File size: 3,956 Bytes
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 | import OAI.Probability.FreeIsing.Estimates.RelabelMeasure
import OAI.Probability.FreeIsing.Factors.DiagonalRecovery
import OAI.Probability.FreeIsing.Factors.RecoveryEquivariance
import OAI.Probability.FreeIsing.Factors.Transfer
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
namespace Problem367
theorem exists_ae_equivariant_recovery_of_conditionalCopies
{W X G : Type*} [MeasurableSpace W] [MeasurableSpace X]
[MeasurableEq X] [Nonempty X]
(μ : Measure W) [SFinite μ] (κ : Kernel W X) [IsMarkovKernel κ]
(T : G → W → W) (S : G → X → X)
(hT : ∀ g, Measurable (T g)) (hS : ∀ g, Measurable (S g))
(hjoint : ∀ g, Measure.QuasiMeasurePreserving
(fun p : W × X => (T g p.1, S g p.2)) (μ ⊗ₘ κ) (μ ⊗ₘ κ))
(hcopies : ∀ᵐ p ∂μ ⊗ₘ (κ ×ₖ κ), p.2.1 = p.2.2) :
∃ F : W → X, Measurable F ∧
(∀ᵐ w ∂μ, κ w = Measure.dirac (F w)) ∧
Measure.map F μ = Measure.map Prod.snd (μ ⊗ₘ κ) ∧
∀ g, ∀ᵐ w ∂μ, F (T g w) = S g (F w) := by
obtain ⟨F, hF, hdirac, hgraph⟩ :=
exists_measurable_recovery_of_conditionalCopies μ κ hcopies
have hfst : Measure.map Prod.fst (μ ⊗ₘ κ) = μ :=
Measure.fst_compProd μ κ
exact ⟨F, hF, hdirac,
map_eq_second_marginal_of_joint_recovery hfst hF hgraph,
ae_equivariant_family_of_joint_recovery hfst hT hS hF hjoint hgraph⟩
theorem isFactorOfIID_of_invariant_conditionalCopies {d : ℕ} {theta : ℝ}
(κ : Kernel (TreeVertex d → ℝ) (TreeVertex d → IsingSpin))
[IsMarkovKernel κ]
(hLaw : HasFreeIsingLaw theta
(Measure.map Prod.snd ((labelMeasure d) ⊗ₘ κ)))
(hjoint : ∀ g : TreeAut d, Measure.QuasiMeasurePreserving
(fun p : (TreeVertex d → ℝ) × (TreeVertex d → IsingSpin) =>
(relabel g p.1, relabel g p.2))
((labelMeasure d) ⊗ₘ κ) ((labelMeasure d) ⊗ₘ κ))
(hcopies : ∀ᵐ p ∂(labelMeasure d) ⊗ₘ (κ ×ₖ κ), p.2.1 = p.2.2) :
IsFactorOfIID d theta := by
let : Nonempty (TreeVertex d → IsingSpin) := ⟨fun _ => IsingSpin.pos⟩
let : IsProbabilityMeasure (labelMeasure d) := by
unfold labelMeasure
infer_instance
obtain ⟨F, hF, _, hmap, heq⟩ :=
exists_ae_equivariant_recovery_of_conditionalCopies (labelMeasure d) κ
(fun g : TreeAut d => relabel g) (fun g : TreeAut d => relabel g)
measurable_relabel measurable_relabel hjoint hcopies
refine ⟨F, hF, ?_, heq⟩
rw [hmap]
exact hLaw
theorem isFactorOfIID_of_invariant_product_conditionalCopies
{d : ℕ} {theta : ℝ} {α : Type*} [MeasurableSpace α]
(ν : Measure α) [IsProbabilityMeasure ν]
(ξ : ℝ → α) (hξ : Measurable ξ) (hξlaw : Measure.map ξ uniform01 = ν)
(κ : Kernel (TreeVertex d → α) (TreeVertex d → IsingSpin))
[IsMarkovKernel κ]
(hLaw : HasFreeIsingLaw theta
(Measure.map Prod.snd ((Measure.infinitePi (fun _ : TreeVertex d => ν)) ⊗ₘ κ)))
(hjoint : ∀ g : TreeAut d, Measure.QuasiMeasurePreserving
(fun p : (TreeVertex d → α) × (TreeVertex d → IsingSpin) =>
(relabel g p.1, relabel g p.2))
((Measure.infinitePi (fun _ : TreeVertex d => ν)) ⊗ₘ κ)
((Measure.infinitePi (fun _ : TreeVertex d => ν)) ⊗ₘ κ))
(hcopies : ∀ᵐ p ∂(Measure.infinitePi (fun _ : TreeVertex d => ν)) ⊗ₘ (κ ×ₖ κ),
p.2.1 = p.2.2) :
IsFactorOfIID d theta := by
let : Nonempty (TreeVertex d → IsingSpin) := ⟨fun _ => IsingSpin.pos⟩
obtain ⟨F, hF, _, hmap, heq⟩ :=
exists_ae_equivariant_recovery_of_conditionalCopies
(Measure.infinitePi (fun _ : TreeVertex d => ν)) κ
(fun g : TreeAut d => relabel g) (fun g : TreeAut d => relabel g)
measurable_relabel measurable_relabel hjoint hcopies
apply isFactorOfIID_of_product_sampler ν ξ hξ hξlaw F hF _ heq
rw [hmap]
exact hLaw
end Problem367
end
end OAI
|