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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 | import OAI.Probability.FreeIsing.Posterior.PlantedEdgeRegression
import OAI.Probability.FreeIsing.Trees.NoisyBranchSeparation
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
namespace Problem367
theorem HasFreeIsingLaw.condExp_relabel_spin_of_noisy_root_others_observable
{d : ℕ} {theta : ℝ} {A C : Type*}
[MeasurableSpace A] [StandardBorelSpace A]
[mC : MeasurableSpace C] [StandardBorelSpace C] [Nonempty C]
{μ : Measure (TreeVertex d → IsingSpin)} [IsFiniteMeasure μ]
(hμ : HasFreeIsingLaw theta μ) (h0 : -1 ≤ theta) (h1 : theta ≤ 1)
(ν : Measure A) [IsProbabilityMeasure ν] (g : TreeAut d) (c : Fin d)
(F : (IsingSpin × A) × (∀ b : {b : Fin d // b ≠ c},
(RootedBroadcast.BranchVertex b.val → IsingSpin) ×
(RootedBroadcast.BranchVertex b.val → A)) → C)
(hF : Measurable F) :
(μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν)))[
(fun z : (TreeVertex d → IsingSpin) × (TreeVertex d → A) =>
spinSign (z.1 (g.toEquiv.symm ⟨[c], List.IsChain.singleton c⟩))) |
mC.comap (F ∘ noisyRootOtherArray c ∘ Prod.map (relabel g) (relabel g))] =ᵐ[
μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν))]
fun z => theta * (μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν)))[
(fun z : (TreeVertex d → IsingSpin) × (TreeVertex d → A) =>
spinSign (z.1 (g.toEquiv.symm (RootedBroadcast.root d)))) |
mC.comap (F ∘ noisyRootOtherArray c ∘ Prod.map (relabel g) (relabel g))] z := by
have hadj : treeAdj (g.toEquiv.symm ⟨[c], List.IsChain.singleton c⟩)
(g.toEquiv.symm (RootedBroadcast.root d)) := by
apply (g.map_adj _ _).mp
simp only [Equiv.apply_symm_apply]
exact Or.inr ⟨c, rfl⟩
have hS : Measurable (F ∘ noisyRootOtherArray c ∘
Prod.map (relabel g) (relabel g)) :=
hF.comp ((measurable_noisyRootOtherArray c).comp
((measurable_relabel g).prodMap (measurable_relabel g)))
apply hμ.condExp_planted_spin_of_separator h0
(Measure.infinitePi (fun _ : TreeVertex d => ν)) hadj hS
let vc : RootedBroadcast.BranchVertex c :=
⟨⟨[c], List.IsChain.singleton c⟩, List.prefix_refl _⟩
have hparent : Measurable (fun p :
(RootedBroadcast.BranchVertex c → IsingSpin) ×
(RootedBroadcast.BranchVertex c → A) => spinSign (p.1 vc)) :=
measurable_spinToReal.comp ((measurable_pi_apply vc).comp measurable_fst)
have h := (hμ.condIndepFun_relabel_noisy_root_others_branch h0 h1 ν g c).comp hF hparent
simpa only [Function.comp_def, noisyBranchArray, Prod.map, relabel, vc] using h
end Problem367
end
end OAI
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