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f61e547 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 | import OAI.Probability.FreeIsing.Conditioning.CopyInvariance
import OAI.Probability.FreeIsing.Posterior.ObservationFactorRecovery
import OAI.Probability.FreeIsing.Trees.Transitivity
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
namespace Problem367.ObservationCopyStationarity
def arrayRelabelEquiv {d : ℕ} {A : Type*} [MeasurableSpace A]
(g : TreeAut d) : (TreeVertex d → A) ≃ᵐ (TreeVertex d → A) where
toFun := relabel g
invFun := fun x v => x (g.toEquiv v)
left_inv := by intro x; funext v; simp [relabel]
right_inv := by intro x; funext v; simp [relabel]
measurable_toFun := measurable_relabel g
measurable_invFun := Measurable.of_eval (fun _ => measurable_pi_apply _)
theorem canonical_conditionalCopies_invariant
{d : ℕ} {A : Type*} [MeasurableSpace A]
(nu : Measure A) [IsProbabilityMeasure nu]
(rho : Measure ((TreeVertex d → A) × (TreeVertex d → ℕ → ℝ)))
[IsProbabilityMeasure rho]
(hfirst : rho.fst = Measure.infinitePi (fun _ : TreeVertex d => nu))
(g : TreeAut d)
(hjoint : rho.map (Prod.map (relabel g) (relabel g)) = rho) :
(rho.fst ⊗ₘ (rho.condKernel ×ₖ rho.condKernel)).map
(Prod.map (relabel g) (Prod.map (relabel g) (relabel g))) =
rho.fst ⊗ₘ (rho.condKernel ×ₖ rho.condKernel) := by
have hT : MeasurePreserving (arrayRelabelEquiv (A := A) g) rho.fst rho.fst := by
refine ⟨measurable_relabel g, ?_⟩
rw [hfirst]
exact Measure.map_infinitePi_infinitePi_of_inj g.toEquiv.symm.injective
have hdis : rho.fst ⊗ₘ rho.condKernel = rho :=
Measure.disintegrate rho rho.condKernel
apply conditionalCopies_invariant rho.fst rho.condKernel (arrayRelabelEquiv g)
(measurable_relabel g) hT
simpa [hdis, arrayRelabelEquiv] using hjoint
theorem ae_eq_arrays_of_reference_vertex
{d : ℕ} {A : Type*} [MeasurableSpace A]
(mu : Measure ((TreeVertex d → A) ×
((TreeVertex d → ℕ → ℝ) × (TreeVertex d → ℕ → ℝ))))
(o : TreeVertex d)
(hinvariant : ∀ g : TreeAut d,
mu.map (Prod.map (relabel g) (Prod.map (relabel g) (relabel g))) = mu)
(hroot : ∀ n, ∀ᵐ p ∂mu, p.2.1 o n = p.2.2 o n) :
∀ᵐ p ∂mu, p.2.1 = p.2.2 := by
have hcoord : ∀ v n, ∀ᵐ p ∂mu, p.2.1 v n = p.2.2 v n := by
intro v n
obtain ⟨g, hg⟩ := exists_treeAut_map v o
have hgo : g.toEquiv.symm o = v := by
rw [← hg, Equiv.symm_apply_apply]
have hpres : MeasurePreserving
(Prod.map (relabel g) (Prod.map (relabel g) (relabel g))) mu mu :=
⟨(measurable_relabel g).prodMap
((measurable_relabel g).prodMap (measurable_relabel g)), hinvariant g⟩
filter_upwards [hpres.quasiMeasurePreserving.ae (hroot n)] with p hp
change p.2.1 (g.toEquiv.symm o) n = p.2.2 (g.toEquiv.symm o) n at hp
simpa only [hgo] using hp
have hall : ∀ᵐ p ∂mu, ∀ v n, p.2.1 v n = p.2.2 v n :=
ae_all_iff.2 (fun v => ae_all_iff.2 (hcoord v))
filter_upwards [hall] with p hp
exact funext (fun v => funext (hp v))
theorem isFactorOfIID_of_joint_observation_reference_collapse
{d : ℕ} {theta : ℝ} {A : Type*} [MeasurableSpace A]
[StandardBorelSpace A] [Nonempty A]
(nu : Measure A) [IsProbabilityMeasure nu]
(rho : Measure ((TreeVertex d → A) × (TreeVertex d → ℕ → ℝ)))
[IsProbabilityMeasure rho] (o : TreeVertex d)
(hfirst : rho.fst = Measure.infinitePi (fun _ : TreeVertex d => nu))
(hLaw : HasFreeIsingLaw theta
(Measure.map (fun p => SpinRecovery.decodeArray p.2) rho))
(hjoint : ∀ g : TreeAut d, rho.map (Prod.map (relabel g) (relabel g)) = rho)
(hroot : ∀ n, ∀ᵐ p ∂rho.fst ⊗ₘ (rho.condKernel ×ₖ rho.condKernel),
p.2.1 o n = p.2.2 o n) :
IsFactorOfIID d theta := by
have hcopies := ae_eq_arrays_of_reference_vertex
(rho.fst ⊗ₘ (rho.condKernel ×ₖ rho.condKernel)) o
(fun g => canonical_conditionalCopies_invariant nu rho hfirst g (hjoint g)) hroot
have hdis : (Measure.infinitePi (fun _ : TreeVertex d => nu)) ⊗ₘ
rho.condKernel = rho := by
rw [← hfirst]
exact Measure.disintegrate rho rho.condKernel
apply isFactorOfIID_of_equal_observation_conditionalCopies nu rho.condKernel
· simpa only [hdis] using hLaw
· intro g
rw [hdis]
exact (show MeasurePreserving (Prod.map (relabel g) (relabel g)) rho rho from
⟨(measurable_relabel g).prodMap (measurable_relabel g), hjoint g⟩).quasiMeasurePreserving
· simpa only [hfirst] using hcopies
theorem isFactorOfIID_of_joint_observation_reference_zero
{d : ℕ} {theta : ℝ} {A : Type*} [MeasurableSpace A]
[StandardBorelSpace A] [Nonempty A]
(nu : Measure A) [IsProbabilityMeasure nu]
(rho : Measure ((TreeVertex d → A) × (TreeVertex d → ℕ → ℝ)))
[IsProbabilityMeasure rho] (o : TreeVertex d)
(hfirst : rho.fst = Measure.infinitePi (fun _ : TreeVertex d => nu))
(hLaw : HasFreeIsingLaw theta
(Measure.map (fun p => SpinRecovery.decodeArray p.2) rho))
(hjoint : ∀ g : TreeAut d, rho.map (Prod.map (relabel g) (relabel g)) = rho)
(hInt : ∀ n, Integrable (fun p => (p.2.1 o n - p.2.2 o n) ^ 2)
(rho.fst ⊗ₘ (rho.condKernel ×ₖ rho.condKernel)))
(hzero : ∀ n, (∫ p, (p.2.1 o n - p.2.2 o n) ^ 2
∂rho.fst ⊗ₘ (rho.condKernel ×ₖ rho.condKernel)) = 0) :
IsFactorOfIID d theta := by
apply isFactorOfIID_of_joint_observation_reference_collapse nu rho o hfirst hLaw hjoint
exact fun n => ae_eq_of_integral_sq_sub_eq_zero (hInt n) (hzero n)
end Problem367.ObservationCopyStationarity
end
end OAI
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