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b317718 | 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 | import OAI.Probability.FreeIsing.Conditioning.FactorRecovery
import OAI.Probability.FreeIsing.Estimates.UniformSampler
import OAI.Probability.FreeIsing.Factors.ContinuousRecovery
import OAI.Probability.FreeIsing.Spins.Recovery
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
namespace Problem367
theorem ae_eq_array_of_coordinate_integral_sq_sub_eq_zero
{O I J : Type*} [MeasurableSpace O] [Countable I] [Countable J]
{mu : Measure O} {f g : O → I → J → ℝ}
(hInt : ∀ i j, Integrable (fun z => (f z i j - g z i j) ^ 2) mu)
(hzero : ∀ i j, (∫ z, (f z i j - g z i j) ^ 2 ∂mu) = 0) :
f =ᵐ[mu] g := by
have hcoord : ∀ i j, ∀ᵐ z ∂mu, f z i j = g z i j :=
fun i j => ae_eq_of_integral_sq_sub_eq_zero (hInt i j) (hzero i j)
have hall : ∀ᵐ z ∂mu, ∀ i j, f z i j = g z i j :=
ae_all_iff.2 (fun i => ae_all_iff.2 (hcoord i))
filter_upwards [hall] with z hz
exact funext (fun i => funext (hz i))
theorem isFactorOfIID_of_equal_observation_conditionalCopies
{d : ℕ} {theta : ℝ} {A : Type*} [MeasurableSpace A]
[StandardBorelSpace A] [Nonempty A]
(nu : Measure A) [IsProbabilityMeasure nu]
(kappa : Kernel (TreeVertex d → A) (TreeVertex d → ℕ → ℝ))
[IsMarkovKernel kappa]
(hLaw : HasFreeIsingLaw theta
(Measure.map (fun p => SpinRecovery.decodeArray p.2)
((Measure.infinitePi (fun _ : TreeVertex d => nu)) ⊗ₘ kappa)))
(hjoint : ∀ g : TreeAut d, Measure.QuasiMeasurePreserving
(fun p : (TreeVertex d → A) × (TreeVertex d → ℕ → ℝ) =>
(relabel g p.1, relabel g p.2))
((Measure.infinitePi (fun _ : TreeVertex d => nu)) ⊗ₘ kappa)
((Measure.infinitePi (fun _ : TreeVertex d => nu)) ⊗ₘ kappa))
(hcopies : ∀ᵐ p ∂(Measure.infinitePi (fun _ : TreeVertex d => nu)) ⊗ₘ
(kappa ×ₖ kappa), p.2.1 = p.2.2) :
IsFactorOfIID d theta := by
obtain ⟨F, hF, _, hmap, heq⟩ :=
exists_ae_equivariant_recovery_of_conditionalCopies
(Measure.infinitePi (fun _ : TreeVertex d => nu)) kappa
(fun g : TreeAut d => relabel g) (fun g : TreeAut d => relabel g)
measurable_relabel measurable_relabel hjoint hcopies
obtain ⟨sample, hsample, hsampleLaw⟩ := exists_uniform01_sampler nu
apply isFactorOfIID_of_product_sampler nu sample hsample hsampleLaw
(SpinRecovery.decodeArray ∘ F) (SpinRecovery.measurable_decodeArray.comp hF)
· rw [← Measure.map_map SpinRecovery.measurable_decodeArray hF, hmap,
Measure.map_map SpinRecovery.measurable_decodeArray measurable_snd]
exact hLaw
· intro g
filter_upwards [heq g] with w hw
change SpinRecovery.decodeArray (F (relabel g w)) =
relabel g (SpinRecovery.decodeArray (F w))
rw [hw, SpinRecovery.decodeArray_relabel]
theorem isFactorOfIID_of_observation_conditionalCopies_zero
{d : ℕ} {theta : ℝ} {A : Type*} [MeasurableSpace A]
[StandardBorelSpace A] [Nonempty A]
(nu : Measure A) [IsProbabilityMeasure nu]
(kappa : Kernel (TreeVertex d → A) (TreeVertex d → ℕ → ℝ))
[IsMarkovKernel kappa]
(hLaw : HasFreeIsingLaw theta
(Measure.map (fun p => SpinRecovery.decodeArray p.2)
((Measure.infinitePi (fun _ : TreeVertex d => nu)) ⊗ₘ kappa)))
(hjoint : ∀ g : TreeAut d, Measure.QuasiMeasurePreserving
(fun p : (TreeVertex d → A) × (TreeVertex d → ℕ → ℝ) =>
(relabel g p.1, relabel g p.2))
((Measure.infinitePi (fun _ : TreeVertex d => nu)) ⊗ₘ kappa)
((Measure.infinitePi (fun _ : TreeVertex d => nu)) ⊗ₘ kappa))
(hInt : ∀ v n, Integrable (fun p => (p.2.1 v n - p.2.2 v n) ^ 2)
((Measure.infinitePi (fun _ : TreeVertex d => nu)) ⊗ₘ (kappa ×ₖ kappa)))
(hzero : ∀ v n, (∫ p, (p.2.1 v n - p.2.2 v n) ^ 2
∂(Measure.infinitePi (fun _ : TreeVertex d => nu)) ⊗ₘ (kappa ×ₖ kappa)) = 0) :
IsFactorOfIID d theta := by
apply isFactorOfIID_of_equal_observation_conditionalCopies nu kappa hLaw hjoint
exact ae_eq_array_of_coordinate_integral_sq_sub_eq_zero hInt hzero
theorem isFactorOfIID_of_joint_observation_conditionalCopies_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]
(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, Measure.QuasiMeasurePreserving
(fun p : (TreeVertex d → A) × (TreeVertex d → ℕ → ℝ) =>
(relabel g p.1, relabel g p.2)) rho rho)
(hInt : ∀ v n, Integrable (fun p => (p.2.1 v n - p.2.2 v n) ^ 2)
(rho.fst ⊗ₘ (rho.condKernel ×ₖ rho.condKernel)))
(hzero : ∀ v n, (∫ p, (p.2.1 v n - p.2.2 v n) ^ 2
∂rho.fst ⊗ₘ (rho.condKernel ×ₖ rho.condKernel)) = 0) :
IsFactorOfIID d theta := by
have hdis : (Measure.infinitePi (fun _ : TreeVertex d => nu)) ⊗ₘ
rho.condKernel = rho := by
rw [← hfirst]
exact Measure.disintegrate rho rho.condKernel
apply isFactorOfIID_of_observation_conditionalCopies_zero nu rho.condKernel
· simpa only [hdis] using hLaw
· simpa only [hdis] using hjoint
· simpa only [hfirst] using hInt
· simpa only [hfirst] using hzero
end Problem367
end
end OAI
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