 1 (v 1)) /
(1 + (1 / 10 : ℝ) *
cancellationPrimitive (reflectedCoordinate s 0 (v 0)) *
centeredCoordinate (reflectedCoordinate s 1 (v 1))))) := by
simp [cancellationWitness, sparseQ, cancellationP]
theorem cancellation_density_difference_eq_reflected_integrand
The full density difference against a ratio-law test function is exactly the reflected two-coordinate cancellation integrand. the stated conclusion follows.
Formal statement
hypotheses
conclusion
(interventionalDensity (cancellationWitness s) 0 v - observationalDensity (cancellationWitness s) v) * psi ((cancellationWitness s).q 1 (v 1) / (cancellationWitness s).p 1 v)
= (exponentialInterventionDensity (reflectedCoordinate s 0 (v 0)) - 1) * ((1 + (1 / 10 : ℝ) * cancellationPrimitive (reflectedCoordinate s 0 (v 0)) * centeredCoordinate (reflectedCoordinate s 1 (v 1))) * psi (exponentialInterventionDensity (reflectedCoordinate s 1 (v 1)) / (1 + (1 / 10 : ℝ) * cancellationPrimitive (reflectedCoordinate s 0 (v 0)) * centeredCoordinate (reflectedCoordinate s 1 (v 1)))))
Proof (Lean source)
lemma cancellation_density_difference_eq_reflected_integrand (s : SignVector 3)
(v : LatentState 3) (psi : ℝ → ℝ) :
(interventionalDensity (cancellationWitness s) 0 v -
observationalDensity (cancellationWitness s) v) *
psi ((cancellationWitness s).q 1 (v 1) /
(cancellationWitness s).p 1 v) =
(exponentialInterventionDensity (reflectedCoordinate s 0 (v 0)) - 1) *
((1 + (1 / 10 : ℝ) *
cancellationPrimitive (reflectedCoordinate s 0 (v 0)) *
centeredCoordinate (reflectedCoordinate s 1 (v 1))) *
psi (exponentialInterventionDensity (reflectedCoordinate s 1 (v 1)) /
(1 + (1 / 10 : ℝ) *
cancellationPrimitive (reflectedCoordinate s 0 (v 0)) *
centeredCoordinate (reflectedCoordinate s 1 (v 1))))) := by
rw [cancellation_density_difference_mul_test,
cancellation_factored_test_eq_reflected_integrand]
theorem cancellation_reflected_iterated_test_integral_zero
Coordinate reflections preserve the complete cancellation integral, so it vanishes for every prescribed sign vector. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
∫ y in Icc (0 : ℝ) 1, ∫ x in Icc (0 : ℝ) 1, (exponentialInterventionDensity (reflectedCoordinate s 0 x) - 1) * ((1 + (1 / 10 : ℝ) * cancellationPrimitive (reflectedCoordinate s 0 x) * centeredCoordinate (reflectedCoordinate s 1 y)) * psi (exponentialInterventionDensity (reflectedCoordinate s 1 y) / (1 + (1 / 10 : ℝ) * cancellationPrimitive (reflectedCoordinate s 0 x) * centeredCoordinate (reflectedCoordinate s 1 y))))
= 0
Proof (Lean source)
lemma cancellation_reflected_iterated_test_integral_zero (s : SignVector 3)
(psi : ℝ → ℝ) (hpsi : Continuous psi) :
∫ y in Icc (0 : ℝ) 1,
∫ x in Icc (0 : ℝ) 1,
(exponentialInterventionDensity (reflectedCoordinate s 0 x) - 1) *
((1 + (1 / 10 : ℝ) *
cancellationPrimitive (reflectedCoordinate s 0 x) *
centeredCoordinate (reflectedCoordinate s 1 y)) *
psi (exponentialInterventionDensity (reflectedCoordinate s 1 y) /
(1 + (1 / 10 : ℝ) *
cancellationPrimitive (reflectedCoordinate s 0 x) *
centeredCoordinate (reflectedCoordinate s 1 y)))) = 0 := by
let J : ℝ → ℝ → ℝ := fun x y =>
(exponentialInterventionDensity x - 1) *
((1 + (1 / 10 : ℝ) * cancellationPrimitive x * centeredCoordinate y) *
psi (exponentialInterventionDensity y /
(1 + (1 / 10 : ℝ) * cancellationPrimitive x * centeredCoordinate y)))
change (∫ y in Icc (0 : ℝ) 1, ∫ x in Icc (0 : ℝ) 1,
J (reflectedCoordinate s 0 x) (reflectedCoordinate s 1 y)) = 0
calc
_ = ∫ y in Icc (0 : ℝ) 1, ∫ x in Icc (0 : ℝ) 1,
J x (reflectedCoordinate s 1 y) := by
apply integral_congr_ae
filter_upwards with y
exact integral_reflectedCoordinate s 0
(fun x => J x (reflectedCoordinate s 1 y))
_ = ∫ y in Icc (0 : ℝ) 1, ∫ x in Icc (0 : ℝ) 1, J x y :=
integral_reflectedCoordinate s 1
(fun y => ∫ x in Icc (0 : ℝ) 1, J x y)
_ = 0 := cancellation_iterated_test_integral_zero psi hpsi
theorem measurable_sparseQ
The reflected exponential intervention slots are globally measurable.
Formal statement
Proof (Lean source)
@[fun_prop] lemma measurable_sparseQ (s : SignVector 3) (i : Fin 3) :
Measurable (sparseQ s i) := by
rcases s.signed i with hi | hi
· unfold sparseQ exponentialInterventionDensity reflectedCoordinate reflect
rw [hi]
norm_num
fun_prop
· unfold sparseQ exponentialInterventionDensity reflectedCoordinate reflect
rw [hi]
simp only [if_pos]
fun_prop
theorem measurable_cancellationP
Every cancellation-witness observational mechanism slot is globally measurable.
Formal statement
Proof (Lean source)
@[fun_prop] lemma measurable_cancellationP (s : SignVector 3) (i : Fin 3) :
Measurable (cancellationP s i) := by
fin_cases i
· change Measurable (fun _ : LatentState 3 => (1 : ℝ))
fun_prop
· rcases s.signed 0 with h0 | h0 <;>
rcases s.signed 1 with h1 | h1 <;>
change Measurable (fun v : LatentState 3 => 1 + (1 / 10 : ℝ) *
cancellationPrimitive (reflectedCoordinate s 0 (v 0)) *
centeredCoordinate (reflectedCoordinate s 1 (v 1))) <;>
simp [cancellationPrimitive, centeredCoordinate, reflectedCoordinate, reflect, h0, h1,
show (-1 : ℝ) ≠ 1 by norm_num] <;> fun_prop
· change Measurable (fun _ : LatentState 3 => (1 : ℝ))
fun_prop
theorem measurable_cancellationWitness_observationalDensity
The cancellation witness's observational joint density is globally measurable.
Formal statement
Proof (Lean source)
@[fun_prop] lemma measurable_cancellationWitness_observationalDensity (s : SignVector 3) :
Measurable (observationalDensity (cancellationWitness s)) := by
unfold observationalDensity
apply Finset.measurable_prod
intro i _
exact measurable_cancellationP s i
theorem measurable_cancellationWitness_interventionalDensity
Every cancellation-witness interventional joint density is globally measurable.
Formal statement
Proof (Lean source)
@[fun_prop] lemma measurable_cancellationWitness_interventionalDensity
(s : SignVector 3) (j : Fin 3) :
Measurable (interventionalDensity (cancellationWitness s) j) := by
unfold interventionalDensity
apply (measurable_sparseQ s j).comp (measurable_pi_apply j) |>.mul
apply Finset.measurable_prod
intro i _
exact measurable_cancellationP s i
The finite latent unit cube is measurable. the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma measurableSet_latentCube (n : ℕ) : MeasurableSet (latentCube n) := by
rw [latentCube]
exact MeasurableSet.univ_pi fun _ => measurableSet_Icc
theorem cancellationWitness_ratioLaws_eq
The cancellation witness has exactly the same child-ratio law before and after intervening on its parent. the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma cancellationWitness_ratioLaws_eq (s : SignVector 3) :
let W := canonicalObservedWorld threeNodeDAG (cancellationWitness s)
(refl (Fin 3))
observationalRatioLaw W 1 = interventionalRatioLaw W 0 1 := by
dsimp only
let θ := cancellationWitness s
let W := canonicalObservedWorld threeNodeDAG θ (refl (Fin 3))
have hpos : PositiveNormalizedSmoothMechanisms threeNodeDAG θ :=
cancellationWitness_positive_normalized_smooth s
letI : IsProbabilityMeasure (observationalLaw θ) :=
observationalLaw_isProbabilityMeasure hpos
letI : IsProbabilityMeasure (interventionalLaw θ ((refl (Fin 3)) 0)) :=
interventionalLaw_isProbabilityMeasure W hpos 0
letI : IsFiniteMeasure (W.law 0) := by
change IsFiniteMeasure (observationalLaw θ)
infer_instance
letI : IsFiniteMeasure (W.law (succ 0)) := by
change IsFiniteMeasure (interventionalLaw θ ((refl (Fin 3)) 0))
infer_instance
have hratio : Measurable (fun v : LatentState 3 =>
θ.q ((refl (Fin 3)) 1) (v ((refl (Fin 3)) 1)) /
θ.p ((refl (Fin 3)) 1) v) := by
dsimp only [θ]
change Measurable ((sparseQ s 1 ∘ fun v : LatentState 3 => v 1) /
cancellationP s 1)
exact ((measurable_sparseQ s 1).comp (measurable_pi_apply 1)).div
(measurable_cancellationP s 1)
apply ratioLaws_eq_of_boundedContinuous_integrals_eq W 0 1
intro ψ
rw [canonical_observationalRatioLaw_eq_mechanismRatio_map hpos (refl (Fin 3)) 1,
canonical_interventionalRatioLaw_eq_mechanismRatio_map hpos (refl (Fin 3)) 0 1]
rw [MeasureTheory.integral_map]
· rw [MeasureTheory.integral_map]
· simp only [Equiv.refl_apply]
let ratio : LatentState 3 → ℝ := fun v => θ.q 1 (v 1) / θ.p 1 v
let μ : Measure (LatentState 3) := volume.restrict (latentCube 3)
have hobs : (∫ v, ψ (ratio v) ∂observationalLaw θ) =
∫ v, observationalDensity θ v * ψ (ratio v) ∂μ := by
unfold observationalLaw
rw [integral_withDensity_eq_integral_toReal_smul]
· apply integral_congr_ae
filter_upwards [ae_restrict_mem (measurableSet_latentCube 3)] with v hv
rw [ENNReal.toReal_ofReal (le_of_lt (by
unfold observationalDensity
exact Finset.prod_pos fun i _ => hpos.1 i v hv))]
rfl
· exact (measurable_cancellationWitness_observationalDensity s).ennreal_ofReal
· filter_upwards with v
exact ENNReal.ofReal_lt_top
have hint : (∫ v, ψ (ratio v) ∂interventionalLaw θ 0) =
∫ v, interventionalDensity θ 0 v * ψ (ratio v) ∂μ := by
unfold interventionalLaw
rw [integral_withDensity_eq_integral_toReal_smul]
· apply integral_congr_ae
filter_upwards [ae_restrict_mem (measurableSet_latentCube 3)] with v hv
rw [ENNReal.toReal_ofReal (le_of_lt (by
unfold interventionalDensity
exact mul_pos (hpos.2.1 0 (v 0) (hv 0 (Set.mem_univ 0)))
(Finset.prod_pos fun i _ => hpos.1 i v hv)))]
rfl
· exact (measurable_cancellationWitness_interventionalDensity s 0).ennreal_ofReal
· filter_upwards with v
exact ENNReal.ofReal_lt_top
rw [hobs, hint]
have hcubeCompact : IsCompact (latentCube 3) := by
rw [latentCube]
exact isCompact_univ_pi fun _ => isCompact_Icc
have hratioCont : ContinuousOn ratio (latentCube 3) := by
apply ((hpos.2.2.2.1 1).continuousOn.comp
((continuous_apply 1).continuousOn)
(fun (v : LatentState 3) (hv : v ∈ latentCube 3) =>
hv 1 (Set.mem_univ 1))).div
((hpos.2.2.1 1).continuousOn)
intro v hv
exact ne_of_gt (hpos.1 1 v hv)
have hobsDensityCont : ContinuousOn (observationalDensity θ) (latentCube 3) := by
unfold observationalDensity
exact continuousOn_finset_prod _ fun i _ => (hpos.2.2.1 i).continuousOn
have hintDensityCont : ContinuousOn (interventionalDensity θ 0) (latentCube 3) := by
unfold interventionalDensity
apply ((hpos.2.2.2.1 0).continuousOn.comp
((continuous_apply 0).continuousOn)
(fun (v : LatentState 3) (hv : v ∈ latentCube 3) =>
hv 0 (Set.mem_univ 0))).mul
exact continuousOn_finset_prod _ fun i _ => (hpos.2.2.1 i).continuousOn
have hψratioCont : ContinuousOn (fun v => ψ (ratio v)) (latentCube 3) :=
ψ.continuous.continuousOn.comp hratioCont (fun _ _ => Set.mem_univ _)
have hobsInt : Integrable
(fun v => observationalDensity θ v * ψ (ratio v)) μ := by
dsimp only [μ]
exact (hobsDensityCont.mul hψratioCont).integrableOn_compact hcubeCompact
have hintInt : Integrable
(fun v => interventionalDensity θ 0 v * ψ (ratio v)) μ := by
dsimp only [μ]
exact (hintDensityCont.mul hψratioCont).integrableOn_compact hcubeCompact
rw [eq_comm, ← sub_eq_zero, ← integral_sub hintInt hobsInt]
let f : LatentState 3 → ℝ := fun v =>
(exponentialInterventionDensity (reflectedCoordinate s 0 (v 0)) - 1) *
((1 + (1 / 10 : ℝ) *
cancellationPrimitive (reflectedCoordinate s 0 (v 0)) *
centeredCoordinate (reflectedCoordinate s 1 (v 1))) *
ψ (exponentialInterventionDensity (reflectedCoordinate s 1 (v 1)) /
(1 + (1 / 10 : ℝ) *
cancellationPrimitive (reflectedCoordinate s 0 (v 0)) *
centeredCoordinate (reflectedCoordinate s 1 (v 1)))))
have hdiff : (fun v => interventionalDensity θ 0 v * ψ (ratio v) -
observationalDensity θ v * ψ (ratio v)) =ᵐ[μ] f := by
filter_upwards with v
dsimp only [θ, ratio, f]
rw [← cancellation_density_difference_eq_reflected_integrand s v ψ]
ring
rw [integral_congr_ae hdiff]
have hμ : μ = Measure.pi (fun _ : Fin 3 => volume.restrict (Icc (0 : ℝ) 1)) := by
dsimp only [μ]
change volume.restrict (Set.univ.pi fun _ : Fin 3 => Icc (0 : ℝ) 1) = _
rw [MeasureTheory.volume_pi, Measure.restrict_pi_pi]
have hf : Integrable f
(Measure.pi (fun _ : Fin 3 => volume.restrict (Icc (0 : ℝ) 1))) := by
rw [← hμ]
exact (hintInt.sub hobsInt).congr hdiff
let J : ℝ → ℝ → ℝ := fun x y =>
(exponentialInterventionDensity (reflectedCoordinate s 0 x) - 1) *
((1 + (1 / 10 : ℝ) * cancellationPrimitive (reflectedCoordinate s 0 x) *
centeredCoordinate (reflectedCoordinate s 1 y)) *
ψ (exponentialInterventionDensity (reflectedCoordinate s 1 y) /
(1 + (1 / 10 : ℝ) * cancellationPrimitive (reflectedCoordinate s 0 x) *
centeredCoordinate (reflectedCoordinate s 1 y))))
let embed : ℝ × ℝ → LatentState 3 := fun z => ![z.1, z.2, 0]
have hembedCont : Continuous embed := by
fun_prop
have hembedCube : ∀ z ∈ Icc (0 : ℝ) 1 ×ˢ Icc (0 : ℝ) 1,
embed z ∈ latentCube 3 := by
intro z hz i _
fin_cases i
· simpa [embed] using hz.1
· simpa [embed] using hz.2
· simp [embed]
have hdiffCont : ContinuousOn (fun v =>
interventionalDensity θ 0 v * ψ (ratio v) -
observationalDensity θ v * ψ (ratio v)) (latentCube 3) :=
(hintDensityCont.mul hψratioCont).sub (hobsDensityCont.mul hψratioCont)
have hJCont : ContinuousOn (uncurry J)
(Icc (0 : ℝ) 1 ×ˢ Icc (0 : ℝ) 1) := by
apply (hdiffCont.comp hembedCont.continuousOn hembedCube).congr
intro z hz
dsimp only [uncurry, J, embed, θ, ratio]
simp only [Function.comp_apply]
rw [← sub_mul,
cancellation_density_difference_eq_reflected_integrand s ![z.1, z.2, 0] ψ]
simp only [Matrix.cons_val_zero, Matrix.cons_val_one]
have hJInt : Integrable (uncurry J)
((volume.restrict (Icc (0 : ℝ) 1)).prod
(volume.restrict (Icc (0 : ℝ) 1))) := by
rw [Measure.prod_restrict]
exact hJCont.integrableOn_compact (isCompact_Icc.prod isCompact_Icc)
have hunit : (volume.restrict (Icc (0 : ℝ) 1)).real univ = 1 := by
simp [Measure.real, Real.volume_Icc]
rw [hμ, integral_fin_three_pi_eq_iterated _ f hf]
dsimp only [f]
simp only [Matrix.cons_val_zero, Matrix.cons_val_one, integral_const,
hunit, one_smul]
change (∫ x in Icc (0 : ℝ) 1, ∫ y in Icc (0 : ℝ) 1, J x y) = 0
rw [MeasureTheory.integral_integral_swap hJInt]
exact cancellation_reflected_iterated_test_integral_zero s ψ ψ.continuous
· exact hratio.aemeasurable
· exact ψ.continuous.measurable.aestronglyMeasurable
· exact hratio.aemeasurable
· exact ψ.continuous.measurable.aestronglyMeasurable
theorem cancellationWitness_populationDiscrepancy_eq_zero
Exact cancellation of the ratio laws makes the cancellation witness's Gaussian population discrepancy vanish. the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma cancellationWitness_populationDiscrepancy_eq_zero (s : SignVector 3) :
let W := canonicalObservedWorld threeNodeDAG (cancellationWitness s)
(refl (Fin 3))
populationDiscrepancy gaussianFeatureMap W 0 1 = 0 := by
dsimp only
apply populationDiscrepancy_eq_zero_of_ratioLaws_eq
exact cancellationWitness_ratioLaws_eq s
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.WitnessCancellation 15 declarations This file isolates the fundamental-theorem-of-calculus identity behind equality of the cancellation witness's observational and parent-interventional ratio laws.
Cancellation-witness calculus
This file isolates the fundamental-theorem-of-calculus identity behind equality of the cancellation witness's observational and parent-interventional ratio laws.
theorem canonicalObservedWorld_observedLawRatio_ae_eq_mechanismRatio
In the canonical world, the law-defined Radon--Nikodym ratio agrees almost everywhere under the observational law with the explicit mechanism ratio. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
hpos :
i :
Fin n
conclusion
observedLawRatio (canonicalObservedWorld G θ π).law i
=ᵐ[observationalLaw θ] fun v => θ.q (π i) (v (π i)) / θ.p (π i) v
Proof (Lean source)
lemma canonicalObservedWorld_observedLawRatio_ae_eq_mechanismRatio
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(hpos : PositiveNormalizedSmoothMechanisms G θ) (π : Perm (Fin n))
(i : Fin n) :
observedLawRatio (canonicalObservedWorld G θ π).law i =ᵐ[observationalLaw θ]
fun v => θ.q (π i) (v (π i)) / θ.p (π i) v := by
let W := canonicalObservedWorld G θ π
have hone := canonicalObservedWorld_onePerfectInterventionPerNode hpos π
filter_upwards [hone.2.2.1 i, observationalLaw_ae_mem_latentCube hpos] with v hrv hv
change ((interventionalLaw θ (π i)).rnDeriv (observationalLaw θ) v).toReal = _
calc
_ = (ofReal (W.ratio i v)).toReal := congrArg toReal hrv.symm
_ = W.ratio i v := ENNReal.toReal_ofReal
(div_nonneg (hpos.2.1 _ _ (hv (π i) (Set.mem_univ _))).le
(hpos.1 _ _ hv).le)
_ = _ := by rfl
theorem canonical_observationalRatioLaw_eq_mechanismRatio_map
The canonical observational ratio law is the pushforward of the observational latent law by the explicit mechanism ratio. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
hpos :
i :
Fin n
conclusion
observationalRatioLaw (canonicalObservedWorld G θ π) i
= Measure.map (fun v => θ.q (π i) (v (π i)) / θ.p (π i) v) (observationalLaw θ)
Proof (Lean source)
lemma canonical_observationalRatioLaw_eq_mechanismRatio_map
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(hpos : PositiveNormalizedSmoothMechanisms G θ) (π : Perm (Fin n))
(i : Fin n) :
observationalRatioLaw (canonicalObservedWorld G θ π) i =
Measure.map (fun v => θ.q (π i) (v (π i)) / θ.p (π i) v)
(observationalLaw θ) := by
unfold observationalRatioLaw
exact Measure.map_congr
(canonicalObservedWorld_observedLawRatio_ae_eq_mechanismRatio hpos π i)
theorem canonical_interventionalRatioLaw_eq_mechanismRatio_map
The canonical interventional ratio law is the pushforward of the corresponding latent interventional law by the same explicit mechanism ratio. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
hpos :
j i :
Fin n
conclusion
interventionalRatioLaw (canonicalObservedWorld G θ π) j i
= Measure.map (fun v => θ.q (π i) (v (π i)) / θ.p (π i) v) (interventionalLaw θ (π j))
Proof (Lean source)
lemma canonical_interventionalRatioLaw_eq_mechanismRatio_map
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(hpos : PositiveNormalizedSmoothMechanisms G θ) (π : Perm (Fin n))
(j i : Fin n) :
interventionalRatioLaw (canonicalObservedWorld G θ π) j i =
Measure.map (fun v => θ.q (π i) (v (π i)) / θ.p (π i) v)
(interventionalLaw θ (π j)) := by
let W := canonicalObservedWorld G θ π
have hac : interventionalLaw θ (π j) ≪ observationalLaw θ := by
simpa only [W, canonicalObservedWorld] using
interventionalLaw_absolutelyContinuous_observational W hpos j
unfold interventionalRatioLaw
change Measure.map (observedLawRatio W.law i) (interventionalLaw θ (π j)) = _
exact Measure.map_congr
(hac.ae_eq
(canonicalObservedWorld_observedLawRatio_ae_eq_mechanismRatio hpos π i))
theorem canonical_observationalRatioLaw_secondMoment_eq
The second moment of the canonical observational ratio law is the latent observational integral of the square of the explicit mechanism ratio. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
hpos :
i :
Fin n
conclusion
(∫ r, r ^ 2 ∂observationalRatioLaw (canonicalObservedWorld G θ π) i)
= ∫ v, (θ.q (π i) (v (π i)) / θ.p (π i) v) ^ 2 ∂observationalLaw θ
Proof (Lean source)
lemma canonical_observationalRatioLaw_secondMoment_eq
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(hpos : PositiveNormalizedSmoothMechanisms G θ) (π : Perm (Fin n))
(i : Fin n) :
(∫ r, r ^ 2 ∂observationalRatioLaw (canonicalObservedWorld G θ π) i) =
∫ v, (θ.q (π i) (v (π i)) / θ.p (π i) v) ^ 2 ∂observationalLaw θ := by
let W := canonicalObservedWorld G θ π
rw [observationalRatioLaw,
MeasureTheory.integral_map (measurable_observedLawRatio W.law i).aemeasurable
(by fun_prop)]
apply integral_congr_ae
filter_upwards
[canonicalObservedWorld_observedLawRatio_ae_eq_mechanismRatio hpos π i] with v hv
rw [hv]
theorem canonical_interventionalRatioLaw_secondMoment_eq
The second moment of a canonical interventional ratio law is the corresponding latent interventional integral of the square of the explicit mechanism ratio. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
hpos :
j i :
Fin n
conclusion
(∫ r, r ^ 2 ∂interventionalRatioLaw (canonicalObservedWorld G θ π) j i)
= ∫ v, (θ.q (π i) (v (π i)) / θ.p (π i) v) ^ 2 ∂interventionalLaw θ (π j)
Proof (Lean source)
lemma canonical_interventionalRatioLaw_secondMoment_eq
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(hpos : PositiveNormalizedSmoothMechanisms G θ) (π : Perm (Fin n))
(j i : Fin n) :
(∫ r, r ^ 2 ∂interventionalRatioLaw (canonicalObservedWorld G θ π) j i) =
∫ v, (θ.q (π i) (v (π i)) / θ.p (π i) v) ^ 2
∂interventionalLaw θ (π j) := by
let W := canonicalObservedWorld G θ π
have hac : interventionalLaw θ (π j) ≪ observationalLaw θ := by
simpa only [W, canonicalObservedWorld] using
interventionalLaw_absolutelyContinuous_observational W hpos j
rw [interventionalRatioLaw,
MeasureTheory.integral_map (measurable_observedLawRatio W.law i).aemeasurable
(by fun_prop)]
apply integral_congr_ae
filter_upwards
[hac.ae_eq
(canonicalObservedWorld_observedLawRatio_ae_eq_mechanismRatio hpos π i)] with v hv
rw [hv]
The cancellation primitive has derivative equal to the intervention-density tilt. the stated conclusion follows.
Formal statement
hypotheses
x :
ℝ
conclusion
Proof (Lean source)
lemma cancellationPrimitive_hasDerivAt (x : ℝ) :
HasDerivAt cancellationPrimitive (exponentialInterventionDensity x - 1) x := by
change HasDerivAt
(fun y : ℝ => (exp (4 * y) - 1) / (exp 4 - 1) - y)
(exponentialInterventionDensity x - 1) x
have hexp : HasDerivAt (fun y : ℝ => exp (4 * y))
(4 * exp (4 * x)) x := by
simpa [Function.comp_def, mul_comm] using
(Real.hasDerivAt_exp (4 * x)).comp x (hasDerivAt_const_mul (x := x) 4)
have hfrac : HasDerivAt
(fun y : ℝ => (exp (4 * y) - 1) / (exp 4 - 1))
(exponentialInterventionDensity x) x := by
simpa [exponentialInterventionDensity] using
(hexp.sub_const 1).div_const (exp 4 - 1)
exact hfrac.sub (hasDerivAt_id x)
theorem cancellation_tilt_integral_comp_zero Lemma cancellation_tilt_integral_comp_zero in the paper ↗
Integrating the cancellation tilt against any continuous function of the cancellation primitive gives zero. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
B :
ℝ → ℝ
hB :
conclusion
= 0
Proof (Lean source)
lemma cancellation_tilt_integral_comp_zero (B : ℝ → ℝ) (hB : Continuous B) :
∫ x in Icc (0 : ℝ) 1,
(exponentialInterventionDensity x - 1) * B (cancellationPrimitive x) = 0 := by
let F : ℝ → ℝ := fun u => ∫ z in (0 : ℝ)..u, B z
have hF (u : ℝ) : HasDerivAt F (B u) u := by
exact intervalIntegral.integral_hasDerivAt_right
(hB.intervalIntegrable 0 u) (hB.stronglyMeasurableAtFilter volume (nhds u))
hB.continuousAt
have hcomp (x : ℝ) : HasDerivAt (fun x => F (cancellationPrimitive x))
((exponentialInterventionDensity x - 1) * B (cancellationPrimitive x)) x := by
change HasDerivAt (F ∘ cancellationPrimitive)
((exponentialInterventionDensity x - 1) * B (cancellationPrimitive x)) x
simpa only [mul_comm] using
(hF (cancellationPrimitive x)).comp x (cancellationPrimitive_hasDerivAt x)
have hint : IntervalIntegrable
(fun x => (exponentialInterventionDensity x - 1) * B (cancellationPrimitive x))
volume 0 1 := by
apply Continuous.intervalIntegrable
apply Continuous.mul
· unfold exponentialInterventionDensity
fun_prop
· apply hB.comp
unfold cancellationPrimitive
fun_prop
have hFTC := intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le
(show (0 : ℝ) ≤ 1 by norm_num)
(show ContinuousOn (fun x => F (cancellationPrimitive x)) (Icc 0 1) by
exact (continuous_iff_continuousAt.mpr fun x => (hcomp x).continuousAt).continuousOn)
(fun x _ => hcomp x) hint
rw [MeasureTheory.integral_Icc_eq_integral_Ioc]
rw [← intervalIntegral.integral_of_le (show (0 : ℝ) ≤ 1 by norm_num)]
rw [hFTC]
simp [cancellationPrimitive_zero]
Clamp a real argument to the range of the cancellation primitive.
theorem clampCancellationRange_eq_self
If the argument already lies in the cancellation range, then clamping leaves it unchanged.
Formal statement
Proof (Lean source)
@[simp] lemma clampCancellationRange_eq_self {u : ℝ} (hu : u ∈ Icc (-1 : ℝ) 0) :
clampCancellationRange u = u := by
simp [clampCancellationRange, hu.1, hu.2]
A globally continuous extension of the one-dimensional test integrand used to identify the cancellation ratio law.
Definition (Lean source)
parameters
ψ :
ℝ → ℝ
y u :
ℝ
def
cancellationTestIntegrand ψ y u :
ℝ
given by
d :
1 + (1 / 10 : ℝ) * clampCancellationRange u * centeredCoordinate y
d * ψ (exponentialInterventionDensity y / d)
theorem cancellationTestIntegrand_continuous
The extended cancellation test integrand is continuous when the test function is continuous and the child coordinate lies in the unit interval. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma cancellationTestIntegrand_continuous (ψ : ℝ → ℝ) (hψ : Continuous ψ) {y : ℝ}
(hy : y ∈ Icc (0 : ℝ) 1) : Continuous (cancellationTestIntegrand ψ y) := by
have hcy := abs_centeredCoordinate_le_one hy
rw [abs_le] at hcy
have hclamp (u : ℝ) : clampCancellationRange u ∈ Icc (-1 : ℝ) 0 := by
simp only [clampCancellationRange, Set.mem_Icc]
constructor
· exact le_max_left _ _
· exact max_le (by norm_num) (min_le_left _ _)
have hden (u : ℝ) :
1 + (1 / 10 : ℝ) * clampCancellationRange u * centeredCoordinate y ≠ 0 := by
have hu := hclamp u
have hab : -1 ≤ clampCancellationRange u * centeredCoordinate y := by
apply hu.1.trans
simpa using mul_le_mul_of_nonpos_left hcy.2 hu.2
nlinarith
unfold cancellationTestIntegrand
dsimp only
apply Continuous.mul
· unfold clampCancellationRange centeredCoordinate
fun_prop
· apply hψ.comp
apply Continuous.div continuous_const
· unfold clampCancellationRange centeredCoordinate
fun_prop
· exact hden
theorem cancellation_test_integrand_tilt_zero Lemma cancellation_test_integrand_tilt_zero in the paper ↗
The parent intervention tilt integrates to zero against the exact test integrand that appears after fixing the cancellation witness's child coordinate. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
∫ x in Icc (0 : ℝ) 1, (exponentialInterventionDensity x - 1) * ((1 + (1 / 10 : ℝ) * cancellationPrimitive x * centeredCoordinate y) * ψ (exponentialInterventionDensity y / (1 + (1 / 10 : ℝ) * cancellationPrimitive x * centeredCoordinate y)))
= 0
Proof (Lean source)
lemma cancellation_test_integrand_tilt_zero (ψ : ℝ → ℝ) (hψ : Continuous ψ)
{y : ℝ} (hy : y ∈ Icc (0 : ℝ) 1) :
∫ x in Icc (0 : ℝ) 1,
(exponentialInterventionDensity x - 1) *
((1 + (1 / 10 : ℝ) * cancellationPrimitive x * centeredCoordinate y) *
ψ (exponentialInterventionDensity y /
(1 + (1 / 10 : ℝ) * cancellationPrimitive x * centeredCoordinate y))) = 0 := by
calc
_ = ∫ x in Icc (0 : ℝ) 1,
(exponentialInterventionDensity x - 1) *
cancellationTestIntegrand ψ y (cancellationPrimitive x) := by
apply integral_congr_ae
filter_upwards [ae_restrict_mem measurableSet_Icc] with x hx
simp only [cancellationTestIntegrand]
rw [show clampCancellationRange (cancellationPrimitive x) = cancellationPrimitive x by
exact clampCancellationRange_eq_self (cancellationPrimitive_mem_negUnitInterval hx)]
_ = 0 := cancellation_tilt_integral_comp_zero (cancellationTestIntegrand ψ y)
(cancellationTestIntegrand_continuous ψ hψ hy)
theorem cancellation_iterated_test_integral_zero Lemma cancellation_iterated_test_integral_zero in the paper ↗
The complete two-coordinate cancellation integral vanishes for every continuous test function. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
ψ :
ℝ → ℝ
hψ :
conclusion
∫ y in Icc (0 : ℝ) 1, ∫ x in Icc (0 : ℝ) 1, (exponentialInterventionDensity x - 1) * ((1 + (1 / 10 : ℝ) * cancellationPrimitive x * centeredCoordinate y) * ψ (exponentialInterventionDensity y / (1 + (1 / 10 : ℝ) * cancellationPrimitive x * centeredCoordinate y)))
= 0
Proof (Lean source)
lemma cancellation_iterated_test_integral_zero (ψ : ℝ → ℝ) (hψ : Continuous ψ) :
∫ y in Icc (0 : ℝ) 1,
∫ x in Icc (0 : ℝ) 1,
(exponentialInterventionDensity x - 1) *
((1 + (1 / 10 : ℝ) * cancellationPrimitive x * centeredCoordinate y) *
ψ (exponentialInterventionDensity y /
(1 + (1 / 10 : ℝ) * cancellationPrimitive x * centeredCoordinate y))) = 0 := by
calc
_ = ∫ _y in Icc (0 : ℝ) 1, (0 : ℝ) := by
apply integral_congr_ae
filter_upwards [ae_restrict_mem measurableSet_Icc] with y hy
exact cancellation_test_integrand_tilt_zero ψ hψ hy
_ = 0 := by simp
theorem ratioLaws_eq_of_boundedContinuous_integrals_eq
Two finite ratio laws coincide when every bounded continuous real test function has the same integral under both laws. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
W :
ObservedWorld G θ
j i :
Fin n
IsFiniteMeasure (W.law 0)
IsFiniteMeasure (W.law j.succ)
h :
∀ ψ : BoundedContinuousFunction ℝ ℝ,
(∫ r, ψ r ∂observationalRatioLaw W i) = ∫ r, ψ r ∂interventionalRatioLaw W j i
conclusion
observationalRatioLaw W i = interventionalRatioLaw W j i
Proof (Lean source)
lemma ratioLaws_eq_of_boundedContinuous_integrals_eq
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(W : ObservedWorld G θ) (j i : Fin n)
[IsFiniteMeasure (W.law 0)] [IsFiniteMeasure (W.law j.succ)]
(h : ∀ ψ : BoundedContinuousFunction ℝ ℝ,
(∫ r, ψ r ∂observationalRatioLaw W i) =
∫ r, ψ r ∂interventionalRatioLaw W j i) :
observationalRatioLaw W i = interventionalRatioLaw W j i := by
letI : IsFiniteMeasure (observationalRatioLaw W i) := by
unfold observationalRatioLaw
exact Measure.isFiniteMeasure_map _ _
letI : IsFiniteMeasure (interventionalRatioLaw W j i) := by
unfold interventionalRatioLaw
exact Measure.isFiniteMeasure_map _ _
exact MeasureTheory.ext_of_forall_integral_eq_of_IsFiniteMeasure h
theorem populationDiscrepancy_eq_zero_of_ratioLaws_eq
Equality of the observational and interventional ratio laws forces their kernel mean discrepancy to vanish. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
H :
U :
W :
ObservedWorld G θ
j i :
Fin n
h :
observationalRatioLaw W i = interventionalRatioLaw W j i
conclusion
populationDiscrepancy U W j i = 0
Proof (Lean source)
lemma populationDiscrepancy_eq_zero_of_ratioLaws_eq
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
{H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℝ H] [CompleteSpace H]
(U : UnitNormFeatureMap H) (W : ObservedWorld G θ) (j i : Fin n)
(h : observationalRatioLaw W i = interventionalRatioLaw W j i) :
populationDiscrepancy U W j i = 0 := by
simp only [populationDiscrepancy, h, sub_self, norm_zero]
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.WitnessCancellationFaithfulness 9 declarations This file proves dependence of the cancellation witness's unique edge from a strictly positive covariance, and then invokes the three-node graph reduction.
Cancellation-witness faithfulness
This file proves dependence of the cancellation witness's unique edge from a strictly positive covariance, and then invokes the three-node graph reduction.
theorem continuous_reflectedCancellationPrimitive
A fixed signed reflection followed by the cancellation primitive is continuous.
Formal statement
Proof (Lean source)
@[fun_prop] lemma continuous_reflectedCancellationPrimitive
(s : SignVector 3) (i : Fin 3) :
Continuous (fun z : ℝ => cancellationPrimitive (reflectedCoordinate s i z)) := by
rcases s.signed i with hi | hi <;>
simp [cancellationPrimitive, reflectedCoordinate, reflect, hi,
show (-1 : ℝ) ≠ 1 by norm_num] <;> fun_prop
theorem cancellationWitness_integral_eq_iterated
A continuous test function under the cancellation observational law is its explicit three-coordinate iterated density integral. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
(∫ v, f v ∂observationalLaw (cancellationWitness s))
= ∫ x in Icc (0 : ℝ) 1, ∫ y in Icc (0 : ℝ) 1, ∫ z in Icc (0 : ℝ) 1, observationalDensity (cancellationWitness s) ![x, y, z] * f ![x, y, z]
Proof (Lean source)
lemma cancellationWitness_integral_eq_iterated (s : SignVector 3)
(f : LatentState 3 → ℝ) (hf : ContinuousOn f (latentCube 3)) :
(∫ v, f v ∂observationalLaw (cancellationWitness s)) =
∫ x in Icc (0 : ℝ) 1,
∫ y in Icc (0 : ℝ) 1,
∫ z in Icc (0 : ℝ) 1,
observationalDensity (cancellationWitness s) ![x, y, z] * f ![x, y, z] := by
let μ : Measure (LatentState 3) := volume.restrict (latentCube 3)
have hpos := cancellationWitness_positive_normalized_smooth s
have hcubeCompact : IsCompact (latentCube 3) := by
rw [latentCube]
exact isCompact_univ_pi fun _ => isCompact_Icc
have hdensityCont : ContinuousOn (observationalDensity (cancellationWitness s))
(latentCube 3) := by
unfold observationalDensity
exact continuousOn_finset_prod _ fun i _ => (hpos.2.2.1 i).continuousOn
have hprodInt : Integrable
(fun v => observationalDensity (cancellationWitness s) v * f v) μ := by
dsimp only [μ]
exact (hdensityCont.mul hf).integrableOn_compact hcubeCompact
have hμ : μ = Measure.pi (fun _ : Fin 3 =>
volume.restrict (Icc (0 : ℝ) 1)) := by
dsimp only [μ]
change volume.restrict (Set.univ.pi fun _ : Fin 3 => Icc (0 : ℝ) 1) = _
rw [MeasureTheory.volume_pi, Measure.restrict_pi_pi]
calc
(∫ v, f v ∂observationalLaw (cancellationWitness s)) =
∫ v, observationalDensity (cancellationWitness s) v * f v ∂μ := by
unfold observationalLaw
rw [integral_withDensity_eq_integral_toReal_smul]
· apply integral_congr_ae
filter_upwards [ae_restrict_mem (measurableSet_latentCube 3)] with v hv
rw [ENNReal.toReal_ofReal (le_of_lt (by
unfold observationalDensity
exact Finset.prod_pos fun i _ => hpos.1 i v hv))]
rfl
· exact (measurable_cancellationWitness_observationalDensity s).ennreal_ofReal
· filter_upwards with v
exact ENNReal.ofReal_lt_top
_ = _ := by
rw [hμ, integral_fin_three_pi_eq_iterated _ _ (by simpa [hμ] using hprodInt)]
theorem cancellationWitness_parent_inner_integral
Integrating over the child leaves the cancellation coefficient unchanged. the stated conclusion follows.
Formal statement
hypotheses
s :
x :
ℝ
conclusion
(∫ y in Icc (0 : ℝ) 1, (1 + (1 / 10 : ℝ) * cancellationPrimitive (reflectedCoordinate s 0 x) * centeredCoordinate (reflectedCoordinate s 1 y)) * cancellationPrimitive (reflectedCoordinate s 0 x))
= cancellationPrimitive (reflectedCoordinate s 0 x)
Proof (Lean source)
lemma cancellationWitness_parent_inner_integral (s : SignVector 3) (x : ℝ) :
(∫ y in Icc (0 : ℝ) 1,
(1 + (1 / 10 : ℝ) *
cancellationPrimitive (reflectedCoordinate s 0 x) *
centeredCoordinate (reflectedCoordinate s 1 y)) *
cancellationPrimitive (reflectedCoordinate s 0 x)) =
cancellationPrimitive (reflectedCoordinate s 0 x) := by
let A := cancellationPrimitive (reflectedCoordinate s 0 x)
let B : ℝ → ℝ := fun y => centeredCoordinate (reflectedCoordinate s 1 y)
have hB : Continuous B := continuous_reflectedCenteredCoordinate s 1
have h2 : IntegrableOn (fun y => (1 / 10 : ℝ) * A ^ 2 * B y)
(Icc (0 : ℝ) 1) :=
(continuous_const.mul hB).continuousOn.integrableOn_Icc
have hconst : IntegrableOn (fun _ : ℝ => A) (Icc (0 : ℝ) 1) :=
continuous_const.continuousOn.integrableOn_Icc
rw [show (fun y => (1 + (1 / 10 : ℝ) * A * B y) * A) =
fun y => A + (1 / 10 : ℝ) * A ^ 2 * B y by funext y; ring,
integral_add hconst h2, MeasureTheory.integral_const,
MeasureTheory.integral_const_mul, centeredCoordinate_reflected_integral,
mul_zero, add_zero]
simp [A, Real.volume_Icc]
theorem cancellationWitness_child_inner_integral
Integrating the centered child coordinate gives one thirtieth of the parent coefficient. the stated conclusion follows.
Formal statement
hypotheses
s :
x :
ℝ
conclusion
(∫ y in Icc (0 : ℝ) 1, (1 + (1 / 10 : ℝ) * cancellationPrimitive (reflectedCoordinate s 0 x) * centeredCoordinate (reflectedCoordinate s 1 y)) * centeredCoordinate (reflectedCoordinate s 1 y))
= (1 / 30 : ℝ) * cancellationPrimitive (reflectedCoordinate s 0 x)
Proof (Lean source)
lemma cancellationWitness_child_inner_integral (s : SignVector 3) (x : ℝ) :
(∫ y in Icc (0 : ℝ) 1,
(1 + (1 / 10 : ℝ) *
cancellationPrimitive (reflectedCoordinate s 0 x) *
centeredCoordinate (reflectedCoordinate s 1 y)) *
centeredCoordinate (reflectedCoordinate s 1 y)) =
(1 / 30 : ℝ) * cancellationPrimitive (reflectedCoordinate s 0 x) := by
let A := cancellationPrimitive (reflectedCoordinate s 0 x)
let B : ℝ → ℝ := fun y => centeredCoordinate (reflectedCoordinate s 1 y)
have hB : Continuous B := continuous_reflectedCenteredCoordinate s 1
have h1 : IntegrableOn B (Icc (0 : ℝ) 1) :=
hB.continuousOn.integrableOn_Icc
have h2 : IntegrableOn (fun y => (1 / 10 : ℝ) * A * B y ^ 2)
(Icc (0 : ℝ) 1) :=
(continuous_const.mul (hB.pow 2)).continuousOn.integrableOn_Icc
rw [show (fun y => (1 + (1 / 10 : ℝ) * A * B y) * B y) =
fun y => B y + (1 / 10 : ℝ) * A * B y ^ 2 by funext y; ring,
integral_add h1 h2, MeasureTheory.integral_const_mul,
centeredCoordinate_reflected_integral,
centeredCoordinate_reflected_sq_integral]
dsimp only [A]
ring
theorem cancellationWitness_joint_inner_integral
The child-parent test-product integral is one thirtieth of the squared coefficient. the stated conclusion follows.
Formal statement
hypotheses
s :
x :
ℝ
conclusion
(∫ y in Icc (0 : ℝ) 1, (1 + (1 / 10 : ℝ) * cancellationPrimitive (reflectedCoordinate s 0 x) * centeredCoordinate (reflectedCoordinate s 1 y)) * (cancellationPrimitive (reflectedCoordinate s 0 x) * centeredCoordinate (reflectedCoordinate s 1 y)))
= (1 / 30 : ℝ) * cancellationPrimitive (reflectedCoordinate s 0 x) ^ 2
Proof (Lean source)
lemma cancellationWitness_joint_inner_integral (s : SignVector 3) (x : ℝ) :
(∫ y in Icc (0 : ℝ) 1,
(1 + (1 / 10 : ℝ) *
cancellationPrimitive (reflectedCoordinate s 0 x) *
centeredCoordinate (reflectedCoordinate s 1 y)) *
(cancellationPrimitive (reflectedCoordinate s 0 x) *
centeredCoordinate (reflectedCoordinate s 1 y))) =
(1 / 30 : ℝ) * cancellationPrimitive (reflectedCoordinate s 0 x) ^ 2 := by
let A := cancellationPrimitive (reflectedCoordinate s 0 x)
let B : ℝ → ℝ := fun y => centeredCoordinate (reflectedCoordinate s 1 y)
have hB : Continuous B := continuous_reflectedCenteredCoordinate s 1
have h1 : IntegrableOn (fun y => A * B y) (Icc (0 : ℝ) 1) :=
(continuous_const.mul hB).continuousOn.integrableOn_Icc
have h2 : IntegrableOn (fun y => (1 / 10 : ℝ) * A ^ 2 * B y ^ 2)
(Icc (0 : ℝ) 1) :=
(continuous_const.mul (hB.pow 2)).continuousOn.integrableOn_Icc
rw [show (fun y => (1 + (1 / 10 : ℝ) * A * B y) * (A * B y)) =
fun y => A * B y + (1 / 10 : ℝ) * A ^ 2 * B y ^ 2 by funext y; ring,
integral_add h1 h2, MeasureTheory.integral_const_mul,
MeasureTheory.integral_const_mul, centeredCoordinate_reflected_integral,
centeredCoordinate_reflected_sq_integral]
dsimp only [A]
ring
theorem reflectedCancellationPrimitive_variance_pos
The reflected cancellation coefficient has strictly positive variance under unit Lebesgue law. the stated conclusion follows.
Formal statement
hypotheses
s :
conclusion
Proof (Lean source)
lemma reflectedCancellationPrimitive_variance_pos (s : SignVector 3) :
0 < (∫ x in Icc (0 : ℝ) 1,
cancellationPrimitive (reflectedCoordinate s 0 x) ^ 2) -
(∫ x in Icc (0 : ℝ) 1,
cancellationPrimitive (reflectedCoordinate s 0 x)) ^ 2 := by
let K : ℝ → ℝ := fun x => cancellationPrimitive (reflectedCoordinate s 0 x)
let m : ℝ := ∫ x in Icc (0 : ℝ) 1, K x
have hK : Continuous K := continuous_reflectedCancellationPrimitive s 0
have hx : ∃ x ∈ Icc (0 : ℝ) 1, K x ≠ m := by
have hmid : K (1 / 2) = cancellationPrimitive (1 / 2) := by
rcases s.signed 0 with hi | hi <;>
norm_num [K, reflectedCoordinate, reflect, hi,
show (-1 : ℝ) ≠ 1 by norm_num]
by_cases hm : K (1 / 2) = m
· rcases s.signed 0 with hi | hi
· refine ⟨1, by norm_num, ?_⟩
have hend : K 1 = 0 := by
simp only [K, reflectedCoordinate, hi, reflect,
if_neg (show (-1 : ℝ) ≠ 1 by norm_num)]
norm_num
exact cancellationPrimitive_zero.1
rw [← hm, hmid]
rw [hend]
exact ne_of_gt cancellationPrimitive_half_neg
· refine ⟨0, by norm_num, ?_⟩
have hend : K 0 = 0 := by
simp [K, reflectedCoordinate, reflect, hi, cancellationPrimitive_zero.1]
rw [← hm, hmid]
rw [hend]
exact ne_of_gt cancellationPrimitive_half_neg
· exact ⟨1 / 2, by norm_num, hm⟩
have hposInt : 0 < ∫ x in Icc (0 : ℝ) 1, (K x - m) ^ 2 := by
rw [MeasureTheory.integral_Icc_eq_integral_Ioc,
← intervalIntegral.integral_of_le (by norm_num : (0 : ℝ) ≤ 1)]
have hlt := intervalIntegral.integral_lt_integral_of_continuousOn_of_le_of_exists_lt
(f := fun _ : ℝ => 0) (g := fun x => (K x - m) ^ 2)
(by norm_num : (0 : ℝ) < 1) continuousOn_const
((hK.sub continuous_const).pow 2).continuousOn
(fun x hx' => sq_nonneg _)
(by
rcases hx with ⟨x, hxI, hxm⟩
exact ⟨x, hxI, sq_pos_of_ne_zero (sub_ne_zero.mpr hxm)⟩)
simpa using hlt
have hexpand : (∫ x in Icc (0 : ℝ) 1, (K x - m) ^ 2) =
(∫ x in Icc (0 : ℝ) 1, K x ^ 2) - m ^ 2 := by
have hKint : IntegrableOn K (Icc (0 : ℝ) 1) :=
hK.continuousOn.integrableOn_Icc
have hKsq : IntegrableOn (fun x => K x ^ 2) (Icc (0 : ℝ) 1) :=
(hK.pow 2).continuousOn.integrableOn_Icc
have hcross : IntegrableOn (fun x => 2 * m * K x) (Icc (0 : ℝ) 1) :=
hKint.const_mul (2 * m)
have hconst : IntegrableOn (fun _ : ℝ => m ^ 2) (Icc (0 : ℝ) 1) :=
continuous_const.continuousOn.integrableOn_Icc
calc
_ = ∫ x in Icc (0 : ℝ) 1,
(K x ^ 2 - 2 * m * K x) + m ^ 2 := by
apply integral_congr_ae
filter_upwards with x
ring
_ = (∫ x in Icc (0 : ℝ) 1, K x ^ 2 - 2 * m * K x) +
∫ _x in Icc (0 : ℝ) 1, m ^ 2 := by
simpa only [Pi.add_apply, Pi.sub_apply] using
integral_add (hKsq.sub hcross) hconst
_ = ((∫ x in Icc (0 : ℝ) 1, K x ^ 2) -
∫ x in Icc (0 : ℝ) 1, 2 * m * K x) +
∫ _x in Icc (0 : ℝ) 1, m ^ 2 := by
rw [integral_sub hKsq hcross]
_ = _ := by
rw [MeasureTheory.integral_const_mul, MeasureTheory.integral_const]
have hvol : (volume.restrict (Icc (0 : ℝ) 1)).real univ = 1 := by
simp [Measure.real, Real.volume_Icc]
rw [hvol]
dsimp only [m]
ring
rw [hexpand] at hposInt
simpa only [K, m] using hposInt
theorem cancellationWitness_not_condIndep_edge
Positive coefficient-child covariance rules out independence of the cancellation edge. the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma cancellationWitness_not_condIndep_edge (s : SignVector 3) :
¬ CondIndepCoordinates (cancellationWitness s) {0} {1} ∅ := by
intro hCI
let K : ℝ → ℝ := fun x => cancellationPrimitive (reflectedCoordinate s 0 x)
let H : ℝ → ℝ := fun y => centeredCoordinate (reflectedCoordinate s 1 y)
let m : ℝ := ∫ x in Icc (0 : ℝ) 1, K x
let v2 : ℝ := ∫ x in Icc (0 : ℝ) 1, K x ^ 2
have hparent : (∫ v, K (v 0) ∂observationalLaw (cancellationWitness s)) = m := by
rw [cancellationWitness_integral_eq_iterated]
· simp [observationalDensity, cancellationWitness, cancellationP]
apply integral_congr_ae
filter_upwards with x
simpa [K, one_div] using cancellationWitness_parent_inner_integral s x
· exact (continuous_reflectedCancellationPrimitive s 0).comp
(continuous_apply 0) |>.continuousOn
have hchild : (∫ v, H (v 1) ∂observationalLaw (cancellationWitness s)) =
(1 / 30 : ℝ) * m := by
rw [cancellationWitness_integral_eq_iterated]
· simp [observationalDensity, cancellationWitness, cancellationP]
calc
_ = ∫ x in Icc (0 : ℝ) 1, (1 / 30 : ℝ) * K x := by
apply integral_congr_ae
filter_upwards with x
simpa [K, H, one_div] using cancellationWitness_child_inner_integral s x
_ = _ := by
rw [MeasureTheory.integral_const_mul]
norm_num
change (∫ x in Icc (0 : ℝ) 1, K x) =
∫ x in Icc (0 : ℝ) 1, K x
rfl
· exact (continuous_reflectedCenteredCoordinate s 1).comp
(continuous_apply 1) |>.continuousOn
have hjoint : (∫ v, K (v 0) * H (v 1)
∂observationalLaw (cancellationWitness s)) = (1 / 30 : ℝ) * v2 := by
rw [cancellationWitness_integral_eq_iterated]
· simp [observationalDensity, cancellationWitness, cancellationP]
calc
_ = ∫ x in Icc (0 : ℝ) 1, (1 / 30 : ℝ) * K x ^ 2 := by
apply integral_congr_ae
filter_upwards with x
simpa [K, H, one_div] using cancellationWitness_joint_inner_integral s x
_ = _ := by
rw [MeasureTheory.integral_const_mul]
norm_num
change (∫ x in Icc (0 : ℝ) 1, K x ^ 2) =
∫ x in Icc (0 : ℝ) 1, K x ^ 2
rfl
· exact ((continuous_reflectedCancellationPrimitive s 0).comp
(continuous_apply 0)).mul
((continuous_reflectedCenteredCoordinate s 1).comp
(continuous_apply 1)) |>.continuousOn
letI : IsProbabilityMeasure (observationalLaw (cancellationWitness s)) :=
observationalLaw_isProbabilityMeasure (cancellationWitness_positive_normalized_smooth s)
have hblocks := indepFun_of_condIndepCoordinates_empty
({0} : Finset (Fin 3)) ({1} : Finset (Fin 3)) hCI
let z0 : {j // j ∈ ({0} : Finset (Fin 3))} := ⟨0, by simp⟩
let z1 : {j // j ∈ ({1} : Finset (Fin 3))} := ⟨1, by simp⟩
have hind : IndepFun (fun v : LatentState 3 => K (v 0))
(fun v : LatentState 3 => H (v 1))
(observationalLaw (cancellationWitness s)) := by
let left := fun z : ((j : {j // j ∈ ({0} : Finset (Fin 3))}) → ℝ) => K (z z0)
let right := fun z : ((j : {j // j ∈ ({1} : Finset (Fin 3))}) → ℝ) => H (z z1)
have hraw := hblocks.comp
(show Measurable left by unfold left; fun_prop)
(show Measurable right by unfold right; fun_prop)
convert hraw using 1 <;> funext v <;> rfl
have hmul := hind.integral_fun_mul_eq_mul_integral
((continuous_reflectedCancellationPrimitive s 0).measurable.comp
(measurable_pi_apply 0)).aestronglyMeasurable
((continuous_reflectedCenteredCoordinate s 1).measurable.comp
(measurable_pi_apply 1)).aestronglyMeasurable
rw [hjoint, hparent, hchild] at hmul
have hvar := reflectedCancellationPrimitive_variance_pos s
nlinarith
theorem cancellationWitness_causalMinimality
The cancellation witness is causally minimal for the one-edge three-node DAG. the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma cancellationWitness_causalMinimality (s : SignVector 3) :
CausalMinimality threeNodeDAG (cancellationWitness s) := by
intro j i hji
rcases hji with ⟨rfl, rfl⟩
intro hCI
apply cancellationWitness_not_condIndep_edge s
have hparents : (threeNodeDAG.parents 1).erase 0 = ∅ := by
ext k
fin_cases k <;> simp [threeNodeDAG, threeNodeEdge, parents]
rw [hparents] at hCI
exact condIndepCoordinates_symm hCI
theorem cancellationWitness_faithfulness
The cancellation witness is faithful to 0 → 1 with isolated node 2. the stated conclusion follows.
Formal statement
Proof (Lean source)
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.WitnessFaithfulness 9 declarations This file isolates the finite graph calculation used to reduce faithfulness of the explicit three-node witnesses to dependence of the unique adjacent pair.
Graph reduction for explicit-witness faithfulness
This file isolates the finite graph calculation used to reduce faithfulness of the explicit three-node witnesses to dependence of the unique adjacent pair.
Conditional independence given the trivial sigma algebra is ordinary independence on a probability space. This is the reverse of Causalean's existing trivial-conditioning bridge and is used to expose dependence of the explicit witness edge. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma indepFun_of_condIndepFun_bot
{Ω : Type*} [MeasurableSpace Ω] [StandardBorelSpace Ω]
{β γ : Type*} [MeasurableSpace β] [MeasurableSpace γ]
{f : Ω → β} {g : Ω → γ} (hf : Measurable f) (hg : Measurable g)
{μ : Measure Ω} [IsProbabilityMeasure μ]
(h : CondIndepFun ⊥ bot_le f g μ) :
IndepFun f g μ := by
rw [indepFun_iff_measure_inter_preimage_eq_mul]
intro s t hs ht
have hfs : MeasurableSet (f ⁻¹' s) := hf hs
have hgt : MeasurableSet (g ⁻¹' t) := hg ht
have hfgst : MeasurableSet (f ⁻¹' s ∩ g ⁻¹' t) := hfs.inter hgt
rw [ProbabilityTheory.condIndepFun_iff_condExp_inter_preimage_eq_mul hf hg] at h
have heq := h s t hs ht
change condExp ⊥ μ
(indicator (f ⁻¹' s ∩ g ⁻¹' t) (1 : Ω → ℝ)) =ᵐ[μ]
fun ω =>
condExp ⊥ μ (indicator (f ⁻¹' s) (1 : Ω → ℝ)) ω *
condExp ⊥ μ (indicator (g ⁻¹' t) (1 : Ω → ℝ)) ω at heq
rw [MeasureTheory.condExp_bot (μ := μ) _, MeasureTheory.condExp_bot (μ := μ) _,
MeasureTheory.condExp_bot (μ := μ) _] at heq
have hpoint := heq.exists
rcases hpoint with ⟨ω, hω⟩
rw [← ENNReal.toReal_eq_toReal_iff' (measure_ne_top μ _)
(ENNReal.mul_ne_top (measure_ne_top μ _) (measure_ne_top μ _))]
simpa only [MeasureTheory.integral_indicator_one hfgst,
MeasureTheory.integral_indicator_one hfs,
MeasureTheory.integral_indicator_one hgt, MeasureTheory.measureReal_def,
ENNReal.toReal_mul] using hω
theorem indepFun_of_condIndepCoordinates_empty
Coordinate-block conditional independence given the empty block implies ordinary independence whenever the observational law is a probability measure. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
hCI :
CondIndepCoordinates θ X Y ∅
conclusion
IndepFun (coordinateProjection X) (coordinateProjection Y) (observationalLaw θ)
Proof (Lean source)
lemma indepFun_of_condIndepCoordinates_empty
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(X Y : Finset (Fin n)) [IsProbabilityMeasure (observationalLaw θ)]
(hCI : CondIndepCoordinates θ X Y ∅) :
IndepFun (coordinateProjection X) (coordinateProjection Y) (observationalLaw θ) := by
rcases hCI with ⟨hμ, hX, hY, hZ, hCI⟩
let _ := hμ
have hbot : comap (coordinateProjection (∅ : Finset (Fin n)))
inferInstance = (⊥ : MeasurableSpace (LatentState n)) :=
comap_eq_bot_of_subsingleton _
rw [condIndepFun_iff_condExp_inter_preimage_eq_mul hX hY] at hCI
rw [hbot] at hCI
apply indepFun_of_condIndepFun_bot hX hY
rw [condIndepFun_iff_condExp_inter_preimage_eq_mul hX hY]
exact hCI
theorem threeNodeDAG_not_dSep_iff_cross_edge Lemma threeNodeDAG_not_dSep_iff_cross_edge in the paper ↗
For pairwise-disjoint blocks in the three-node witness DAG, d-separation fails exactly when the endpoints of its unique edge occur in opposite query blocks. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
Proof (Lean source)
theorem condIndepCoordinates_singletons_of_mem
Conditional independence of two coordinate blocks descends to any chosen singleton coordinate from each block. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma condIndepCoordinates_singletons_of_mem
{n : ℕ} {G : DAG (Fin n)} (θ : Mechanism n G)
(X Y Z : Finset (Fin n)) {a b : Fin n} (ha : a ∈ X) (hb : b ∈ Y)
(hCI : CondIndepCoordinates θ X Y Z) :
CondIndepCoordinates θ {a} {b} Z := by
rcases hCI with ⟨hμ, hX, hY, hZ, hCI⟩
let left : ((j : {j // j ∈ X}) → ℝ) →
((j : {j // j ∈ ({a} : Finset (Fin n))}) → ℝ) :=
fun v _ => v ⟨a, ha⟩
let right : ((j : {j // j ∈ Y}) → ℝ) →
((j : {j // j ∈ ({b} : Finset (Fin n))}) → ℝ) :=
fun v _ => v ⟨b, hb⟩
have hleft : coordinateProjection ({a} : Finset (Fin n)) =
left ∘ coordinateProjection X := by
funext v j
have hj : (j : Fin n) = a := Finset.mem_singleton.mp j.prop
change v j = v a
exact congrArg v hj
have hright : coordinateProjection ({b} : Finset (Fin n)) =
right ∘ coordinateProjection Y := by
funext v j
have hj : (j : Fin n) = b := Finset.mem_singleton.mp j.prop
change v j = v b
exact congrArg v hj
refine ⟨hμ, ?_, ?_, hZ, ?_⟩
· rw [hleft]
fun_prop
· rw [hright]
fun_prop
have hc := hCI.comp (show Measurable left by fun_prop)
(show Measurable right by fun_prop)
simpa only [← hleft, ← hright] using hc
theorem condIndepCoordinates_symm
Coordinate-block conditional independence is symmetric in its two query blocks. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
theorem threeNodeDAG_faithful_of_edge_dependence
For the three-node witness DAG, dependence of the unique adjacent pair under every admissible conditioning block suffices for full faithfulness. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
θ :
conclusion
Proof (Lean source)
lemma threeNodeDAG_faithful_of_edge_dependence
{θ : Mechanism 3 threeNodeDAG}
(hdep : ∀ Z : Finset (Fin 3), Disjoint ({0} : Finset (Fin 3)) Z →
Disjoint ({1} : Finset (Fin 3)) Z →
¬ CondIndepCoordinates θ {0} {1} Z) :
Faithfulness threeNodeDAG θ := by
intro X Y Z hXY hXZ hYZ hCI
by_contra hdsep
rcases (threeNodeDAG_not_dSep_iff_cross_edge X Y Z hXY hXZ hYZ).mp hdsep with
hcross | hcross
· exact hdep Z (Finset.disjoint_singleton_left.mpr fun h0Z =>
(Finset.disjoint_left.mp hXZ) hcross.1 h0Z)
(Finset.disjoint_singleton_left.mpr fun h1Z =>
(Finset.disjoint_left.mp hYZ) hcross.2 h1Z)
(condIndepCoordinates_singletons_of_mem θ X Y Z hcross.1 hcross.2 hCI)
· exact hdep Z (Finset.disjoint_singleton_left.mpr fun h0Z =>
(Finset.disjoint_left.mp hYZ) hcross.2 h0Z)
(Finset.disjoint_singleton_left.mpr fun h1Z =>
(Finset.disjoint_left.mp hXZ) hcross.1 h1Z)
(condIndepCoordinates_symm
(condIndepCoordinates_singletons_of_mem θ X Y Z hcross.1 hcross.2 hCI))
theorem threeNodeDAG_faithful_of_unconditional_edge_dependence
For the three-node witness DAG, unconditional dependence of the unique edge and independence of its first endpoint from the isolated node imply full faithfulness. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
θ :
hdep :
¬ CondIndepCoordinates θ {0} {1} ∅
hisolated :
CondIndepCoordinates θ {0} {2} ∅
conclusion
Proof (Lean source)
lemma threeNodeDAG_faithful_of_unconditional_edge_dependence
{θ : Mechanism 3 threeNodeDAG}
(hdep : ¬ CondIndepCoordinates θ {0} {1} ∅)
(hisolated : CondIndepCoordinates θ {0} {2} ∅) :
Faithfulness threeNodeDAG θ := by
apply threeNodeDAG_faithful_of_edge_dependence
intro Z h0Z h1Z hCI
classical
have hZ : Z = ∅ ∨ Z = {2} := by
apply Finset.subset_singleton_iff.mp
intro k hk
fin_cases k
· exact (Finset.disjoint_singleton_left.mp h0Z hk).elim
· exact (Finset.disjoint_singleton_left.mp h1Z hk).elim
· simp
rcases hZ with rfl | rfl
· exact hdep hCI
· apply hdep
rcases hCI with ⟨hμ, hX, hY, hW, hXY⟩
letI := hμ
rcases hisolated with ⟨_, hX', hW', hEmpty, hXW⟩
have hbot : comap (coordinateProjection (∅ : Finset (Fin 3)))
inferInstance = (⊥ : MeasurableSpace (LatentState 3)) :=
Causalean.comap_eq_bot_of_subsingleton _
have hXY' : CondIndepFun
((⊥ : MeasurableSpace (LatentState 3)) ⊔
comap (coordinateProjection ({2} : Finset (Fin 3))) inferInstance)
(sup_le bot_le hW.comap_le) (coordinateProjection {0})
(coordinateProjection {1}) (observationalLaw θ) := by
simpa using hXY
have hXW' : CondIndepFun
(⊥ : MeasurableSpace (LatentState 3)) bot_le
(coordinateProjection {0}) (coordinateProjection {2}) (observationalLaw θ) := by
simpa only [hbot] using hXW
have hpair := Causalean.condIndepFun_contraction_of_prodMk bot_le
hX hY hW hXY' hXW'
have hplain : CondIndepFun
(⊥ : MeasurableSpace (LatentState 3)) bot_le
(coordinateProjection {0}) (coordinateProjection {1}) (observationalLaw θ) := by
convert hpair.comp measurable_id measurable_fst using 1 <;> funext ω <;> rfl
exact ⟨hμ, hX, hY, hEmpty, by simpa only [hbot] using hplain⟩
theorem threeNodeDAG_isolated_independence
Every positive normalized mechanism factorizing over the three-node witness DAG makes the first endpoint of the unique edge independent of the isolated third coordinate. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
θ :
conclusion
CondIndepCoordinates θ {0} {2} ∅
Proof (Lean source)
lemma threeNodeDAG_isolated_independence
{θ : Mechanism 3 threeNodeDAG}
(hpos : PositiveNormalizedSmoothMechanisms threeNodeDAG θ) :
CondIndepCoordinates θ {0} {2} ∅ := by
let τ : TopologicalRanking threeNodeDAG := {
rank := fun i ↦ i.val
injective_rank := Fin.val_injective
edge_lt := by
intro i j hij
fin_cases i <;> fin_cases j <;> simp [threeNodeDAG, threeNodeEdge] at hij ⊢ }
have hpred : predecessors τ 2 = {0, 1} := by
ext k
fin_cases k <;> decide
have hraw := mechanism_condIndepGiven_orderedLocalMarkov hpos τ 2 ∅
(by simp [hpred]) (by simp [threeNodeDAG, threeNodeEdge, parents])
rw [hpred] at hraw
rcases hraw with ⟨hμ, h2, h01, hE, hci⟩
letI := hμ
have h0 : Measurable (coordinateProjection ({0} : Finset (Fin 3))) := by
unfold coordinateProjection
fun_prop
have h2local : Measurable (coordinateProjection ({2} : Finset (Fin 3))) := by
unfold coordinateProjection
fun_prop
have hleft : Measurable (fun r : ℝ ↦
fun _ : {j // j ∈ ({2} : Finset (Fin 3))} ↦ r) := by
apply measurable_pi_lambda
intro j
exact measurable_id
have hright : Measurable
(fun z : (j : {j // j ∈ ({0, 1} : Finset (Fin 3))}) → ℝ ↦
fun _ : {j // j ∈ ({0} : Finset (Fin 3))} ↦ z ⟨0, by simp⟩) := by
apply measurable_pi_lambda
intro j
exact measurable_pi_apply _
have hci' := hci.comp hleft hright
have hEmptyLocal : Measurable (coordinateProjection (∅ : Finset (Fin 3))) := by
unfold coordinateProjection
fun_prop
refine condIndepCoordinates_symm ⟨hμ, h2local, h0, hEmptyLocal, ?_⟩
change CondIndepFun
(comap
(coordinateProjection (∅ : Finset (Fin 3)))
inferInstance) _ (coordinateProjection {2}) (coordinateProjection {0})
(observationalLaw θ)
convert hci' using 1
· funext v j
have hj : (j : Fin 3) = 2 := Finset.mem_singleton.mp j.prop
change v j = v 2
exact congrArg v hj
· funext v j
have hj : (j : Fin 3) = 0 := Finset.mem_singleton.mp j.prop
change v j = v 0
exact congrArg v hj
theorem threeNodeDAG_faithful_of_causalMinimality
On the explicit three-node DAG, positivity and causal minimality already imply faithfulness because the only nontrivial d-connection is its unique edge. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
θ :
hminimal :
conclusion
Proof (Lean source)
lemma threeNodeDAG_faithful_of_causalMinimality
{θ : Mechanism 3 threeNodeDAG}
(hpos : PositiveNormalizedSmoothMechanisms threeNodeDAG θ)
(hminimal : CausalMinimality threeNodeDAG θ) :
Faithfulness threeNodeDAG θ := by
apply threeNodeDAG_faithful_of_unconditional_edge_dependence
· intro h01
have h10 := condIndepCoordinates_symm h01
have hm := hminimal (j := 0) (i := 1) (by
simp [threeNodeDAG, threeNodeEdge])
apply hm
convert h10 using 1
ext k
fin_cases k <;> simp [threeNodeDAG, threeNodeEdge, parents]
· exact threeNodeDAG_isolated_independence hpos
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.WitnessFaithfulnessAssembly 14 declarations This file turns the explicit sparse density calculation into dependence of the unique adjacent coordinate pair, and hence into faithfulness of the sparse three-node witness.
Explicit-witness faithfulness assembly
This file turns the explicit sparse density calculation into dependence of the unique adjacent coordinate pair, and hence into faithfulness of the sparse three-node witness.
The squared centered coordinate has unit-interval integral 1/3. the stated conclusion follows.
Formal statement
conclusion
(∫ z in Icc (0 : ℝ) 1, centeredCoordinate z ^ 2) = 1 / 3
Proof (Lean source)
lemma centeredCoordinate_sq_integral :
(∫ z in Icc (0 : ℝ) 1, centeredCoordinate z ^ 2) = 1 / 3 := by
rw [MeasureTheory.integral_Icc_eq_integral_Ioc]
rw [← intervalIntegral.integral_of_le (by norm_num : (0 : ℝ) ≤ 1)]
have hfun : (fun z : ℝ => centeredCoordinate z ^ 2) =
fun z => 4 * z ^ 2 - 4 * z + 1 := by
funext z
simp only [centeredCoordinate]
ring
have hpow : IntervalIntegrable (fun z : ℝ => 4 * z ^ 2) volume 0 1 := by
exact (intervalIntegral.intervalIntegrable_pow 2).const_mul 4
have hid : IntervalIntegrable (fun z : ℝ => 4 * z) volume 0 1 := by
exact intervalIntegral.intervalIntegrable_id.const_mul 4
rw [hfun, intervalIntegral.integral_add (hpow.sub hid) intervalIntegrable_const,
intervalIntegral.integral_sub hpow hid,
intervalIntegral.integral_const_mul, integral_pow,
intervalIntegral.integral_const_mul, integral_id,
intervalIntegral.integral_const]
norm_num
theorem centeredCoordinate_reflected_sq_integral
Coordinate reflection preserves the squared centered-coordinate integral. the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma centeredCoordinate_reflected_sq_integral (s : SignVector 3) (i : Fin 3) :
(∫ z in Icc (0 : ℝ) 1,
centeredCoordinate (reflectedCoordinate s i z) ^ 2) = 1 / 3 := by
calc
_ = ∫ z in Icc (0 : ℝ) 1, centeredCoordinate z ^ 2 :=
integral_reflectedCoordinate s i (fun z => centeredCoordinate z ^ 2)
_ = 1 / 3 := centeredCoordinate_sq_integral
theorem measurable_sparseP
Every sparse-witness observational mechanism slot is globally measurable.
Formal statement
Proof (Lean source)
@[fun_prop] lemma measurable_sparseP (s : SignVector 3) (i : Fin 3) :
Measurable (sparseP s i) := by
fin_cases i
· change Measurable (fun _ : LatentState 3 => (1 : ℝ))
fun_prop
· rcases s.signed 0 with h0 | h0 <;>
rcases s.signed 1 with h1 | h1 <;>
change Measurable (fun v : LatentState 3 => 1 + (1 / 10 : ℝ) *
centeredCoordinate (reflectedCoordinate s 0 (v 0)) *
centeredCoordinate (reflectedCoordinate s 1 (v 1))) <;>
simp [centeredCoordinate, reflectedCoordinate, reflect, h0, h1,
show (-1 : ℝ) ≠ 1 by norm_num] <;> fun_prop
· change Measurable (fun _ : LatentState 3 => (1 : ℝ))
fun_prop
theorem measurable_sparseWitness_observationalDensity
The sparse witness's observational joint density is globally measurable.
Formal statement
Proof (Lean source)
@[fun_prop] lemma measurable_sparseWitness_observationalDensity (s : SignVector 3) :
Measurable (observationalDensity (sparseWitness s)) := by
unfold observationalDensity
apply Finset.measurable_prod
intro i _
exact measurable_sparseP s i
theorem continuous_reflectedCenteredCoordinate
A fixed signed reflection followed by centering is continuous.
Formal statement
Proof (Lean source)
@[fun_prop] lemma continuous_reflectedCenteredCoordinate
(s : SignVector 3) (i : Fin 3) :
Continuous (fun z : ℝ => centeredCoordinate (reflectedCoordinate s i z)) := by
rcases s.signed i with hi | hi
· simp [centeredCoordinate, reflectedCoordinate, reflect, hi,
show (-1 : ℝ) ≠ 1 by norm_num]
fun_prop
· simp [centeredCoordinate, reflectedCoordinate, reflect, hi]
fun_prop
theorem sparseWitness_integral_eq_iterated
A continuous test function under the sparse observational law can be evaluated as the explicit three-coordinate iterated density integral. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
(∫ v, f v ∂observationalLaw (sparseWitness s))
= ∫ x in Icc (0 : ℝ) 1, ∫ y in Icc (0 : ℝ) 1, ∫ z in Icc (0 : ℝ) 1, observationalDensity (sparseWitness s) ![x, y, z] * f ![x, y, z]
Proof (Lean source)
lemma sparseWitness_integral_eq_iterated (s : SignVector 3)
(f : LatentState 3 → ℝ) (hf : ContinuousOn f (latentCube 3)) :
(∫ v, f v ∂observationalLaw (sparseWitness s)) =
∫ x in Icc (0 : ℝ) 1,
∫ y in Icc (0 : ℝ) 1,
∫ z in Icc (0 : ℝ) 1,
observationalDensity (sparseWitness s) ![x, y, z] * f ![x, y, z] := by
let μ : Measure (LatentState 3) := volume.restrict (latentCube 3)
have hpos := sparseWitness_positive_normalized_smooth s
have hcubeCompact : IsCompact (latentCube 3) := by
rw [latentCube]
exact isCompact_univ_pi fun _ => isCompact_Icc
have hdensityCont : ContinuousOn (observationalDensity (sparseWitness s))
(latentCube 3) := by
unfold observationalDensity
exact continuousOn_finset_prod _ fun i _ => (hpos.2.2.1 i).continuousOn
have hprodInt : Integrable
(fun v => observationalDensity (sparseWitness s) v * f v) μ := by
dsimp only [μ]
exact (hdensityCont.mul hf).integrableOn_compact hcubeCompact
have hμ : μ = Measure.pi (fun _ : Fin 3 =>
volume.restrict (Icc (0 : ℝ) 1)) := by
dsimp only [μ]
change volume.restrict (Set.univ.pi fun _ : Fin 3 => Icc (0 : ℝ) 1) = _
rw [MeasureTheory.volume_pi, Measure.restrict_pi_pi]
calc
(∫ v, f v ∂observationalLaw (sparseWitness s)) =
∫ v, observationalDensity (sparseWitness s) v * f v ∂μ := by
unfold observationalLaw
rw [integral_withDensity_eq_integral_toReal_smul]
· apply integral_congr_ae
filter_upwards [ae_restrict_mem (measurableSet_latentCube 3)] with v hv
rw [ENNReal.toReal_ofReal (le_of_lt (by
unfold observationalDensity
exact Finset.prod_pos fun i _ => hpos.1 i v hv))]
rfl
· exact (measurable_sparseWitness_observationalDensity s).ennreal_ofReal
· filter_upwards with v
exact ENNReal.ofReal_lt_top
_ = _ := by
rw [hμ, integral_fin_three_pi_eq_iterated _ _ (by simpa [hμ] using hprodInt)]
theorem sparseWitness_parent_inner_integral
Integrating the sparse density times its centered parent coordinate over the child coordinate leaves the centered parent coordinate unchanged. the stated conclusion follows.
Formal statement
hypotheses
s :
x :
ℝ
conclusion
(∫ y in Icc (0 : ℝ) 1, (1 + (1 / 10 : ℝ) * centeredCoordinate (reflectedCoordinate s 0 x) * centeredCoordinate (reflectedCoordinate s 1 y)) * centeredCoordinate (reflectedCoordinate s 0 x))
= centeredCoordinate (reflectedCoordinate s 0 x)
Proof (Lean source)
lemma sparseWitness_parent_inner_integral (s : SignVector 3) (x : ℝ) :
(∫ y in Icc (0 : ℝ) 1,
(1 + (1 / 10 : ℝ) * centeredCoordinate (reflectedCoordinate s 0 x) *
centeredCoordinate (reflectedCoordinate s 1 y)) *
centeredCoordinate (reflectedCoordinate s 0 x)) =
centeredCoordinate (reflectedCoordinate s 0 x) := by
let A := centeredCoordinate (reflectedCoordinate s 0 x)
let B : ℝ → ℝ := fun y => centeredCoordinate (reflectedCoordinate s 1 y)
have hB : Continuous B := continuous_reflectedCenteredCoordinate s 1
have h1 : IntegrableOn (fun y => A * B y) (Icc (0 : ℝ) 1) :=
(continuous_const.mul hB).continuousOn.integrableOn_Icc
have h2 : IntegrableOn (fun y => (1 / 10 : ℝ) * A ^ 2 * B y)
(Icc (0 : ℝ) 1) :=
(continuous_const.mul hB).continuousOn.integrableOn_Icc
have hconst : IntegrableOn (fun _ : ℝ => A) (Icc (0 : ℝ) 1) :=
continuous_const.continuousOn.integrableOn_Icc
rw [show (fun y => (1 + (1 / 10 : ℝ) * A * B y) * A) =
fun y => A + (1 / 10 : ℝ) * A ^ 2 * B y by funext y; ring,
integral_add hconst h2, MeasureTheory.integral_const,
MeasureTheory.integral_const_mul, centeredCoordinate_reflected_integral,
mul_zero, add_zero]
simp [A, Real.volume_Icc]
theorem sparseWitness_child_inner_integral
Integrating the sparse density times the centered child coordinate produces one thirtieth of the centered parent coordinate. the stated conclusion follows.
Formal statement
hypotheses
s :
x :
ℝ
conclusion
(∫ y in Icc (0 : ℝ) 1, (1 + (1 / 10 : ℝ) * centeredCoordinate (reflectedCoordinate s 0 x) * centeredCoordinate (reflectedCoordinate s 1 y)) * centeredCoordinate (reflectedCoordinate s 1 y))
= (1 / 30 : ℝ) * centeredCoordinate (reflectedCoordinate s 0 x)
Proof (Lean source)
lemma sparseWitness_child_inner_integral (s : SignVector 3) (x : ℝ) :
(∫ y in Icc (0 : ℝ) 1,
(1 + (1 / 10 : ℝ) * centeredCoordinate (reflectedCoordinate s 0 x) *
centeredCoordinate (reflectedCoordinate s 1 y)) *
centeredCoordinate (reflectedCoordinate s 1 y)) =
(1 / 30 : ℝ) * centeredCoordinate (reflectedCoordinate s 0 x) := by
let A := centeredCoordinate (reflectedCoordinate s 0 x)
let B : ℝ → ℝ := fun y => centeredCoordinate (reflectedCoordinate s 1 y)
have hB : Continuous B := continuous_reflectedCenteredCoordinate s 1
have h1 : IntegrableOn B (Icc (0 : ℝ) 1) :=
hB.continuousOn.integrableOn_Icc
have h2 : IntegrableOn (fun y => (1 / 10 : ℝ) * A * B y ^ 2)
(Icc (0 : ℝ) 1) :=
(continuous_const.mul (hB.pow 2)).continuousOn.integrableOn_Icc
rw [show (fun y => (1 + (1 / 10 : ℝ) * A * B y) * B y) =
fun y => B y + (1 / 10 : ℝ) * A * B y ^ 2 by funext y; ring,
integral_add h1 h2, MeasureTheory.integral_const_mul,
centeredCoordinate_reflected_integral,
centeredCoordinate_reflected_sq_integral]
dsimp only [A]
ring
theorem sparseWitness_joint_inner_integral
Integrating the sparse density times both centered edge coordinates over the child coordinate produces one thirtieth of the squared parent coordinate. the stated conclusion follows.
Formal statement
hypotheses
s :
x :
ℝ
conclusion
(∫ y in Icc (0 : ℝ) 1, (1 + (1 / 10 : ℝ) * centeredCoordinate (reflectedCoordinate s 0 x) * centeredCoordinate (reflectedCoordinate s 1 y)) * (centeredCoordinate (reflectedCoordinate s 0 x) * centeredCoordinate (reflectedCoordinate s 1 y)))
= (1 / 30 : ℝ) * centeredCoordinate (reflectedCoordinate s 0 x) ^ 2
Proof (Lean source)
lemma sparseWitness_joint_inner_integral (s : SignVector 3) (x : ℝ) :
(∫ y in Icc (0 : ℝ) 1,
(1 + (1 / 10 : ℝ) * centeredCoordinate (reflectedCoordinate s 0 x) *
centeredCoordinate (reflectedCoordinate s 1 y)) *
(centeredCoordinate (reflectedCoordinate s 0 x) *
centeredCoordinate (reflectedCoordinate s 1 y))) =
(1 / 30 : ℝ) * centeredCoordinate (reflectedCoordinate s 0 x) ^ 2 := by
let A := centeredCoordinate (reflectedCoordinate s 0 x)
let B : ℝ → ℝ := fun y => centeredCoordinate (reflectedCoordinate s 1 y)
have hB : Continuous B := continuous_reflectedCenteredCoordinate s 1
have h1 : IntegrableOn (fun y => A * B y) (Icc (0 : ℝ) 1) :=
(continuous_const.mul hB).continuousOn.integrableOn_Icc
have h2 : IntegrableOn (fun y => (1 / 10 : ℝ) * A ^ 2 * B y ^ 2)
(Icc (0 : ℝ) 1) :=
(continuous_const.mul (hB.pow 2)).continuousOn.integrableOn_Icc
rw [show (fun y => (1 + (1 / 10 : ℝ) * A * B y) * (A * B y)) =
fun y => A * B y + (1 / 10 : ℝ) * A ^ 2 * B y ^ 2 by funext y; ring,
integral_add h1 h2, MeasureTheory.integral_const_mul,
MeasureTheory.integral_const_mul,
centeredCoordinate_reflected_integral,
centeredCoordinate_reflected_sq_integral]
dsimp only [A]
ring
theorem sparseWitness_centered_marginal_zero
Each reflected centered endpoint has zero mean under the sparse observational law. the stated conclusion follows.
Formal statement
hypotheses
s :
i :
Fin 2
conclusion
(∫ v, centeredCoordinate (reflectedCoordinate s i.castSucc (v i.castSucc)) ∂observationalLaw (sparseWitness s))
= 0
Proof (Lean source)
lemma sparseWitness_centered_marginal_zero (s : SignVector 3) (i : Fin 2) :
(∫ v, centeredCoordinate (reflectedCoordinate s i.castSucc (v i.castSucc))
∂observationalLaw (sparseWitness s)) = 0 := by
rw [sparseWitness_integral_eq_iterated]
· fin_cases i
· simp [observationalDensity, sparseWitness, sparseP]
calc
_ = ∫ x in Icc (0 : ℝ) 1,
centeredCoordinate (reflectedCoordinate s 0 x) := by
apply integral_congr_ae
filter_upwards with x
simpa [one_div] using sparseWitness_parent_inner_integral s x
_ = 0 := centeredCoordinate_reflected_integral s 0
· simp [observationalDensity, sparseWitness, sparseP]
calc
_ = ∫ x in Icc (0 : ℝ) 1,
(1 / 30 : ℝ) * centeredCoordinate (reflectedCoordinate s 0 x) := by
apply integral_congr_ae
filter_upwards with x
simpa [one_div] using sparseWitness_child_inner_integral s x
_ = 0 := by
rw [MeasureTheory.integral_const_mul,
centeredCoordinate_reflected_integral, mul_zero]
· apply Continuous.continuousOn
exact (continuous_reflectedCenteredCoordinate s i.castSucc).comp
(continuous_apply i.castSucc)
theorem sparseWitness_centered_jointMoment
The two reflected centered edge coordinates have the exact positive sparse-law joint moment 1/90. the stated conclusion follows.
Formal statement
hypotheses
s :
conclusion
(∫ v, centeredCoordinate (reflectedCoordinate s 0 (v 0)) * centeredCoordinate (reflectedCoordinate s 1 (v 1)) ∂observationalLaw (sparseWitness s))
= 1 / 90
Proof (Lean source)
lemma sparseWitness_centered_jointMoment (s : SignVector 3) :
(∫ v,
centeredCoordinate (reflectedCoordinate s 0 (v 0)) *
centeredCoordinate (reflectedCoordinate s 1 (v 1))
∂observationalLaw (sparseWitness s)) = 1 / 90 := by
rw [sparseWitness_integral_eq_iterated]
· simp [observationalDensity, sparseWitness, sparseP]
calc
_ = ∫ x in Icc (0 : ℝ) 1,
(1 / 30 : ℝ) * centeredCoordinate (reflectedCoordinate s 0 x) ^ 2 := by
apply integral_congr_ae
filter_upwards with x
simpa [one_div] using sparseWitness_joint_inner_integral s x
_ = 1 / 90 := by
rw [MeasureTheory.integral_const_mul,
centeredCoordinate_reflected_sq_integral]
norm_num
norm_num
· apply Continuous.continuousOn
exact ((continuous_reflectedCenteredCoordinate s 0).comp (continuous_apply 0)).mul
((continuous_reflectedCenteredCoordinate s 1).comp (continuous_apply 1))
theorem sparseWitness_not_condIndep_edge
The exact nonzero centered cross-moment rules out independence of the sparse witness's unique adjacent coordinate pair. the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma sparseWitness_not_condIndep_edge (s : SignVector 3) :
¬ CondIndepCoordinates (sparseWitness s) {0} {1} ∅ := by
intro hCI
letI : IsProbabilityMeasure (observationalLaw (sparseWitness s)) :=
observationalLaw_isProbabilityMeasure (sparseWitness_positive_normalized_smooth s)
have hblocks := indepFun_of_condIndepCoordinates_empty
({0} : Finset (Fin 3)) ({1} : Finset (Fin 3)) hCI
let z0 : {j // j ∈ ({0} : Finset (Fin 3))} := ⟨0, by simp⟩
let z1 : {j // j ∈ ({1} : Finset (Fin 3))} := ⟨1, by simp⟩
let left : ((j : {j // j ∈ ({0} : Finset (Fin 3))}) → ℝ) → ℝ :=
fun z => centeredCoordinate (reflectedCoordinate s 0 (z z0))
let right : ((j : {j // j ∈ ({1} : Finset (Fin 3))}) → ℝ) → ℝ :=
fun z => centeredCoordinate (reflectedCoordinate s 1 (z z1))
have hleft : Measurable left := by
unfold left
exact (continuous_reflectedCenteredCoordinate s 0).measurable.comp
(measurable_pi_apply z0)
have hright : Measurable right := by
unfold right
exact (continuous_reflectedCenteredCoordinate s 1).measurable.comp
(measurable_pi_apply z1)
have hindRaw := hblocks.comp hleft hright
have hind : IndepFun
(fun v : LatentState 3 => centeredCoordinate (reflectedCoordinate s 0 (v 0)))
(fun v : LatentState 3 => centeredCoordinate (reflectedCoordinate s 1 (v 1)))
(observationalLaw (sparseWitness s)) := by
convert hindRaw using 1 <;> funext v <;> rfl
have hmul := hind.integral_fun_mul_eq_mul_integral
((continuous_reflectedCenteredCoordinate s 0).measurable.comp
(measurable_pi_apply 0)).aestronglyMeasurable
((continuous_reflectedCenteredCoordinate s 1).measurable.comp
(measurable_pi_apply 1)).aestronglyMeasurable
have hm0 : (∫ v, centeredCoordinate (reflectedCoordinate s 0 (v 0))
∂observationalLaw (sparseWitness s)) = 0 := by
simpa using sparseWitness_centered_marginal_zero s (0 : Fin 2)
have hm1 : (∫ v, centeredCoordinate (reflectedCoordinate s 1 (v 1))
∂observationalLaw (sparseWitness s)) = 0 := by
simpa using sparseWitness_centered_marginal_zero s (1 : Fin 2)
rw [sparseWitness_centered_jointMoment, hm0, hm1] at hmul
norm_num at hmul
theorem sparseWitness_causalMinimality
The sparse witness is causally minimal for the one-edge three-node DAG. the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma sparseWitness_causalMinimality (s : SignVector 3) :
CausalMinimality threeNodeDAG (sparseWitness s) := by
intro j i hji
rcases hji with ⟨rfl, rfl⟩
intro hCI
apply sparseWitness_not_condIndep_edge s
have hparents : (threeNodeDAG.parents 1).erase 0 = ∅ := by
ext k
fin_cases k <;> simp [threeNodeDAG, threeNodeEdge, parents]
rw [hparents] at hCI
have hsymm := condIndepCoordinates_symm hCI
exact hsymm
theorem sparseWitness_faithfulness
The sparse witness is faithful to 0 → 1 with isolated node 2. the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma sparseWitness_faithfulness (s : SignVector 3) :
Faithfulness threeNodeDAG (sparseWitness s) := by
exact threeNodeDAG_faithful_of_causalMinimality
(sparseWitness_positive_normalized_smooth s)
(sparseWitness_causalMinimality s)
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.WitnessMomentMmdAssembly 10 declarations
For a parent coordinate, the sparse child-weight integral averages the squared intervention density against the sparse denominator.
Definition (Lean source)
parameters
x :
ℝ
def
sparseA x :
ℝ
given by
∫ y in Icc (0 : ℝ) 1, exponentialInterventionDensity y ^ 2 / (1 + (centeredCoordinate x / 10) * centeredCoordinate y)
theorem sparseA_hasDerivAt
At a point of the unit interval, the sparse child-weight integral has the stated derivative.
Formal statement
hypotheses
x :
ℝ
hx :
x ∈ Icc (0 : ℝ) 1
conclusion
HasDerivAt sparseA (-(1 / 5 : ℝ) * ∫ y in Icc (0 : ℝ) 1, centeredCoordinate y * sparseWeight (centeredCoordinate x / 10) y) x
Proof (Lean source)
lemma sparseA_hasDerivAt {x : ℝ} (hx : x ∈ Icc (0 : ℝ) 1) :
HasDerivAt sparseA
(-(1 / 5 : ℝ) * ∫ y in Icc (0 : ℝ) 1,
centeredCoordinate y * sparseWeight (centeredCoordinate x / 10) y) x := by
change HasDerivAt (fun x : ℝ => ∫ y in Icc (0 : ℝ) 1,
exponentialInterventionDensity y ^ 2 /
(1 + (centeredCoordinate x / 10) * centeredCoordinate y)) _ x
let F : ℝ → ℝ → ℝ := fun x y =>
exponentialInterventionDensity y ^ 2 /
(1 + (centeredCoordinate x / 10) * centeredCoordinate y)
let F' : ℝ → ℝ → ℝ := fun x y =>
-(1 / 5 : ℝ) * centeredCoordinate y * sparseWeight
(centeredCoordinate x / 10) y
let μ : Measure ℝ := volume.restrict (Icc (0 : ℝ) 1)
have hs : Ioo (-1 / 4 : ℝ) (5 / 4) ∈ 𝓝 x :=
IsOpen.mem_nhds isOpen_Ioo (by constructor <;> nlinarith [hx.1, hx.2])
have hden : ∀ z ∈ Ioo (-1 / 4 : ℝ) (5 / 4), ∀ y ∈ Icc (0 : ℝ) 1,
0 < 1 + centeredCoordinate z / 10 * centeredCoordinate y := by
intro z hz y hy
have hh := abs_centeredCoordinate_le_one hy
rw [abs_le] at hh
have hzcenter : (-3 / 2 : ℝ) < centeredCoordinate z ∧
centeredCoordinate z < 3 / 2 := by
unfold centeredCoordinate
constructor <;> linarith [hz.1, hz.2]
have hzabs : |centeredCoordinate z| < 3 / 2 := (abs_lt).2 ⟨by linarith, hzcenter.2⟩
have hyabs : |centeredCoordinate y| ≤ 1 := (abs_le).2 hh
have habs : |centeredCoordinate z / 10 * centeredCoordinate y| < 3 / 20 := by
rw [abs_mul, abs_div, abs_of_pos (by norm_num : (0 : ℝ) < 10)]
calc
|centeredCoordinate z| / 10 * |centeredCoordinate y| ≤
|centeredCoordinate z| / 10 * 1 := by gcongr
_ < 3 / 20 := by nlinarith
linarith [neg_lt_of_abs_lt habs]
have hqcont : Continuous exponentialInterventionDensity := by
unfold exponentialInterventionDensity
fun_prop
have hccont : Continuous centeredCoordinate := by
unfold centeredCoordinate
fun_prop
have hF_meas : ∀ᶠ z in 𝓝 x, AEStronglyMeasurable (F z) μ := by
filter_upwards [hs] with z hz
have hzcont : ContinuousOn (F z) (Icc (0 : ℝ) 1) := by
dsimp only [F]
apply ContinuousOn.div (hqcont.pow 2).continuousOn
(continuous_const.add (continuous_const.mul hccont)).continuousOn
intro y hy
exact (hden z hz y hy).ne'
exact hzcont.aestronglyMeasurable measurableSet_Icc
have hF_int : Integrable (F x) μ := by
apply ContinuousOn.integrableOn_compact isCompact_Icc
dsimp only [F]
apply ContinuousOn.div
· exact (hqcont.pow 2).continuousOn
· exact (continuous_const.add (continuous_const.mul hccont)).continuousOn
· intro y hy
exact (hden x (by constructor <;> nlinarith [hx.1, hx.2]) y hy).ne'
have hF'_meas : AEStronglyMeasurable (F' x) μ := by
have hcont : ContinuousOn (F' x) (Icc (0 : ℝ) 1) := by
dsimp only [F', sparseWeight]
apply ContinuousOn.mul (continuous_const.mul hccont).continuousOn
apply ContinuousOn.div (hqcont.pow 2).continuousOn
((continuous_const.add (continuous_const.mul hccont)).pow 2).continuousOn
intro y hy
exact pow_ne_zero 2 (hden x (by constructor <;> nlinarith [hx.1, hx.2]) y hy).ne'
exact hcont.aestronglyMeasurable measurableSet_Icc
have h_bound : ∀ᵐ y ∂μ, ∀ z ∈ Ioo (-1 / 4 : ℝ) (5 / 4), ‖F' z y‖ ≤ (10 : ℝ) := by
filter_upwards [ae_restrict_mem measurableSet_Icc] with y hy z hz
have hh := abs_centeredCoordinate_le_one hy
have hq := exponentialInterventionDensity_lt_thirteen_div_three hy
have hqp := exponentialInterventionDensity_pos y
have hd := hden z hz y hy
have hdlower : (17 / 20 : ℝ) < 1 + centeredCoordinate z / 10 * centeredCoordinate y := by
rw [abs_le] at hh
have hzcenter : (-3 / 2 : ℝ) < centeredCoordinate z ∧
centeredCoordinate z < 3 / 2 := by
unfold centeredCoordinate
constructor <;> linarith [hz.1, hz.2]
have hzabs : |centeredCoordinate z| < 3 / 2 := (abs_lt).2 ⟨by linarith, hzcenter.2⟩
have hyabs : |centeredCoordinate y| ≤ 1 := (abs_le).2 hh
have habs : |centeredCoordinate z / 10 * centeredCoordinate y| < 3 / 20 := by
rw [abs_mul, abs_div, abs_of_pos (by norm_num : (0 : ℝ) < 10)]
calc
|centeredCoordinate z| / 10 * |centeredCoordinate y| ≤
|centeredCoordinate z| / 10 * 1 := by gcongr
_ < 3 / 20 := by nlinarith
linarith [neg_lt_of_abs_lt habs]
dsimp only [F', sparseWeight]
rw [Real.norm_eq_abs, abs_mul, abs_mul, abs_div, abs_pow, abs_of_pos hqp]
norm_num only [abs_neg, abs_of_nonneg, one_div]
rw [show |(1 + centeredCoordinate z / 10 * centeredCoordinate y) ^ 2| =
(1 + centeredCoordinate z / 10 * centeredCoordinate y) ^ 2 by
exact abs_of_nonneg (sq_nonneg _)]
have hhs : |centeredCoordinate y| ≤ 1 := hh
have hnum : exponentialInterventionDensity y ^ 2 < (13 / 3 : ℝ) ^ 2 := by nlinarith
have hdsq : (17 / 20 : ℝ) ^ 2 <
(1 + centeredCoordinate z / 10 * centeredCoordinate y) ^ 2 := by nlinarith
rw [show (1 / 5 : ℝ) * |centeredCoordinate y| *
(exponentialInterventionDensity y ^ 2 /
(1 + centeredCoordinate z / 10 * centeredCoordinate y) ^ 2) =
((1 / 5 : ℝ) * |centeredCoordinate y| *
exponentialInterventionDensity y ^ 2) /
(1 + centeredCoordinate z / 10 * centeredCoordinate y) ^ 2 by ring]
rw [div_le_iff₀ (sq_pos_of_pos hd)]
nlinarith [sq_nonneg (1 + centeredCoordinate z / 10 * centeredCoordinate y)]
have hbound_int : Integrable (fun _ : ℝ => (10 : ℝ)) μ := by
dsimp only [μ]
exact continuous_const.continuousOn.integrableOn_compact isCompact_Icc
have hdiff : ∀ᵐ y ∂μ, ∀ z ∈ Ioo (-1 / 4 : ℝ) (5 / 4),
HasDerivAt (fun z => F z y) (F' z y) z := by
filter_upwards [ae_restrict_mem measurableSet_Icc] with y hy z hz
have hd := (hden z hz y hy).ne'
have hh : HasDerivAt (fun z : ℝ => centeredCoordinate z / 10) (1 / 5) z := by
have hp0 : HasDerivAt (fun u : ℝ => 2 * u - 1) 2 z := by
convert (hasDerivAt_const_mul (x := z) 2).sub_const 1 using 1
have hh' := hp0.div_const 10
apply hh'.congr_deriv
norm_num
have hdenDeriv : HasDerivAt
(fun z : ℝ => 1 + centeredCoordinate z / 10 * centeredCoordinate y)
((1 / 5 : ℝ) * centeredCoordinate y) z := by
simpa using (hh.mul_const (centeredCoordinate y)).const_add 1
have hnum : HasDerivAt (fun _ : ℝ => exponentialInterventionDensity y ^ 2) 0 z :=
hasDerivAt_const (x := z) (c := exponentialInterventionDensity y ^ 2)
have hraw : HasDerivAt
(fun z : ℝ => exponentialInterventionDensity y ^ 2 /
(1 + centeredCoordinate z / 10 * centeredCoordinate y))
((0 * (1 + centeredCoordinate z / 10 * centeredCoordinate y) -
exponentialInterventionDensity y ^ 2 *
((1 / 5 : ℝ) * centeredCoordinate y)) /
(1 + centeredCoordinate z / 10 * centeredCoordinate y) ^ 2) z :=
hnum.div hdenDeriv hd
dsimp only [F, F', sparseWeight]
apply hraw.congr_deriv
field_simp [hd]
ring
have h := hasDerivAt_integral_of_dominated_loc_of_deriv_le (𝕜 := ℝ) hs hF_meas hF_int
hF'_meas h_bound hbound_int hdiff
apply h.2.congr_deriv
dsimp only [F', μ]
rw [show (fun y : ℝ => -(1 / 5 : ℝ) * centeredCoordinate y *
sparseWeight (centeredCoordinate x / 10) y) =
fun y => -(1 / 5 : ℝ) * (centeredCoordinate y *
sparseWeight (centeredCoordinate x / 10) y) by funext y; ring]
rw [MeasureTheory.integral_const_mul]
theorem sparseA_deriv_eq
At a point of the unit interval, the derivative of the sparse child-weight integral equals its centered-weight formula.
Formal statement
hypotheses
x :
ℝ
hx :
x ∈ Icc (0 : ℝ) 1
conclusion
= -(1 / 5 : ℝ) * ∫ y in Icc (0 : ℝ) 1, centeredCoordinate y * sparseWeight (centeredCoordinate x / 10) y
Proof (Lean source)
lemma sparseA_deriv_eq {x : ℝ} (hx : x ∈ Icc (0 : ℝ) 1) :
deriv sparseA x =
-(1 / 5 : ℝ) * ∫ y in Icc (0 : ℝ) 1,
centeredCoordinate y * sparseWeight (centeredCoordinate x / 10) y :=
(sparseA_hasDerivAt hx).deriv
theorem sparseA_continuousOn
The sparse child-weight integral is continuous on the unit interval.
Formal statement
conclusion
Proof (Lean source)
lemma sparseA_continuousOn : ContinuousOn sparseA (Icc (0 : ℝ) 1) := by
exact fun x hx => (sparseA_hasDerivAt hx).continuousAt.continuousWithinAt
theorem sparseA_deriv_intervalIntegrable
The derivative of the sparse child-weight integral is interval-integrable.
Formal statement
conclusion
Proof (Lean source)
lemma sparseA_deriv_intervalIntegrable :
IntervalIntegrable (deriv sparseA) volume 0 1 := by
rw [intervalIntegrable_iff]
refine IntegrableOn.of_bound (by simp)
((stronglyMeasurable_deriv sparseA).aestronglyMeasurable.restrict) 10 ?_
filter_upwards [ae_restrict_mem measurableSet_uIoc] with x hx
all_goals
have hx' : x ∈ Icc (0 : ℝ) 1 := by
rw [uIoc_of_le (by norm_num : (0 : ℝ) ≤ 1)] at hx
exact ⟨le_of_lt hx.1, hx.2⟩
rw [sparseA_deriv_eq hx']
rw [norm_mul, Real.norm_eq_abs]
norm_num only [abs_neg, abs_of_nonneg, one_div]
have hi : IntegrableOn (fun y => centeredCoordinate y *
sparseWeight (centeredCoordinate x / 10) y) (Icc (0 : ℝ) 1) := by
have ha : centeredCoordinate x / 10 ∈ Icc (-1 / 10 : ℝ) (1 / 10) := by
have hh := abs_centeredCoordinate_le_one hx'
rw [abs_le] at hh
exact ⟨by nlinarith [hh.1], by nlinarith [hh.2]⟩
have hden : ∀ y ∈ Icc (0 : ℝ) 1,
1 + centeredCoordinate x / 10 * centeredCoordinate y ≠ 0 := by
intro y hy
have hh := abs_centeredCoordinate_le_one hy
rw [abs_le] at hh
rcases ha with ⟨ha1, ha2⟩
nlinarith
apply ContinuousOn.integrableOn_Icc
apply ContinuousOn.mul (by unfold centeredCoordinate; fun_prop)
unfold sparseWeight
apply ContinuousOn.div
· unfold exponentialInterventionDensity; fun_prop
· unfold centeredCoordinate; fun_prop
· intro y hy
exact pow_ne_zero 2 (hden y hy)
calc
(1 / 5 : ℝ) * ‖∫ y in Icc (0 : ℝ) 1,
centeredCoordinate y * sparseWeight (centeredCoordinate x / 10) y‖ ≤
(1 / 5 : ℝ) * ∫ y in Icc (0 : ℝ) 1,
‖centeredCoordinate y * sparseWeight (centeredCoordinate x / 10) y‖ := by
gcongr
exact norm_integral_le_integral_norm _
_ ≤ (1 / 5 : ℝ) * 50 := by
gcongr
calc
(∫ y in Icc (0 : ℝ) 1,
‖centeredCoordinate y * sparseWeight (centeredCoordinate x / 10) y‖) ≤
∫ _y in Icc (0 : ℝ) 1, (50 : ℝ) := by
apply integral_mono_ae hi.norm
(continuous_const.continuousOn.integrableOn_compact isCompact_Icc)
filter_upwards [ae_restrict_mem measurableSet_Icc] with y hy
have hh := abs_centeredCoordinate_le_one hy
have hq := exponentialInterventionDensity_lt_thirteen_div_three hy
have hqp := exponentialInterventionDensity_pos y
have ha : centeredCoordinate x / 10 ∈ Icc (-1 / 10 : ℝ) (1 / 10) := by
have hhx := abs_centeredCoordinate_le_one hx'
rw [abs_le] at hhx
exact ⟨by nlinarith [hhx.1], by nlinarith [hhx.2]⟩
have hd : (9 / 10 : ℝ) ≤
1 + centeredCoordinate x / 10 * centeredCoordinate y := by
rw [abs_le] at hh
rcases ha with ⟨ha1, ha2⟩
nlinarith
rw [Real.norm_eq_abs, abs_mul]
simp only [sparseWeight]
rw [abs_div, abs_pow, abs_of_pos hqp]
rw [show |(1 + centeredCoordinate x / 10 * centeredCoordinate y) ^ 2| =
(1 + centeredCoordinate x / 10 * centeredCoordinate y) ^ 2 by
exact abs_of_nonneg (sq_nonneg _)]
rw [show |centeredCoordinate y| *
(exponentialInterventionDensity y ^ 2 /
(1 + centeredCoordinate x / 10 * centeredCoordinate y) ^ 2) =
(|centeredCoordinate y| * exponentialInterventionDensity y ^ 2) /
(1 + centeredCoordinate x / 10 * centeredCoordinate y) ^ 2 by ring]
rw [div_le_iff₀ (sq_pos_of_pos (by linarith))]
have hqsq : exponentialInterventionDensity y ^ 2 < (13 / 3 : ℝ) ^ 2 := by
nlinarith
nlinarith [sq_nonneg (1 + centeredCoordinate x / 10 * centeredCoordinate y)]
_ = 50 := by simp [Measure.real, Real.volume_Icc]
_ = 10 := by norm_num
theorem neg_cancellationPrimitive_integral_eq_centeredMean Lemma neg_cancellationPrimitive_integral_eq_centeredMean in the paper ↗
The integral of the negated cancellation primitive equals its explicit centered mean.
Formal statement
conclusion
Proof (Lean source)
lemma neg_cancellationPrimitive_integral_eq_centeredMean :
(∫ x in Icc (0 : ℝ) 1, -cancellationPrimitive x) =
1 / (1 - exp (-4)) - 3 / 4 := by
have hip := intervalIntegral.integral_mul_deriv_eq_deriv_mul
(a := (0 : ℝ)) (b := 1)
(u := fun x : ℝ => x) (v := cancellationPrimitive)
(u' := fun _ => (1 : ℝ))
(v' := fun x => exponentialInterventionDensity x - 1)
(fun x _ => hasDerivAt_id x)
(fun x _ => cancellationPrimitive_hasDerivAt x)
(continuous_const.intervalIntegrable 0 1)
((by unfold exponentialInterventionDensity; fun_prop :
Continuous (fun x => exponentialInterventionDensity x - 1)).intervalIntegrable 0 1)
simp only [one_mul, one_mul, zero_mul, sub_zero, cancellationPrimitive_zero] at hip
rw [MeasureTheory.integral_Icc_eq_integral_Ioc,
← intervalIntegral.integral_of_le (by norm_num : (0 : ℝ) ≤ 1)]
rw [intervalIntegral.integral_neg]
have hxq : IntervalIntegrable
(fun x => x * exponentialInterventionDensity x) volume 0 1 :=
(by unfold exponentialInterventionDensity; fun_prop :
Continuous (fun x => x * exponentialInterventionDensity x)).intervalIntegrable 0 1
have hxone : IntervalIntegrable (fun x : ℝ => x) volume 0 1 :=
continuous_id.intervalIntegrable 0 1
have hsplit : (∫ x in (0 : ℝ)..1,
x * (exponentialInterventionDensity x - 1)) =
(∫ x in (0 : ℝ)..1, x * exponentialInterventionDensity x) -
∫ x in (0 : ℝ)..1, x := by
rw [show (fun x : ℝ => x * (exponentialInterventionDensity x - 1)) =
fun x => x * exponentialInterventionDensity x - x by funext x; ring,
intervalIntegral.integral_sub hxq hxone]
rw [hsplit] at hip
have hxqeq : (∫ x in (0 : ℝ)..1, x * exponentialInterventionDensity x) =
1 / (1 - exp (-4)) - 1 / 4 := by
rw [intervalIntegral.integral_of_le (by norm_num : (0 : ℝ) ≤ 1),
← MeasureTheory.integral_Icc_eq_integral_Ioc]
exact exponentialInterventionDensity_firstMoment
have hid : (∫ x in (0 : ℝ)..1, x) = 1 / 2 := by rw [integral_id]; norm_num
rw [hxqeq, hid] at hip
linarith
theorem sparse_unreflected_moment_gap
The unreflected sparse construction has a strictly positive quantitative moment gap.
Formal statement
conclusion
Proof (Lean source)
lemma sparse_unreflected_moment_gap :
(3 / 10000 : ℝ) < ∫ x in Icc (0 : ℝ) 1,
(1 - exponentialInterventionDensity x) * sparseA x := by
let C : ℝ := (1 / 6 : ℝ) * ((68 / 9 : ℝ) * (4 / 729) * (100 / 121))
have hHcont : Continuous cancellationPrimitive := by
unfold cancellationPrimitive
fun_prop
have hqcont : Continuous exponentialInterventionDensity := by
unfold exponentialInterventionDensity
fun_prop
have hip := intervalIntegral.integral_mul_deriv_eq_deriv_mul
(a := (0 : ℝ)) (b := 1)
(u := cancellationPrimitive) (v := sparseA)
(u' := fun x => exponentialInterventionDensity x - 1)
(v' := deriv sparseA)
(fun x _ => cancellationPrimitive_hasDerivAt x)
(fun x hx => by
have hx' : x ∈ Icc (0 : ℝ) 1 := by
simpa only [uIcc_of_le (by norm_num : (0 : ℝ) ≤ 1)] using hx
simpa only [sparseA_deriv_eq hx'] using sparseA_hasDerivAt hx')
((hqcont.sub continuous_const).intervalIntegrable 0 1)
sparseA_deriv_intervalIntegrable
simp only [cancellationPrimitive_zero, zero_mul, sub_self] at hip
have hgapEq : (∫ x in Icc (0 : ℝ) 1,
(1 - exponentialInterventionDensity x) * sparseA x) =
∫ x in Icc (0 : ℝ) 1, cancellationPrimitive x * deriv sparseA x := by
rw [MeasureTheory.integral_Icc_eq_integral_Ioc,
MeasureTheory.integral_Icc_eq_integral_Ioc,
← intervalIntegral.integral_of_le (by norm_num : (0 : ℝ) ≤ 1),
← intervalIntegral.integral_of_le (by norm_num : (0 : ℝ) ≤ 1)]
rw [show (fun x : ℝ => (1 - exponentialInterventionDensity x) * sparseA x) =
fun x => -((exponentialInterventionDensity x - 1) * sparseA x) by funext x; ring,
intervalIntegral.integral_neg]
linarith
rw [hgapEq]
have hright : IntegrableOn (fun x => cancellationPrimitive x * deriv sparseA x)
(Icc (0 : ℝ) 1) := by
rw [integrableOn_Icc_iff_integrableOn_Ioc]
rw [← intervalIntegrable_iff_integrableOn_Ioc_of_le (by norm_num : (0 : ℝ) ≤ 1)]
exact sparseA_deriv_intervalIntegrable.continuousOn_mul hHcont.continuousOn
have hleft : IntegrableOn (fun x => (1 / 5 : ℝ) * C * (-cancellationPrimitive x))
(Icc (0 : ℝ) 1) := by
exact (continuous_const.mul hHcont.neg).continuousOn.integrableOn_Icc
have hmono : (∫ x in Icc (0 : ℝ) 1,
(1 / 5 : ℝ) * C * (-cancellationPrimitive x)) ≤
∫ x in Icc (0 : ℝ) 1, cancellationPrimitive x * deriv sparseA x := by
apply integral_mono_ae hleft hright
filter_upwards [ae_restrict_mem measurableSet_Icc] with x hx
have hH : cancellationPrimitive x ≤ 0 := (cancellationPrimitive_mem_negUnitInterval hx).2
have ha : centeredCoordinate x / 10 ∈ Icc (-1 / 10 : ℝ) (1 / 10) := by
have hh := abs_centeredCoordinate_le_one hx
rw [abs_le] at hh
exact ⟨by nlinarith [hh.1], by nlinarith [hh.2]⟩
have hI := sparseWeight_centered_integral_lower_bound ha
rw [sparseA_deriv_eq hx]
dsimp only [C]
nlinarith
rw [MeasureTheory.integral_const_mul, neg_cancellationPrimitive_integral_eq_centeredMean] at hmono
have hcert := sparseWeight_moment_certificate (a := 0) (by norm_num)
dsimp only [C] at hmono
have hI0 := sparseWeight_centered_integral_lower_bound
(show (0 : ℝ) ∈ Icc (-1 / 10) (1 / 10) by norm_num)
have hgap := exponentialTiltMeanGap_gt
have hrat := sparseMomentRationalCertificate
have hexact := sparseMomentCoefficient_exact
nlinarith
theorem sparse_secondMomentContrast_eq_neg_gap
The sparse witness's second-moment contrast equals the negative analytic gap.
Formal statement
hypotheses
s :
conclusion
Proof (Lean source)
lemma sparse_secondMomentContrast_eq_neg_gap (s : SignVector 3) :
secondMomentContrast
(canonicalObservedWorld threeNodeDAG (sparseWitness s) (refl (Fin 3))) 0 1 =
-(∫ x in Icc (0 : ℝ) 1,
(1 - exponentialInterventionDensity x) * sparseA x) := by
have hpos := sparseWitness_positive_normalized_smooth s
rw [canonical_secondMomentContrast_eq_integral_for_witness hpos
(by norm_num : (0 : Fin 3) ≠ 1)]
let f : LatentState 3 → ℝ := fun v =>
((sparseWitness s).q 1 (v 1)) ^ 2 *
((sparseWitness s).q 0 (v 0) - (sparseWitness s).p 0 v) *
(∏ l ∈ (Finset.univ.erase 1).erase 0, (sparseWitness s).p l v) /
(sparseWitness s).p 1 v
have hfcont : ContinuousOn f (latentCube 3) := by
apply ContinuousOn.div
· apply ContinuousOn.mul
· apply ContinuousOn.mul
· exact ((hpos.2.2.2.1 1).continuousOn.comp
((continuous_apply 1).continuousOn)
(fun (v : LatentState 3) (hv : v ∈ latentCube 3) =>
hv 1 (Set.mem_univ 1))).pow 2
· exact ((hpos.2.2.2.1 0).continuousOn.comp
((continuous_apply 0).continuousOn)
(fun (v : LatentState 3) (hv : v ∈ latentCube 3) =>
hv 0 (Set.mem_univ 0))).sub
(hpos.2.2.1 0).continuousOn
· exact continuousOn_finsetProd _ fun l _ => (hpos.2.2.1 l).continuousOn
· exact (hpos.2.2.1 1).continuousOn
· intro v hv
exact ne_of_gt (hpos.1 1 v hv)
have hf : Integrable f (volume.restrict (latentCube 3)) :=
hfcont.integrableOn_compact (by
rw [latentCube]
exact isCompact_univ_pi fun _ => isCompact_Icc)
have hμ : volume.restrict (latentCube 3) =
Measure.pi (fun _ : Fin 3 => volume.restrict (Icc (0 : ℝ) 1)) := by
change volume.restrict (Set.univ.pi fun _ : Fin 3 => Icc (0 : ℝ) 1) = _
rw [MeasureTheory.volume_pi, Measure.restrict_pi_pi]
change (∫ v in latentCube 3, f v) = _
rw [hμ, integral_fin_three_pi_eq_iterated _ f (by simpa [hμ] using hf)]
have hone : (volume.restrict (Icc (0 : ℝ) 1)).real univ = 1 := by
simp [Measure.real, Real.volume_Icc]
simp only [f, sparseWitness, sparseQ, sparseP,
Matrix.cons_val_zero, Matrix.cons_val_one, integral_const, hone, one_smul]
have herase : (Finset.univ.erase (1 : Fin 3)).erase 0 = {2} := by decide
rw [herase]
norm_num
have hrefY (x : ℝ) :
(∫ y in Icc (0 : ℝ) 1,
exponentialInterventionDensity (reflectedCoordinate s 1 y) ^ 2 *
(exponentialInterventionDensity (reflectedCoordinate s 0 x) - 1) /
(1 + (10 : ℝ)⁻¹ * centeredCoordinate (reflectedCoordinate s 0 x) *
centeredCoordinate (reflectedCoordinate s 1 y))) =
(exponentialInterventionDensity (reflectedCoordinate s 0 x) - 1) *
sparseA (reflectedCoordinate s 0 x) := by
change (∫ y in Icc (0 : ℝ) 1,
(fun y => exponentialInterventionDensity y ^ 2 *
(exponentialInterventionDensity (reflectedCoordinate s 0 x) - 1) /
(1 + (10 : ℝ)⁻¹ * centeredCoordinate (reflectedCoordinate s 0 x) *
centeredCoordinate y)) (reflectedCoordinate s 1 y)) = _
have href := integral_reflectedCoordinate s 1
(fun y => exponentialInterventionDensity y ^ 2 *
(exponentialInterventionDensity (reflectedCoordinate s 0 x) - 1) /
(1 + (10 : ℝ)⁻¹ * centeredCoordinate (reflectedCoordinate s 0 x) *
centeredCoordinate y))
rw [href]
unfold sparseA
rw [show (fun y : ℝ =>
exponentialInterventionDensity y ^ 2 *
(exponentialInterventionDensity (reflectedCoordinate s 0 x) - 1) /
(1 + (10 : ℝ)⁻¹ * centeredCoordinate (reflectedCoordinate s 0 x) *
centeredCoordinate y)) =
fun y => (exponentialInterventionDensity (reflectedCoordinate s 0 x) - 1) *
(exponentialInterventionDensity y ^ 2 /
(1 + centeredCoordinate (reflectedCoordinate s 0 x) / 10 *
centeredCoordinate y)) by funext y; ring,
MeasureTheory.integral_const_mul]
simp only [Fin.isValue, Fin.reduceEq, ↓reduceIte, one_div]
simp_rw [hrefY]
change (∫ x in Icc (0 : ℝ) 1,
(fun x => (exponentialInterventionDensity x - 1) * sparseA x)
(reflectedCoordinate s 0 x)) = _
rw [show (∫ x in Icc (0 : ℝ) 1,
(fun x => (exponentialInterventionDensity x - 1) * sparseA x)
(reflectedCoordinate s 0 x)) =
∫ x in Icc (0 : ℝ) 1,
(exponentialInterventionDensity x - 1) * sparseA x by
simpa only using integral_reflectedCoordinate s 0
(fun x => (exponentialInterventionDensity x - 1) * sparseA x)]
rw [← MeasureTheory.integral_neg]
apply integral_congr_ae
filter_upwards with x
ring
theorem sparse_witness_moment_gap
The explicit sparse witness has the required strictly positive observed-ratio moment gap.
Formal statement
hypotheses
s :
conclusion
(3 / 10000 : ℝ) < (∫ x, ((canonicalObservedWorld threeNodeDAG (sparseWitness s) (refl (Fin 3))).ratio 1 x) ^ 2 ∂(canonicalObservedWorld threeNodeDAG (sparseWitness s) (refl (Fin 3))).law 0)
- ∫ x, ((canonicalObservedWorld threeNodeDAG (sparseWitness s) (refl (Fin 3))).ratio 1 x) ^ 2 ∂(canonicalObservedWorld threeNodeDAG (sparseWitness s) (refl (Fin 3))).law (succ 0)
Proof (Lean source)
lemma sparse_witness_moment_gap (s : SignVector 3) :
(3 / 10000 : ℝ) <
(∫ x, ((canonicalObservedWorld threeNodeDAG (sparseWitness s)
(refl (Fin 3))).ratio 1 x) ^ 2
∂(canonicalObservedWorld threeNodeDAG (sparseWitness s)
(refl (Fin 3))).law 0) -
∫ x, ((canonicalObservedWorld threeNodeDAG (sparseWitness s)
(refl (Fin 3))).ratio 1 x) ^ 2
∂(canonicalObservedWorld threeNodeDAG (sparseWitness s)
(refl (Fin 3))).law (succ 0) := by
have hgap := sparse_unreflected_moment_gap
have hcontrast := sparse_secondMomentContrast_eq_neg_gap s
unfold secondMomentContrast at hcontrast
dsimp only [canonicalObservedWorld, Equiv.refl_apply] at hcontrast ⊢
linarith
theorem sparse_witness_mmd_gap
The explicit sparse witness has the required strictly positive Gaussian-kernel discrepancy.
Formal statement
hypotheses
s :
conclusion
(5 / 100000000 : ℝ) < populationDiscrepancy gaussianFeatureMap (canonicalObservedWorld threeNodeDAG (sparseWitness s) (refl (Fin 3))) 0 1
Proof (Lean source)
lemma sparse_witness_mmd_gap (s : SignVector 3) :
(5 / 100000000 : ℝ) < populationDiscrepancy gaussianFeatureMap
(canonicalObservedWorld threeNodeDAG (sparseWitness s) (refl (Fin 3))) 0 1 := by
let θ := sparseWitness s
let W := canonicalObservedWorld threeNodeDAG θ (refl (Fin 3))
let μ := observationalRatioLaw W 1
let ν := interventionalRatioLaw W 0 1
have hpos : PositiveNormalizedSmoothMechanisms threeNodeDAG θ :=
sparseWitness_positive_normalized_smooth s
letI : IsProbabilityMeasure (observationalLaw θ) :=
observationalLaw_isProbabilityMeasure hpos
letI : IsProbabilityMeasure (interventionalLaw θ 0) := by
simpa only [W, canonicalObservedWorld, Equiv.refl_apply] using
interventionalLaw_isProbabilityMeasure W hpos 0
letI : IsProbabilityMeasure μ := by
change IsProbabilityMeasure
(Measure.map (observedLawRatio W.law 1) (observationalLaw θ))
exact Measure.isProbabilityMeasure_map
(measurable_observedLawRatio W.law 1).aemeasurable
letI : IsProbabilityMeasure ν := by
change IsProbabilityMeasure
(Measure.map (observedLawRatio W.law 1) (interventionalLaw θ 0))
exact Measure.isProbabilityMeasure_map
(measurable_observedLawRatio W.law 1).aemeasurable
have hobsCube : ∀ᵐ v ∂observationalLaw θ, v ∈ latentCube 3 :=
observationalLaw_ae_mem_latentCube hpos
have hac : interventionalLaw θ 0 ≪ observationalLaw θ := by
simpa only [W, canonicalObservedWorld, Equiv.refl_apply] using
interventionalLaw_absolutelyContinuous_observational W hpos 0
have hintCube : ∀ᵐ v ∂interventionalLaw θ 0, v ∈ latentCube 3 :=
hac.ae_le hobsCube
have hratioObs := canonicalObservedWorld_observedLawRatio_ae_eq_mechanismRatio
hpos (refl (Fin 3)) 1
have hratioInt : observedLawRatio W.law 1 =ᵐ[interventionalLaw θ 0]
fun v => θ.q 1 (v 1) / θ.p 1 v := by
exact hac.ae_eq (by simpa only [W, Equiv.refl_apply] using hratioObs)
have hμsupport : μ (Icc (0 : ℝ) 5)ᶜ = 0 := by
dsimp only [μ, observationalRatioLaw]
rw [Measure.map_apply (measurable_observedLawRatio W.law 1) measurableSet_Icc.compl]
apply ae_iff.mp
filter_upwards [hobsCube, hratioObs] with v hv hr
rw [hr]
exact ⟨(sparseWitness_child_ratio_bounds s v hv).1.le,
le_of_lt (sparseWitness_child_ratio_bounds s v hv).2⟩
have hνsupport : ν (Icc (0 : ℝ) 5)ᶜ = 0 := by
dsimp only [ν, interventionalRatioLaw]
change Measure.map (observedLawRatio W.law 1) (interventionalLaw θ 0)
(Icc (0 : ℝ) 5)ᶜ = 0
rw [Measure.map_apply (measurable_observedLawRatio W.law 1) measurableSet_Icc.compl]
apply ae_iff.mp
filter_upwards [hintCube, hratioInt] with v hv hr
rw [hr]
exact ⟨(sparseWitness_child_ratio_bounds s v hv).1.le,
le_of_lt (sparseWitness_child_ratio_bounds s v hv).2⟩
have hmom : (3 / 10000 : ℝ) <
(∫ r, r ^ 2 ∂μ) - ∫ r, r ^ 2 ∂ν := by
rw [canonical_observationalRatioLaw_secondMoment_eq hpos (refl (Fin 3)) 1,
canonical_interventionalRatioLaw_secondMoment_eq hpos (refl (Fin 3)) 0 1]
have hbase := sparse_witness_moment_gap s
simp only [canonicalObservedWorld, Fin.cases_zero, Fin.cases_succ,
Equiv.refl_apply] at hbase
simpa only [μ, ν, θ, W, Equiv.refl_apply] using hbase
have habs : (3 / 10000 : ℝ) <
|(∫ r, r ^ 2 ∂μ) - ∫ r, r ^ 2 ∂ν| := by
rw [abs_of_pos (lt_trans (by norm_num) hmom)]
exact hmom
have hlower :=
CausalSmith.Substrate.GaussianMeanEmbeddingQuantitativeMomentStability.gaussian_meanEmbedding_norm_lowerBound_of_secondMoment_gap
μ ν hμsupport hνsupport (lt_trans (by norm_num) habs)
have hrat := sparseMmdRationalCertificate
rw [← meanEmbedding_gaussianFeatureMap_eq_recovery μ,
← meanEmbedding_gaussianFeatureMap_eq_recovery ν] at hlower
unfold populationDiscrepancy
nlinarith [hlower.2]
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.WitnessPath 3 declarations This file records the elementary one-dimensional facts about the child-mechanism coefficient along the path from the cancellation witness to the sparse witness.
Explicit cancellation-to-sparse witness path
This file records the elementary one-dimensional facts about the child-mechanism coefficient along the path from the cancellation witness to the sparse witness.
The child-mechanism coefficient on the affine path from the cancellation witness (t = 0) to the sparse witness (t = 1).
Definition (Lean source)
parameters
t x :
ℝ
def
cancellationSparsePathPrimitive t x :
ℝ
given by
(1 - t) * cancellationPrimitive x + t * centeredCoordinate x
theorem cancellationSparsePathPrimitive_mem_unitInterval
Convex interpolation preserves the coefficient bound on the unit interval. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
cancellationSparsePathPrimitive t x ∈ Icc (-1 : ℝ) 1
Proof (Lean source)
lemma cancellationSparsePathPrimitive_mem_unitInterval
{t x : ℝ} (ht : t ∈ Icc (0 : ℝ) 1) (hx : x ∈ Icc (0 : ℝ) 1) :
cancellationSparsePathPrimitive t x ∈ Icc (-1 : ℝ) 1 := by
have hH := cancellationPrimitive_mem_unitInterval_sub hx
have hh := abs_centeredCoordinate_le_one hx
rw [abs_le] at hh
unfold cancellationSparsePathPrimitive
constructor <;> nlinarith [ht.1, ht.2, hH.1, hH.2, hh.1, hh.2]
theorem cancellationSparsePathPrimitive_nonconstant
Every coefficient on the closed cancellation-to-sparse segment is nonconstant. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
t :
ℝ
ht :
t ∈ Icc (0 : ℝ) 1
conclusion
∃ x, cancellationSparsePathPrimitive t x ≠ cancellationSparsePathPrimitive t 0
Proof (Lean source)
lemma cancellationSparsePathPrimitive_nonconstant
{t : ℝ} (ht : t ∈ Icc (0 : ℝ) 1) :
∃ x, cancellationSparsePathPrimitive t x ≠
cancellationSparsePathPrimitive t 0 := by
by_cases ht0 : t = 0
· subst t
simpa [cancellationSparsePathPrimitive] using cancellationPrimitive_nonconstant
· refine ⟨1, ?_⟩
have hH := cancellationPrimitive_zero
simp only [cancellationSparsePathPrimitive, hH.1, hH.2, centeredCoordinate]
intro heq
apply ht0
linarith
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.WitnessPathRegularity 1 declarations This file records the positive, normalized, smooth part of stratum preservation along the closed affine path.
Affine witness-path regularity
This file records the positive, normalized, smooth part of stratum preservation along the closed affine path. Causal minimality and the fixed-sign cell are handled separately by the analytic perturbation argument.
theorem affinePathExtension_positive_normalized_smooth
Convex interpolation with the embedded sparse endpoint preserves positivity, normalization, and C³ smoothness throughout the closed unit interval. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
PositiveNormalizedSmoothMechanisms G (affinePathExtension s θ hji t)
Proof (Lean source)
lemma affinePathExtension_positive_normalized_smooth
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(θ : StratumPoint G s) {j i : Fin n} (hji : G.edge j i)
{t : ℝ} (ht : t ∈ Icc (0 : ℝ) 1) :
PositiveNormalizedSmoothMechanisms G (affinePathExtension s θ hji t) := by
let θstar := embeddedSparseWitness s hji
have hθ := θ.property.positiveSmooth
have hstar := embeddedSparseWitness_positive_normalized_smooth s hji
refine ⟨?_, ?_, ?_, ?_, ?_, ?_⟩
· intro l v hv
change 0 < (1 - t) * θ.1.p l v + t * θstar.p l v
have hp := hθ.1 l v hv
have hpstar := hstar.1 l v hv
by_cases ht0 : t = 0
· simp [ht0, hp]
· have htpos : 0 < t := lt_of_le_of_ne ht.1 (Ne.symm ht0)
exact add_pos_of_nonneg_of_pos
(mul_nonneg (sub_nonneg.mpr ht.2) hp.le) (mul_pos htpos hpstar)
· intro l z hz
change 0 < (1 - t) * θ.1.q l z + t * θstar.q l z
have hq := hθ.2.1 l z hz
have hqstar := hstar.2.1 l z hz
by_cases ht0 : t = 0
· simp [ht0, hq]
· have htpos : 0 < t := lt_of_le_of_ne ht.1 (Ne.symm ht0)
exact add_pos_of_nonneg_of_pos
(mul_nonneg (sub_nonneg.mpr ht.2) hq.le) (mul_pos htpos hqstar)
· intro l
exact (hθ.2.2.1 l).const_smul (1 - t) |>.add ((hstar.2.2.1 l).const_smul t)
· intro l
exact (hθ.2.2.2.1 l).const_smul (1 - t) |>.add
((hstar.2.2.2.1 l).const_smul t)
· intro l v hv
have hupdate : Continuous (fun z : ℝ => update v l z) := by fun_prop
have hmaps : MapsTo (fun z : ℝ => update v l z)
(Icc (0 : ℝ) 1) (latentCube n) := by
intro z hz k _
by_cases hkl : k = l
· subst k
simpa using hz
· simpa only [update, dif_neg hkl] using hv k (Set.mem_univ k)
have hpInt : IntegrableOn (fun z : ℝ => (1 - t) * θ.1.p l (update v l z))
(Icc (0 : ℝ) 1) := by
exact ((hθ.2.2.1 l).continuousOn.comp hupdate.continuousOn hmaps |>.const_mul _)
|>.integrableOn_Icc
have hpstarInt : IntegrableOn (fun z : ℝ => t * θstar.p l (update v l z))
(Icc (0 : ℝ) 1) := by
exact ((hstar.2.2.1 l).continuousOn.comp hupdate.continuousOn hmaps |>.const_mul _)
|>.integrableOn_Icc
change ∫ z in Icc (0 : ℝ) 1,
((1 - t) * θ.1.p l (update v l z) +
t * θstar.p l (update v l z)) = 1
rw [MeasureTheory.integral_add,
MeasureTheory.integral_const_mul, MeasureTheory.integral_const_mul,
hθ.2.2.2.2.1 l v hv, hstar.2.2.2.2.1 l v hv]
· ring
· exact hpInt
· exact hpstarInt
· intro l
change ∫ z in Icc (0 : ℝ) 1,
((1 - t) * θ.1.q l z + t * θstar.q l z) = 1
rw [MeasureTheory.integral_add,
MeasureTheory.integral_const_mul, MeasureTheory.integral_const_mul,
hθ.2.2.2.2.2 l, hstar.2.2.2.2.2 l]
· ring
· exact ((hθ.2.2.2.1 l).continuousOn.const_mul _).integrableOn_Icc
· exact ((hstar.2.2.2.1 l).continuousOn.const_mul _).integrableOn_Icc
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.WitnessQuantitative 25 declarations This file isolates the elementary exponential and rational estimates used in the quantitative moment and Gaussian-MMD certificate.
Rational bounds for the explicit sparse witness
This file isolates the elementary exponential and rational estimates used in the quantitative moment and Gaussian-MMD certificate.
The one-dimensional weight appearing after differentiating the sparse second-moment integrand with respect to its parent coefficient.
Definition (Lean source)
parameters
a y :
ℝ
def
sparseWeight a y :
ℝ
given by
exponentialInterventionDensity y ^ 2 / (1 + a * centeredCoordinate y) ^ 2
theorem sparseWeight_log_hasDerivAt
The logarithmic derivative of the sparse weight has the explicit form used in the quantitative moment-gap argument. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
Proof (Lean source)
-- @node: sparseWeight_log_hasDerivAt
lemma sparseWeight_log_hasDerivAt {a y : ℝ}
(hden : 1 + a * centeredCoordinate y ≠ 0) :
HasDerivAt (fun z => log (sparseWeight a z))
(8 - 4 * a / (1 + a * centeredCoordinate y)) y := by
have hq : HasDerivAt exponentialInterventionDensity
(4 * exponentialInterventionDensity y) y := by
simpa [mul_comm] using exponentialInterventionDensity_hasDerivAt y
have hp0 : HasDerivAt (fun z : ℝ => 2 * z - 1) 2 y := by
convert (hasDerivAt_const_mul (x := y) 2).sub_const 1 using 1
have hp : HasDerivAt (fun z : ℝ => 1 + a * centeredCoordinate z) (2 * a) y := by
simpa only [centeredCoordinate, mul_comm] using (hp0.const_mul a).const_add 1
have hw := (hq.pow 2).div (hp.pow 2) (pow_ne_zero 2 hden)
have hwne : sparseWeight a y ≠ 0 := by
unfold sparseWeight
exact div_ne_zero
(pow_ne_zero 2 (ne_of_gt (exponentialInterventionDensity_pos y)))
(pow_ne_zero 2 hden)
have hlog := hw.log hwne
have hqne : exponentialInterventionDensity y ≠ 0 :=
ne_of_gt (exponentialInterventionDensity_pos y)
convert hlog using 1
· rfl
· simp only [Pi.pow_apply, Pi.div_apply]
field_simp [hqne, hden]
ring
theorem sparseWeight_log_deriv_lower_bound
For every parent coefficient in [-1/10,1/10], the logarithmic derivative of the sparse weight is at least 68/9 on the unit interval. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: sparseWeight_log_deriv_lower_bound
lemma sparseWeight_log_deriv_lower_bound {a y : ℝ}
(ha : a ∈ Icc (-1 / 10 : ℝ) (1 / 10))
(hy : y ∈ Icc (0 : ℝ) 1) :
(68 / 9 : ℝ) ≤ deriv (fun z => log (sparseWeight a z)) y := by
have hh := abs_centeredCoordinate_le_one hy
have ha' : |a| ≤ (1 / 10 : ℝ) := by
rw [abs_le]
constructor <;> norm_num at ha ⊢
· exact ha.1
· exact ha.2
have haprod : |a * centeredCoordinate y| ≤ (1 / 10 : ℝ) := by
rw [abs_mul]
calc
|a| * |centeredCoordinate y| ≤ (1 / 10 : ℝ) * 1 :=
mul_le_mul ha' hh (abs_nonneg _) (by norm_num)
_ = 1 / 10 := by norm_num
rw [abs_le] at haprod
have hden_pos : 0 < 1 + a * centeredCoordinate y := by
nlinarith [haprod.1]
have hfactor_nonneg : 0 ≤ 9 - centeredCoordinate y := by
rw [abs_le] at hh
linarith
have hmul := mul_le_mul_of_nonneg_right ha.2 hfactor_nonneg
have hratio : a / (1 + a * centeredCoordinate y) ≤ (1 / 9 : ℝ) := by
rw [div_le_iff₀ hden_pos]
nlinarith [hmul]
have hratio4 := mul_le_mul_of_nonneg_left hratio (by norm_num : (0 : ℝ) ≤ 4)
rw [(sparseWeight_log_hasDerivAt hden_pos.ne').deriv]
rw [show 4 * a / (1 + a * centeredCoordinate y) =
4 * (a / (1 + a * centeredCoordinate y)) by ring]
norm_num at hratio4 ⊢
nlinarith
theorem sparseWeight_hasDerivAt
The sparse weight derivative is its value times the explicit logarithmic derivative factor. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma sparseWeight_hasDerivAt {a y : ℝ}
(hden : 1 + a * centeredCoordinate y ≠ 0) :
HasDerivAt (sparseWeight a)
(sparseWeight a y * (8 - 4 * a / (1 + a * centeredCoordinate y))) y := by
have hq : HasDerivAt exponentialInterventionDensity
(4 * exponentialInterventionDensity y) y := by
simpa [mul_comm] using exponentialInterventionDensity_hasDerivAt y
have hp0 : HasDerivAt (fun z : ℝ => 2 * z - 1) 2 y := by
convert (hasDerivAt_const_mul (x := y) 2).sub_const 1 using 1
have hp : HasDerivAt (fun z : ℝ => 1 + a * centeredCoordinate z) (2 * a) y := by
simpa only [centeredCoordinate, mul_comm] using (hp0.const_mul a).const_add 1
have hw := (hq.pow 2).div (hp.pow 2) (pow_ne_zero 2 hden)
change HasDerivAt (sparseWeight a) _ y at hw
apply hw.congr_deriv
simp only [Pi.pow_apply]
unfold sparseWeight
field_simp [hden]
ring
The exponential normalizer in the witness is strictly below 55. the stated conclusion follows.
Formal statement
conclusion
exp 4 < 55
theorem exponentialNormalizer_gt_two_div_twenty_seven Lemma exponentialNormalizer_gt_two_div_twenty_seven in the paper ↗
The normalizing constant 4 / (exp 4 - 1) exceeds 2/27. the stated conclusion follows.
Formal statement
conclusion
(2 / 27 : ℝ) < 4 / (exp 4 - 1)
Proof (Lean source)
lemma exponentialNormalizer_gt_two_div_twenty_seven :
(2 / 27 : ℝ) < 4 / (exp 4 - 1) := by
have hden : 0 < exp 4 - 1 := by linarith [thirteen_lt_exp_four]
rw [div_lt_div_iff₀ (by norm_num : (0 : ℝ) < 27) hden]
nlinarith [exp_four_lt_fifty_five]
theorem sparseWeight_gt_rational_floor
On the coefficient strip used by the witness, the sparse weight is strictly above the rational floor obtained from c > 2/27 and p₂ ≤ 11/10. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
(4 / 729 : ℝ) * (100 / 121) < sparseWeight a y
Proof (Lean source)
lemma sparseWeight_gt_rational_floor {a y : ℝ}
(ha : a ∈ Icc (-1 / 10 : ℝ) (1 / 10))
(hy : y ∈ Icc (0 : ℝ) 1) :
(4 / 729 : ℝ) * (100 / 121) < sparseWeight a y := by
have hh := abs_centeredCoordinate_le_one hy
rw [abs_le] at hh
have hden : 0 < 1 + a * centeredCoordinate y := by
have ha1 : (-1 / 10 : ℝ) ≤ a := ha.1
have ha2 : a ≤ (1 / 10 : ℝ) := ha.2
nlinarith
have hdenUpper : 1 + a * centeredCoordinate y ≤ (11 / 10 : ℝ) := by
have ha1 : (-1 / 10 : ℝ) ≤ a := ha.1
have ha2 : a ≤ (1 / 10 : ℝ) := ha.2
nlinarith
have hexp : 1 ≤ exp (4 * y) := Real.one_le_exp (by nlinarith [hy.1])
have hc := exponentialNormalizer_gt_two_div_twenty_seven
have hq : (2 / 27 : ℝ) < exponentialInterventionDensity y := by
unfold exponentialInterventionDensity
have hnormalizer_pos : 0 < 4 / (exp 4 - 1) := lt_trans (by norm_num) hc
rw [show 4 * exp (4 * y) / (exp 4 - 1) =
(4 / (exp 4 - 1)) * exp (4 * y) by ring]
nlinarith
have hdenSq : 0 < (1 + a * centeredCoordinate y) ^ 2 := sq_pos_of_pos hden
have hupperSq : (1 + a * centeredCoordinate y) ^ 2 ≤ (11 / 10 : ℝ) ^ 2 := by
nlinarith
have hqSq : (2 / 27 : ℝ) ^ 2 < exponentialInterventionDensity y ^ 2 := by
nlinarith [exponentialInterventionDensity_pos y]
unfold sparseWeight
calc
(4 / 729 : ℝ) * (100 / 121) =
(2 / 27 : ℝ) ^ 2 / (11 / 10 : ℝ) ^ 2 := by norm_num
_ < exponentialInterventionDensity y ^ 2 /
(1 + a * centeredCoordinate y) ^ 2 := by
rw [div_lt_div_iff₀ (by norm_num : (0 : ℝ) < (11 / 10) ^ 2) hdenSq]
have hleft := mul_le_mul_of_nonneg_left hupperSq
(sq_nonneg (2 / 27 : ℝ))
have hright := mul_lt_mul_of_pos_right hqSq
(by norm_num : (0 : ℝ) < (11 / 10) ^ 2)
exact lt_of_le_of_lt hleft hright
theorem sparseWeight_deriv_gt_rational_floor
The derivative of the sparse weight has the uniform rational lower bound used in the moment certificate. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
(68 / 9 : ℝ) * ((4 / 729) * (100 / 121)) < deriv (sparseWeight a) y
Proof (Lean source)
lemma sparseWeight_deriv_gt_rational_floor {a y : ℝ}
(ha : a ∈ Icc (-1 / 10 : ℝ) (1 / 10))
(hy : y ∈ Icc (0 : ℝ) 1) :
(68 / 9 : ℝ) * ((4 / 729) * (100 / 121)) < deriv (sparseWeight a) y := by
have hden : 0 < 1 + a * centeredCoordinate y := by
have hh := abs_centeredCoordinate_le_one hy
rw [abs_le] at hh
have ha1 : (-1 / 10 : ℝ) ≤ a := ha.1
have ha2 : a ≤ (1 / 10 : ℝ) := ha.2
nlinarith
have hw := sparseWeight_gt_rational_floor ha hy
have hlog := sparseWeight_log_deriv_lower_bound ha hy
have hfactor : (68 / 9 : ℝ) ≤
8 - 4 * a / (1 + a * centeredCoordinate y) := by
simpa [(sparseWeight_log_hasDerivAt hden.ne').deriv] using hlog
rw [(sparseWeight_hasDerivAt hden.ne').deriv]
have hfloor : 0 < (4 / 729 : ℝ) * (100 / 121) := by norm_num
have hfactorPos : 0 < 8 - 4 * a / (1 + a * centeredCoordinate y) :=
lt_of_lt_of_le (by norm_num) hfactor
nlinarith
theorem sparseWeight_secant_lower_bound
The derivative certificate integrates to a uniform secant lower bound on the unit interval. This is the monotonicity input for the covariance step in the sparse moment calculation. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
a z y :
ℝ
ha :
a ∈ Icc (-1 / 10 : ℝ) (1 / 10)
hz :
z ∈ Icc (0 : ℝ) 1
hy :
y ∈ Icc (0 : ℝ) 1
hzy :
z ≤ y
conclusion
(68 / 9 : ℝ) * (4 / 729) * (100 / 121) * (y - z) ≤ sparseWeight a y - sparseWeight a z
Proof (Lean source)
lemma sparseWeight_secant_lower_bound {a z y : ℝ}
(ha : a ∈ Icc (-1 / 10 : ℝ) (1 / 10))
(hz : z ∈ Icc (0 : ℝ) 1) (hy : y ∈ Icc (0 : ℝ) 1)
(hzy : z ≤ y) :
(68 / 9 : ℝ) * (4 / 729) * (100 / 121) * (y - z) ≤
sparseWeight a y - sparseWeight a z := by
have ha' : -(1 / 10 : ℝ) ≤ a ∧ a ≤ 1 / 10 := by
constructor
· norm_num at ha ⊢
exact ha.1
· exact ha.2
have hdiff : ∀ x ∈ Icc (0 : ℝ) 1, DifferentiableAt ℝ (sparseWeight a) x := by
intro x hx
have hc : |centeredCoordinate x| ≤ 1 := abs_centeredCoordinate_le_one hx
have habs : |a| ≤ 1 / 10 := by rw [abs_le]; exact ha'
have hp : |a * centeredCoordinate x| ≤ 1 / 10 := by
rw [abs_mul]
exact (mul_le_mul habs hc (abs_nonneg _) (by norm_num)).trans_eq (by norm_num)
rw [abs_le] at hp
exact (sparseWeight_hasDerivAt (by nlinarith)).differentiableAt
apply (convex_Icc (0 : ℝ) 1).mul_sub_le_image_sub_of_le_deriv
· exact fun x hx => (hdiff x hx).continuousAt.continuousWithinAt
· intro x hx
apply (hdiff x ?_).differentiableWithinAt
rw [interior_Icc] at hx
exact ⟨le_of_lt hx.1, le_of_lt hx.2⟩
· intro x hx
have hx' : x ∈ Icc (0 : ℝ) 1 := by
rw [interior_Icc] at hx
exact ⟨le_of_lt hx.1, le_of_lt hx.2⟩
simpa only [mul_assoc] using
le_of_lt (sparseWeight_deriv_gt_rational_floor ha hx')
· exact hz
· exact hy
· exact hzy
theorem sparseWeight_centered_product_lower_bound
Centering at the midpoint turns the secant estimate into the pointwise quadratic lower bound whose integral is the 1/6 covariance factor. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
(68 / 9 : ℝ) * (4 / 729) * (100 / 121) * (2 * (y - 1 / 2) ^ 2)
Proof (Lean source)
lemma sparseWeight_centered_product_lower_bound {a y : ℝ}
(ha : a ∈ Icc (-1 / 10 : ℝ) (1 / 10))
(hy : y ∈ Icc (0 : ℝ) 1) :
(68 / 9 : ℝ) * (4 / 729) * (100 / 121) *
(2 * (y - 1 / 2) ^ 2) ≤
centeredCoordinate y * (sparseWeight a y - sparseWeight a (1 / 2)) := by
by_cases hmid : (1 / 2 : ℝ) ≤ y
· have hsec := sparseWeight_secant_lower_bound ha
(by norm_num : (1 / 2 : ℝ) ∈ Icc 0 1) hy hmid
simp only [centeredCoordinate]
nlinarith
· have hsec := sparseWeight_secant_lower_bound ha hy
(by norm_num : (1 / 2 : ℝ) ∈ Icc 0 1) (le_of_not_ge hmid)
simp only [centeredCoordinate]
nlinarith
theorem sparseWeight_centered_integral_lower_bound
Integrating the centered secant estimate supplies the exact 1/6 factor used in the sparse second-moment certificate. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
a :
ℝ
ha :
a ∈ Icc (-1 / 10 : ℝ) (1 / 10)
conclusion
(1 / 6 : ℝ) * ((68 / 9 : ℝ) * (4 / 729) * (100 / 121))
Proof (Lean source)
lemma sparseWeight_centered_integral_lower_bound {a : ℝ}
(ha : a ∈ Icc (-1 / 10 : ℝ) (1 / 10)) :
(1 / 6 : ℝ) * ((68 / 9 : ℝ) * (4 / 729) * (100 / 121)) ≤
∫ y in Icc (0 : ℝ) 1, centeredCoordinate y * sparseWeight a y := by
let C : ℝ := (68 / 9 : ℝ) * (4 / 729) * (100 / 121)
have hden : ∀ y ∈ Icc (0 : ℝ) 1,
1 + a * centeredCoordinate y ≠ 0 := by
intro y hy
have hh := abs_centeredCoordinate_le_one hy
rw [abs_le] at hh
have ha1 : (-1 / 10 : ℝ) ≤ a := ha.1
have ha2 : a ≤ (1 / 10 : ℝ) := ha.2
nlinarith
have hc : Continuous centeredCoordinate := by
unfold centeredCoordinate
fun_prop
have hwcont : ContinuousOn (sparseWeight a) (Icc (0 : ℝ) 1) := by
unfold sparseWeight
apply ContinuousOn.div
· unfold exponentialInterventionDensity
fun_prop
· exact (continuous_const.add (continuous_const.mul hc)).pow 2 |>.continuousOn
· intro y hy
exact pow_ne_zero 2 (hden y hy)
have hwint : IntegrableOn
(fun y => centeredCoordinate y * sparseWeight a y) (Icc (0 : ℝ) 1) :=
(hc.continuousOn.mul hwcont).integrableOn_Icc
have hconst : IntegrableOn
(fun y => sparseWeight a (1 / 2) * centeredCoordinate y) (Icc (0 : ℝ) 1) :=
(continuous_const.mul hc).continuousOn.integrableOn_Icc
have hright : IntegrableOn
(fun y => centeredCoordinate y *
(sparseWeight a y - sparseWeight a (1 / 2))) (Icc (0 : ℝ) 1) :=
(hc.continuousOn.mul (hwcont.sub continuousOn_const)).integrableOn_Icc
have hleft : IntegrableOn (fun y : ℝ => C * (2 * (y - 1 / 2) ^ 2))
(Icc (0 : ℝ) 1) := by
apply ContinuousOn.integrableOn_Icc
fun_prop
have hmono :
(∫ y in Icc (0 : ℝ) 1, C * (2 * (y - 1 / 2) ^ 2)) ≤
∫ y in Icc (0 : ℝ) 1,
centeredCoordinate y * (sparseWeight a y - sparseWeight a (1 / 2)) := by
apply integral_mono_ae hleft hright
filter_upwards [ae_restrict_mem measurableSet_Icc] with y hy
exact sparseWeight_centered_product_lower_bound ha hy
have hquad : (∫ y in Icc (0 : ℝ) 1, 2 * (y - 1 / 2) ^ 2) = 1 / 6 := by
have hfun : (fun y : ℝ => 2 * (y - 1 / 2) ^ 2) =
fun y => 2 * y ^ 2 - 2 * y ^ 1 + 1 / 2 := by
funext y
ring
rw [hfun, MeasureTheory.integral_Icc_eq_integral_Ioc]
rw [← intervalIntegral.integral_of_le (by norm_num : (0 : ℝ) ≤ 1)]
rw [intervalIntegral.integral_add
((intervalIntegral.intervalIntegrable_pow 2).const_mul 2 |>.sub
((intervalIntegral.intervalIntegrable_pow 1).const_mul 2))
intervalIntegrable_const,
intervalIntegral.integral_sub
((intervalIntegral.intervalIntegrable_pow 2).const_mul 2)
((intervalIntegral.intervalIntegrable_pow 1).const_mul 2),
intervalIntegral.integral_const_mul,
integral_pow,
intervalIntegral.integral_const_mul,
integral_pow,
intervalIntegral.integral_const]
norm_num
have hcentered :
(∫ y in Icc (0 : ℝ) 1,
centeredCoordinate y * (sparseWeight a y - sparseWeight a (1 / 2))) =
∫ y in Icc (0 : ℝ) 1, centeredCoordinate y * sparseWeight a y := by
have hfun : (fun y => centeredCoordinate y *
(sparseWeight a y - sparseWeight a (1 / 2))) =
fun y => centeredCoordinate y * sparseWeight a y -
sparseWeight a (1 / 2) * centeredCoordinate y := by
funext y
ring
rw [hfun, integral_sub hwint hconst, MeasureTheory.integral_const_mul,
centeredCoordinate_integral, mul_zero, sub_zero]
rw [MeasureTheory.integral_const_mul, hquad, hcentered] at hmono
dsimp only [C] at hmono ⊢
nlinarith
theorem cancellationPrimitive_mem_negUnitInterval Lemma cancellationPrimitive_mem_negUnitInterval in the paper ↗
Convexity of the exponential puts the cancellation primitive in [-1,0] on the unit interval. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma cancellationPrimitive_mem_negUnitInterval {x : ℝ}
(hx : x ∈ Icc (0 : ℝ) 1) :
cancellationPrimitive x ∈ Icc (-1 : ℝ) 0 := by
have hden : 0 < exp 4 - 1 := by linarith [thirteen_lt_exp_four]
have hfrac0 : 0 ≤ (exp (4 * x) - 1) / (exp 4 - 1) := by
apply div_nonneg
· have : 1 ≤ exp (4 * x) := Real.one_le_exp (by nlinarith [hx.1])
linarith
· exact hden.le
have hconv : exp (4 * x) ≤ 1 + x * (exp 4 - 1) := by
rw [mul_comm 4 x]
have h := convexOn_exp.2 (Set.mem_univ (0 : ℝ)) (Set.mem_univ (4 : ℝ))
(sub_nonneg.mpr hx.2) hx.1 (by ring : (1 - x) + x = 1)
calc
exp (x * 4) ≤ 1 - x + x * exp 4 := by simpa [smul_eq_mul] using h
_ = 1 + x * (exp 4 - 1) := by ring
have hfracx : (exp (4 * x) - 1) / (exp 4 - 1) ≤ x := by
rw [div_le_iff₀ hden]
nlinarith
unfold cancellationPrimitive
constructor <;> linarith [hx.1, hx.2]
theorem reflectedCenteredCoordinate_nonconstant_on_unitInterval
Reflection preserves the nonconstancy of the centered coordinate on the unit interval. the stated conclusion follows.
Formal statement
hypotheses
s :
i :
Fin 3
conclusion
Proof (Lean source)
lemma reflectedCenteredCoordinate_nonconstant_on_unitInterval (s : SignVector 3) (i : Fin 3) :
∃ x ∈ Icc (0 : ℝ) 1, ∃ x' ∈ Icc (0 : ℝ) 1,
centeredCoordinate (reflectedCoordinate s i x) ≠
centeredCoordinate (reflectedCoordinate s i x') := by
rcases s.signed i with hi | hi
· refine ⟨0, by norm_num, 1, by norm_num, ?_⟩
norm_num [centeredCoordinate, reflectedCoordinate, reflect, hi,
show (-1 : ℝ) ≠ 1 by norm_num]
· refine ⟨0, by norm_num, 1, by norm_num, ?_⟩
norm_num [centeredCoordinate, reflectedCoordinate, reflect, hi]
The cancellation primitive is strictly negative at the midpoint of the unit interval. the stated conclusion follows.
Formal statement
conclusion
cancellationPrimitive (1 / 2 : ℝ) < 0
Proof (Lean source)
lemma cancellationPrimitive_half_neg : cancellationPrimitive (1 / 2 : ℝ) < 0 := by
have he2 : 1 < exp 2 := Real.one_lt_exp_iff.mpr (by norm_num)
have hfac : exp 4 - 1 = (exp 2 - 1) * (exp 2 + 1) := by
rw [show (4 : ℝ) = 2 + 2 by norm_num, Real.exp_add]
ring
have hne1 : exp 2 - 1 ≠ 0 := ne_of_gt (sub_pos.mpr he2)
have hne2 : exp 2 + 1 ≠ 0 := by positivity
unfold cancellationPrimitive
rw [show (4 : ℝ) * (1 / 2) = 2 by norm_num, hfac]
have hcancel :
(exp 2 - 1) / ((exp 2 - 1) * (exp 2 + 1)) =
1 / (exp 2 + 1) := by
field_simp
rw [hcancel]
have hhalf : 1 / (exp 2 + 1) < (1 / 2 : ℝ) := by
rw [div_lt_div_iff₀ (by positivity : 0 < exp 2 + 1) (by norm_num : (0 : ℝ) < 2)]
nlinarith
linarith
theorem reflectedCancellationPrimitive_nonconstant_on_unitInterval
Reflection preserves the nonconstancy of the cancellation coefficient on the unit interval. the stated conclusion follows.
Formal statement
hypotheses
s :
i :
Fin 3
conclusion
Proof (Lean source)
lemma reflectedCancellationPrimitive_nonconstant_on_unitInterval
(s : SignVector 3) (i : Fin 3) :
∃ x ∈ Icc (0 : ℝ) 1, ∃ x' ∈ Icc (0 : ℝ) 1,
cancellationPrimitive (reflectedCoordinate s i x) ≠
cancellationPrimitive (reflectedCoordinate s i x') := by
refine ⟨1 / 2, by norm_num, ?_⟩
rcases s.signed i with hi | hi
· refine ⟨1, by norm_num, ?_⟩
simp only [reflectedCoordinate, reflect, hi,
if_neg (show (-1 : ℝ) ≠ 1 by norm_num)]
norm_num [cancellationPrimitive_zero.1]
exact ne_of_lt cancellationPrimitive_half_neg
· refine ⟨0, by norm_num, ?_⟩
simp [reflectedCoordinate, reflect, hi, cancellationPrimitive_zero.1]
simpa only [one_div] using ne_of_lt cancellationPrimitive_half_neg
theorem reflectedCoefficient_child_nonconstant
A nonconstant reflected parent coefficient makes the affine child conditional genuinely depend on that parent somewhere on the unit square. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma reflectedCoefficient_child_nonconstant (s : SignVector 3) (K : ℝ → ℝ)
(hK : ∃ x ∈ Icc (0 : ℝ) 1, ∃ x' ∈ Icc (0 : ℝ) 1,
K (reflectedCoordinate s 0 x) ≠ K (reflectedCoordinate s 0 x')) :
∃ x ∈ Icc (0 : ℝ) 1, ∃ x' ∈ Icc (0 : ℝ) 1,
∃ y ∈ Icc (0 : ℝ) 1,
1 + (1 / 10 : ℝ) * K (reflectedCoordinate s 0 x) *
centeredCoordinate (reflectedCoordinate s 1 y) ≠
1 + (1 / 10 : ℝ) * K (reflectedCoordinate s 0 x') *
centeredCoordinate (reflectedCoordinate s 1 y) := by
rcases hK with ⟨x, hx, x', hx', hxx'⟩
refine ⟨x, hx, x', hx', 0, by norm_num, ?_⟩
rcases s.signed 1 with hi | hi
· simp only [centeredCoordinate, reflectedCoordinate, reflect, hi,
if_neg (show (-1 : ℝ) ≠ 1 by norm_num)]
norm_num
intro heq
apply hxx'
simpa [reflectedCoordinate, reflect] using heq
· simp only [centeredCoordinate, reflectedCoordinate, reflect, hi]
norm_num
intro heq
apply hxx'
simpa [reflectedCoordinate, reflect] using heq
theorem explicitWitness_child_parent_dependence
Both explicit witness child conditionals genuinely depend on the reflected parent coordinate on the unit square. the stated conclusion follows.
Formal statement
hypotheses
s :
conclusion
conclusion 1
∃ x ∈ Icc (0 : ℝ) 1,
∃ x' ∈ Icc (0 : ℝ) 1,
∃ y ∈ Icc (0 : ℝ) 1,
1
+ (1 / 10 : ℝ) * centeredCoordinate (reflectedCoordinate s 0 x) * centeredCoordinate (reflectedCoordinate s 1 y)
≠ 1
+ (1 / 10 : ℝ) * centeredCoordinate (reflectedCoordinate s 0 x') * centeredCoordinate (reflectedCoordinate s 1 y)
conclusion 2
∃ x ∈ Icc (0 : ℝ) 1,
∃ x' ∈ Icc (0 : ℝ) 1,
∃ y ∈ Icc (0 : ℝ) 1,
1
+ (1 / 10 : ℝ) * cancellationPrimitive (reflectedCoordinate s 0 x) * centeredCoordinate (reflectedCoordinate s 1 y)
≠ 1
+ (1 / 10 : ℝ) * cancellationPrimitive (reflectedCoordinate s 0 x') * centeredCoordinate (reflectedCoordinate s 1 y)
Proof (Lean source)
lemma explicitWitness_child_parent_dependence (s : SignVector 3) :
(∃ x ∈ Icc (0 : ℝ) 1, ∃ x' ∈ Icc (0 : ℝ) 1,
∃ y ∈ Icc (0 : ℝ) 1,
1 + (1 / 10 : ℝ) * centeredCoordinate (reflectedCoordinate s 0 x) *
centeredCoordinate (reflectedCoordinate s 1 y) ≠
1 + (1 / 10 : ℝ) * centeredCoordinate (reflectedCoordinate s 0 x') *
centeredCoordinate (reflectedCoordinate s 1 y)) ∧
(∃ x ∈ Icc (0 : ℝ) 1, ∃ x' ∈ Icc (0 : ℝ) 1,
∃ y ∈ Icc (0 : ℝ) 1,
1 + (1 / 10 : ℝ) * cancellationPrimitive (reflectedCoordinate s 0 x) *
centeredCoordinate (reflectedCoordinate s 1 y) ≠
1 + (1 / 10 : ℝ) * cancellationPrimitive (reflectedCoordinate s 0 x') *
centeredCoordinate (reflectedCoordinate s 1 y)) := by
exact ⟨reflectedCoefficient_child_nonconstant s centeredCoordinate
(reflectedCenteredCoordinate_nonconstant_on_unitInterval s 0),
reflectedCoefficient_child_nonconstant s cancellationPrimitive
(reflectedCancellationPrimitive_nonconstant_on_unitInterval s 0)⟩
theorem sparseP_bounds
The sparse child conditional lies between 9/10 and 11/10 on the latent cube. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
Proof (Lean source)
lemma sparseP_bounds (s : SignVector 3) (v : LatentState 3) (hv : v ∈ latentCube 3) :
(9 / 10 : ℝ) ≤ sparseP s 1 v ∧ sparseP s 1 v ≤ (11 / 10 : ℝ) := by
have hv0 : v 0 ∈ Icc (0 : ℝ) 1 := hv 0 (Set.mem_univ 0)
have hv1 : v 1 ∈ Icc (0 : ℝ) 1 := hv 1 (Set.mem_univ 1)
have h0 := abs_centeredCoordinate_le_one
(reflectedCoordinate_mem_unitInterval s 0 hv0)
have h1 := abs_centeredCoordinate_le_one
(reflectedCoordinate_mem_unitInterval s 1 hv1)
rw [abs_le] at h0 h1
simp only [sparseP, if_true]
constructor <;> nlinarith
theorem sparseWitness_child_ratio_bounds
On the latent cube the sparse child ratio is strictly between zero and five. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
conclusion 1
0 < (sparseWitness s).q 1 (v 1) / (sparseWitness s).p 1 v
conclusion 2
(sparseWitness s).q 1 (v 1) / (sparseWitness s).p 1 v < 5
Proof (Lean source)
lemma sparseWitness_child_ratio_bounds (s : SignVector 3) (v : LatentState 3)
(hv : v ∈ latentCube 3) :
0 < (sparseWitness s).q 1 (v 1) / (sparseWitness s).p 1 v ∧
(sparseWitness s).q 1 (v 1) / (sparseWitness s).p 1 v < 5 := by
have hv1 : v 1 ∈ Icc (0 : ℝ) 1 := hv 1 (Set.mem_univ 1)
have href : reflectedCoordinate s 1 (v 1) ∈ Icc (0 : ℝ) 1 :=
reflectedCoordinate_mem_unitInterval s 1 hv1
have hqpos : 0 < (sparseWitness s).q 1 (v 1) := by
exact exponentialInterventionDensity_pos _
have hq : (sparseWitness s).q 1 (v 1) < (13 / 3 : ℝ) := by
exact exponentialInterventionDensity_lt_thirteen_div_three href
have hp := sparseP_bounds s v hv
change (9 / 10 : ℝ) ≤ (sparseWitness s).p 1 v ∧
(sparseWitness s).p 1 v ≤ (11 / 10 : ℝ) at hp
have hppos : 0 < (sparseWitness s).p 1 v := lt_of_lt_of_le (by norm_num) hp.1
constructor
· positivity
· rw [div_lt_iff₀ hppos]
nlinarith
The rational lower bound for the exponential-tilt mean excess used in the sparse moment calculation. the stated conclusion follows.
Formal statement
conclusion
(29 / 108 : ℝ) < 1 / (1 - exp (-4)) - 3 / 4
Proof (Lean source)
lemma exponentialTiltMeanGap_gt :
(29 / 108 : ℝ) < 1 / (1 - exp (-4)) - 3 / 4 := by
have hEpos : 0 < exp 4 := Real.exp_pos 4
have hden : 0 < exp 4 - 1 := by linarith [thirteen_lt_exp_four]
rw [Real.exp_neg, inv_eq_one_div]
field_simp
nlinarith [exp_four_lt_fifty_five]
The product of rational lower bounds in the sparse moment argument is the stated exact certificate. the stated conclusion follows.
Formal statement
conclusion
(1 / 30 : ℝ) * (68 / 9) * (4 / 729) * (100 / 121) * (29 / 108) = 19720 / 64304361
Proof (Lean source)
lemma sparseMomentCoefficient_exact :
(1 / 30 : ℝ) * (68 / 9) * (4 / 729) * (100 / 121) * (29 / 108) =
19720 / 64304361 := by
norm_num
The exact rational lower certificate in the sparse moment calculation is strictly larger than 3 / 10000. the stated conclusion follows.
Formal statement
conclusion
(3 / 10000 : ℝ) < 19720 / 64304361
Proof (Lean source)
lemma sparseMomentRationalCertificate :
(3 / 10000 : ℝ) < 19720 / 64304361 := by norm_num
theorem sparseWeight_moment_certificate
The integrated sparse-weight covariance and exponential-tilt mean gap combine to exceed the paper's 3 / 10000 moment threshold. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
a :
ℝ
ha :
a ∈ Icc (-1 / 10 : ℝ) (1 / 10)
conclusion
(3 / 10000 : ℝ) < (1 / 5) * (∫ y in Icc (0 : ℝ) 1, centeredCoordinate y * sparseWeight a y) * (1 / (1 - exp (-4)) - 3 / 4)
Proof (Lean source)
lemma sparseWeight_moment_certificate {a : ℝ}
(ha : a ∈ Icc (-1 / 10 : ℝ) (1 / 10)) :
(3 / 10000 : ℝ) <
(1 / 5) * (∫ y in Icc (0 : ℝ) 1,
centeredCoordinate y * sparseWeight a y) *
(1 / (1 - exp (-4)) - 3 / 4) := by
have hI := sparseWeight_centered_integral_lower_bound ha
have hgap := exponentialTiltMeanGap_gt
have hcert := sparseMomentRationalCertificate
have hexact := sparseMomentCoefficient_exact
have hC : 0 < (68 / 9 : ℝ) * (4 / 729) * (100 / 121) := by norm_num
have hgapPos : 0 < 1 / (1 - exp (-4)) - 3 / 4 := by
linarith
have hIPos : 0 < ∫ y in Icc (0 : ℝ) 1,
centeredCoordinate y * sparseWeight a y := by
nlinarith
nlinarith
After the coordinate and tail estimates, the remaining numerical MMD comparison is strictly larger than 5 / 10^8. the stated conclusion follows.
Formal statement
conclusion
(5 / 100000000 : ℝ) < ((3 / 10000 : ℝ) - 1 / 10 ^ 15) / 5151
Proof (Lean source)
lemma sparseMmdRationalCertificate :
(5 / 100000000 : ℝ) < ((3 / 10000 : ℝ) - 1 / 10 ^ 15) / 5151 := by
norm_num
The geometric majorant for the factorial tail beyond degree one hundred is below 10⁻¹⁵. the stated conclusion follows.
Formal statement
conclusion
(50 : ℝ) * (25 ^ 101 / ((101 : ℕ).factorial : ℝ)) * (102 / 77) < 1 / 10 ^ 15
Proof (Lean source)
lemma sparseMmdTailRationalCertificate :
(50 : ℝ) * (25 ^ 101 / ((101 : ℕ).factorial : ℝ)) * (102 / 77) <
1 / 10 ^ 15 := by
norm_num [factorial]
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.WitnessSigns 6 declarations This file proves the elementary logarithmic-derivative bounds that give the sparse witness its prescribed own-coordinate signs after reflection.
Derivative signs for the explicit witnesses
This file proves the elementary logarithmic-derivative bounds that give the sparse witness its prescribed own-coordinate signs after reflection.
theorem exponentialInterventionDensity_hasDerivAt Lemma exponentialInterventionDensity_hasDerivAt in the paper ↗
The normalized exponential intervention density has logarithmic slope four. the stated conclusion follows.
Formal statement
hypotheses
x :
ℝ
conclusion
Proof (Lean source)
lemma exponentialInterventionDensity_hasDerivAt (x : ℝ) :
HasDerivAt exponentialInterventionDensity (4 * exponentialInterventionDensity x) x := by
have hexp : HasDerivAt (fun y : ℝ => exp (4 * y))
(4 * exp (4 * x)) x := by
simpa [Function.comp_def, mul_comm] using
(Real.hasDerivAt_exp (4 * x)).comp x (hasDerivAt_const_mul (x := x) 4)
unfold exponentialInterventionDensity
convert (hexp.const_mul 4).div_const (exp 4 - 1) using 1 <;>
first | rfl | ring
The unreflected child log ratio has slope 4 - K / (5p). Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
K x :
ℝ
hden :
1 + (1 / 10 : ℝ) * K * centeredCoordinate x ≠ 0
conclusion
HasDerivAt (fun z : ℝ => log (exponentialInterventionDensity z / (1 + (1/10 : ℝ) * K * centeredCoordinate z))) (4 - (1/5 : ℝ) * K / (1 + (1/10 : ℝ) * K * centeredCoordinate x)) x
Proof (Lean source)
lemma exponentialLogRatio_hasDerivAt (K x : ℝ)
(hden : 1 + (1 / 10 : ℝ) * K * centeredCoordinate x ≠ 0) :
HasDerivAt (fun z : ℝ => log
(exponentialInterventionDensity z /
(1 + (1/10 : ℝ) * K * centeredCoordinate z)))
(4 - (1/5 : ℝ) * K /
(1 + (1/10 : ℝ) * K * centeredCoordinate x)) x := by
have hq := exponentialInterventionDensity_hasDerivAt x
have hd : HasDerivAt (fun z : ℝ => 1 + (1/10 : ℝ) * K * centeredCoordinate z)
((1/5 : ℝ) * K) x := by
unfold centeredCoordinate
convert ((((hasDerivAt_id x).const_mul 2).sub_const 1).const_mul
((1/10 : ℝ) * K)).const_add 1 using 1 <;> first | rfl | ring
have hquot := hq.div hd hden
have hqpos := exponentialInterventionDensity_pos x
have hh := hquot.log (div_ne_zero (ne_of_gt hqpos) hden)
simp only [Pi.div_apply] at hh
change HasDerivAt (fun z : ℝ => log
(exponentialInterventionDensity z /
(1 + (1 / 10 : ℝ) * K * centeredCoordinate z))) _ x at hh
convert hh using 1
have hden' : 10 + K * centeredCoordinate x ≠ 0 := by
intro h
apply hden
nlinarith
field_simp [hden', ne_of_gt hqpos]
theorem reflectedExponentialLogRatio_hasDerivAt Lemma reflectedExponentialLogRatio_hasDerivAt in the paper ↗
Reflection reverses the child log-ratio derivative. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
K x :
ℝ
hden :
1 + (1 / 10 : ℝ) * K * centeredCoordinate (1 - x) ≠ 0
conclusion
HasDerivAt (fun z : ℝ => log (exponentialInterventionDensity (1-z) / (1 + (1/10 : ℝ) * K * centeredCoordinate (1-z)))) (-4 + (1/5 : ℝ) * K / (1 + (1/10 : ℝ) * K * centeredCoordinate (1-x))) x
Proof (Lean source)
lemma reflectedExponentialLogRatio_hasDerivAt (K x : ℝ)
(hden : 1 + (1 / 10 : ℝ) * K * centeredCoordinate (1 - x) ≠ 0) :
HasDerivAt (fun z : ℝ => log
(exponentialInterventionDensity (1-z) /
(1 + (1/10 : ℝ) * K * centeredCoordinate (1-z))))
(-4 + (1/5 : ℝ) * K /
(1 + (1/10 : ℝ) * K * centeredCoordinate (1-x))) x := by
have hh := (exponentialLogRatio_hasDerivAt K (1 - x) hden).comp x
((hasDerivAt_const (x := x) (1 : ℝ)).sub (hasDerivAt_id x))
convert hh using 1 <;> first | rfl | ring
theorem sparseWitness_fixedOwnDerivativeSign
The reflected sparse witness has the prescribed strict own-coordinate derivative signs. the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma sparseWitness_fixedOwnDerivativeSign (s : SignVector 3) :
FixedOwnDerivativeSign threeNodeDAG s (sparseWitness s) := by
intro i v hv
unfold ownLogRatioDerivative
have hud : UniqueDiffWithinAt ℝ (Icc (0 : ℝ) 1) (v i) :=
(uniqueDiffOn_Icc (by norm_num)).uniqueDiffWithinAt (hv i (Set.mem_univ i))
by_cases hi : i = 1
· subst i
have hv0 : v 0 ∈ Icc (0 : ℝ) 1 := hv 0 (Set.mem_univ 0)
have hK := abs_centeredCoordinate_le_one
(reflectedCoordinate_mem_unitInterval s 0 hv0)
rw [abs_le] at hK
rcases s.signed 1 with hs | hs
· rw [hs]
have href (z : ℝ) : reflectedCoordinate s 1 z = 1-z := by
simp [reflectedCoordinate, reflect, hs, show (-1 : ℝ) ≠ 1 by norm_num]
have hup (z : ℝ) : reflectedCoordinate s 0 ((update v 1 z) 0) =
reflectedCoordinate s 0 (v 0) := by simp [update]
simp only [sparseWitness, sparseQ, sparseP, if_true]
simp_rw [href, hup]
simp only [Function.update_self]
change 0 < (-1 : ℝ) * derivWithin (fun z : ℝ => log
(exponentialInterventionDensity (1-z) /
(1 + (1/10 : ℝ) * centeredCoordinate (reflectedCoordinate s 0 (v 0)) *
centeredCoordinate (1-z)))) (Icc 0 1) (v 1)
have hp := sparseP_pos s 1 v hv
have hden : 1 + (1 / 10 : ℝ) *
centeredCoordinate (reflectedCoordinate s 0 (v 0)) *
centeredCoordinate (1 - v 1) ≠ 0 := by
simpa [sparseP, reflectedCoordinate, reflect, hs,
show (-1 : ℝ) ≠ 1 by norm_num] using ne_of_gt hp
rw [(reflectedExponentialLogRatio_hasDerivAt _ _ hden).hasDerivWithinAt.derivWithin
hud]
have hp' : 0 < 1 + (1 / 10 : ℝ) * centeredCoordinate (reflectedCoordinate s 0 (v 0)) *
centeredCoordinate (1 - v 1) := by
have hv1 : 0 ≤ v 1 ∧ v 1 ≤ 1 := hv 1 (Set.mem_univ 1)
have h1 := abs_centeredCoordinate_le_one
(show 1 - v 1 ∈ Icc (0 : ℝ) 1 by constructor <;> linarith)
rw [abs_le] at h1
nlinarith
have hv1 : 0 ≤ v 1 ∧ v 1 ≤ 1 := hv 1 (Set.mem_univ 1)
have h1 := abs_centeredCoordinate_le_one
(show 1 - v 1 ∈ Icc (0 : ℝ) 1 by constructor <;> linarith)
rw [abs_le] at h1
have hden_lb : (9 / 10 : ℝ) ≤ 1 + (1 / 10 : ℝ) *
centeredCoordinate (reflectedCoordinate s 0 (v 0)) *
centeredCoordinate (1 - v 1) := by
nlinarith
have hfrac : centeredCoordinate (reflectedCoordinate s 0 (v 0)) /
(1 + (1 / 10 : ℝ) * centeredCoordinate
(reflectedCoordinate s 0 (v 0)) * centeredCoordinate (1 - v 1)) ≤ 10 / 9 :=
(div_le_iff₀ hp').2 (by nlinarith [hden_lb])
norm_num at hfrac ⊢
have hsmall : (1 / 5 : ℝ) *
(centeredCoordinate (reflectedCoordinate s 0 (v 0)) /
(1 + (1 / 10 : ℝ) * centeredCoordinate (reflectedCoordinate s 0 (v 0)) *
centeredCoordinate (1 - v 1))) ≤ 2 / 9 :=
calc
_ ≤ (1 / 5 : ℝ) * (10 / 9) :=
mul_le_mul_of_nonneg_left hfrac (by norm_num)
_ = 2 / 9 := by norm_num
norm_num at hsmall
rw [mul_div_assoc]
linarith
· rw [hs]
have href (z : ℝ) : reflectedCoordinate s 1 z = z := by
simp [reflectedCoordinate, reflect, hs]
have hup (z : ℝ) : reflectedCoordinate s 0 ((update v 1 z) 0) =
reflectedCoordinate s 0 (v 0) := by simp [update]
simp only [sparseWitness, sparseQ, sparseP, if_true]
simp_rw [href, hup]
simp only [Function.update_self]
change 0 < (1 : ℝ) * derivWithin (fun z : ℝ => log
(exponentialInterventionDensity z /
(1 + (1/10 : ℝ) * centeredCoordinate (reflectedCoordinate s 0 (v 0)) *
centeredCoordinate z))) (Icc 0 1) (v 1)
have hp := sparseP_pos s 1 v hv
have hden : 1 + (1 / 10 : ℝ) *
centeredCoordinate (reflectedCoordinate s 0 (v 0)) *
centeredCoordinate (v 1) ≠ 0 := by
simpa [sparseP, reflectedCoordinate, reflect, hs] using ne_of_gt hp
rw [(exponentialLogRatio_hasDerivAt _ _ hden).hasDerivWithinAt.derivWithin hud]
have hv1 : 0 ≤ v 1 ∧ v 1 ≤ 1 := hv 1 (Set.mem_univ 1)
have h1 := abs_centeredCoordinate_le_one hv1
rw [abs_le] at h1
have hp' : 0 < 1 + (1 / 10 : ℝ) *
centeredCoordinate (reflectedCoordinate s 0 (v 0)) *
centeredCoordinate (v 1) := by
nlinarith
have hden_lb : (9 / 10 : ℝ) ≤ 1 + (1 / 10 : ℝ) *
centeredCoordinate (reflectedCoordinate s 0 (v 0)) *
centeredCoordinate (v 1) := by
nlinarith
have hfrac : centeredCoordinate (reflectedCoordinate s 0 (v 0)) /
(1 + (1 / 10 : ℝ) * centeredCoordinate
(reflectedCoordinate s 0 (v 0)) * centeredCoordinate (v 1)) ≤ 10 / 9 :=
(div_le_iff₀ hp').2 (by nlinarith [hden_lb])
norm_num at hfrac ⊢
have hsmall : (1 / 5 : ℝ) *
(centeredCoordinate (reflectedCoordinate s 0 (v 0)) /
(1 + (1 / 10 : ℝ) * centeredCoordinate (reflectedCoordinate s 0 (v 0)) *
centeredCoordinate (v 1))) ≤ 2 / 9 :=
calc
_ ≤ (1 / 5 : ℝ) * (10 / 9) :=
mul_le_mul_of_nonneg_left hfrac (by norm_num)
_ = 2 / 9 := by norm_num
norm_num at hsmall
rw [mul_div_assoc]
linarith
· rcases s.signed i with hs | hs
· rw [hs]
have href (z : ℝ) : reflectedCoordinate s i z = 1 - z := by
simp [reflectedCoordinate, reflect, hs, show (-1 : ℝ) ≠ 1 by norm_num]
simp only [sparseWitness, sparseQ, sparseP, hi]
simp_rw [href]
simp only [if_false, div_one]
have hd := reflectedExponentialLogRatio_hasDerivAt (0 : ℝ) (v i) (by norm_num)
have hd' : HasDerivAt (fun z : ℝ => log
(exponentialInterventionDensity (1 - z))) (-4) (v i) := by
simpa using hd
rw [hd'.hasDerivWithinAt.derivWithin hud]
norm_num
· rw [hs]
have href (z : ℝ) : reflectedCoordinate s i z = z := by
simp [reflectedCoordinate, reflect, hs]
simp only [sparseWitness, sparseQ, sparseP, hi]
simp_rw [href]
simp only [if_false, div_one]
have hd := exponentialLogRatio_hasDerivAt (0 : ℝ) (v i) (by norm_num)
have hd' : HasDerivAt (fun z : ℝ => log
(exponentialInterventionDensity z)) 4 (v i) := by
simpa using hd
rw [hd'.hasDerivWithinAt.derivWithin hud]
norm_num
theorem cancellationWitness_fixedOwnDerivativeSign
The reflected cancellation witness has the prescribed strict own-coordinate derivative signs. the stated conclusion follows.
Formal statement
hypotheses
s :
conclusion
Proof (Lean source)
lemma cancellationWitness_fixedOwnDerivativeSign (s : SignVector 3) :
FixedOwnDerivativeSign threeNodeDAG s (cancellationWitness s) := by
intro i v hv
unfold ownLogRatioDerivative
have hud : UniqueDiffWithinAt ℝ (Icc (0 : ℝ) 1) (v i) :=
(uniqueDiffOn_Icc (by norm_num)).uniqueDiffWithinAt (hv i (Set.mem_univ i))
by_cases hi : i = 1
· subst i
have hv0 : v 0 ∈ Icc (0 : ℝ) 1 := hv 0 (Set.mem_univ 0)
have hK := cancellationPrimitive_mem_unitInterval_sub
(reflectedCoordinate_mem_unitInterval s 0 hv0)
rcases s.signed 1 with hs | hs
· rw [hs]
have href (z : ℝ) : reflectedCoordinate s 1 z = 1-z := by
simp [reflectedCoordinate, reflect, hs, show (-1 : ℝ) ≠ 1 by norm_num]
have hup (z : ℝ) : reflectedCoordinate s 0 ((update v 1 z) 0) =
reflectedCoordinate s 0 (v 0) := by simp [update]
simp only [cancellationWitness, sparseQ, cancellationP, if_true]
simp_rw [href, hup]
simp only [Function.update_self]
change 0 < (-1 : ℝ) * derivWithin (fun z : ℝ => log
(exponentialInterventionDensity (1-z) /
(1 + (1/10 : ℝ) * cancellationPrimitive (reflectedCoordinate s 0 (v 0)) *
centeredCoordinate (1-z)))) (Icc 0 1) (v 1)
have hp := cancellationP_pos s 1 v hv
have hden : 1 + (1 / 10 : ℝ) *
cancellationPrimitive (reflectedCoordinate s 0 (v 0)) *
centeredCoordinate (1 - v 1) ≠ 0 := by
simpa [cancellationP, reflectedCoordinate, reflect, hs,
show (-1 : ℝ) ≠ 1 by norm_num] using ne_of_gt hp
rw [(reflectedExponentialLogRatio_hasDerivAt _ _ hden).hasDerivWithinAt.derivWithin
hud]
have hv1 : 0 ≤ v 1 ∧ v 1 ≤ 1 := hv 1 (Set.mem_univ 1)
have h1 := abs_centeredCoordinate_le_one
(show 1 - v 1 ∈ Icc (0 : ℝ) 1 by constructor <;> linarith)
rw [abs_le] at h1
have hp' : 0 < 1 + (1 / 10 : ℝ) *
cancellationPrimitive (reflectedCoordinate s 0 (v 0)) *
centeredCoordinate (1 - v 1) := by
nlinarith [hK.1, hK.2]
have hden_lb : (9 / 10 : ℝ) ≤ 1 + (1 / 10 : ℝ) *
cancellationPrimitive (reflectedCoordinate s 0 (v 0)) *
centeredCoordinate (1 - v 1) := by
nlinarith [hK.1, hK.2]
have hfrac : cancellationPrimitive (reflectedCoordinate s 0 (v 0)) /
(1 + (1 / 10 : ℝ) * cancellationPrimitive
(reflectedCoordinate s 0 (v 0)) * centeredCoordinate (1 - v 1)) ≤ 10 / 9 :=
(div_le_iff₀ hp').2 (by nlinarith [hK.2, hden_lb])
norm_num at hfrac ⊢
have hsmall : (1 / 5 : ℝ) *
(cancellationPrimitive (reflectedCoordinate s 0 (v 0)) /
(1 + (1 / 10 : ℝ) * cancellationPrimitive (reflectedCoordinate s 0 (v 0)) *
centeredCoordinate (1 - v 1))) ≤ 2 / 9 :=
calc
_ ≤ (1 / 5 : ℝ) * (10 / 9) :=
mul_le_mul_of_nonneg_left hfrac (by norm_num)
_ = 2 / 9 := by norm_num
norm_num at hsmall
rw [mul_div_assoc]
linarith
· rw [hs]
have href (z : ℝ) : reflectedCoordinate s 1 z = z := by
simp [reflectedCoordinate, reflect, hs]
have hup (z : ℝ) : reflectedCoordinate s 0 ((update v 1 z) 0) =
reflectedCoordinate s 0 (v 0) := by simp [update]
simp only [cancellationWitness, sparseQ, cancellationP, if_true]
simp_rw [href, hup]
simp only [Function.update_self]
change 0 < (1 : ℝ) * derivWithin (fun z : ℝ => log
(exponentialInterventionDensity z /
(1 + (1/10 : ℝ) * cancellationPrimitive (reflectedCoordinate s 0 (v 0)) *
centeredCoordinate z))) (Icc 0 1) (v 1)
have hp := cancellationP_pos s 1 v hv
have hden : 1 + (1 / 10 : ℝ) *
cancellationPrimitive (reflectedCoordinate s 0 (v 0)) *
centeredCoordinate (v 1) ≠ 0 := by
simpa [cancellationP, reflectedCoordinate, reflect, hs] using ne_of_gt hp
rw [(exponentialLogRatio_hasDerivAt _ _ hden).hasDerivWithinAt.derivWithin hud]
have hv1 : 0 ≤ v 1 ∧ v 1 ≤ 1 := hv 1 (Set.mem_univ 1)
have h1 := abs_centeredCoordinate_le_one hv1
rw [abs_le] at h1
have hp' : 0 < 1 + (1 / 10 : ℝ) *
cancellationPrimitive (reflectedCoordinate s 0 (v 0)) *
centeredCoordinate (v 1) := by
nlinarith [hK.1, hK.2]
have hden_lb : (9 / 10 : ℝ) ≤ 1 + (1 / 10 : ℝ) *
cancellationPrimitive (reflectedCoordinate s 0 (v 0)) *
centeredCoordinate (v 1) := by
nlinarith [hK.1, hK.2]
have hfrac : cancellationPrimitive (reflectedCoordinate s 0 (v 0)) /
(1 + (1 / 10 : ℝ) * cancellationPrimitive
(reflectedCoordinate s 0 (v 0)) * centeredCoordinate (v 1)) ≤ 10 / 9 :=
(div_le_iff₀ hp').2 (by nlinarith [hK.2, hden_lb])
norm_num at hfrac ⊢
have hsmall : (1 / 5 : ℝ) *
(cancellationPrimitive (reflectedCoordinate s 0 (v 0)) /
(1 + (1 / 10 : ℝ) * cancellationPrimitive (reflectedCoordinate s 0 (v 0)) *
centeredCoordinate (v 1))) ≤ 2 / 9 :=
calc
_ ≤ (1 / 5 : ℝ) * (10 / 9) :=
mul_le_mul_of_nonneg_left hfrac (by norm_num)
_ = 2 / 9 := by norm_num
norm_num at hsmall
rw [mul_div_assoc]
linarith
· rcases s.signed i with hs | hs
· rw [hs]
have href (z : ℝ) : reflectedCoordinate s i z = 1 - z := by
simp [reflectedCoordinate, reflect, hs, show (-1 : ℝ) ≠ 1 by norm_num]
simp only [cancellationWitness, sparseQ, cancellationP, hi]
simp_rw [href]
simp only [if_false, div_one]
have hd := reflectedExponentialLogRatio_hasDerivAt (0 : ℝ) (v i) (by norm_num)
have hd' : HasDerivAt (fun z : ℝ => log
(exponentialInterventionDensity (1 - z))) (-4) (v i) := by
simpa using hd
rw [hd'.hasDerivWithinAt.derivWithin hud]
norm_num
· rw [hs]
have href (z : ℝ) : reflectedCoordinate s i z = z := by
simp [reflectedCoordinate, reflect, hs]
simp only [cancellationWitness, sparseQ, cancellationP, hi]
simp_rw [href]
simp only [if_false, div_one]
have hd := exponentialLogRatio_hasDerivAt (0 : ℝ) (v i) (by norm_num)
have hd' : HasDerivAt (fun z : ℝ => log
(exponentialInterventionDensity z)) 4 (v i) := by
simpa using hd
rw [hd'.hasDerivWithinAt.derivWithin hud]
norm_num
theorem explicitWitness_regularities
Both explicit witnesses are positive, normalized, smooth to every finite order, and obey their prescribed own-coordinate derivative signs. the stated conclusion follows.
Formal statement
hypotheses
s :
conclusion
conclusion 1
conclusion 2
conclusion 5
conclusion 6
Proof (Lean source)
lemma explicitWitness_regularities (s : SignVector 3) :
PositiveNormalizedSmoothMechanisms threeNodeDAG (sparseWitness s) ∧
FixedOwnDerivativeSign threeNodeDAG s (sparseWitness s) ∧
(∀ k i, ContDiffOn ℝ k ((sparseWitness s).p i) (latentCube 3)) ∧
(∀ k i, ContDiffOn ℝ k ((sparseWitness s).q i) (Icc (0 : ℝ) 1)) ∧
PositiveNormalizedSmoothMechanisms threeNodeDAG (cancellationWitness s) ∧
FixedOwnDerivativeSign threeNodeDAG s (cancellationWitness s) ∧
(∀ k i, ContDiffOn ℝ k ((cancellationWitness s).p i) (latentCube 3)) ∧
(∀ k i, ContDiffOn ℝ k ((cancellationWitness s).q i) (Icc (0 : ℝ) 1)) := by
rcases sparseWitness_contDiff_all s with ⟨hsp, hsq⟩
rcases cancellationWitness_contDiff_all s with ⟨hcp, hcq⟩
exact ⟨sparseWitness_positive_normalized_smooth s,
sparseWitness_fixedOwnDerivativeSign s, hsp, hsq,
cancellationWitness_positive_normalized_smooth s,
cancellationWitness_fixedOwnDerivativeSign s, hcp, hcq⟩
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.WitnessTransport 4 declarations This file records the distribution-function and first-moment calculations used to transport the sparse witness's derivative bound along the exponential tilt.
Exponential-tilt transport identities
This file records the distribution-function and first-moment calculations used to transport the sparse witness's derivative bound along the exponential tilt.
theorem exponentialInterventionDensity_intervalIntegral Lemma exponentialInterventionDensity_intervalIntegral in the paper ↗
The exponential intervention density has its stated closed-form distribution function. the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma exponentialInterventionDensity_intervalIntegral (x : ℝ) :
(∫ z in (0 : ℝ)..x, exponentialInterventionDensity z) =
(exp (4 * x) - 1) / (exp 4 - 1) := by
have hden : exp 4 - 1 ≠ 0 := by
linarith [thirteen_lt_exp_four]
have hd (z : ℝ) : HasDerivAt
(fun u : ℝ => (exp (4 * u) - 1) / (exp 4 - 1))
(exponentialInterventionDensity z) z := by
have he : HasDerivAt (fun u : ℝ => exp (4 * u))
(4 * exp (4 * z)) z := by
simpa [Function.comp_def, mul_comm] using
(Real.hasDerivAt_exp (4 * z)).comp z (hasDerivAt_const_mul (x := z) 4)
simpa [exponentialInterventionDensity] using
(he.sub_const 1).div_const (exp 4 - 1)
rw [intervalIntegral.integral_eq_sub_of_hasDerivAt (fun z _ => hd z)
(by apply Continuous.intervalIntegrable; unfold exponentialInterventionDensity; fun_prop)]
simp
theorem exponentialInterventionCDF_le_identity Lemma exponentialInterventionCDF_le_identity in the paper ↗
The exponential-tilt distribution function lies below the uniform distribution function. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
x :
ℝ
hx :
x ∈ Icc (0 : ℝ) 1
conclusion
(∫ z in (0 : ℝ)..x, exponentialInterventionDensity z) ≤ x
Proof (Lean source)
lemma exponentialInterventionCDF_le_identity {x : ℝ} (hx : x ∈ Icc (0 : ℝ) 1) :
(∫ z in (0 : ℝ)..x, exponentialInterventionDensity z) ≤ x := by
rw [exponentialInterventionDensity_intervalIntegral]
have hH := (cancellationPrimitive_mem_negUnitInterval hx).2
unfold cancellationPrimitive at hH
linarith
theorem exponentialInterventionDensity_firstMoment Lemma exponentialInterventionDensity_firstMoment in the paper ↗
The exponential intervention density has the exact first moment used in the sparse gap. the stated conclusion follows.
Formal statement
conclusion
(∫ z in Icc (0 : ℝ) 1, z * exponentialInterventionDensity z)
= 1 / (1 - exp (-4)) - 1 / 4
Proof (Lean source)
lemma exponentialInterventionDensity_firstMoment :
(∫ z in Icc (0 : ℝ) 1, z * exponentialInterventionDensity z) =
1 / (1 - exp (-4)) - 1 / 4 := by
have hden : exp 4 - 1 ≠ 0 := by
linarith [thirteen_lt_exp_four]
let F : ℝ → ℝ := fun z => exp (4 * z) * (z - 1 / 4) / (exp 4 - 1)
have hd (z : ℝ) : HasDerivAt F (z * exponentialInterventionDensity z) z := by
have he : HasDerivAt (fun u : ℝ => exp (4 * u))
(4 * exp (4 * z)) z := by
simpa [Function.comp_def, mul_comm] using
(Real.hasDerivAt_exp (4 * z)).comp z (hasDerivAt_const_mul (x := z) 4)
have hz : HasDerivAt (fun u : ℝ => u - 1 / 4) 1 z :=
(hasDerivAt_id z).sub_const (1 / 4)
have hraw : HasDerivAt F
((4 * exp (4 * z) * (z - 1 / 4) + exp (4 * z)) /
(exp 4 - 1)) z := by
simpa only [F, Pi.mul_apply, mul_one] using
(he.mul hz).div_const (exp 4 - 1)
apply hraw.congr_deriv
unfold exponentialInterventionDensity
field_simp
ring
rw [MeasureTheory.integral_Icc_eq_integral_Ioc]
rw [← intervalIntegral.integral_of_le (by norm_num : (0 : ℝ) ≤ 1)]
rw [intervalIntegral.integral_eq_sub_of_hasDerivAt (fun z _ => hd z)
(by apply Continuous.intervalIntegrable; unfold exponentialInterventionDensity; fun_prop)]
dsimp only [F]
simp only [mul_one, mul_zero, Real.exp_zero, zero_sub, one_mul]
rw [Real.exp_neg]
field_simp [hden, Real.exp_ne_zero]
ring
theorem exponentialInterventionDensity_centeredMean Lemma exponentialInterventionDensity_centeredMean in the paper ↗
The exponential-tilt mean excess over the uniform mean has its exact closed form. the stated conclusion follows.
Formal statement
conclusion
(∫ z in Icc (0 : ℝ) 1, z * exponentialInterventionDensity z) - 1 / 2
= 1 / (1 - exp (-4)) - 3 / 4
Proof (Lean source)
lemma exponentialInterventionDensity_centeredMean :
(∫ z in Icc (0 : ℝ) 1, z * exponentialInterventionDensity z) - 1 / 2 =
1 / (1 - exp (-4)) - 3 / 4 := by
rw [exponentialInterventionDensity_firstMoment]
ring
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.Helpers.Witnesses 46 declarations This file contains the elementary compact-cube mechanisms used by the genericity and cancellation arguments.
Explicit sparse, cancellation, and affine-path mechanisms
This file contains the elementary compact-cube mechanisms used by the genericity and cancellation arguments.
Reflection of a coordinate when its prescribed sign is negative.
Definition (Lean source)
parameters
σ z :
ℝ
def
reflect σ z :
ℝ
given by
if σ = 1 then z else 1 - z
def centeredCoordinate
The centered affine function 2z-1.
Definition (Lean source)
parameters
z :
ℝ
def
centeredCoordinate z :
ℝ
given by
2 * z - 1
def exponentialInterventionDensity
The normalized exponential intervention density.
def cancellationPrimitive
The cancellation primitive H.
def threeNodeEdge
The three-node graph 0 → 1 with node 2 isolated.
Definition (Lean source)
theorem threeNodeEdge_acyclic
The three-node edge relation has no directed cycle.
Formal statement
Proof (Lean source)
lemma threeNodeEdge_acyclic (v : Fin 3) : ¬ TransGen threeNodeEdge v v := by
intro h
have path_shape : ∀ {a b : Fin 3}, TransGen threeNodeEdge a b →
a = 0 ∧ b = 1 := by
intro a b hab
induction hab with
| single hab => exact hab
| tail hab hbc ih => exact ⟨ih.1, hbc.2⟩
rcases path_shape h with ⟨hv0, hv1⟩
simp_all
def threeNodeDAG
The fixed three-node witness DAG.
Definition (Lean source)
given by
clause 1
clause 2
decEdge := fun i j => by unfold threeNodeEdge; infer_instance
clause 3
acyclic := threeNodeEdge_acyclic
def reflectedCoordinate
Reflected coordinate selected by a sign vector.
Definition (Lean source)
def sparseP
Sparse witness observational conditional mechanism.
Definition (Lean source)
parameters
def
sparseP s i v :
ℝ
given by
if i
= 1 then 1
+ (1 / 10 : ℝ) * centeredCoordinate (reflectedCoordinate s 0 (v 0)) * centeredCoordinate (reflectedCoordinate s 1 (v 1)) else 1
def sparseQ
Sparse witness intervention mechanism.
Definition (Lean source)
parameters
def
sparseQ s i z :
ℝ
given by
theorem sparseP_parent_local
The sparse observational factor depends only on its own coordinate and graph parents.
Formal statement
hypotheses
Proof (Lean source)
lemma sparseP_parent_local (s : SignVector 3) :
∀ i v w, v i = w i →
(∀ j ∈ threeNodeDAG.parents i, v j = w j) → sparseP s i v = sparseP s i w := by
intro i v w hi hp
simp only [sparseP]
split
· rename_i h_i
have h0 : v 0 = w 0 := hp 0 (by simp [threeNodeDAG, parents,
threeNodeEdge, h_i])
have h1 : v 1 = w 1 := by simpa [h_i] using hi
rw [h0, h1]
· rfl
def sparseWitness
The explicit reflected sparse witness on 0 → 1 plus an isolated node.
Definition (Lean source)
parameters
s :
def
def cancellationP
Cancellation witness observational conditional mechanism.
Definition (Lean source)
parameters
def
cancellationP s i v :
ℝ
given by
if i
= 1 then 1
+ (1 / 10 : ℝ) * cancellationPrimitive (reflectedCoordinate s 0 (v 0)) * centeredCoordinate (reflectedCoordinate s 1 (v 1)) else 1
theorem cancellationP_parent_local
The cancellation observational factor depends only on its own coordinate and graph parents.
Formal statement
hypotheses
conclusion
cancellationP s i v = cancellationP s i w
Proof (Lean source)
lemma cancellationP_parent_local (s : SignVector 3) :
∀ i v w, v i = w i →
(∀ j ∈ threeNodeDAG.parents i, v j = w j) →
cancellationP s i v = cancellationP s i w := by
intro i v w hi hp
simp only [cancellationP]
split
· rename_i h_i
have h0 : v 0 = w 0 := hp 0 (by simp [threeNodeDAG, parents,
threeNodeEdge, h_i])
have h1 : v 1 = w 1 := by simpa [h_i] using hi
rw [h0, h1]
· rfl
def cancellationWitness
The explicit reflected faithful cancellation mechanism.
Definition (Lean source)
parameters
s :
def
given by
clause 1
p := cancellationP s
clause 2
q := sparseQ s
clause 3
parent_local := cancellationP_parent_local s
def embeddedSparseP
Edge-specific sparse endpoint embedded in an arbitrary DAG.
Definition (Lean source)
parameters
def
embeddedSparseP s j i l v :
ℝ
given by
if l
= i then 1
+ (1 / 10 : ℝ) * centeredCoordinate (reflectedCoordinate s j (v j)) * centeredCoordinate (reflectedCoordinate s i (v i)) else 1
theorem embeddedSparseP_parent_local
Given the selected directed edge, the embedded sparse factor depends only on its own coordinate and graph parents.
Formal statement
hypotheses
conclusion
embeddedSparseP s j i l v = embeddedSparseP s j i l w
Proof (Lean source)
lemma embeddedSparseP_parent_local {n : ℕ} {G : DAG (Fin n)}
(s : SignVector n) {j i : Fin n} (hji : G.edge j i) :
∀ l v w, v l = w l → (∀ k ∈ G.parents l, v k = w k) →
embeddedSparseP s j i l v = embeddedSparseP s j i l w := by
intro l v w hl hp
simp only [embeddedSparseP]
split
· rename_i hli
have hj : v j = w j := hp j (by simpa [parents, hli] using hji)
have hi : v i = w i := by simpa [hli] using hl
rw [hj, hi]
· rfl
def embeddedSparseWitness
The edge-specific sparse endpoint used by the affine perturbation.
Definition (Lean source)
parameters
def
given by
clause 1
p := embeddedSparseP s j i
clause 2
q := fun l z => exponentialInterventionDensity (reflectedCoordinate s l z)
clause 3
parent_local := embeddedSparseP_parent_local s hji
theorem affinePath_parent_local
The affine interpolation of two mechanisms remains local to each node and its parents.
Formal statement
Proof (Lean source)
lemma affinePath_parent_local {n : ℕ} {G : DAG (Fin n)}
(θ endpoint : Mechanism n G) (t : ℝ) :
∀ i v w, v i = w i → (∀ j ∈ G.parents i, v j = w j) →
((1 - t) * θ.p i v + t * endpoint.p i v) =
((1 - t) * θ.p i w + t * endpoint.p i w) := by
intro i v w hi hp
rw [θ.parent_local i v w hi hp, endpoint.parent_local i v w hi hp]
def affinePathExtension
For a finite dimension, DAG, sign pattern, stratum point, directed edge endpoints, edge certificate, and real path parameter, the unrestricted affine mechanism path interpolates toward the embedded sparse witness.
Definition (Lean source)
parameters
def
given by
clause 1
p := fun l v => (1 - t) * θ.1.p l v + t * (embeddedSparseWitness s hji).p l v
clause 2
q := fun l z => (1 - t) * θ.1.q l z + t * (embeddedSparseWitness s hji).q l z
clause 3
parent_local := affinePath_parent_local θ.1 (embeddedSparseWitness s hji) t
def affinePath
Nodewise normalized affine interpolation, indexed exactly by t ∈ [0,1].
Definition (Lean source)
parameters
def
given by
affinePathExtension s θ hji t.1
theorem exponentialInterventionDensity_integral
The exponential intervention density integrates to one on the unit interval.
Formal statement
conclusion
∫ z in Icc (0 : ℝ) 1, exponentialInterventionDensity z = 1
Proof (Lean source)
lemma exponentialInterventionDensity_integral :
∫ z in Icc (0 : ℝ) 1, exponentialInterventionDensity z = 1 := by
rw [MeasureTheory.integral_Icc_eq_integral_Ioc]
rw [← intervalIntegral.integral_of_le (by norm_num : (0 : ℝ) ≤ 1)]
unfold exponentialInterventionDensity
have hfun : (fun z : ℝ => 4 * exp (4 * z) / (exp 4 - 1)) =
fun z => (exp 4 - 1)⁻¹ * (4 * exp (4 * z)) := by
funext z
field_simp
rw [hfun, intervalIntegral.integral_const_mul, intervalIntegral.integral_const_mul]
rw [intervalIntegral.integral_comp_mul_left exp (by norm_num : (4 : ℝ) ≠ 0)]
rw [integral_exp]
have hden : exp 4 - 1 ≠ 0 := by
have h : 1 < exp 4 := Real.one_lt_exp_iff.mpr (by norm_num)
linarith
simp only [mul_zero, mul_one, Real.exp_zero, inv_eq_one_div, smul_eq_mul]
field_simp
theorem centeredCoordinate_integral
The centered coordinate integrates to zero on the unit interval.
Formal statement
conclusion
∫ z in Icc (0 : ℝ) 1, centeredCoordinate z = 0
Proof (Lean source)
lemma centeredCoordinate_integral :
∫ z in Icc (0 : ℝ) 1, centeredCoordinate z = 0 := by
rw [MeasureTheory.integral_Icc_eq_integral_Ioc]
rw [← intervalIntegral.integral_of_le (by norm_num : (0 : ℝ) ≤ 1)]
unfold centeredCoordinate
rw [intervalIntegral.integral_sub
(intervalIntegral.intervalIntegrable_id.const_mul 2)
intervalIntegrable_const,
intervalIntegral.integral_const_mul, integral_id, integral_one]
norm_num
theorem integral_reflectedCoordinate
Reflection about the midpoint preserves integrals over the unit interval. the stated conclusion follows.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma integral_reflectedCoordinate {n : ℕ} (s : SignVector n) (i : Fin n)
(f : ℝ → ℝ) :
∫ z in Icc (0 : ℝ) 1, f (reflectedCoordinate s i z) =
∫ z in Icc (0 : ℝ) 1, f z := by
rcases s.signed i with hi | hi
· have hne : (-1 : ℝ) ≠ 1 := by norm_num
simp only [reflectedCoordinate, reflect, hi, if_neg hne]
rw [MeasureTheory.integral_Icc_eq_integral_Ioc,
MeasureTheory.integral_Icc_eq_integral_Ioc]
rw [← intervalIntegral.integral_of_le (by norm_num : (0 : ℝ) ≤ 1),
← intervalIntegral.integral_of_le (by norm_num : (0 : ℝ) ≤ 1)]
simpa using intervalIntegral.integral_comp_sub_left f (1 : ℝ)
· simp [reflectedCoordinate, reflect, hi]
theorem sparseQ_normalized
Every reflected exponential intervention density is normalized. the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma sparseQ_normalized (s : SignVector 3) (i : Fin 3) :
∫ z in Icc (0 : ℝ) 1, sparseQ s i z = 1 := by
unfold sparseQ
rw [integral_reflectedCoordinate]
exact exponentialInterventionDensity_integral
theorem centeredCoordinate_reflected_integral
Every reflected centered coordinate has zero integral. the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma centeredCoordinate_reflected_integral (s : SignVector 3) (i : Fin 3) :
∫ z in Icc (0 : ℝ) 1,
centeredCoordinate (reflectedCoordinate s i z) = 0 := by
rw [integral_reflectedCoordinate]
exact centeredCoordinate_integral
theorem sparseP_normalized
Every sparse observational conditional is normalized in its own coordinate. the stated conclusion follows.
Formal statement
hypotheses
Proof (Lean source)
lemma sparseP_normalized (s : SignVector 3) (i : Fin 3) (v : LatentState 3) :
∫ z in Icc (0 : ℝ) 1, sparseP s i (update v i z) = 1 := by
by_cases hi : i = 1
· subst i
simp only [sparseP, if_true, Function.update_self]
simp only [update, dif_neg (by decide : ¬(0 : Fin 3) = 1)]
rw [integral_add]
· simp only [integral_const, Measure.restrict_apply_univ,
Measure.real, measureReal_def, Real.volume_Icc, sub_zero, ENNReal.toReal_one,
one_smul]
rw [show (∫ a in Icc (0 : ℝ) 1,
(1 / 10 * centeredCoordinate (reflectedCoordinate s 0 (v 0))) *
centeredCoordinate (reflectedCoordinate s 1 a)) =
(1 / 10 * centeredCoordinate (reflectedCoordinate s 0 (v 0))) *
∫ a in Icc (0 : ℝ) 1,
centeredCoordinate (reflectedCoordinate s 1 a) by
rw [integral_const_mul]]
rw [centeredCoordinate_reflected_integral]
ring
norm_num
· exact integrableOn_const (ne_of_lt measure_Icc_lt_top)
· have hc : Continuous (fun z : ℝ =>
1 / 10 * centeredCoordinate (reflectedCoordinate s 0 (v 0)) *
centeredCoordinate (reflectedCoordinate s 1 z)) := by
rcases s.signed 1 with hs | hs
· simp only [centeredCoordinate, reflectedCoordinate, reflect, hs,
if_neg (by norm_num : (-1 : ℝ) ≠ 1)]
fun_prop
· simp only [centeredCoordinate, reflectedCoordinate, reflect, hs]
simp only [if_true]
fun_prop
exact hc.integrableOn_Icc
· simp [sparseP, hi]
theorem cancellationP_normalized
Every cancellation observational conditional is normalized in its own coordinate. the stated conclusion follows.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma cancellationP_normalized (s : SignVector 3) (i : Fin 3) (v : LatentState 3) :
∫ z in Icc (0 : ℝ) 1, cancellationP s i (update v i z) = 1 := by
by_cases hi : i = 1
· subst i
simp only [cancellationP, if_true, Function.update_self]
simp only [update, dif_neg (by decide : ¬(0 : Fin 3) = 1)]
rw [integral_add]
· simp only [integral_const, Measure.restrict_apply_univ,
Measure.real, measureReal_def, Real.volume_Icc, sub_zero, ENNReal.toReal_one,
one_smul]
rw [show (∫ a in Icc (0 : ℝ) 1,
(1 / 10 * cancellationPrimitive (reflectedCoordinate s 0 (v 0))) *
centeredCoordinate (reflectedCoordinate s 1 a)) =
(1 / 10 * cancellationPrimitive (reflectedCoordinate s 0 (v 0))) *
∫ a in Icc (0 : ℝ) 1,
centeredCoordinate (reflectedCoordinate s 1 a) by
rw [integral_const_mul]]
rw [centeredCoordinate_reflected_integral]
ring
norm_num
· exact integrableOn_const (ne_of_lt measure_Icc_lt_top)
· have hc : Continuous (fun z : ℝ =>
1 / 10 * cancellationPrimitive (reflectedCoordinate s 0 (v 0)) *
centeredCoordinate (reflectedCoordinate s 1 z)) := by
rcases s.signed 1 with hs | hs
· simp only [centeredCoordinate, reflectedCoordinate, reflect, hs,
if_neg (by norm_num : (-1 : ℝ) ≠ 1)]
fun_prop
· simp only [centeredCoordinate, reflectedCoordinate, reflect, hs]
simp only [if_true]
fun_prop
exact hc.integrableOn_Icc
· simp [cancellationP, hi]
theorem cancellationPrimitive_zero
The cancellation primitive vanishes at both endpoints of the unit interval.
Formal statement
Proof (Lean source)
theorem cancellationPrimitive_deriv
The derivative of the cancellation primitive is the intervention density minus one.
Formal statement
Proof (Lean source)
lemma cancellationPrimitive_deriv (x : ℝ) :
deriv cancellationPrimitive x = exponentialInterventionDensity x - 1 := by
change deriv (fun y : ℝ => (exp (4 * y) - 1) / (exp 4 - 1) - y) x =
exponentialInterventionDensity x - 1
have hexp : HasDerivAt (fun y : ℝ => exp (4 * y))
(4 * exp (4 * x)) x := by
simpa [Function.comp_def, mul_comm] using
(Real.hasDerivAt_exp (4 * x)).comp x (hasDerivAt_const_mul (x := x) 4)
have hF : HasDerivAt
(fun y : ℝ => (exp (4 * y) - 1) / (exp 4 - 1))
(exponentialInterventionDensity x) x := by
simpa [exponentialInterventionDensity] using
(hexp.sub_const 1).div_const (exp 4 - 1)
exact (hF.sub (hasDerivAt_id x)).deriv
The cancellation primitive genuinely varies, as witnessed by its nonzero derivative at zero. the stated conclusion follows.
Formal statement
conclusion
∃ x, cancellationPrimitive x ≠ cancellationPrimitive 0
Proof (Lean source)
lemma cancellationPrimitive_nonconstant :
∃ x, cancellationPrimitive x ≠ cancellationPrimitive 0 := by
by_contra hconst
push Not at hconst
have hd0 : deriv cancellationPrimitive 0 = 0 := by
rw [show cancellationPrimitive = fun _ => cancellationPrimitive 0 by
funext x
exact hconst x]
simp
rw [cancellationPrimitive_deriv] at hd0
have hexp : 5 < exp 4 := by
nlinarith [Real.add_one_lt_exp (by norm_num : (4 : ℝ) ≠ 0)]
have hden : 0 < exp 4 - 1 := by linarith
have hq0 : exponentialInterventionDensity 0 < 1 := by
simp only [exponentialInterventionDensity, mul_zero, Real.exp_zero]
apply (div_lt_one hden).2
linarith
linarith
theorem cancellationPrimitive_mem_unitInterval_sub Lemma cancellationPrimitive_mem_unitInterval_sub in the paper ↗
On the unit interval the cancellation primitive lies between minus one and one. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma cancellationPrimitive_mem_unitInterval_sub {x : ℝ} (hx : x ∈ Icc (0 : ℝ) 1) :
cancellationPrimitive x ∈ Icc (-1 : ℝ) 1 := by
have hden : 0 < exp 4 - 1 := by
have h := Real.one_lt_exp_iff.mpr (by norm_num : (0 : ℝ) < 4)
linarith
have hexp0 : 1 ≤ exp (4 * x) := by
simpa using Real.one_le_exp (by nlinarith [hx.1] : 0 ≤ 4 * x)
have hexp4 : exp (4 * x) ≤ exp 4 := by
apply Real.exp_le_exp.mpr
nlinarith [hx.2]
have hfrac0 : 0 ≤ (exp (4 * x) - 1) / (exp 4 - 1) := div_nonneg (by linarith) hden.le
have hfrac1 : (exp (4 * x) - 1) / (exp 4 - 1) ≤ 1 := by
apply (div_le_one hden).2
linarith
unfold cancellationPrimitive
constructor <;> linarith [hx.1, hx.2]
theorem reflectedCoordinate_mem_unitInterval
A prescribed coordinate reflection preserves the closed unit interval. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
reflectedCoordinate s i z ∈ Icc (0 : ℝ) 1
Proof (Lean source)
-- @node: reflectedCoordinate_mem_unitInterval
lemma reflectedCoordinate_mem_unitInterval {n : ℕ} (s : SignVector n) (i : Fin n)
{z : ℝ} (hz : z ∈ Icc (0 : ℝ) 1) :
reflectedCoordinate s i z ∈ Icc (0 : ℝ) 1 := by
rcases s.signed i with hi | hi
· simp only [reflectedCoordinate, reflect, hi, if_neg (by norm_num : (-1 : ℝ) ≠ 1)]
constructor <;> linarith [hz.1, hz.2]
· rw [reflectedCoordinate, reflect, hi, if_pos rfl]
exact hz
The centered coordinate has absolute value at most one on the unit interval. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
Proof (Lean source)
-- @node: abs_centeredCoordinate_le_one
lemma abs_centeredCoordinate_le_one {z : ℝ} (hz : z ∈ Icc (0 : ℝ) 1) :
|centeredCoordinate z| ≤ 1 := by
rw [abs_le]
constructor <;> simp only [centeredCoordinate] <;> linarith [hz.1, hz.2]
theorem embeddedSparseP_bounds
The embedded sparse conditional stays uniformly between 9/10 and 11/10 on the latent cube. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
conclusion 1
(9 / 10 : ℝ) ≤ embeddedSparseP s j i l v
conclusion 2
embeddedSparseP s j i l v ≤ (11 / 10 : ℝ)
Proof (Lean source)
lemma embeddedSparseP_bounds {n : ℕ} {G : DAG (Fin n)}
(s : SignVector n) {j i : Fin n} (hji : G.edge j i)
(l : Fin n) (v : LatentState n) (hv : v ∈ latentCube n) :
(9 / 10 : ℝ) ≤ embeddedSparseP s j i l v ∧
embeddedSparseP s j i l v ≤ (11 / 10 : ℝ) := by
by_cases hli : l = i
· subst l
have hj : v j ∈ Icc (0 : ℝ) 1 := hv j (Set.mem_univ j)
have hi : v i ∈ Icc (0 : ℝ) 1 := hv i (Set.mem_univ i)
have hbj := abs_centeredCoordinate_le_one
(reflectedCoordinate_mem_unitInterval s j hj)
have hbi := abs_centeredCoordinate_le_one
(reflectedCoordinate_mem_unitInterval s i hi)
rw [abs_le] at hbj hbi
simp only [embeddedSparseP, if_true]
constructor <;> nlinarith
· simp [embeddedSparseP, hli]
norm_num
theorem embeddedSparseP_normalized
Every embedded sparse observational conditional is normalized in its own coordinate. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma embeddedSparseP_normalized {n : ℕ} {G : DAG (Fin n)}
(s : SignVector n) {j i : Fin n} (hji : G.edge j i)
(l : Fin n) (v : LatentState n) :
∫ z in Icc (0 : ℝ) 1, embeddedSparseP s j i l (update v l z) = 1 := by
by_cases hli : l = i
· subst l
have hji_ne : j ≠ i := by
intro h
subst j
exact G.irrefl i hji
simp only [embeddedSparseP, if_true, Function.update_self]
simp only [update, dif_neg hji_ne]
rw [integral_add]
· simp only [integral_const, Measure.restrict_apply_univ,
Measure.real, measureReal_def, Real.volume_Icc, sub_zero, ENNReal.toReal_one,
one_smul]
rw [show (∫ a in Icc (0 : ℝ) 1,
(1 / 10 * centeredCoordinate (reflectedCoordinate s j (v j))) *
centeredCoordinate (reflectedCoordinate s i a)) =
(1 / 10 * centeredCoordinate (reflectedCoordinate s j (v j))) *
∫ a in Icc (0 : ℝ) 1,
centeredCoordinate (reflectedCoordinate s i a) by
rw [integral_const_mul]]
rw [integral_reflectedCoordinate, centeredCoordinate_integral]
ring
norm_num
· exact integrableOn_const (ne_of_lt measure_Icc_lt_top)
· have hc : Continuous (fun z : ℝ =>
1 / 10 * centeredCoordinate (reflectedCoordinate s j (v j)) *
centeredCoordinate (reflectedCoordinate s i z)) := by
rcases s.signed i with hs | hs
· simp only [centeredCoordinate, reflectedCoordinate, reflect, hs,
if_neg (by norm_num : (-1 : ℝ) ≠ 1)]
fun_prop
· simp only [centeredCoordinate, reflectedCoordinate, reflect, hs, if_true]
fun_prop
exact hc.integrableOn_Icc
· simp [embeddedSparseP, hli]
theorem embeddedSparseWitness_positive_normalized_smooth
The edge-specific sparse endpoint is positive, normalized, and smooth on every finite DAG, including DAGs with additional unused parents and edges. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma embeddedSparseWitness_positive_normalized_smooth
{n : ℕ} {G : DAG (Fin n)} (s : SignVector n)
{j i : Fin n} (hji : G.edge j i) :
PositiveNormalizedSmoothMechanisms G (embeddedSparseWitness s hji) := by
refine ⟨?_, ?_, ?_, ?_, ?_, ?_⟩
· intro l v hv
exact lt_of_lt_of_le (by norm_num : (0 : ℝ) < 9 / 10)
(embeddedSparseP_bounds s hji l v hv).1
· intro l z hz
change 0 < exponentialInterventionDensity (reflectedCoordinate s l z)
unfold exponentialInterventionDensity
have hden : 0 < exp 4 - 1 := by
have := Real.one_lt_exp_iff.mpr (by norm_num : (0 : ℝ) < 4)
linarith
positivity
· intro l
change ContDiffOn ℝ 3 (embeddedSparseP s j i l) (latentCube n)
by_cases hli : l = i
· subst l
rw [show embeddedSparseP s j i i = fun v : LatentState n =>
1 + (1 / 10 : ℝ) * centeredCoordinate (reflectedCoordinate s j (v j)) *
centeredCoordinate (reflectedCoordinate s i (v i)) by
funext v
simp [embeddedSparseP]]
rcases s.signed j with hj | hj <;> rcases s.signed i with hi | hi <;>
simp [reflectedCoordinate, reflect, hj, hi, centeredCoordinate,
show (-1 : ℝ) ≠ 1 by norm_num] <;> fun_prop
· rw [show embeddedSparseP s j i l = fun _ => (1 : ℝ) by
funext v
simp [embeddedSparseP, hli]]
fun_prop
· intro l
change ContDiffOn ℝ 3 (fun z : ℝ =>
exponentialInterventionDensity (reflectedCoordinate s l z)) (Icc 0 1)
unfold exponentialInterventionDensity
rcases s.signed l with hl | hl
· simp [reflectedCoordinate, reflect, hl, show (-1 : ℝ) ≠ 1 by norm_num]
fun_prop
· simp [reflectedCoordinate, reflect, hl]
fun_prop
· intro l v hv
exact embeddedSparseP_normalized s hji l v
· intro l
change ∫ z in Icc (0 : ℝ) 1,
exponentialInterventionDensity (reflectedCoordinate s l z) = 1
rw [integral_reflectedCoordinate]
exact exponentialInterventionDensity_integral
The normalized exponential intervention density is strictly positive. the stated conclusion follows.
Formal statement
Proof (Lean source)
-- @node: exponentialInterventionDensity_pos
lemma exponentialInterventionDensity_pos (z : ℝ) :
0 < exponentialInterventionDensity z := by
unfold exponentialInterventionDensity
have hden : 0 < exp 4 - 1 := by
have := Real.one_lt_exp_iff.mpr (by norm_num : (0 : ℝ) < 4)
linarith
positivity
theorem thirteen_lt_exp_four
A rational lower bound on the exponential normalizing constant used by the quantitative sparse certificate. the stated conclusion follows.
Formal statement
conclusion
(13 : ℝ) < exp 4
Proof (Lean source)
lemma thirteen_lt_exp_four : (13 : ℝ) < exp 4 := by
have h := Real.sum_le_exp_of_nonneg (x := (4 : ℝ)) (by norm_num) 4
norm_num [Finset.sum_range_succ] at h ⊢
linarith
theorem exponentialInterventionDensity_lt_thirteen_div_three
On the unit interval the intervention density is strictly below 13/3. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma exponentialInterventionDensity_lt_thirteen_div_three {z : ℝ}
(hz : z ∈ Icc (0 : ℝ) 1) :
exponentialInterventionDensity z < (13 / 3 : ℝ) := by
have hden : 0 < exp 4 - 1 := by linarith [thirteen_lt_exp_four]
have he : exp (4 * z) ≤ exp 4 := by
exact Real.exp_le_exp.mpr (by nlinarith [hz.2])
unfold exponentialInterventionDensity
calc
4 * exp (4 * z) / (exp 4 - 1) ≤
4 * exp 4 / (exp 4 - 1) :=
div_le_div_of_nonneg_right (by nlinarith) hden.le
_ < 13 / 3 := (div_lt_iff₀ hden).2 (by nlinarith [thirteen_lt_exp_four])
theorem sparseP_pos
Every sparse observational mechanism is strictly positive on the latent cube. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
0 < sparseP s i v
Proof (Lean source)
-- @node: sparseP_pos
lemma sparseP_pos (s : SignVector 3) (i : Fin 3) (v : LatentState 3)
(hv : v ∈ latentCube 3) : 0 < sparseP s i v := by
by_cases hi : i = 1
· subst i
have hv0 : v 0 ∈ Icc (0 : ℝ) 1 := hv 0 (Set.mem_univ 0)
have hv1 : v 1 ∈ Icc (0 : ℝ) 1 := hv 1 (Set.mem_univ 1)
have h0 := abs_centeredCoordinate_le_one
(reflectedCoordinate_mem_unitInterval s 0 hv0)
have h1 := abs_centeredCoordinate_le_one
(reflectedCoordinate_mem_unitInterval s 1 hv1)
have hprod : -1 ≤ centeredCoordinate (reflectedCoordinate s 0 (v 0)) *
centeredCoordinate (reflectedCoordinate s 1 (v 1)) := by
rw [abs_le] at h0 h1
nlinarith
simp only [sparseP, if_true]
nlinarith
· simp [sparseP, hi]
theorem sparseWitness_positive_normalized_smooth
The sparse witness is positive, normalized, and C³ on every mechanism domain. the stated conclusion follows.
Formal statement
hypotheses
s :
conclusion
Proof (Lean source)
-- @node: sparseWitness_positive_normalized_smooth
lemma sparseWitness_positive_normalized_smooth (s : SignVector 3) :
PositiveNormalizedSmoothMechanisms threeNodeDAG (sparseWitness s) := by
refine ⟨?_, ?_, ?_, ?_, ?_, ?_⟩
· intro i v hv
exact sparseP_pos s i v hv
· intro i z hz
exact exponentialInterventionDensity_pos _
· intro i
by_cases hi : i = 1
· subst i
change ContDiffOn ℝ 3 (fun v : LatentState 3 =>
1 + (1 / 10 : ℝ) * centeredCoordinate (reflectedCoordinate s 0 (v 0)) *
centeredCoordinate (reflectedCoordinate s 1 (v 1))) (latentCube 3)
rcases s.signed 0 with h0 | h0 <;> rcases s.signed 1 with h1 | h1 <;>
simp [reflectedCoordinate, reflect, h0, h1, centeredCoordinate,
show (-1 : ℝ) ≠ 1 by norm_num] <;> fun_prop
· change ContDiffOn ℝ 3 (sparseP s i) (latentCube 3)
rw [show sparseP s i = fun _ => (1 : ℝ) by
funext v
simp [sparseP, hi]]
fun_prop
· intro i
change ContDiffOn ℝ 3 (fun z : ℝ =>
exponentialInterventionDensity (reflectedCoordinate s i z)) (Icc 0 1)
unfold exponentialInterventionDensity
rcases s.signed i with hi | hi
· simp [reflectedCoordinate, reflect, hi, show (-1 : ℝ) ≠ 1 by norm_num]
fun_prop
· simp [reflectedCoordinate, reflect, hi]
fun_prop
· intro i v hv
exact sparseP_normalized s i v
· intro i
exact sparseQ_normalized s i
theorem cancellationP_pos
Every cancellation observational mechanism is strictly positive on the latent cube. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
0 < cancellationP s i v
Proof (Lean source)
lemma cancellationP_pos (s : SignVector 3) (i : Fin 3) (v : LatentState 3)
(hv : v ∈ latentCube 3) : 0 < cancellationP s i v := by
by_cases hi : i = 1
· subst i
have hv0 : v 0 ∈ Icc (0 : ℝ) 1 := hv 0 (Set.mem_univ 0)
have hv1 : v 1 ∈ Icc (0 : ℝ) 1 := hv 1 (Set.mem_univ 1)
have h0 := cancellationPrimitive_mem_unitInterval_sub
(reflectedCoordinate_mem_unitInterval s 0 hv0)
have h1 := abs_centeredCoordinate_le_one
(reflectedCoordinate_mem_unitInterval s 1 hv1)
rw [abs_le] at h1
simp only [cancellationP, if_true]
nlinarith [h0.1, h0.2]
· simp [cancellationP, hi]
theorem cancellationWitness_positive_normalized_smooth
The cancellation witness is positive, normalized, and C³ on every mechanism domain. the stated conclusion follows.
Formal statement
hypotheses
s :
conclusion
Proof (Lean source)
lemma cancellationWitness_positive_normalized_smooth (s : SignVector 3) :
PositiveNormalizedSmoothMechanisms threeNodeDAG (cancellationWitness s) := by
refine ⟨?_, ?_, ?_, ?_, ?_, ?_⟩
· intro i v hv
exact cancellationP_pos s i v hv
· intro i z hz
exact exponentialInterventionDensity_pos _
· intro i
by_cases hi : i = 1
· subst i
change ContDiffOn ℝ 3 (fun v : LatentState 3 =>
1 + (1 / 10 : ℝ) * cancellationPrimitive (reflectedCoordinate s 0 (v 0)) *
centeredCoordinate (reflectedCoordinate s 1 (v 1))) (latentCube 3)
rcases s.signed 0 with h0 | h0 <;> rcases s.signed 1 with h1 | h1 <;>
simp [cancellationPrimitive, reflectedCoordinate, reflect, h0, h1, centeredCoordinate,
show (-1 : ℝ) ≠ 1 by norm_num] <;> fun_prop
· change ContDiffOn ℝ 3 (cancellationP s i) (latentCube 3)
rw [show cancellationP s i = fun _ => (1 : ℝ) by
funext v
simp [cancellationP, hi]]
fun_prop
· intro i
change ContDiffOn ℝ 3 (fun z : ℝ =>
exponentialInterventionDensity (reflectedCoordinate s i z)) (Icc 0 1)
unfold exponentialInterventionDensity
rcases s.signed i with hi | hi
· simp [reflectedCoordinate, reflect, hi, show (-1 : ℝ) ≠ 1 by norm_num]
fun_prop
· simp [reflectedCoordinate, reflect, hi]
fun_prop
· intro i v hv
exact cancellationP_normalized s i v
· intro i
exact sparseQ_normalized s i
theorem sparseWitness_contDiff_all
Every sparse-witness mechanism component is smooth to every finite order. the stated conclusion follows.
Formal statement
hypotheses
s :
conclusion
Proof (Lean source)
lemma sparseWitness_contDiff_all (s : SignVector 3) :
(∀ k i, ContDiffOn ℝ k ((sparseWitness s).p i) (latentCube 3)) ∧
∀ k i, ContDiffOn ℝ k ((sparseWitness s).q i) (Icc (0 : ℝ) 1) := by
constructor
· intro k i
by_cases hi : i = 1
· subst i
change ContDiffOn ℝ k (fun v : LatentState 3 =>
1 + (1 / 10 : ℝ) * centeredCoordinate (reflectedCoordinate s 0 (v 0)) *
centeredCoordinate (reflectedCoordinate s 1 (v 1))) (latentCube 3)
rcases s.signed 0 with h0 | h0 <;> rcases s.signed 1 with h1 | h1 <;>
simp [reflectedCoordinate, reflect, h0, h1, centeredCoordinate,
show (-1 : ℝ) ≠ 1 by norm_num] <;> fun_prop
· change ContDiffOn ℝ k (sparseP s i) (latentCube 3)
rw [show sparseP s i = fun _ => (1 : ℝ) by
funext v
simp [sparseP, hi]]
fun_prop
· intro k i
change ContDiffOn ℝ k (fun z : ℝ =>
exponentialInterventionDensity (reflectedCoordinate s i z)) (Icc 0 1)
unfold exponentialInterventionDensity
rcases s.signed i with hi | hi
· simp [reflectedCoordinate, reflect, hi, show (-1 : ℝ) ≠ 1 by norm_num]
fun_prop
· simp [reflectedCoordinate, reflect, hi]
fun_prop
theorem cancellationWitness_contDiff_all
Every cancellation-witness mechanism component is smooth to every finite order. the stated conclusion follows.
Formal statement
hypotheses
s :
conclusion
Proof (Lean source)
lemma cancellationWitness_contDiff_all (s : SignVector 3) :
(∀ k i, ContDiffOn ℝ k ((cancellationWitness s).p i) (latentCube 3)) ∧
∀ k i, ContDiffOn ℝ k ((cancellationWitness s).q i) (Icc (0 : ℝ) 1) := by
constructor
· intro k i
by_cases hi : i = 1
· subst i
change ContDiffOn ℝ k (fun v : LatentState 3 =>
1 + (1 / 10 : ℝ) * cancellationPrimitive (reflectedCoordinate s 0 (v 0)) *
centeredCoordinate (reflectedCoordinate s 1 (v 1))) (latentCube 3)
rcases s.signed 0 with h0 | h0 <;> rcases s.signed 1 with h1 | h1 <;>
simp [cancellationPrimitive, reflectedCoordinate, reflect, h0, h1, centeredCoordinate,
show (-1 : ℝ) ≠ 1 by norm_num] <;> fun_prop
· change ContDiffOn ℝ k (cancellationP s i) (latentCube 3)
rw [show cancellationP s i = fun _ => (1 : ℝ) by
funext v
simp [cancellationP, hi]]
fun_prop
· intro k i
change ContDiffOn ℝ k (fun z : ℝ =>
exponentialInterventionDensity (reflectedCoordinate s i z)) (Icc 0 1)
unfold exponentialInterventionDensity
rcases s.signed i with hi | hi
· simp [reflectedCoordinate, reflect, hi, show (-1 : ℝ) ≠ 1 by norm_num]
fun_prop
· simp [reflectedCoordinate, reflect, hi]
fun_prop
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.TExactRatioDecoder 11 declarations This file states the population transitive-closure, continuous conditional-rank, parent-pruning, ordering-independence, and representation-uniqueness conclusions.
Exact ratio decoder
This file states the population transitive-closure, continuous conditional-rank, parent-pruning, ordering-independence, and representation-uniqueness conclusions. Faithfulness is confined to a separate overlap corollary.
The componentwise C² ambiguity relating two compatible representations.
Definition (Lean source)
parameters
def
ComponentwiseC2Equivalent W₁ W₂ :
Prop
given by
clause 1
observedSupport G₁ W₁ = observedSupport G₂ W₂
clause 2
j i :
G₂.edge (σ j) (σ i) ↔ G₁.edge j i
e :
W₂.targetPerm e = σ (W₁.targetPerm e)
∃ h hinv : Fin n → ℝ → ℝ,
i :
ContDiffOn ℝ 2 (h i) (Icc (0 : ℝ) 1)
i :
ContDiffOn ℝ 2 (hinv i) (Icc (0 : ℝ) 1)
i z :
z ∈ Icc (0 : ℝ) 1
hinv i (h i z) = z
i z :
z ∈ Icc (0 : ℝ) 1
h i (hinv i z) = z
x ∈ observedSupport G₁ W₁ :
i :
W₂.unmix x (σ i) = h i (W₁.unmix x i)
theorem componentwiseC2Equivalent_of_common_rank_and_aligned_edges
Common law-selected scalar ranks and aligned edges assemble the componentwise C² equivalence, including explicit inverse charts on the closed latent interval. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
n :
W₁ :
ObservedWorld G₁ θ₁
W₂ :
ObservedWorld G₂ θ₂
hpos₁ :
hmix₁ :
SharedDiffeomorphicMixing G₁ θ₁ W₁
hpos₂ :
hmix₂ :
SharedDiffeomorphicMixing G₂ θ₂ W₂
laws :
order :
Fin n → ℕ
hrank₁ :
∀ e v
if
v ∈ latentCube n
then
(observedLawRankCoordinate laws order e (W₁.mix v) : ℝ)
= signedInterventionCDF s θ₁ (W₁.targetPerm e) (v (W₁.targetPerm e))
hrank₂ :
∀ e v
if
v ∈ latentCube n
then
(observedLawRankCoordinate laws order e (W₂.mix v) : ℝ)
= signedInterventionCDF s θ₂ (W₂.targetPerm e) (v (W₂.targetPerm e))
hsupport :
observedSupport G₁ W₁ = observedSupport G₂ W₂
hedge :
∀ j i,
G₂.edge ((W₁.targetPerm.symm.trans W₂.targetPerm) j) ((W₁.targetPerm.symm.trans W₂.targetPerm) i)
↔ G₁.edge j i
conclusion
Proof (Lean source)
lemma componentwiseC2Equivalent_of_common_rank_and_aligned_edges
{n : ℕ} {G₁ G₂ : DAG (Fin n)}
{s : SignVector n} {θ₁ : Mechanism n G₁} {θ₂ : Mechanism n G₂}
(W₁ : ObservedWorld G₁ θ₁) (W₂ : ObservedWorld G₂ θ₂)
(hpos₁ : PositiveNormalizedSmoothMechanisms G₁ θ₁)
(hmix₁ : SharedDiffeomorphicMixing G₁ θ₁ W₁)
(hpos₂ : PositiveNormalizedSmoothMechanisms G₂ θ₂)
(hmix₂ : SharedDiffeomorphicMixing G₂ θ₂ W₂)
(laws : ObservedProbabilityLawFamily n) (order : Fin n → ℕ)
(hrank₁ : ∀ e v, v ∈ latentCube n →
(observedLawRankCoordinate laws order e (W₁.mix v) : ℝ) =
signedInterventionCDF s θ₁ (W₁.targetPerm e) (v (W₁.targetPerm e)))
(hrank₂ : ∀ e v, v ∈ latentCube n →
(observedLawRankCoordinate laws order e (W₂.mix v) : ℝ) =
signedInterventionCDF s θ₂ (W₂.targetPerm e) (v (W₂.targetPerm e)))
(hsupport : observedSupport G₁ W₁ = observedSupport G₂ W₂)
(hedge : ∀ j i,
G₂.edge ((W₁.targetPerm.symm.trans W₂.targetPerm) j)
((W₁.targetPerm.symm.trans W₂.targetPerm) i) ↔ G₁.edge j i) :
ComponentwiseC2Equivalent W₁ W₂ := by
let σ : Perm (Fin n) := W₁.targetPerm.symm.trans W₂.targetPerm
let base : LatentState n := 0
let h : Fin n → ℝ → ℝ := fun i z =>
W₂.unmix (W₁.mix (update base i z)) (σ i)
let hinv : Fin n → ℝ → ℝ := fun i z =>
W₁.unmix (W₂.mix (update base (σ i) z)) i
have hbase (i : Fin n) (z : ℝ) (hz : z ∈ Icc (0 : ℝ) 1) :
update base i z ∈ latentCube n := by
intro k hk
by_cases hki : k = i
· subst k; simpa [update]
· simp [base, update, hki]
have hsigma (i : Fin n) : σ i = W₂.targetPerm (W₁.targetPerm.symm i) := rfl
have hmaps (i : Fin n) : Set.MapsTo (h i) (Icc (0 : ℝ) 1) (Icc 0 1) := by
intro z hz
have hy₁ : W₁.mix (update base i z) ∈ observedSupport G₁ W₁ :=
⟨_, hbase i z hz, rfl⟩
have hy₂ : W₁.mix (update base i z) ∈ observedSupport G₂ W₂ := by
rw [← hsupport]; exact hy₁
rcases hy₂ with ⟨v, hv, heq⟩
dsimp only [h]
rw [← heq, hmix₂.2.2.1 v hv]
exact hv (σ i) (Set.mem_univ _)
have hinvmaps (i : Fin n) : Set.MapsTo (hinv i) (Icc (0 : ℝ) 1) (Icc 0 1) := by
intro z hz
have hy₂ : W₂.mix (update base (σ i) z) ∈ observedSupport G₂ W₂ :=
⟨_, hbase (σ i) z hz, rfl⟩
have hy₁ : W₂.mix (update base (σ i) z) ∈ observedSupport G₁ W₁ := by
rw [hsupport]; exact hy₂
rcases hy₁ with ⟨v, hv, heq⟩
dsimp only [hinv]
rw [← heq, hmix₁.2.2.1 v hv]
exact hv i (Set.mem_univ _)
have hdepend : ∀ x ∈ observedSupport G₁ W₁, ∀ i,
W₂.unmix x (σ i) = h i (W₁.unmix x i) := by
intro x hx i
have hx₂ : x ∈ observedSupport G₂ W₂ := by rw [← hsupport]; exact hx
let y := W₁.mix (update base i (W₁.unmix x i))
have hxi : W₁.unmix x i ∈ Icc (0 : ℝ) 1 := by
rcases hx with ⟨v, hv, rfl⟩
simpa only [hmix₁.2.2.1 v hv] using hv i (Set.mem_univ _)
have hy₁ : y ∈ observedSupport G₁ W₁ := ⟨_, hbase i _ hxi, rfl⟩
have hy₂ : y ∈ observedSupport G₂ W₂ := by rw [← hsupport]; exact hy₁
have hcoord : W₁.unmix x (W₁.targetPerm (W₁.targetPerm.symm i)) =
W₁.unmix y (W₁.targetPerm (W₁.targetPerm.symm i)) := by
simp only [W₁.targetPerm.apply_symm_apply]
rw [show W₁.unmix y = update base i (W₁.unmix x i) by
exact hmix₁.2.2.1 _ (hbase i _ hxi)]
simp
have hc := alignedCoordinate_eq_of_common_rank W₁ W₂ hmix₁ hpos₂ hmix₂
laws order hrank₁ hrank₂ x y hx hx₂ hy₁ hy₂ (W₁.targetPerm.symm i) hcoord
simpa only [W₁.targetPerm.apply_symm_apply, hsigma, h] using hc
refine ⟨hsupport, σ, hedge, ?_, h, hinv, ?_, ?_, hmaps, hinvmaps, ?_, ?_, hdepend⟩
· intro e
simp [σ]
· intro i
have hup : ContDiffOn ℝ 2 (update base i) (Icc (0 : ℝ) 1) :=
(contDiff_update 2 base i).contDiffOn
have hf : ContDiffOn ℝ 2 (W₁.mix ∘ update base i) (Icc 0 1) :=
hmix₁.1.comp hup (hbase i)
have hfmap : Set.MapsTo (W₁.mix ∘ update base i) (Icc (0 : ℝ) 1)
(observedSupport G₂ W₂) := by
intro z hz
rw [← hsupport]
exact ⟨_, hbase i z hz, rfl⟩
have hg : ContDiffOn ℝ 2
(W₂.unmix ∘ W₁.mix ∘ update base i) (Icc (0 : ℝ) 1) := by
convert hmix₂.2.1.comp hf hfmap using 1
change ContDiffOn ℝ 2
((fun v : LatentState n ↦ v (σ i)) ∘
(W₂.unmix ∘ W₁.mix ∘ update base i)) (Icc (0 : ℝ) 1)
exact (contDiff_apply ℝ ℝ (σ i)).contDiffOn.comp hg (Set.mapsTo_univ _ _)
· intro i
have hup : ContDiffOn ℝ 2 (update base (σ i)) (Icc (0 : ℝ) 1) :=
(contDiff_update 2 base (σ i)).contDiffOn
have hf : ContDiffOn ℝ 2 (W₂.mix ∘ update base (σ i)) (Icc 0 1) :=
hmix₂.1.comp hup (hbase (σ i))
have hfmap : Set.MapsTo (W₂.mix ∘ update base (σ i)) (Icc (0 : ℝ) 1)
(observedSupport G₁ W₁) := by
intro z hz
rw [hsupport]
exact ⟨_, hbase (σ i) z hz, rfl⟩
have hg : ContDiffOn ℝ 2
(W₁.unmix ∘ W₂.mix ∘ update base (σ i)) (Icc (0 : ℝ) 1) := by
convert hmix₁.2.1.comp hf hfmap using 1
change ContDiffOn ℝ 2
((fun v : LatentState n ↦ v i) ∘
(W₁.unmix ∘ W₂.mix ∘ update base (σ i))) (Icc (0 : ℝ) 1)
exact (contDiff_apply ℝ ℝ i).contDiffOn.comp hg (Set.mapsTo_univ _ _)
· intro i z hz
let y₁ := W₁.mix (update base i z)
let y₂ := W₂.mix (update base (σ i) (h i z))
have hy₁₁ : y₁ ∈ observedSupport G₁ W₁ := ⟨_, hbase i z hz, rfl⟩
have hy₁₂ : y₁ ∈ observedSupport G₂ W₂ := by rw [← hsupport]; exact hy₁₁
have hhmem := hmaps i hz
have hy₂₂ : y₂ ∈ observedSupport G₂ W₂ := ⟨_, hbase (σ i) _ hhmem, rfl⟩
have hy₂₁ : y₂ ∈ observedSupport G₁ W₁ := by rw [hsupport]; exact hy₂₂
let e := W₂.targetPerm.symm (σ i)
have he : e = W₁.targetPerm.symm i := by
apply W₂.targetPerm.injective
simp [e, hsigma]
have hcoord : W₂.unmix y₁ (W₂.targetPerm e) =
W₂.unmix y₂ (W₂.targetPerm e) := by
rw [W₂.targetPerm.apply_symm_apply]
rw [show W₂.unmix y₂ = update base (σ i) (h i z) by
exact hmix₂.2.2.1 _ (hbase (σ i) _ hhmem)]
simp only [Function.update_self]
rfl
have hc := alignedCoordinate_eq_of_common_rank W₂ W₁ hmix₂ hpos₁ hmix₁
laws order hrank₂ hrank₁ y₁ y₂ hy₁₂ hy₁₁ hy₂₂ hy₂₁ e hcoord
rw [he, W₁.targetPerm.apply_symm_apply] at hc
rw [show W₁.unmix y₁ = update base i z by
exact hmix₁.2.2.1 _ (hbase i z hz)] at hc
simpa only [Function.update_self, hinv, y₂] using hc.symm
· intro i z hz
let y₂ := W₂.mix (update base (σ i) z)
let y₁ := W₁.mix (update base i (hinv i z))
have hy₂₂ : y₂ ∈ observedSupport G₂ W₂ := ⟨_, hbase (σ i) z hz, rfl⟩
have hy₂₁ : y₂ ∈ observedSupport G₁ W₁ := by rw [hsupport]; exact hy₂₂
have himem := hinvmaps i hz
have hy₁₁ : y₁ ∈ observedSupport G₁ W₁ := ⟨_, hbase i _ himem, rfl⟩
have hy₁₂ : y₁ ∈ observedSupport G₂ W₂ := by rw [← hsupport]; exact hy₁₁
let e := W₁.targetPerm.symm i
have hcoord : W₁.unmix y₂ (W₁.targetPerm e) =
W₁.unmix y₁ (W₁.targetPerm e) := by
rw [W₁.targetPerm.apply_symm_apply]
rw [show W₁.unmix y₁ = update base i (hinv i z) by
exact hmix₁.2.2.1 _ (hbase i _ himem)]
simp only [Function.update_self]
rfl
have hc := alignedCoordinate_eq_of_common_rank W₁ W₂ hmix₁ hpos₂ hmix₂
laws order hrank₁ hrank₂ y₂ y₁ hy₂₁ hy₂₂ hy₁₁ hy₁₂ e hcoord
have hsige : W₂.targetPerm e = σ i := by simp [e, σ]
rw [hsige] at hc
rw [show W₂.unmix y₂ = update base (σ i) z by
exact hmix₂.2.2.1 _ (hbase (σ i) z hz)] at hc
simpa only [Function.update_self, h, y₁] using hc.symm
theorem componentwiseC2Equivalent_of_common_exactPruning
Exact law-only pruning for two compatible causal-minimal representations first aligns their graphs; their common selected rank coordinates then assemble the componentwise C² ambiguity. No cover-separation condition is imposed on the competitor. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
s :
θ₁ :
Mechanism n G₁
θ₂ :
Mechanism n G₂
W₁ :
ObservedWorld G₁ θ₁
W₂ :
ObservedWorld G₂ θ₂
hpos₁ :
hmix₁ :
SharedDiffeomorphicMixing G₁ θ₁ W₁
hone₁ :
OnePerfectInterventionPerNode G₁ θ₁ W₁
hsign₁ :
FixedOwnDerivativeSign G₁ s θ₁
hpos₂ :
hmix₂ :
SharedDiffeomorphicMixing G₂ θ₂ W₂
hone₂ :
OnePerfectInterventionPerNode G₂ θ₂ W₂
hsign₂ :
FixedOwnDerivativeSign G₂ s θ₂
hlaw :
W₂.law = W₁.law
htc₁ :
= TransGen (permutedGraph G₁ W₁)
hprune₁ :
∀ i
if
MinimalAdmissibleParentSet (observedProbabilityLawFamily W₁.law) (selectedTopologicalOrder (observedProbabilityLawFamily W₁.law)) i (environmentParentSet W₁ i) ∧
∀ A,
MinimalAdmissibleParentSet (observedProbabilityLawFamily W₁.law) (selectedTopologicalOrder (observedProbabilityLawFamily W₁.law)) i A
then
A = environmentParentSet W₁ i
hprune₂ :
∀ i
if
MinimalAdmissibleParentSet (observedProbabilityLawFamily W₁.law) (selectedTopologicalOrder (observedProbabilityLawFamily W₁.law)) i (environmentParentSet W₂ i) ∧
∀ A,
MinimalAdmissibleParentSet (observedProbabilityLawFamily W₁.law) (selectedTopologicalOrder (observedProbabilityLawFamily W₁.law)) i A
then
A = environmentParentSet W₂ i
conclusion
Proof (Lean source)
lemma componentwiseC2Equivalent_of_common_exactPruning
{n : ℕ} {G₁ G₂ : DAG (Fin n)} (s : SignVector n)
{θ₁ : Mechanism n G₁} {θ₂ : Mechanism n G₂}
(W₁ : ObservedWorld G₁ θ₁) (W₂ : ObservedWorld G₂ θ₂)
(hpos₁ : PositiveNormalizedSmoothMechanisms G₁ θ₁)
(hmix₁ : SharedDiffeomorphicMixing G₁ θ₁ W₁)
(hone₁ : OnePerfectInterventionPerNode G₁ θ₁ W₁)
(hsign₁ : FixedOwnDerivativeSign G₁ s θ₁)
(hpos₂ : PositiveNormalizedSmoothMechanisms G₂ θ₂)
(hmix₂ : SharedDiffeomorphicMixing G₂ θ₂ W₂)
(hone₂ : OnePerfectInterventionPerNode G₂ θ₂ W₂)
(hsign₂ : FixedOwnDerivativeSign G₂ s θ₂)
(hlaw : W₂.law = W₁.law)
(htc₁ : TransGen (observedLawRatioGraph gaussianFeatureMap W₁.law) =
TransGen (permutedGraph G₁ W₁))
(hprune₁ : ∀ i,
MinimalAdmissibleParentSet (observedProbabilityLawFamily W₁.law)
(selectedTopologicalOrder (observedProbabilityLawFamily W₁.law)) i
(environmentParentSet W₁ i) ∧
∀ A, MinimalAdmissibleParentSet (observedProbabilityLawFamily W₁.law)
(selectedTopologicalOrder (observedProbabilityLawFamily W₁.law)) i A →
A = environmentParentSet W₁ i)
(hprune₂ : ∀ i,
MinimalAdmissibleParentSet (observedProbabilityLawFamily W₁.law)
(selectedTopologicalOrder (observedProbabilityLawFamily W₁.law)) i
(environmentParentSet W₂ i) ∧
∀ A, MinimalAdmissibleParentSet (observedProbabilityLawFamily W₁.law)
(selectedTopologicalOrder (observedProbabilityLawFamily W₁.law)) i A →
A = environmentParentSet W₂ i) :
ComponentwiseC2Equivalent W₁ W₂ := by
let laws := observedProbabilityLawFamily W₁.law
let order := selectedTopologicalOrder laws
have hgraph₁ : (selectedParentDAG laws order).edge = permutedGraph G₁ W₁ :=
selectedParentDAG_eq_permutedGraph_of_exactPruning W₁ laws hprune₁
have hgraph₂ : (selectedParentDAG laws order).edge = permutedGraph G₂ W₂ :=
selectedParentDAG_eq_permutedGraph_of_exactPruning W₂ laws hprune₂
have hperm : permutedGraph G₁ W₁ = permutedGraph G₂ W₂ :=
hgraph₁.symm.trans hgraph₂
have htc₂ : TransGen (observedLawRatioGraph gaussianFeatureMap W₂.law) =
TransGen (permutedGraph G₂ W₂) := by
rw [hlaw, htc₁, hperm]
have horder : IsTopologicalOrdering
(observedLawRatioGraph gaussianFeatureMap W₁.law) order := by
exact selectedTopologicalOrder_valid_of_assumptions
(s := s) W₁ hpos₁ hmix₁ hone₁
have horder₂ : IsTopologicalOrdering
(observedLawRatioGraph gaussianFeatureMap W₂.law) order := by
simpa only [hlaw] using horder
have hrank₁ : ∀ e v, v ∈ latentCube n →
(observedLawRankCoordinate laws order e (W₁.mix v) : ℝ) =
signedInterventionCDF s θ₁ (W₁.targetPerm e) (v (W₁.targetPerm e)) := by
intro e v hv
simpa only [laws, order, signedInterventionCDF] using
(exactRatioDecoder_continuousRankClauses s W₁ hpos₁ hmix₁ hone₁ hsign₁
htc₁ order horder e).2.2.2.2 v hv
have hrank₂ : ∀ e v, v ∈ latentCube n →
(observedLawRankCoordinate laws order e (W₂.mix v) : ℝ) =
signedInterventionCDF s θ₂ (W₂.targetPerm e) (v (W₂.targetPerm e)) := by
intro e v hv
simpa only [laws, order, hlaw, signedInterventionCDF] using
(exactRatioDecoder_continuousRankClauses s W₂ hpos₂ hmix₂ hone₂ hsign₂
htc₂ order horder₂ e).2.2.2.2 v hv
have hsupport := observedSupport_eq_of_observationalLaw_eq W₁ W₂
hpos₁ hmix₁ hone₁ hpos₂ hmix₂ hone₂ hlaw
exact componentwiseC2Equivalent_of_common_rank_and_aligned_edges W₁ W₂
hpos₁ hmix₁ hpos₂ hmix₂ laws order hrank₁ hrank₂ hsupport
(aligned_edge_iff_of_permutedGraph_eq W₁ W₂ hperm)
def VonKugelgenTwoVersusOneComparison Definition aux_VonKugelgenTwoVersusOneComparison in the paper ↗
On the faithful overlap with von Kügelgen et al. (2023), the present law-only decoder uses one unknown-target intervention per node, strictly fewer than the comparator's two interventions per node in arbitrary dimension, while retaining representation uniqueness.
Definition (Lean source)
parameters
def
VonKugelgenTwoVersusOneComparison s θ W :
Prop
given by
clause 1
= "von Kugelgen et al. (2023), Nonparametric Identifiability of Causal Representations from Unknown Interventions, Theorems 3.2 and 3.4 and Section 7 (vonKugelgenEtAl2023UnknownInterventions; https://papers.nips.cc/paper_files/paper/2023/file/97fe251c25b6f99a2a23b330a75b11d4-Paper-Conference.pdf): Theorem 3.2 is bivariate with one unknown-target perfect intervention per node and a continuous witness genericity condition; Theorem 3.4 uses two paired perfect interventions per node in arbitrary dimension; the one-intervention extension for n greater than two is stated as a conjecture."
clause 2
Faithfulness G θ
(1 : ℕ) < 2
θ' :
Mechanism n G'
W' :
ObservedWorld G' θ'
CausalMinimality G' θ'
FixedOwnDerivativeSign G' s θ'
SharedDiffeomorphicMixing G' θ' W'
OnePerfectInterventionPerNode G' θ' W'
W'.law = W.law
Explicit scope boundary for the comparison with von Kügelgen et al. (2023): the present result assumes positive C³ densities on the compact latent cube and C² mixing. In positive dimension that cube is strictly smaller than the full Euclidean latent space, and derivative order two is not the comparator's C¹ order.
Definition (Lean source)
parameters
def
FullRnC1NoncoverageClause θ W :
Prop
given by
clause 1
= "von Kugelgen et al. (2023), Nonparametric Identifiability of Causal Representations from Unknown Interventions, Theorems 3.2 and 3.4 and Section 7 (vonKugelgenEtAl2023UnknownInterventions; https://papers.nips.cc/paper_files/paper/2023/file/97fe251c25b6f99a2a23b330a75b11d4-Paper-Conference.pdf): Theorem 3.2 is bivariate with one unknown-target perfect intervention per node and a continuous witness genericity condition; Theorem 3.4 uses two paired perfect interventions per node in arbitrary dimension; the one-intervention extension for n greater than two is stated as a conjecture."
clause 2
clause 3
clause 5
(2 : ℕ) ≠ 1
def WendongYaoSuppliedStructureComparison Definition aux_WendongYaoSuppliedStructureComparison in the paper ↗
Relative to Wendong et al. (2023) and Yao et al. (2025), the graph, topological order, and environment-to-coordinate alignment are outputs computed from the observed laws, rather than supplied structural inputs.
Definition (Lean source)
parameters
def
WendongYaoSuppliedStructureComparison s θ W :
Prop
given by
laws :
= "Wendong et al. (2023), Causal Component Analysis, Definition 3.2, Theorem 4.2, and Appendix E.2 (WendongEtAl2023CauCA; https://papers.nips.cc/paper_files/paper/2023/file/67089958e98b243d5cc1881ad60418b8-Paper-Conference.pdf): the graph G is assumed known, intervention targets are observed and fixed across candidate models, and one perfect stochastic intervention per node, each satisfying Assumption 4.1, gives identification up to componentwise scaling."
= "Yao et al. (2025), Unifying Causal Representation Learning with the Invariance Principle, Assumption D.1, Corollary D.1, and the immediately following remark (YaoEtAl2025InvariancePrinciple; arXiv:2409.02772v2; https://arxiv.org/abs/2409.02772v2): exactly one imperfect intervention is supplied per node, the target labels preserve a supplied topological order, componentwise identification follows from marginal and score invariances, and identifying the order is explicitly treated as a separate subproblem."
environmentParentSet W i ⊆ predecessorSet order i
i v :
v ∈ latentCube n
(observedLawRankCoordinate laws (selectedTopologicalOrder laws) i (W.mix v) : ℝ)
= if s.value (W.targetPerm i) = 1 then interventionCDF θ (W.targetPerm i) (v (W.targetPerm i)) else 1
- interventionCDF θ (W.targetPerm i) (v (W.targetPerm i))
def ExactRatioDecoderComparisonClauses Definition aux_ExactRatioDecoderComparisonClauses in the paper ↗
Formal scope/comparison payload: the frozen von-Kügelgen intervention-count and noncoverage clauses, recovery of the structure supplied by Wendong/Yao, and the bivariate MMD witness.
Definition (Lean source)
parameters
def
ExactRatioDecoderComparisonClauses s θ W :
Prop
given by
clause 1
clause 2
clause 3
clause 4
n = 2
j i :
G.edge j i
0 < observedLawDiscrepancy gaussianFeatureMap W.law (W.targetPerm.symm j) (W.targetPerm.symm i)
All mathematical conclusions of the exact population decoder theorem.
Definition (Lean source)
parameters
def
ExactRatioDecoderConclusion s θ W :
Prop
given by
laws :
= TransGen (permutedGraph G W)
IsContinuousConditionalRatioCDFVersion laws order i (observedConditionalRatioCDF laws order i)
∃ C,
IsContinuousConditionalRatioCDFVersion laws order i C
t v :
v ∈ latentCube n
= equationElevenConditionalRatioCDF θ (W.targetPerm i) t v
C :
IsContinuousConditionalRatioCDFVersion laws order i C
z ∈ observedConditionalRatioSupport W.law order i :
C z.1 z.2 = observedConditionalRatioCDF laws order i z.1 z.2
t v :
v ∈ latentCube n
(observedConditionalRatioCDF laws order i t (familyProjection (observedLawLogRatio W.law) (predecessorSet order i) (W.mix v)) : ℝ)
= equationElevenConditionalRatioCDF θ (W.targetPerm i) t v
v :
v ∈ latentCube n
(observedLawRankCoordinate laws order i (W.mix v) : ℝ)
= if s.value (W.targetPerm i) = 1 then interventionCDF θ (W.targetPerm i) (v (W.targetPerm i)) else 1
- interventionCDF θ (W.targetPerm i) (v (W.targetPerm i))
environmentParentSet W i ⊆ predecessorSet order i
CondIndepGiven (W.law 0) (observedLawRankCoordinate laws order i) (familyProjection (observedLawRankCoordinate laws order) (predecessorSet order i \ A)) (familyProjection (observedLawRankCoordinate laws order) A)
↔ environmentParentSet W i ⊆ A
MinimalAdmissibleParentSet laws order i (environmentParentSet W i)
A :
MinimalAdmissibleParentSet laws order i A
A = environmentParentSet W i
θ' :
Mechanism n G'
W' :
ObservedWorld G' θ'
CausalMinimality G' θ'
FixedOwnDerivativeSign G' s θ'
SharedDiffeomorphicMixing G' θ' W'
OnePerfectInterventionPerNode G' θ' W'
W'.law = W.law
n = 2
j i :
G.edge j i
0 < observedLawDiscrepancy gaussianFeatureMap W.law (W.targetPerm.symm j) (W.targetPerm.symm i)
theorem exactRatioDecoderComparisonClauses_of_mainConclusions Lemma exactRatioDecoderComparisonClauses_of_mainConclusions in the paper ↗
The formal comparator payload follows from exact graph recovery, representation uniqueness, the bivariate edge witness, and predecessor containment; target uniqueness is supplied by the world's target permutation. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
s :
θ :
Mechanism n G
W :
ObservedWorld G θ
hpos :
hmix :
hgraph :
hunique :
and
CausalMinimality G' θ'
and
FixedOwnDerivativeSign G' s θ'
and
SharedDiffeomorphicMixing G' θ' W'
and
OnePerfectInterventionPerNode G' θ' W'
and
W'.law = W.law
then
hbiv :
n
= 2 → ∀ ⦃j i⦄, G.edge j i → 0 < observedLawDiscrepancy gaussianFeatureMap W.law (W.targetPerm.symm j) (W.targetPerm.symm i)
hpred :
∀ order
if
then
∀ i, environmentParentSet W i ⊆ predecessorSet order i
horder :
hrank :
∀ i v
if
v ∈ latentCube n
then
(observedLawRankCoordinate (observedProbabilityLawFamily W.law) (selectedTopologicalOrder (observedProbabilityLawFamily W.law)) i (W.mix v) : ℝ)
= if s.value (W.targetPerm i) = 1 then interventionCDF θ (W.targetPerm i) (v (W.targetPerm i)) else 1
- interventionCDF θ (W.targetPerm i) (v (W.targetPerm i))
conclusion
Proof (Lean source)
lemma exactRatioDecoderComparisonClauses_of_mainConclusions
{n : ℕ} {G : DAG (Fin n)} (s : SignVector n)
{θ : Mechanism n G} (W : ObservedWorld G θ)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hgraph : (populationDecoder (observedProbabilityLawFamily W.law)).2.2.edge =
permutedGraph G W)
(hunique : ∀ (G' : DAG (Fin n)) (θ' : Mechanism n G')
(W' : ObservedWorld G' θ'),
PositiveNormalizedSmoothMechanisms G' θ' → CausalMinimality G' θ' →
FixedOwnDerivativeSign G' s θ' → SharedDiffeomorphicMixing G' θ' W' →
OnePerfectInterventionPerNode G' θ' W' → W'.law = W.law →
ComponentwiseC2Equivalent W W')
(hbiv : n = 2 → ∀ ⦃j i⦄, G.edge j i →
0 < observedLawDiscrepancy gaussianFeatureMap W.law
(W.targetPerm.symm j) (W.targetPerm.symm i))
(hpred : ∀ order, IsTopologicalOrdering
(observedLawRatioGraph gaussianFeatureMap W.law) order →
∀ i, environmentParentSet W i ⊆ predecessorSet order i)
(horder : IsTopologicalOrdering
(observedLawRatioGraph gaussianFeatureMap W.law)
(selectedTopologicalOrder (observedProbabilityLawFamily W.law)))
(hrank : ∀ i v, v ∈ latentCube n →
(observedLawRankCoordinate (observedProbabilityLawFamily W.law)
(selectedTopologicalOrder (observedProbabilityLawFamily W.law)) i (W.mix v) : ℝ) =
if s.value (W.targetPerm i) = 1
then interventionCDF θ (W.targetPerm i) (v (W.targetPerm i))
else 1 - interventionCDF θ (W.targetPerm i) (v (W.targetPerm i))) :
ExactRatioDecoderComparisonClauses s θ W := by
refine ⟨?_, ?_, ⟨rfl, rfl, horder, hgraph, hpred, hrank⟩, hbiv⟩
· refine ⟨rfl, ?_⟩
intro _hfaith
refine ⟨?_, by norm_num, hunique⟩
intro i
refine ⟨W.targetPerm.symm i, W.targetPerm.apply_symm_apply i, ?_⟩
intro e he
exact W.targetPerm.injective (he.trans (W.targetPerm.apply_symm_apply i).symm)
· refine ⟨rfl, hpos, hmix, ?_, by norm_num⟩
intro hn hcube
have hnpos : 0 < n := Nat.pos_of_ne_zero hn
let v : LatentState n := fun _ ↦ 2
have hv : v ∈ latentCube n := by
rw [hcube]
exact Set.mem_univ v
let i : Fin n := ⟨0, hnpos⟩
have hi := hv i (Set.mem_univ i)
norm_num at hi
Cover-separated positive smooth causal-minimal mechanisms are decoded exactly from their observed environment laws, up to relabeling and componentwise C² diffeomorphisms. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
s :
θ :
Mechanism n G
world :
∀ θ' : Mechanism n G, ObservedWorld G θ'
hpos :
hminimal :
CausalMinimality G θ
hmix :
SharedDiffeomorphicMixing G θ (world θ)
hone :
OnePerfectInterventionPerNode G θ (world θ)
hsign :
FixedOwnDerivativeSign G s θ
hcover :
(⟨θ, ⟨hpos, hminimal, hsign⟩⟩ : StratumPoint G s) ∈ coverSeparatedSet (s := s) gaussianFeatureMap (world θ).targetPerm
conclusion
ExactRatioDecoderConclusion s θ (world θ)
Proof (Lean source)
theorem exact_ratio_decoder
{n : ℕ} {G : DAG (Fin n)} (s : SignVector n)
(θ : Mechanism n G) (world : ∀ θ' : Mechanism n G, ObservedWorld G θ')
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hminimal : CausalMinimality G θ)
(hmix : SharedDiffeomorphicMixing G θ (world θ))
(hone : OnePerfectInterventionPerNode G θ (world θ))
(hsign : FixedOwnDerivativeSign G s θ)
(hcover : (⟨θ, ⟨hpos, hminimal, hsign⟩⟩ : StratumPoint G s) ∈
coverSeparatedSet (s := s) gaussianFeatureMap (world θ).targetPerm) :
ExactRatioDecoderConclusion s θ (world θ) := by
have hcoherent : ObservedWorldLawCoherent (world θ) :=
observedWorldLawCoherent_of_assumptions (world θ) hpos hmix hone
have _hratio : ∀ i v, v ∈ latentCube n →
observedLawLogRatio (world θ).law i ((world θ).mix v) =
log
(θ.q ((world θ).targetPerm i) (v ((world θ).targetPerm i)) /
θ.p ((world θ).targetPerm i) v) := fun i =>
observedLawLogRatio_comp_mix_eq (world θ) hpos hmix hone i
have hcoverCanonical : ∀ ⦃a b : Fin n⦄, ancestralCover G a b →
0 < populationDiscrepancy gaussianFeatureMap
(canonicalObservedWorld G θ (world θ).targetPerm)
((world θ).targetPerm.symm a) ((world θ).targetPerm.symm b) := by
simpa only [coverSeparatedSet, Set.mem_setOf_eq] using hcover
have hcoverWorld := canonical_coverDiscrepancy_to_supplied gaussianFeatureMap
(world θ) hpos hmix hone hcoverCanonical
have hgraph := ratioGraph_reconstruction_order_and_predecessors
(s := s) (world θ) hpos hmix hone hcoverWorld
have _htriangularInverse : ∀ order,
IsTopologicalOrdering
(observedLawRatioGraph gaussianFeatureMap (world θ).law) order →
∀ i, PredecessorLatentCube order i ≃ₜ
range (predecessorScoreMap (world θ) order i) := by
intro order horder i
exact predecessorScoreHomeomorph s (world θ) hpos hmix hone hsign
horder (permutedGraphOrdered_of_transitiveClosure
(world θ) order horder hgraph.1) i
have _hequationTenDensity : ∀ order,
IsTopologicalOrdering
(observedLawRatioGraph gaussianFeatureMap (world θ).law) order →
∀ i,
let B := mechanismUnitCubeFactorization hpos
let q := mechanismInterventionDensity (world θ) hpos i
let A := decoderRetainedLatentSet (world θ) order i
(∫⋯∫⁻_(univ \ A),
B.interventionDensity ((world θ).targetPerm i) q
∂fun _ : Fin n =>
unitIntervalReference) =
fun v => q.density (v ((world θ).targetPerm i)) *
B.partialDensity (A.erase ((world θ).targetPerm i)) v := by
intro order horder i
exact equationTen_interventionMarginalDensity (world θ) hpos horder
(permutedGraphOrdered_of_transitiveClosure (world θ) order horder hgraph.1) i
have _hequationTenFiberProduct : ∀ order,
IsTopologicalOrdering
(observedLawRatioGraph gaussianFeatureMap (world θ).law) order →
∀ i (v : LatentState n) (w : ℝ),
let B := mechanismUnitCubeFactorization hpos
let q := mechanismInterventionDensity (world θ) hpos i
let A := decoderRetainedLatentSet (world θ) order i
(∫⋯∫⁻_(univ \ A),
B.interventionDensity ((world θ).targetPerm i) q
∂fun _ : Fin n ↦
unitIntervalReference)
(update v ((world θ).targetPerm i) w) =
q.density w *
B.partialDensity (A.erase ((world θ).targetPerm i)) v := by
intro order horder i v w
exact equationTen_interventionMarginalDensity_fiber_product
(world θ) hpos horder
(permutedGraphOrdered_of_transitiveClosure (world θ) order horder hgraph.1) i v w
have hcontinuousRankClauses := exactRatioDecoder_continuousRankClauses
s (world θ) hpos hmix hone hsign hgraph.1
have hbivariate : n = 2 → ∀ ⦃j i⦄, G.edge j i →
0 < observedLawDiscrepancy gaussianFeatureMap (world θ).law
((world θ).targetPerm.symm j) ((world θ).targetPerm.symm i) := by
intro hn
subst n
exact bivariate_edge_discrepancy_pos_of_coverSeparated (world θ) hcoverWorld
have hprob : ∀ e, IsProbabilityMeasure ((world θ).law e) :=
observedWorld_laws_isProbabilityMeasure (world θ) hpos hmix hone
let laws := observedProbabilityLawFamily (world θ).law
have hlaws : laws.1 = (world θ).law := by
simp [laws, observedProbabilityLawFamily, hprob]
have hiff : ∀ order, IsTopologicalOrdering
(observedLawRatioGraph gaussianFeatureMap (world θ).law) order →
∀ i A, A ⊆ predecessorSet order i →
(CondIndepGiven ((world θ).law 0) (observedLawRankCoordinate laws order i)
(familyProjection (observedLawRankCoordinate laws order)
(predecessorSet order i \ A))
(familyProjection (observedLawRankCoordinate laws order) A) ↔
environmentParentSet (world θ) i ⊆ A) := by
intro order horder i A hA
have hrank : ∀ e v, v ∈ latentCube n →
(observedLawRankCoordinate laws order e ((world θ).mix v) : ℝ) =
signedInterventionCDF s θ ((world θ).targetPerm e)
(v ((world θ).targetPerm e)) := by
intro e v hv
simpa only [laws, signedInterventionCDF] using
(hcontinuousRankClauses order horder e).2.2.2.2 v hv
exact exactRankCondIndepCharacterization s (world θ) hpos hminimal hmix hone
laws hgraph.1 order horder hrank i A hA
have hprune : ∀ order, IsTopologicalOrdering
(observedLawRatioGraph gaussianFeatureMap (world θ).law) order →
∀ i, MinimalAdmissibleParentSet laws order i (environmentParentSet (world θ) i) ∧
∀ A, MinimalAdmissibleParentSet laws order i A →
A = environmentParentSet (world θ) i := by
have hp := exactParentPruning_of_condIndepCharacterization (world θ) laws
(by simpa only [hlaws] using hgraph.2.2)
(by simpa only [hlaws] using hiff)
simpa only [hlaws] using hp
have hselected : (populationDecoder laws).2.2.edge = permutedGraph G (world θ) :=
selectedParentDAG_eq_permutedGraph_of_exactPruning (world θ) laws
(hprune (selectedTopologicalOrder laws) hgraph.2.1)
have hunique : ∀ (G' : DAG (Fin n)) (θ' : Mechanism n G')
(W' : ObservedWorld G' θ'),
PositiveNormalizedSmoothMechanisms G' θ' →
CausalMinimality G' θ' →
FixedOwnDerivativeSign G' s θ' →
SharedDiffeomorphicMixing G' θ' W' →
OnePerfectInterventionPerNode G' θ' W' →
W'.law = (world θ).law → ComponentwiseC2Equivalent (world θ) W' := by
intro G' θ' W' hpos' hminimal' hsign' hmix' hone' hlaw'
let order : Fin n → ℕ := fun e ↦ G'.topoOrder (W'.targetPerm e)
have hcommon := competitorTopologicalOrder_is_common_of_law_eq
(s := s) (world θ) W' hpos' hmix' hone' hlaw' hgraph.1
have horder : IsTopologicalOrdering
(observedLawRatioGraph gaussianFeatureMap (world θ).law) order := hcommon.1
have horder' : IsTopologicalOrdering
(observedLawRatioGraph gaussianFeatureMap W'.law) order := by
simpa only [hlaw'] using horder
have hgraphOrder : PermutedGraphOrdered (world θ) order := hcommon.2.1
have hgraphOrder' : PermutedGraphOrdered W' order := hcommon.2.2.1
have hrank : ∀ e v, v ∈ latentCube n →
(observedLawRankCoordinate laws order e ((world θ).mix v) : ℝ) =
signedInterventionCDF s θ ((world θ).targetPerm e)
(v ((world θ).targetPerm e)) := by
intro e v hv
simpa only [laws, signedInterventionCDF] using
(exactRatioDecoder_continuousRankClauses_of_order s (world θ)
hpos hmix hone hsign order horder hgraphOrder e).2.2.2.2 v hv
have hrank' : ∀ e v, v ∈ latentCube n →
(observedLawRankCoordinate laws order e (W'.mix v) : ℝ) =
signedInterventionCDF s θ' (W'.targetPerm e) (v (W'.targetPerm e)) := by
intro e v hv
simpa only [laws, hlaw', signedInterventionCDF] using
(exactRatioDecoder_continuousRankClauses_of_order s W'
hpos' hmix' hone' hsign' order horder' hgraphOrder' e).2.2.2.2 v hv
have hiff' := exactRankCondIndepCharacterization_of_order s W'
hpos' hminimal' hmix' hone' laws order horder' hgraphOrder' hrank'
have hprune' : ∀ i,
MinimalAdmissibleParentSet laws order i (environmentParentSet W' i) ∧
∀ A, MinimalAdmissibleParentSet laws order i A →
A = environmentParentSet W' i := by
apply exactParentPruning_at_order W' laws order hcommon.2.2.2.2
intro i A hA
simpa only [hlaws, hlaw'] using hiff' i A hA
have hpruneMain := hprune order horder
have hparents : ∀ i, environmentParentSet W' i = environmentParentSet (world θ) i := by
intro i
exact (hpruneMain i).2 _ (hprune' i).1
have hedge : ∀ j i,
G'.edge (((world θ).targetPerm.symm.trans W'.targetPerm) j)
(((world θ).targetPerm.symm.trans W'.targetPerm) i) ↔
G.edge j i := by
intro j i
have hm := congrArg
(fun A : Finset (Fin n) => (world θ).targetPerm.symm j ∈ A)
(hparents ((world θ).targetPerm.symm i))
simpa [environmentParentSet] using hm
have hsupport := observedSupport_eq_of_observationalLaw_eq (world θ) W'
hpos hmix hone hpos' hmix' hone' hlaw'
exact componentwiseC2Equivalent_of_common_rank_and_aligned_edges
(world θ) W' hpos hmix hpos' hmix' laws order hrank hrank' hsupport hedge
refine ⟨hgraph.1, hgraph.2.1, hcontinuousRankClauses, hgraph.2.2,
hiff, hprune, hcoherent, hselected, hunique, hbivariate, ?_⟩
exact exactRatioDecoderComparisonClauses_of_mainConclusions s (world θ)
hpos hmix hselected hunique hbivariate hgraph.2.2 hgraph.2.1
(fun i => (hcontinuousRankClauses
(selectedTopologicalOrder laws) hgraph.2.1 i).2.2.2.2)
theorem exact_ratio_decoder_faithful
Faithful-overlap corollary; faithfulness is deliberately not a premise of the main theorem. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
s :
θ :
Mechanism n G
world :
∀ θ' : Mechanism n G, ObservedWorld G θ'
hpos :
hminimal :
CausalMinimality G θ
hmix :
SharedDiffeomorphicMixing G θ (world θ)
hone :
OnePerfectInterventionPerNode G θ (world θ)
hsign :
FixedOwnDerivativeSign G s θ
hfaith :
Faithfulness G θ
hcover :
(⟨θ, ⟨hpos, hminimal, hsign⟩⟩ : StratumPoint G s) ∈ coverSeparatedSet (s := s) gaussianFeatureMap (world θ).targetPerm
conclusion
ExactRatioDecoderConclusion s θ (world θ)
Proof (Lean source)
-- keep: explicit faithful-overlap corollary stated in the paper's scope comparison
theorem exact_ratio_decoder_faithful
{n : ℕ} {G : DAG (Fin n)} (s : SignVector n)
(θ : Mechanism n G) (world : ∀ θ' : Mechanism n G, ObservedWorld G θ')
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hminimal : CausalMinimality G θ)
(hmix : SharedDiffeomorphicMixing G θ (world θ))
(hone : OnePerfectInterventionPerNode G θ (world θ))
(hsign : FixedOwnDerivativeSign G s θ)
(hfaith : Faithfulness G θ)
(hcover : (⟨θ, ⟨hpos, hminimal, hsign⟩⟩ : StratumPoint G s) ∈
coverSeparatedSet (s := s) gaussianFeatureMap (world θ).targetPerm) :
ExactRatioDecoderConclusion s θ (world θ) := by
exact exact_ratio_decoder s θ world hpos hminimal hmix hone hsign hcover
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.TGenericCoverSeparation 27 declarations This file states the open-dense direct-edge result and its residual Gaussian-MMD ancestral-cover consequence, including the empty-graph case.
Generic cover separation
This file states the open-dense direct-edge result and its residual Gaussian-MMD ancestral-cover consequence, including the empty-graph case.
theorem canonical_secondMomentContrast_perm_eq
Relabelling the canonical intervention environments does not change the latent direct-edge contrast after transporting the two environment indices back through the permutation. the stated conclusion follows.
Formal statement
hypotheses
conclusion
secondMomentContrast (canonicalObservedWorld G theta pi) (pi.symm j) (pi.symm i)
Proof (Lean source)
lemma canonical_secondMomentContrast_perm_eq
{n : ℕ} {G : DAG (Fin n)} (theta : Mechanism n G)
(pi : Perm (Fin n)) (j i : Fin n) :
secondMomentContrast (canonicalObservedWorld G theta pi) (pi.symm j) (pi.symm i) =
secondMomentContrast (canonicalObservedWorld G theta (refl (Fin n))) j i := by
unfold secondMomentContrast canonicalObservedWorld
simp
theorem edgeContrastNonzeroSet_dense
For each fixed direct edge, the corresponding canonical contrast-nonzero locus is dense in the mechanism stratum, independently of the target permutation. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
Dense {theta : StratumPoint G s | secondMomentContrast (canonicalObservedWorld G theta.1 pi) (pi.symm j) (pi.symm i) ≠ 0}
Proof (Lean source)
lemma edgeContrastNonzeroSet_dense
{n : ℕ} (G : DAG (Fin n)) (s : SignVector n)
(pi : Perm (Fin n)) {j i : Fin n} (hji : G.edge j i) :
Dense {theta : StratumPoint G s |
secondMomentContrast (canonicalObservedWorld G theta.1 pi)
(pi.symm j) (pi.symm i) ≠ 0} := by
rw [dense_iff_inter_open]
intro U hU hUne
rcases hUne with ⟨theta, hthetaU⟩
rcases isOpen_induced_iff.mp hU with ⟨V, hV, hVU⟩
have hthetaV : theta.1 ∈ V := by
have : theta ∈ val ⁻¹' V := by simpa [hVU] using hthetaU
exact this
have hIntervention : OnePerfectInterventionPerNode G theta.1
(canonicalObservedWorld G theta.1 (refl (Fin n))) :=
canonicalObservedWorld_onePerfectInterventionPerNode
theta.property.positiveSmooth (refl (Fin n))
rcases (analytic_edge_perturbation theta hji hIntervention).2.2 V
(hV.mem_nhds hthetaV) with ⟨t, htV, htStratum, htne⟩
let eta : StratumPoint G s :=
⟨affinePath s theta hji ⟨t.1, ⟨le_of_lt t.2.1, le_of_lt t.2.2⟩⟩, htStratum⟩
refine ⟨eta, ?_, ?_⟩
· have : eta ∈ val ⁻¹' V := htV
simpa [hVU] using this
· change secondMomentContrast (canonicalObservedWorld G eta.1 pi)
(pi.symm j) (pi.symm i) ≠ 0
rw [canonical_secondMomentContrast_perm_eq]
exact htne
theorem continuous_stratum_p_valueCoordinate
The observational value coordinate is continuous on the mechanism stratum by construction of the induced product C² topology. the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma continuous_stratum_p_valueCoordinate
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n} (i : Fin n) :
Continuous (fun theta : StratumPoint G s ↦
ofFun {latentCube n} (theta.1.p i)) := by
have hcoordinates : Continuous
(fun theta : StratumPoint G s ↦ mechanismC2Coordinates theta.1) :=
(continuous_induced_dom (f := @mechanismC2Coordinates n G)).comp continuous_subtype_val
unfold mechanismC2Coordinates at hcoordinates
exact (continuous_apply i).comp (continuous_fst.comp hcoordinates)
theorem continuous_stratum_q_valueCoordinate
The intervention value coordinate is continuous on the mechanism stratum by construction of the induced product C² topology. the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma continuous_stratum_q_valueCoordinate
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n} (i : Fin n) :
Continuous (fun theta : StratumPoint G s ↦
ofFun {Icc (0 : ℝ) 1} (theta.1.q i)) := by
have hcoordinates : Continuous
(fun theta : StratumPoint G s ↦ mechanismC2Coordinates theta.1) :=
(continuous_induced_dom (f := @mechanismC2Coordinates n G)).comp continuous_subtype_val
unfold mechanismC2Coordinates at hcoordinates
exact (continuous_apply i).comp
(continuous_fst.comp (continuous_snd.comp
(continuous_snd.comp (continuous_snd.comp hcoordinates))))
theorem continuous_uniformFun_eval₂_of_continuous
Uniform convergence of a continuous family, combined with continuous motion of the argument, gives joint continuity of evaluation. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
Continuous (fun z : X × alpha ↦ toFun (F z.1) z.2)
Proof (Lean source)
lemma continuous_uniformFun_eval₂_of_continuous
{X alpha beta : Type*} [TopologicalSpace X] [TopologicalSpace alpha]
[UniformSpace beta] (F : X → UniformFun alpha beta) (hF : Continuous F)
(hc : ∀ x, Continuous (toFun (F x))) :
Continuous (fun z : X × alpha ↦ toFun (F z.1) z.2) := by
rw [continuous_iff_continuousAt]
intro z
have hFt : Tendsto (F ∘ fst) (nhds z) (nhds (F z.1)) :=
hF.continuousAt.comp continuousAt_fst
have hu := UniformFun.tendsto_iff_tendstoUniformly.mp hFt
exact hu.tendsto_comp (hc z.1).continuousAt continuousAt_snd
theorem continuous_stratum_p_eval
Observational-factor evaluation is jointly continuous in a stratum mechanism and a compact latent state. the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma continuous_stratum_p_eval
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n} (i : Fin n) :
Continuous (fun z : StratumPoint G s × {v : LatentState n // v ∈ latentCube n} ↦
z.1.1.p i z.2.1) := by
let F : StratumPoint G s →
UniformFun {v : LatentState n // v ∈ latentCube n} ℝ := fun theta ↦
UniformFun.ofFun ((latentCube n).domRestrict (theta.1.p i))
have hF : Continuous F := by
have h := (UniformOnFun.continuous_rng_iff.mp
(continuous_stratum_p_valueCoordinate (G := G) (s := s) i)
(latentCube n) (by simp))
simpa [F, Function.comp_def] using h
apply continuous_uniformFun_eval₂_of_continuous F hF
intro theta
exact (theta.property.positiveSmooth.2.2.1 i).continuousOn.restrict
theorem continuous_stratum_q_eval
Intervention-factor evaluation is jointly continuous in a stratum mechanism and a unit coordinate. the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma continuous_stratum_q_eval
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n} (i : Fin n) :
Continuous (fun z : StratumPoint G s × {x : ℝ // x ∈ Icc (0 : ℝ) 1} ↦
z.1.1.q i z.2.1) := by
let F : StratumPoint G s →
UniformFun {x : ℝ // x ∈ Icc (0 : ℝ) 1} ℝ := fun theta ↦
UniformFun.ofFun (Icc (0 : ℝ) 1 |>.domRestrict (theta.1.q i))
have hF : Continuous F := by
have h := (UniformOnFun.continuous_rng_iff.mp
(continuous_stratum_q_valueCoordinate (G := G) (s := s) i)
(Icc (0 : ℝ) 1) (by simp))
simpa [F, Function.comp_def] using h
apply continuous_uniformFun_eval₂_of_continuous F hF
intro theta
exact (theta.property.positiveSmooth.2.2.2.1 i).continuousOn.restrict
theorem continuous_compactCube_projection
Coordinatewise projection supplies a continuous ambient representative of a latent-cube point. the stated conclusion follows.
Formal statement
hypotheses
n :
ℕ
conclusion
Proof (Lean source)
lemma continuous_compactCube_projection (n : ℕ) :
Continuous (fun v : LatentState n ↦
compactCubeInclude n (compactCubeRetract n v)) := by
unfold compactCubeInclude compactCubeRetract
fun_prop
theorem compactCube_projection_mem
The coordinatewise compact-cube projection belongs to the latent cube.
Formal statement
hypotheses
n :
ℕ
v :
conclusion
Proof (Lean source)
lemma compactCube_projection_mem (n : ℕ) (v : LatentState n) :
compactCubeInclude n (compactCubeRetract n v) ∈ latentCube n := by
intro i _
exact (projIcc 0 1 (by norm_num) (v i)).property
theorem continuous_stratum_contrastIntegrand
The compact-cube rational contrast integrand, continuously extended to the ambient latent space by coordinatewise projection, is jointly continuous in the mechanism and latent state. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
n :
hji :
G.edge j i
conclusion
Continuous (fun z : StratumPoint G s × LatentState n ↦ let v := compactCubeInclude n (compactCubeRetract n z.2) (z.1.1.q i (v i)) ^ 2 * (z.1.1.q j (v j) - z.1.1.p j v) * (∏ l ∈ (Finset.univ.erase i).erase j, z.1.1.p l v) / z.1.1.p i v)
Proof (Lean source)
lemma continuous_stratum_contrastIntegrand
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
{j i : Fin n} (hji : G.edge j i) :
Continuous (fun z : StratumPoint G s × LatentState n ↦
let v := compactCubeInclude n (compactCubeRetract n z.2)
(z.1.1.q i (v i)) ^ 2 * (z.1.1.q j (v j) - z.1.1.p j v) *
(∏ l ∈ (Finset.univ.erase i).erase j, z.1.1.p l v) / z.1.1.p i v) := by
let R : LatentState n → LatentState n := fun v ↦
compactCubeInclude n (compactCubeRetract n v)
have hR : Continuous R := continuous_compactCube_projection n
have hRmem : ∀ v, R v ∈ latentCube n := compactCube_projection_mem n
have hp (l : Fin n) : Continuous (fun z : StratumPoint G s × LatentState n ↦
z.1.1.p l (R z.2)) := by
exact (continuous_stratum_p_eval (G := G) (s := s) l).comp
(continuous_fst.prodMk
((hR.comp continuous_snd).subtype_mk (fun z ↦ hRmem z.2)))
have hq (l : Fin n) : Continuous (fun z : StratumPoint G s × LatentState n ↦
z.1.1.q l (R z.2 l)) := by
exact (continuous_stratum_q_eval (G := G) (s := s) l).comp
(continuous_fst.prodMk (((continuous_apply l).comp (hR.comp continuous_snd)).subtype_mk
(fun z ↦ hRmem z.2 l (Set.mem_univ l))))
dsimp only
apply Continuous.div
· exact (((hq i).pow 2).mul ((hq j).sub (hp j))).mul
(continuous_finsetProd _ fun l _ ↦ hp l)
· exact hp i
· intro z
exact ne_of_gt (z.1.property.positiveSmooth.1 i (R z.2) (hRmem z.2))
theorem continuous_canonical_secondMomentContrast
A canonical direct-edge second-moment contrast varies continuously with the mechanism in the induced relative product C² topology. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
Continuous (fun theta : StratumPoint G s ↦ secondMomentContrast (canonicalObservedWorld G theta.1 pi) (pi.symm j) (pi.symm i))
Proof (Lean source)
lemma continuous_canonical_secondMomentContrast
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(pi : Perm (Fin n)) {j i : Fin n} (hji : G.edge j i) :
Continuous (fun theta : StratumPoint G s ↦
secondMomentContrast (canonicalObservedWorld G theta.1 pi)
(pi.symm j) (pi.symm i)) := by
let F : StratumPoint G s → LatentState n → ℝ := fun theta v ↦
(theta.1.q i (v i)) ^ 2 * (theta.1.q j (v j) - theta.1.p j v) *
(∏ l ∈ (Finset.univ.erase i).erase j, theta.1.p l v) / theta.1.p i v
letI : CompactSpace {v : LatentState n // v ∈ latentCube n} :=
isCompact_iff_compactSpace.mp (by
rw [latentCube]
exact isCompact_univ_pi fun _ ↦ isCompact_Icc)
have hFcont (theta : StratumPoint G s) : ContinuousOn (F theta) (latentCube n) := by
have hq (l : Fin n) : ContinuousOn (fun v : LatentState n ↦ theta.1.q l (v l))
(latentCube n) :=
(theta.property.positiveSmooth.2.2.2.1 l).continuousOn.comp
(continuous_apply l).continuousOn (fun v hv ↦ hv l (Set.mem_univ l))
have hp (l : Fin n) := (theta.property.positiveSmooth.2.2.1 l).continuousOn
exact (((hq i).pow 2).mul ((hq j).sub (hp j))).mul
(continuousOn_finsetProd _ fun l _ ↦ hp l) |>.div (hp i)
(fun v hv ↦ ne_of_gt (theta.property.positiveSmooth.1 i v hv))
have hJoint : Continuous (fun z : StratumPoint G s ×
{v : LatentState n // v ∈ latentCube n} ↦ F z.1 z.2.1) := by
have h := (continuous_stratum_contrastIntegrand (G := G) (s := s) hji).comp
(show Continuous (fun z : StratumPoint G s ×
{v : LatentState n // v ∈ latentCube n} ↦ (z.1, z.2.1)) by fun_prop)
convert h using 1
funext z
dsimp [F]
rw [compactCubeInclude_retract_of_mem z.2.property]
let FC : StratumPoint G s → C({v : LatentState n // v ∈ latentCube n}, ℝ) := fun theta ↦
⟨fun v ↦ F theta v.1, (hFcont theta).restrict⟩
have hFC : Continuous FC := by
apply ContinuousMap.continuous_of_continuous_uncurry
change Continuous (fun z : StratumPoint G s ×
{v : LatentState n // v ∈ latentCube n} ↦ F z.1 z.2.1)
exact hJoint
have hUniform : Continuous (fun theta ↦
ofFun {latentCube n} (F theta)) := by
apply (ContinuousOn.continuous_domRestrict_iff_continuous_uniformOnFun hFcont).mp
exact hFC
have hmu : MeasureTheory.volume (latentCube n) < ⊤ := (show IsCompact (latentCube n) by
rw [latentCube]
exact isCompact_univ_pi fun _ ↦ isCompact_Icc).measure_lt_top
have hInt (theta : StratumPoint G s) :
IntegrableOn (F theta) (latentCube n) :=
(hFcont theta).integrableOn_compact (by
rw [latentCube]
exact isCompact_univ_pi fun _ ↦ isCompact_Icc)
have hint : Continuous (fun theta ↦ ∫ v in latentCube n, F theta v) :=
continuous_setIntegral_of_continuous_uniformOn
MeasureTheory.volume
(latentCube n) hmu F hUniform hInt
have hne : j ≠ i := fun h ↦ by subst j; exact G.irrefl i hji
rw [show (fun theta : StratumPoint G s ↦
secondMomentContrast (canonicalObservedWorld G theta.1 pi)
(pi.symm j) (pi.symm i)) = fun theta ↦ ∫ v in latentCube n, F theta v by
funext theta
rw [canonical_secondMomentContrast_perm_eq,
canonical_secondMomentContrast_eq_integral theta.property.positiveSmooth hne]]
exact hint
theorem continuous_gaussianFeature
The explicit Gaussian feature vector depends continuously on its scalar argument. the stated conclusion follows.
Formal statement
conclusion
Proof (Lean source)
lemma continuous_gaussianFeature : Continuous gaussianFeature := by
rw [continuous_iff_continuousAt]
intro r
have hnormsq (x : ℝ) : ‖gaussianFeature x - gaussianFeature r‖ ^ 2 =
2 - 2 * gaussianKernel x r := by
rw [← real_inner_self_eq_norm_sq]
simp only [inner_sub_left, inner_sub_right, gaussianFeature_inner]
simp [gaussianKernel]
ring
have hsquare : ContinuousAt (fun x : ℝ =>
2 - 2 * gaussianKernel x r) r := by
unfold gaussianKernel
fun_prop
have hzero : 2 - 2 * gaussianKernel r r = 0 := by
simp [gaussianKernel]
rw [ContinuousAt, tendsto_iff_norm_sub_tendsto_zero]
have hsqrt := Real.continuous_sqrt.continuousAt.comp hsquare
change Tendsto (fun x : ℝ => sqrt
(2 - 2 * gaussianKernel x r)) (nhds r)
(nhds (sqrt (2 - 2 * gaussianKernel r r))) at hsqrt
have hsqrt' : Tendsto (fun x : ℝ => sqrt
(2 - 2 * gaussianKernel x r)) (nhds r)
(nhds (sqrt (2 - 2 * gaussianKernel r r))) := by
exact hsqrt
rw [hzero, Real.sqrt_zero] at hsqrt'
convert hsqrt' using 1
funext x
rw [← hnormsq x, Real.sqrt_sq_eq_abs, abs_of_nonneg (norm_nonneg _)]
theorem continuous_stratum_observationalGaussianIntegrand
The observational Gaussian-embedding integrand is jointly continuous on the mechanism stratum and compact latent cube. the stated conclusion follows.
Formal statement
hypotheses
n :
i :
Fin n
conclusion
Continuous (fun z : StratumPoint G s × {v : LatentState n // v ∈ latentCube n} ↦ observationalDensity z.1.1 z.2.1 • gaussianFeature (z.1.1.q i (z.2.1 i) / z.1.1.p i z.2.1))
Proof (Lean source)
lemma continuous_stratum_observationalGaussianIntegrand
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n} (i : Fin n) :
Continuous (fun z : StratumPoint G s ×
{v : LatentState n // v ∈ latentCube n} ↦
observationalDensity z.1.1 z.2.1 •
gaussianFeature (z.1.1.q i (z.2.1 i) / z.1.1.p i z.2.1)) := by
have hp (l : Fin n) : Continuous (fun z : StratumPoint G s ×
{v : LatentState n // v ∈ latentCube n} ↦ z.1.1.p l z.2.1) :=
continuous_stratum_p_eval (G := G) (s := s) l
have hq (l : Fin n) : Continuous (fun z : StratumPoint G s ×
{v : LatentState n // v ∈ latentCube n} ↦ z.1.1.q l (z.2.1 l)) := by
have hc : Continuous (fun z : StratumPoint G s ×
{v : LatentState n // v ∈ latentCube n} ↦ z.2.1 l) :=
(continuous_apply l).comp (continuous_subtype_val.comp continuous_snd)
have hc' : Continuous (fun z : StratumPoint G s ×
{v : LatentState n // v ∈ latentCube n} ↦
(⟨z.2.1 l, z.2.property l (Set.mem_univ l)⟩ :
{x : ℝ // x ∈ Icc (0 : ℝ) 1})) :=
hc.subtype_mk _
convert (continuous_stratum_q_eval (G := G) (s := s) l).comp
(continuous_fst.prodMk hc') using 1 <;> rfl
have hr : Continuous (fun z : StratumPoint G s ×
{v : LatentState n // v ∈ latentCube n} ↦
z.1.1.q i (z.2.1 i) / z.1.1.p i z.2.1) := by
exact (hq i).div (hp i) fun z ↦
ne_of_gt (z.1.property.positiveSmooth.1 i z.2.1 z.2.property)
unfold observationalDensity
exact (continuous_finsetProd _ fun l _ ↦ hp l).smul
(continuous_gaussianFeature.comp hr)
theorem continuous_stratum_interventionalGaussianIntegrand
The target-interventional Gaussian-embedding integrand is jointly continuous on the mechanism stratum and compact latent cube. the stated conclusion follows.
Formal statement
hypotheses
n :
j i :
Fin n
conclusion
Continuous (fun z : StratumPoint G s × {v : LatentState n // v ∈ latentCube n} ↦ interventionalDensity z.1.1 j z.2.1 • gaussianFeature (z.1.1.q i (z.2.1 i) / z.1.1.p i z.2.1))
Proof (Lean source)
lemma continuous_stratum_interventionalGaussianIntegrand
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n} (j i : Fin n) :
Continuous (fun z : StratumPoint G s ×
{v : LatentState n // v ∈ latentCube n} ↦
interventionalDensity z.1.1 j z.2.1 •
gaussianFeature (z.1.1.q i (z.2.1 i) / z.1.1.p i z.2.1)) := by
have hp (l : Fin n) : Continuous (fun z : StratumPoint G s ×
{v : LatentState n // v ∈ latentCube n} ↦ z.1.1.p l z.2.1) :=
continuous_stratum_p_eval (G := G) (s := s) l
have hq (l : Fin n) : Continuous (fun z : StratumPoint G s ×
{v : LatentState n // v ∈ latentCube n} ↦ z.1.1.q l (z.2.1 l)) := by
have hc : Continuous (fun z : StratumPoint G s ×
{v : LatentState n // v ∈ latentCube n} ↦ z.2.1 l) :=
(continuous_apply l).comp (continuous_subtype_val.comp continuous_snd)
have hc' : Continuous (fun z : StratumPoint G s ×
{v : LatentState n // v ∈ latentCube n} ↦
(⟨z.2.1 l, z.2.property l (Set.mem_univ l)⟩ :
{x : ℝ // x ∈ Icc (0 : ℝ) 1})) :=
hc.subtype_mk _
convert (continuous_stratum_q_eval (G := G) (s := s) l).comp
(continuous_fst.prodMk hc') using 1 <;> rfl
have hr : Continuous (fun z : StratumPoint G s ×
{v : LatentState n // v ∈ latentCube n} ↦
z.1.1.q i (z.2.1 i) / z.1.1.p i z.2.1) := by
exact (hq i).div (hp i) fun z ↦
ne_of_gt (z.1.property.positiveSmooth.1 i z.2.1 z.2.property)
unfold interventionalDensity
exact ((hq j).mul (continuous_finsetProd _ fun l _ ↦ hp l)).smul
(continuous_gaussianFeature.comp hr)
theorem canonical_observationalMeanEmbedding_eq_setIntegral
The canonical observational ratio-law embedding is the explicit weighted cube integral. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
hpos :
i :
Fin n
conclusion
= ∫ v in latentCube n, observationalDensity θ v • gaussianFeature (θ.q (π i) (v (π i)) / θ.p (π i) v)
Proof (Lean source)
lemma canonical_observationalMeanEmbedding_eq_setIntegral
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(π : Perm (Fin n)) (i : Fin n) :
meanEmbedding gaussianFeatureMap
(observationalRatioLaw (canonicalObservedWorld G θ π) i) =
∫ v in latentCube n, observationalDensity θ v •
gaussianFeature (θ.q (π i) (v (π i)) / θ.p (π i) v) := by
let W := canonicalObservedWorld G θ π
let r : LatentState n → ℝ := fun v ↦
θ.q (π i) (v (π i)) / θ.p (π i) v
let μ : Measure (LatentState n) :=
MeasureTheory.volume.restrict (latentCube n)
have hcube : MeasurableSet (latentCube n) := by
rw [latentCube]
exact MeasurableSet.univ_pi fun _ ↦ measurableSet_Icc
have hobsCube : ∀ᵐ v ∂observationalLaw θ, v ∈ latentCube n :=
observationalLaw_ae_mem_latentCube hpos
have hone := canonicalObservedWorld_onePerfectInterventionPerNode hpos π
have hr : observedLawRatio W.law i =ᵐ[observationalLaw θ] r := by
filter_upwards [hone.2.2.1 i, hobsCube] with v hrv hv
change ((interventionalLaw θ (π i)).rnDeriv (observationalLaw θ) v).toReal = r v
calc
_ = (ofReal (W.ratio i v)).toReal := congrArg toReal hrv.symm
_ = W.ratio i v := ENNReal.toReal_ofReal
(div_nonneg (hpos.2.1 _ _ (hv (π i) (Set.mem_univ _))).le
(hpos.1 _ _ hv).le)
_ = r v := by rfl
have hobscont : ContinuousOn (observationalDensity θ) (latentCube n) := by
unfold observationalDensity
exact continuousOn_finsetProd _ fun l _ ↦ (hpos.2.2.1 l).continuousOn
unfold meanEmbedding observationalRatioLaw
change (∫ x, gaussianFeature x ∂MeasureTheory.Measure.map
(observedLawRatio W.law i) (observationalLaw θ)) = _
rw [MeasureTheory.integral_map
(measurable_observedLawRatio W.law i).aemeasurable
continuous_gaussianFeature.aestronglyMeasurable]
calc
(∫ v, gaussianFeature (observedLawRatio W.law i v) ∂observationalLaw θ) =
∫ v, gaussianFeature (r v) ∂observationalLaw θ := by
apply MeasureTheory.integral_congr_ae
filter_upwards [hr] with v hv
rw [hv]
_ = ∫ v, observationalDensity θ v • gaussianFeature (r v) ∂μ := by
unfold observationalLaw
rw [integral_withDensity_eq_integral_toReal_smul₀]
· apply MeasureTheory.integral_congr_ae
filter_upwards [MeasureTheory.ae_restrict_mem hcube] with v hv
rw [ENNReal.toReal_ofReal (le_of_lt (by
unfold observationalDensity
exact Finset.prod_pos fun l _ ↦ hpos.1 l v hv))]
· exact (hobscont.aestronglyMeasurable hcube).aemeasurable.ennreal_ofReal
· filter_upwards with v
exact ENNReal.ofReal_lt_top
_ = _ := by rfl
theorem canonical_interventionalMeanEmbedding_eq_setIntegral
The canonical interventional ratio-law embedding is the explicit weighted cube integral. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
hpos :
j i :
Fin n
conclusion
= ∫ v in latentCube n, interventionalDensity θ (π j) v • gaussianFeature (θ.q (π i) (v (π i)) / θ.p (π i) v)
Proof (Lean source)
lemma canonical_interventionalMeanEmbedding_eq_setIntegral
{n : ℕ} {G : DAG (Fin n)} {θ : Mechanism n G}
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(π : Perm (Fin n)) (j i : Fin n) :
meanEmbedding gaussianFeatureMap
(interventionalRatioLaw (canonicalObservedWorld G θ π) j i) =
∫ v in latentCube n, interventionalDensity θ (π j) v •
gaussianFeature (θ.q (π i) (v (π i)) / θ.p (π i) v) := by
let W := canonicalObservedWorld G θ π
let r : LatentState n → ℝ := fun v ↦
θ.q (π i) (v (π i)) / θ.p (π i) v
let μ : Measure (LatentState n) :=
MeasureTheory.volume.restrict (latentCube n)
have hcube : MeasurableSet (latentCube n) := by
rw [latentCube]
exact MeasurableSet.univ_pi fun _ ↦ measurableSet_Icc
have hobsCube : ∀ᵐ v ∂observationalLaw θ, v ∈ latentCube n :=
observationalLaw_ae_mem_latentCube hpos
have hone := canonicalObservedWorld_onePerfectInterventionPerNode hpos π
have hrObs : observedLawRatio W.law i =ᵐ[observationalLaw θ] r := by
filter_upwards [hone.2.2.1 i, hobsCube] with v hrv hv
change ((interventionalLaw θ (π i)).rnDeriv (observationalLaw θ) v).toReal = r v
calc
_ = (ofReal (W.ratio i v)).toReal := congrArg toReal hrv.symm
_ = W.ratio i v := ENNReal.toReal_ofReal
(div_nonneg (hpos.2.1 _ _ (hv (π i) (Set.mem_univ _))).le
(hpos.1 _ _ hv).le)
_ = r v := by rfl
have hac : interventionalLaw θ (π j) ≪ observationalLaw θ := by
simpa only [W, canonicalObservedWorld] using
interventionalLaw_absolutelyContinuous_observational W hpos j
have hr : observedLawRatio W.law i =ᵐ[interventionalLaw θ (π j)] r :=
hac.ae_eq hrObs
have hintcont : ContinuousOn (interventionalDensity θ (π j)) (latentCube n) := by
unfold interventionalDensity
have hq : ContinuousOn (fun v : LatentState n ↦ θ.q (π j) (v (π j)))
(latentCube n) :=
(hpos.2.2.2.1 (π j)).continuousOn.comp
((continuous_apply (π j)).continuousOn)
(fun v hv ↦ hv (π j) (Set.mem_univ _))
exact hq.mul (continuousOn_finsetProd _ fun l _ ↦
(hpos.2.2.1 l).continuousOn)
unfold meanEmbedding interventionalRatioLaw
change (∫ x, gaussianFeature x ∂MeasureTheory.Measure.map
(observedLawRatio W.law i) (interventionalLaw θ (π j))) = _
rw [MeasureTheory.integral_map
(measurable_observedLawRatio W.law i).aemeasurable
continuous_gaussianFeature.aestronglyMeasurable]
calc
(∫ v, gaussianFeature (observedLawRatio W.law i v)
∂interventionalLaw θ (π j)) =
∫ v, gaussianFeature (r v) ∂interventionalLaw θ (π j) := by
apply MeasureTheory.integral_congr_ae
filter_upwards [hr] with v hv
rw [hv]
_ = ∫ v, interventionalDensity θ (π j) v • gaussianFeature (r v) ∂μ := by
unfold interventionalLaw
rw [integral_withDensity_eq_integral_toReal_smul₀]
· apply MeasureTheory.integral_congr_ae
filter_upwards [MeasureTheory.ae_restrict_mem hcube] with v hv
rw [ENNReal.toReal_ofReal (le_of_lt (by
unfold interventionalDensity
exact mul_pos (hpos.2.1 (π j) (v (π j))
(hv (π j) (Set.mem_univ _)))
(Finset.prod_pos fun l _ ↦ hpos.1 l v hv)))]
· exact (hintcont.aestronglyMeasurable hcube).aemeasurable.ennreal_ofReal
· filter_upwards with v
exact ENNReal.ofReal_lt_top
_ = _ := by rfl
theorem continuous_canonical_populationDiscrepancy
The canonical Gaussian population discrepancy varies continuously with the mechanism. the stated conclusion follows.
Formal statement
hypotheses
conclusion
Continuous (fun θ : StratumPoint G s ↦ populationDiscrepancy gaussianFeatureMap (canonicalObservedWorld G θ.1 π) (π.symm j) (π.symm i))
Proof (Lean source)
lemma continuous_canonical_populationDiscrepancy
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
(π : Perm (Fin n)) (j i : Fin n) :
Continuous (fun θ : StratumPoint G s ↦
populationDiscrepancy gaussianFeatureMap
(canonicalObservedWorld G θ.1 π) (π.symm j) (π.symm i)) := by
let Fobs := fun θ : StratumPoint G s ↦ fun v : LatentState n ↦
observationalDensity θ.1 v •
gaussianFeature (θ.1.q i (v i) / θ.1.p i v)
let Fint := fun θ : StratumPoint G s ↦ fun v : LatentState n ↦
interventionalDensity θ.1 j v •
gaussianFeature (θ.1.q i (v i) / θ.1.p i v)
letI : CompactSpace {v : LatentState n // v ∈ latentCube n} :=
isCompact_iff_compactSpace.mp (by
rw [latentCube]
exact isCompact_univ_pi fun _ ↦ isCompact_Icc)
have hcompact : IsCompact (latentCube n) := by
rw [latentCube]
exact isCompact_univ_pi fun _ ↦ isCompact_Icc
have hobsJoint : Continuous (fun z : StratumPoint G s ×
{v : LatentState n // v ∈ latentCube n} ↦ Fobs z.1 z.2.1) := by
exact continuous_stratum_observationalGaussianIntegrand
(G := G) (s := s) i
have hintJoint : Continuous (fun z : StratumPoint G s ×
{v : LatentState n // v ∈ latentCube n} ↦ Fint z.1 z.2.1) := by
exact continuous_stratum_interventionalGaussianIntegrand
(G := G) (s := s) j i
have hobsOn (θ : StratumPoint G s) : ContinuousOn (Fobs θ) (latentCube n) := by
apply continuousOn_iff_continuous_restrict.mpr
exact hobsJoint.comp (continuous_const.prodMk continuous_id)
have hintOn (θ : StratumPoint G s) : ContinuousOn (Fint θ) (latentCube n) := by
apply continuousOn_iff_continuous_restrict.mpr
exact hintJoint.comp (continuous_const.prodMk continuous_id)
let Cobs : StratumPoint G s →
C({v : LatentState n // v ∈ latentCube n}, lp (fun _ : ℕ => ℝ) 2) :=
fun θ ↦ ⟨fun v ↦ Fobs θ v.1, (hobsOn θ).restrict⟩
let Cint : StratumPoint G s →
C({v : LatentState n // v ∈ latentCube n}, lp (fun _ : ℕ => ℝ) 2) :=
fun θ ↦ ⟨fun v ↦ Fint θ v.1, (hintOn θ).restrict⟩
have hCobs : Continuous Cobs := by
apply ContinuousMap.continuous_of_continuous_uncurry
exact hobsJoint
have hCint : Continuous Cint := by
apply ContinuousMap.continuous_of_continuous_uncurry
exact hintJoint
have hobsUniform : Continuous (fun θ ↦
ofFun {latentCube n} (Fobs θ)) := by
apply (ContinuousOn.continuous_domRestrict_iff_continuous_uniformOnFun hobsOn).mp
exact hCobs
have hintUniform : Continuous (fun θ ↦
ofFun {latentCube n} (Fint θ)) := by
apply (ContinuousOn.continuous_domRestrict_iff_continuous_uniformOnFun hintOn).mp
exact hCint
have hmu : volume (latentCube n) < ⊤ := hcompact.measure_lt_top
have hobsInt (θ : StratumPoint G s) : IntegrableOn (Fobs θ) (latentCube n) :=
(hobsOn θ).integrableOn_compact hcompact
have hintInt (θ : StratumPoint G s) : IntegrableOn (Fint θ) (latentCube n) :=
(hintOn θ).integrableOn_compact hcompact
have hobsIntegral : Continuous (fun θ ↦
∫ v in latentCube n, Fobs θ v) :=
continuous_setIntegral_of_continuous_uniformOn
volume (latentCube n) hmu
Fobs hobsUniform hobsInt
have hintIntegral : Continuous (fun θ ↦
∫ v in latentCube n, Fint θ v) :=
continuous_setIntegral_of_continuous_uniformOn
volume (latentCube n) hmu
Fint hintUniform hintInt
rw [show (fun θ : StratumPoint G s ↦
populationDiscrepancy gaussianFeatureMap
(canonicalObservedWorld G θ.1 π) (π.symm j) (π.symm i)) =
fun θ ↦ ‖(∫ v in latentCube n, Fobs θ v) -
∫ v in latentCube n, Fint θ v‖ by
funext θ
unfold populationDiscrepancy
rw [canonical_observationalMeanEmbedding_eq_setIntegral
θ.property.positiveSmooth π (π.symm i),
canonical_interventionalMeanEmbedding_eq_setIntegral
θ.property.positiveSmooth π (π.symm j) (π.symm i)]
simp only [Equiv.apply_symm_apply, Fobs, Fint]]
exact (hobsIntegral.sub hintIntegral).norm
theorem dense_biInter_finset_of_open
A finite intersection of open dense sets is dense, without any Baire-space assumption. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma dense_biInter_finset_of_open
{X I : Type*} [TopologicalSpace X] [DecidableEq I]
(S : Finset I) (A : I → Set X)
(hopen : ∀ i ∈ S, IsOpen (A i)) (hdense : ∀ i ∈ S, Dense (A i)) :
Dense (⋂ i ∈ S, A i) := by
classical
induction S using Finset.induction_on with
| empty => simp
| @insert a S ha ih =>
rw [show (⋂ i ∈ insert a S, A i) = A a ∩ ⋂ i ∈ S, A i by
ext x
simp [ha]]
exact (hdense a (by simp)).inter_of_isOpen_left
(ih (fun i hi ↦ hopen i (by simp [hi])) (fun i hi ↦ hdense i (by simp [hi])))
(hopen a (by simp))
theorem edgeSeparatedSet_isOpen
Direct-edge contrast separation is open in the stratum topology. the stated conclusion follows.
Formal statement
hypotheses
conclusion
IsOpen (edgeSeparatedSet (G := G) (s := s) pi)
Proof (Lean source)
lemma edgeSeparatedSet_isOpen
{n : ℕ} (G : DAG (Fin n)) (s : SignVector n)
(pi : Perm (Fin n)) :
IsOpen (edgeSeparatedSet (G := G) (s := s) pi) := by
rw [isOpen_iff_mem_nhds]
intro theta htheta
have hall : ∀ᶠ eta in nhds theta, ∀ j i, G.edge j i →
secondMomentContrast (canonicalObservedWorld G eta.1 pi)
(pi.symm j) (pi.symm i) ≠ 0 := by
have hpairs : ∀ᶠ eta in nhds theta, ∀ ji : Fin n × Fin n,
ji ∈ (Set.univ : Set (Fin n × Fin n)) → G.edge ji.1 ji.2 →
secondMomentContrast (canonicalObservedWorld G eta.1 pi)
(pi.symm ji.1) (pi.symm ji.2) ≠ 0 :=
(Filter.eventually_all_finite
(l := nhds theta)
(p := fun ji eta ↦ G.edge ji.1 ji.2 →
secondMomentContrast (canonicalObservedWorld G eta.1 pi)
(pi.symm ji.1) (pi.symm ji.2) ≠ 0)
(Set.toFinite (Set.univ : Set (Fin n × Fin n)))).2
(fun ji _ ↦ by
by_cases hedge : G.edge ji.1 ji.2
· have hc := continuous_canonical_secondMomentContrast
(G := G) (s := s) pi hedge
filter_upwards [hc.continuousAt (isOpen_compl_singleton.mem_nhds
(by simpa [edgeSeparatedSet] using htheta hedge))] with eta heta
exact fun _ ↦ heta
· exact Filter.Eventually.of_forall fun _ h ↦ (hedge h).elim)
filter_upwards [hpairs] with eta heta
exact fun j i hji ↦ heta (j, i) (Set.mem_univ _) hji
filter_upwards [hall] with eta heta
exact heta
theorem edgeSeparatedSet_dense
Direct-edge contrast separation is dense in the stratum topology. the stated conclusion follows.
Formal statement
hypotheses
conclusion
Dense (edgeSeparatedSet (G := G) (s := s) pi)
Proof (Lean source)
lemma edgeSeparatedSet_dense
{n : ℕ} (G : DAG (Fin n)) (s : SignVector n)
(pi : Perm (Fin n)) :
Dense (edgeSeparatedSet (G := G) (s := s) pi) := by
let A : (Fin n × Fin n) → Set (StratumPoint G s) := fun ji ↦
if h : G.edge ji.1 ji.2 then
{theta | secondMomentContrast (canonicalObservedWorld G theta.1 pi)
(pi.symm ji.1) (pi.symm ji.2) ≠ 0}
else Set.univ
have hopen : ∀ ji ∈ (Finset.univ : Finset (Fin n × Fin n)), IsOpen (A ji) := by
intro ji _
dsimp [A]
split_ifs with h
· exact (continuous_canonical_secondMomentContrast
(G := G) (s := s) pi h).isOpen_preimage {0}ᶜ isOpen_compl_singleton
· exact isOpen_univ
have hdense : ∀ ji ∈ (Finset.univ : Finset (Fin n × Fin n)), Dense (A ji) := by
intro ji _
dsimp [A]
split_ifs with h
· exact edgeContrastNonzeroSet_dense G s pi h
· exact dense_univ
have h := dense_biInter_finset_of_open (Finset.univ : Finset (Fin n × Fin n))
A hopen hdense
apply h.mono
intro theta htheta j i hji
have hmem : theta ∈ A (j, i) :=
Set.mem_iInter.mp (Set.mem_iInter.mp htheta (j, i)) (Finset.mem_univ _)
simpa [A, hji] using hmem
theorem coverSeparatedSet_isOpen
Ancestral-cover Gaussian-MMD separation is open in the mechanism stratum. the stated conclusion follows.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma coverSeparatedSet_isOpen
{n : ℕ} (G : DAG (Fin n)) (s : SignVector n)
(π : Perm (Fin n)) :
IsOpen (coverSeparatedSet (G := G) (s := s) gaussianFeatureMap π) := by
rw [isOpen_iff_mem_nhds]
intro θ hθ
have hpairs : ∀ᶠ η in nhds θ, ∀ ji : Fin n × Fin n,
ji ∈ (Set.univ : Set (Fin n × Fin n)) → ancestralCover G ji.1 ji.2 →
0 < populationDiscrepancy gaussianFeatureMap
(canonicalObservedWorld G η.1 π) (π.symm ji.1) (π.symm ji.2) :=
(Filter.eventually_all_finite
(l := nhds θ)
(p := fun ji η ↦ ancestralCover G ji.1 ji.2 →
0 < populationDiscrepancy gaussianFeatureMap
(canonicalObservedWorld G η.1 π) (π.symm ji.1) (π.symm ji.2))
(Set.toFinite (Set.univ : Set (Fin n × Fin n)))).2
(fun ji _ ↦ by
by_cases hcover : ancestralCover G ji.1 ji.2
· have hc := continuous_canonical_populationDiscrepancy
(G := G) (s := s) π ji.1 ji.2
filter_upwards [hc.continuousAt
(isOpen_Ioi.mem_nhds (hθ hcover))] with η hη
exact fun _ ↦ hη
· exact Filter.Eventually.of_forall fun _ h ↦ (hcover h).elim)
filter_upwards [hpairs] with η hη
exact fun j i hji ↦ hη (j, i) (Set.mem_univ _) hji
theorem ancestralCover_edge
An ancestral cover in a DAG is necessarily a direct edge. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
Proof (Lean source)
lemma ancestralCover_edge
{n : ℕ} {G : DAG (Fin n)} {j i : Fin n}
(hji : ancestralCover G j i) : G.edge j i := by
have hanc : G.isAncestor j i :=
@CovBy.lt (Fin n) ⟨G.isAncestor⟩ j i hji
rcases G.isAncestor_child hanc with hedge | ⟨k, hjk, hki⟩
· exact hedge
· exact (hji.2 (Causalean.DAG.isAncestor.edge hjk) hki).elim
theorem ancestralCover_false_of_no_edge
An edge-free DAG has no ancestral covers. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma ancestralCover_false_of_no_edge
{n : ℕ} {G : DAG (Fin n)}
(hG : ∀ j i, ¬ G.edge j i) {j i : Fin n}
(hji : ancestralCover G j i) : False := by
exact hG j i (ancestralCover_edge hji)
theorem separatedSets_eq_univ_of_no_edge
Both separation conditions are vacuous for an edge-free DAG. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
Proof (Lean source)
lemma separatedSets_eq_univ_of_no_edge
{n : ℕ} (G : DAG (Fin n)) (s : SignVector n)
(π : Perm (Fin n)) (hG : ∀ j i, ¬ G.edge j i) :
edgeSeparatedSet (G := G) (s := s) π = univ ∧
coverSeparatedSet (G := G) (s := s) gaussianFeatureMap π = univ := by
constructor
· ext θ
simp only [edgeSeparatedSet, Set.mem_ofPred_eq, Set.mem_univ, iff_true]
intro j i hji
exact (hG j i hji).elim
· ext θ
simp only [coverSeparatedSet, Set.mem_ofPred_eq, Set.mem_univ, iff_true]
intro j i hji
exact (ancestralCover_false_of_no_edge hG hji).elim
The complement of an open dense set is closed, nowhere dense, and meagre. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
Proof (Lean source)
lemma openDense_compl_topology
{X : Type*} [TopologicalSpace X] {A : Set X}
(hopen : IsOpen A) (hdense : Dense A) :
IsMeagre Aᶜ ∧ IsClosed Aᶜ ∧ IsNowhereDense Aᶜ := by
have hclosed_nowhere : IsClosed Aᶜ ∧ IsNowhereDense Aᶜ := by
rw [isClosed_isNowhereDense_iff_compl]
simpa using intro hopen hdense
exact ⟨hclosed_nowhere.2.isMeagre, hclosed_nowhere.1, hclosed_nowhere.2⟩
theorem edgeSeparated_subset_coverSeparated
Direct-edge raw second-moment separation implies Gaussian-MMD separation on every ancestral cover. the stated conclusion follows.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma edgeSeparated_subset_coverSeparated
{n : ℕ} (G : DAG (Fin n)) (s : SignVector n)
(π : Perm (Fin n)) :
edgeSeparatedSet (G := G) (s := s) π ⊆
coverSeparatedSet (G := G) (s := s) gaussianFeatureMap π := by
intro θ hθ j i hji
apply canonical_populationDiscrepancy_pos_of_secondMomentContrast_ne
θ.property.positiveSmooth π
exact hθ (ancestralCover_edge hji)
In every nonempty fixed-DAG sign stratum, direct-edge moment separation is open dense and implies an open dense residual ancestral-cover MMD region with closed nowhere-dense complement. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
conclusion
conclusion 1
IsOpen (edgeSeparatedSet (G := G) (s := s) π)
conclusion 2
Dense (edgeSeparatedSet (G := G) (s := s) π)
conclusion 3
conclusion 4
conclusion 5
conclusion 6
conclusion 7
conclusion 8
conclusion 9
(∀ j i, ¬ G.edge j i)
edgeSeparatedSet (G := G) (s := s) π = Set.univ
coverSeparatedSet (G := G) (s := s) gaussianFeatureMap π = Set.univ
Proof (Lean source)
theorem generic_cover_separation
{n : ℕ} (G : DAG (Fin n)) (s : SignVector n)
(π : Perm (Fin n))
(hne : (Set.univ : Set (StratumPoint G s)).Nonempty) :
IsOpen (edgeSeparatedSet (G := G) (s := s) π) ∧
Dense (edgeSeparatedSet (G := G) (s := s) π) ∧
edgeSeparatedSet (G := G) (s := s) π ⊆
coverSeparatedSet (G := G) (s := s) gaussianFeatureMap π ∧
IsOpen (coverSeparatedSet (G := G) (s := s) gaussianFeatureMap π) ∧
Dense (coverSeparatedSet (G := G) (s := s) gaussianFeatureMap π) ∧
IsMeagre ((coverSeparatedSet (G := G) (s := s) gaussianFeatureMap π)ᶜ) ∧
IsClosed ((coverSeparatedSet (G := G) (s := s) gaussianFeatureMap π)ᶜ) ∧
IsNowhereDense ((coverSeparatedSet (G := G) (s := s) gaussianFeatureMap π)ᶜ) ∧
((∀ j i, ¬ G.edge j i) →
edgeSeparatedSet (G := G) (s := s) π = Set.univ ∧
coverSeparatedSet (G := G) (s := s) gaussianFeatureMap π = Set.univ) := by
have hsubset := edgeSeparated_subset_coverSeparated G s π
have htopology :
IsOpen (edgeSeparatedSet (G := G) (s := s) π) ∧
Dense (edgeSeparatedSet (G := G) (s := s) π) ∧
IsOpen (coverSeparatedSet (G := G) (s := s) gaussianFeatureMap π) := by
exact ⟨edgeSeparatedSet_isOpen G s π, edgeSeparatedSet_dense G s π,
coverSeparatedSet_isOpen G s π⟩
have hdenseCover : Dense
(coverSeparatedSet (G := G) (s := s) gaussianFeatureMap π) :=
htopology.2.1.mono hsubset
have hcompl := openDense_compl_topology htopology.2.2 hdenseCover
exact ⟨htopology.1, htopology.2.1, hsubset, htopology.2.2, hdenseCover,
hcompl.1, hcompl.2.1, hcompl.2.2,
separatedSets_eq_univ_of_no_edge G s π⟩
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.TSimultaneousConfidenceEdges 2 declarations This file states the conditional and unconditional simultaneous MMD bounds, soundness of selected edges, and transitive-closure recovery under a cover margin.
Simultaneous confidence edges
This file states the conditional and unconditional simultaneous MMD bounds, soundness of selected edges, and transitive-closure recovery under a cover margin.
The event that every empirical MMD is within the simultaneous confidence radius.
Definition (Lean source)
parameters
def
given by
{ω | ∀ j i : Fin n, j ≠ i → |empiricalDiscrepancy S U j i ω - populationDiscrepancy U W j i| ≤ confidenceRadius S}
Under the simultaneous first-stage event, sample splitting yields familywise MMD confidence bounds. Selected arrows are ancestral, and a two-radius cover margin recovers the true transitive closure. Given the stated inputs and conditions, the stated conclusion follows.
Formal statement
hypotheses
n :
W :
ObservedWorld G θ
Ω :
Type*
S :
SampleSplitWorld W Ω
hpos :
hmix :
hone :
hSampling :
hn :
1 ≤ n
hN :
∀ e, 1 ≤ S.sampleSize e
hα :
S.alpha ∈ Ioo (0 : ℝ) 1
hη :
S.eta ∈ Ioo (0 : ℝ) 1
hfirst :
1 - S.eta ≤ S.probability.real (firstStageL1Event S)
conclusion
conclusion 1
ConditionalProbabilityAtLeast S.probability (comap (trainingFold S) inferInstance) (simultaneousMmdEvent S gaussianFeatureMap) (firstStageL1Event S) (1 - S.alpha)
conclusion 2
1 - S.alpha - S.eta ≤ S.probability.real (simultaneousMmdEvent S gaussianFeatureMap)
conclusion 3
ω ∈ simultaneousMmdEvent S gaussianFeatureMap :
j i :
sampleSplitConfidenceGraph S gaussianFeatureMap hSampling ω j i
G.isAncestor (W.targetPerm j) (W.targetPerm i)
conclusion 4
(∀ j i, ancestralCover G (W.targetPerm j) (W.targetPerm i) → 2 * confidenceRadius S < populationDiscrepancy gaussianFeatureMap W j i)
ω ∈ simultaneousMmdEvent S gaussianFeatureMap :
TransGen (sampleSplitConfidenceGraph S gaussianFeatureMap hSampling ω)
= TransGen (permutedGraph G W)
Proof (Lean source)
theorem simultaneous_confidence_edges
{n : ℕ} {G : DAG (Fin n)} {s : SignVector n}
{θ : Mechanism n G} (W : ObservedWorld G θ)
{Ω : Type*} [MeasurableSpace Ω] (S : SampleSplitWorld W Ω)
(hpos : PositiveNormalizedSmoothMechanisms G θ)
(hmix : SharedDiffeomorphicMixing G θ W)
(hone : OnePerfectInterventionPerNode G θ W)
(hSampling : IndependentEnvironmentSampling S)
(hn : 1 ≤ n) (hN : ∀ e, 1 ≤ S.sampleSize e)
(hα : S.alpha ∈ Ioo (0 : ℝ) 1)
(hη : S.eta ∈ Ioo (0 : ℝ) 1)
(hfirst : 1 - S.eta ≤ S.probability.real (firstStageL1Event S)) :
ConditionalProbabilityAtLeast S.probability
(comap (trainingFold S) inferInstance)
(simultaneousMmdEvent S gaussianFeatureMap) (firstStageL1Event S)
(1 - S.alpha) ∧
1 - S.alpha - S.eta ≤
S.probability.real (simultaneousMmdEvent S gaussianFeatureMap) ∧
(∀ ω ∈ simultaneousMmdEvent S gaussianFeatureMap, ∀ j i,
sampleSplitConfidenceGraph S gaussianFeatureMap hSampling ω j i →
G.isAncestor (W.targetPerm j) (W.targetPerm i)) ∧
((∀ j i, ancestralCover G (W.targetPerm j) (W.targetPerm i) →
2 * confidenceRadius S < populationDiscrepancy gaussianFeatureMap W j i) →
∀ ω ∈ simultaneousMmdEvent S gaussianFeatureMap,
TransGen (sampleSplitConfidenceGraph S gaussianFeatureMap hSampling ω) =
TransGen (permutedGraph G W)) := by
classical
letI : IsProbabilityMeasure S.probability := hSampling.1
letI : MeasurableSpace (lp (fun _ : ℕ => ℝ) 2) := borel _
letI : BorelSpace (lp (fun _ : ℕ => ℝ) 2) := ⟨rfl⟩
let R : ℝ := (1 + sqrt (2 * log
((n : ℝ) * (n + 1 : ℝ) / S.alpha))) /
sqrt (minimumSampleSize S)
let Good : Set Ω := {ω | ∀ i : Fin n, ∀ e : Fin (n + 1),
‖empiricalMeanEmbedding S gaussianFeatureMap e i ω -
fittedMeanEmbedding S gaussianFeatureMap e i ω‖ ≤ R}
have hcondSampling := hSampling.toConditionalEvaluationSampling
have hgood : ConditionalProbabilityAtLeast S.probability
(comap (trainingFold S) inferInstance)
Good univ (1 - S.alpha) := by
simpa only [Good, R] using
bounded_rkhs_empirical_mean S gaussianFeatureMap hcondSampling hn hN hα
have hlaw : ∀ e, IsProbabilityMeasure (W.law e) := hSampling.2.1
have hbridge (ω : Ω) (hω : ω ∈ firstStageL1Event S)
(i : Fin n) (e : Fin (n + 1)) :
‖fittedMeanEmbedding S gaussianFeatureMap e i ω -
meanEmbedding gaussianFeatureMap
(Measure.map (observedLawRatio W.law i) (W.law e))‖ ≤
sqrt 2 * S.firstStageRadius S.eta := by
letI : IsProbabilityMeasure (W.law e) := hlaw e
exact (meanEmbedding_map_sub_le_l1 gaussianFeatureMap (W.law e)
(S.ratioEstimate ω i) (observedLawRatio W.law i)
(S.ratioFit_measurable (trainingFold S ω) i)
(measurable_observedLawRatio W.law i) (hω i e).1).trans
(mul_le_mul_of_nonneg_left (hω i e).2 (Real.sqrt_nonneg _))
have hGoodEvent : Good ∩ firstStageL1Event S ⊆
simultaneousMmdEvent S gaussianFeatureMap := by
intro ω hω
intro j i hji
have hjEval := hω.1 i j.succ
have h0Eval := hω.1 i 0
have hjFit := hbridge ω hω.2 i j.succ
have h0Fit := hbridge ω hω.2 i 0
let Ej := empiricalMeanEmbedding S gaussianFeatureMap j.succ i ω
let E0 := empiricalMeanEmbedding S gaussianFeatureMap 0 i ω
let Fj := fittedMeanEmbedding S gaussianFeatureMap j.succ i ω
let F0 := fittedMeanEmbedding S gaussianFeatureMap 0 i ω
let Pj := meanEmbedding gaussianFeatureMap
(Measure.map (observedLawRatio W.law i) (W.law j.succ))
let P0 := meanEmbedding gaussianFeatureMap
(Measure.map (observedLawRatio W.law i) (W.law 0))
have hvec : ‖(Ej - E0) - (Pj - P0)‖ ≤
2 * R + 2 * (sqrt 2 * S.firstStageRadius S.eta) := by
calc
‖(Ej - E0) - (Pj - P0)‖ = ‖(Ej - Fj) + (Fj - Pj) -
((E0 - F0) + (F0 - P0))‖ := by congr 1 <;> abel
_ ≤ ‖Ej - Fj‖ + ‖Fj - Pj‖ + (‖E0 - F0‖ + ‖F0 - P0‖) := by
exact (norm_sub_le _ _).trans (add_le_add (norm_add_le _ _) (norm_add_le _ _))
_ ≤ R + (sqrt 2 * S.firstStageRadius S.eta) +
(R + (sqrt 2 * S.firstStageRadius S.eta)) := by
gcongr
_ = 2 * R + 2 * (sqrt 2 * S.firstStageRadius S.eta) := by ring
have hnorm : |‖Ej - E0‖ - ‖Pj - P0‖| ≤
2 * R + 2 * (sqrt 2 * S.firstStageRadius S.eta) :=
(abs_norm_sub_norm_le (Ej - E0) (Pj - P0)).trans hvec
have hrev : ‖Pj - P0‖ = ‖P0 - Pj‖ := norm_sub_rev _ _
rw [hrev] at hnorm
have hfinal : |‖Ej - E0‖ - ‖P0 - Pj‖| ≤ confidenceRadius S := by
convert hnorm using 1
all_goals simp only [confidenceRadius, R]
all_goals ring
simpa only [empiricalDiscrepancy, populationDiscrepancy,
observationalRatioLaw, interventionalRatioLaw, Ej, E0, Pj, P0] using hfinal
have hconditional : ConditionalProbabilityAtLeast S.probability
(comap (trainingFold S) inferInstance)
(simultaneousMmdEvent S gaussianFeatureMap) (firstStageL1Event S)
(1 - S.alpha) := by
intro B hB
have hA : @MeasurableSet Ω (comap (trainingFold S) inferInstance)
(firstStageL1Event S) := hSampling.2.2.2.2.2.2.2.2.2.2.2
have hg := hgood (firstStageL1Event S ∩ B) (hA.inter hB)
simp only [Set.univ_inter, Set.inter_assoc] at hg
exact hg.trans (measureReal_mono
(by
intro ω hω
exact ⟨⟨hGoodEvent ⟨hω.1, hω.2.1⟩, hω.2.1⟩, hω.2.2⟩)
(measure_ne_top _ _))
refine ⟨hconditional, ?_, ?_, ?_⟩
· have hc := hconditional univ MeasurableSet.univ
simp only [Set.inter_univ] at hc
have hEA : S.probability.real
(simultaneousMmdEvent S gaussianFeatureMap ∩ firstStageL1Event S) ≤
S.probability.real (simultaneousMmdEvent S gaussianFeatureMap) :=
measureReal_mono Set.inter_subset_left (measure_ne_top _ _)
have hnonneg : 0 ≤ 1 - S.alpha := by linarith [hα.2]
have hmul := mul_le_mul_of_nonneg_left hfirst hnonneg
nlinarith [hη.1, hα.1]
· intro ω hω j i hedge
by_contra hna
have hz := ratio_nonancestor_zero (s := s) W hpos hmix hone hedge.1 hna
have herr := hω j i hedge.1
rw [hz, sub_zero] at herr
have hem : 0 ≤ empiricalDiscrepancy S gaussianFeatureMap j i ω := by
exact norm_nonneg _
rw [abs_of_nonneg hem] at herr
unfold sampleSplitConfidenceGraph at hedge
linarith
· intro hmargin ω hω
have hedge : ∀ {j i},
sampleSplitConfidenceGraph S gaussianFeatureMap hSampling ω j i →
G.isAncestor (W.targetPerm j) (W.targetPerm i) := by
intro j i hji
apply Classical.byContradiction
intro hna
have hz := ratio_nonancestor_zero (s := s) W hpos hmix hone hji.1 hna
have herr := hω j i hji.1
rw [hz, sub_zero] at herr
have hem : 0 ≤ empiricalDiscrepancy S gaussianFeatureMap j i ω := by
exact norm_nonneg _
rw [abs_of_nonneg hem] at herr
unfold sampleSplitConfidenceGraph at hji
linarith
have hliftAncestor : ∀ {a b}, G.isAncestor a b →
TransGen (permutedGraph G W)
(W.targetPerm.symm a) (W.targetPerm.symm b) := by
intro a b hji
induction hji with
| edge h =>
apply Relation.TransGen.single
simpa [permutedGraph] using h
| trans hab hbc ih =>
apply Relation.TransGen.tail ih
simpa [permutedGraph] using hbc
have hancestorPath : ∀ {j i}, G.isAncestor (W.targetPerm j) (W.targetPerm i) →
TransGen (permutedGraph G W) j i := by
intro j i hji
simpa using hliftAncestor hji
have hforward : TransGen
(sampleSplitConfidenceGraph S gaussianFeatureMap hSampling ω) ≤
TransGen (permutedGraph G W) := by
intro j i hji
induction hji with
| single h => exact hancestorPath (hedge h)
| tail hab hbc ih => exact ih.trans (hancestorPath (hedge hbc))
have hcoverEdge : ∀ {a b : Fin n}, ancestralCover G a b →
sampleSplitConfidenceGraph S gaussianFeatureMap hSampling ω
(W.targetPerm.symm a) (W.targetPerm.symm b) := by
intro a b hab
have hne : W.targetPerm.symm a ≠ W.targetPerm.symm b := by
intro heq
have : a = b := W.targetPerm.symm.injective heq
subst b
have hab' : G.isAncestor a a := by
exact (@CovBy.lt (Fin n) ⟨G.isAncestor⟩ a a hab)
exact G.isAncestor_irrefl a hab'
have hD := hmargin (W.targetPerm.symm a) (W.targetPerm.symm b) (by simpa)
have herr := hω (W.targetPerm.symm a) (W.targetPerm.symm b) hne
unfold sampleSplitConfidenceGraph
refine ⟨hne, ?_⟩
have hlow := (abs_le.mp herr).1
linarith
have hmapCoverPath : ∀ {a b}, TransGen (ancestralCover G) a b →
TransGen
(sampleSplitConfidenceGraph S gaussianFeatureMap hSampling ω)
(W.targetPerm.symm a) (W.targetPerm.symm b) := by
intro a b hc
induction hc with
| single h => exact Relation.TransGen.single (hcoverEdge h)
| tail hab hbc ih => exact Relation.TransGen.tail ih (hcoverEdge hbc)
have hpermEdge : ∀ {j i}, permutedGraph G W j i →
TransGen
(sampleSplitConfidenceGraph S gaussianFeatureMap hSampling ω) j i := by
intro j i hji
have ha : G.isAncestor (W.targetPerm j) (W.targetPerm i) :=
Causalean.DAG.isAncestor.edge hji
have hc := isAncestor_transGen_ancestralCover G ha
have mapped := hmapCoverPath hc
simpa using mapped
have hbackward : TransGen (permutedGraph G W) ≤
TransGen
(sampleSplitConfidenceGraph S gaussianFeatureMap hSampling ω) := by
intro j i hji
induction hji with
| single h => exact hpermEdge h
| tail hab hbc ih => exact ih.trans (hpermEdge hbc)
apply le_antisymm
· exact hforward
· exact hbackward
CausalSmith.ExactID.EID_CrlCoverratioMmdGenericity_Research.TSparseWitnessCertificate 2 declarations The theorem certifies both explicit three-node constructions, quantitative sparse separation, and the exact cancellation example.
Sparse and cancellation witness certificate
The theorem certifies both explicit three-node constructions, quantitative sparse separation, and the exact cancellation example.
theorem explicitWitness_structural_cancellation_certificate
The already established witness lemmas assemble both faithfulness assertions, all regularity, and exact cancellation. This leaves only the quantitative sparse moment/MMD estimates to the final assembly. the stated conclusion follows.
Formal statement
hypotheses
s :
conclusion
sparse :
cancel :
Wc :
FixedOwnDerivativeSign threeNodeDAG s sparse
Faithfulness threeNodeDAG sparse
k i :
ContDiffOn ℝ k (sparse.p i) (latentCube 3)
k i :
ContDiffOn ℝ k (sparse.q i) (Icc (0 : ℝ) 1)
FixedOwnDerivativeSign threeNodeDAG s cancel
Faithfulness threeNodeDAG cancel
k i :
ContDiffOn ℝ k (cancel.p i) (latentCube 3)
k i :
ContDiffOn ℝ k (cancel.q i) (Icc (0 : ℝ) 1)
observationalRatioLaw Wc 1 = interventionalRatioLaw Wc 0 1
populationDiscrepancy gaussianFeatureMap Wc 0 1 = 0
Proof (Lean source)
-- @node: explicitWitness_structural_cancellation_certificate
lemma explicitWitness_structural_cancellation_certificate (s : SignVector 3) :
let sparse := sparseWitness s
let cancel := cancellationWitness s
let Wc := canonicalObservedWorld threeNodeDAG cancel (refl (Fin 3))
PositiveNormalizedSmoothMechanisms threeNodeDAG sparse ∧
FixedOwnDerivativeSign threeNodeDAG s sparse ∧
Faithfulness threeNodeDAG sparse ∧
(∀ k i, ContDiffOn ℝ k (sparse.p i) (latentCube 3)) ∧
(∀ k i, ContDiffOn ℝ k (sparse.q i) (Icc (0 : ℝ) 1)) ∧
PositiveNormalizedSmoothMechanisms threeNodeDAG cancel ∧
FixedOwnDerivativeSign threeNodeDAG s cancel ∧
Faithfulness threeNodeDAG cancel ∧
(∀ k i, ContDiffOn ℝ k (cancel.p i) (latentCube 3)) ∧
(∀ k i, ContDiffOn ℝ k (cancel.q i) (Icc (0 : ℝ) 1)) ∧
observationalRatioLaw Wc 1 = interventionalRatioLaw Wc 0 1 ∧
populationDiscrepancy gaussianFeatureMap Wc 0 1 = 0 := by
dsimp only
rcases explicitWitness_regularities s with
⟨hsp, hss, hspd, hsqd, hcp, hcs, hcpd, hcqd⟩
exact ⟨hsp, hss, sparseWitness_faithfulness s, hspd, hsqd, hcp, hcs,
cancellationWitness_faithfulness s, hcpd, hcqd,
cancellationWitness_ratioLaws_eq s,
cancellationWitness_populationDiscrepancy_eq_zero s⟩
The reflected sparse and cancellation mechanisms satisfy the model atoms; the sparse witness has the certified moment and Gaussian-MMD gaps, while the cancellation ratio law is unchanged. the stated conclusion follows.
Formal statement
hypotheses
s :
conclusion
sparse :
cancel :
Ws :
Wc :
FixedOwnDerivativeSign threeNodeDAG s sparse
Faithfulness threeNodeDAG sparse
k i :
ContDiffOn ℝ k (sparse.p i) (latentCube 3)
k i :
ContDiffOn ℝ k (sparse.q i) (Icc (0 : ℝ) 1)
FixedOwnDerivativeSign threeNodeDAG s cancel
Faithfulness threeNodeDAG cancel
k i :
ContDiffOn ℝ k (cancel.p i) (latentCube 3)
k i :
ContDiffOn ℝ k (cancel.q i) (Icc (0 : ℝ) 1)
(3 / 10000 : ℝ) < (∫ x, (Ws.ratio 1 x) ^ 2 ∂Ws.law 0)
- ∫ x, (Ws.ratio 1 x) ^ 2 ∂Ws.law (succ 0)
(5 / 100000000 : ℝ) < populationDiscrepancy gaussianFeatureMap Ws 0 1
observationalRatioLaw Wc 1 = interventionalRatioLaw Wc 0 1
populationDiscrepancy gaussianFeatureMap Wc 0 1 = 0
Proof (Lean source)
-- @node: prop:sparse-witness-certificate
theorem sparse_witness_certificate (s : SignVector 3) :
let sparse := sparseWitness s
let cancel := cancellationWitness s
let Ws := canonicalObservedWorld threeNodeDAG sparse (refl (Fin 3))
let Wc := canonicalObservedWorld threeNodeDAG cancel (refl (Fin 3))
PositiveNormalizedSmoothMechanisms threeNodeDAG sparse ∧
FixedOwnDerivativeSign threeNodeDAG s sparse ∧
Faithfulness threeNodeDAG sparse ∧
(∀ k i, ContDiffOn ℝ k (sparse.p i) (latentCube 3)) ∧
(∀ k i, ContDiffOn ℝ k (sparse.q i) (Icc (0 : ℝ) 1)) ∧
PositiveNormalizedSmoothMechanisms threeNodeDAG cancel ∧
FixedOwnDerivativeSign threeNodeDAG s cancel ∧
Faithfulness threeNodeDAG cancel ∧
(∀ k i, ContDiffOn ℝ k (cancel.p i) (latentCube 3)) ∧
(∀ k i, ContDiffOn ℝ k (cancel.q i) (Icc (0 : ℝ) 1)) ∧
(3 / 10000 : ℝ) <
(∫ x, (Ws.ratio 1 x) ^ 2 ∂Ws.law 0) -
∫ x, (Ws.ratio 1 x) ^ 2 ∂Ws.law (succ 0) ∧
(5 / 100000000 : ℝ) < populationDiscrepancy gaussianFeatureMap Ws 0 1 ∧
observationalRatioLaw Wc 1 = interventionalRatioLaw Wc 0 1 ∧
populationDiscrepancy gaussianFeatureMap Wc 0 1 = 0 := by
dsimp only
rcases explicitWitness_structural_cancellation_certificate s with
⟨hsp, hss, hsf, hspd, hsqd, hcp, hcs, hcf, hcpd, hcqd, hratio, hcMmd⟩
exact ⟨hsp, hss, hsf, hspd, hsqd, hcp, hcs, hcf, hcpd, hcqd,
sparse_witness_moment_gap s, sparse_witness_mmd_gap s, hratio, hcMmd⟩