File size: 3,840 Bytes
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 | import OAI.Probability.FreeIsing.Posterior.BranchObservationRegression
import OAI.Probability.FreeIsing.Posterior.ComponentPosteriorStationarity
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
namespace Problem367
theorem HasFreeIsingLaw.selectedPosterior_root_component
{d : ℕ} {theta : ℝ} {A B : Type*}
[MeasurableSpace A] [StandardBorelSpace A]
[MeasurableSpace B] [StandardBorelSpace B] [Nonempty B]
{μ : Measure (TreeVertex d → IsingSpin)} (hμ : HasFreeIsingLaw theta μ)
(h0 : -1 ≤ theta) (h1 : theta ≤ 1)
(ν : Measure A) [IsProbabilityMeasure ν]
{F : IsingSpin → A → B} (hF : Measurable (Function.uncurry F)) (c : Fin d) :
selectedPosterior (μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν)))
F (componentIndexVertex (c := c)) (RootedBroadcast.root d) =ᵐ[
μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν))]
fun z => theta * selectedPosterior
(μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν)))
F (componentIndexVertex (c := c)) (RootedBroadcast.branchRoot c) z := by
let := hμ.isProbabilityMeasure
let v (u : ComponentIndex c) : RootedBroadcast.BranchVertex c :=
⟨componentIndexVertex u, by simp [componentIndexVertex, componentVertex]⟩
let G : (RootedBroadcast.BranchVertex c → IsingSpin) ×
(RootedBroadcast.BranchVertex c → A) → (ComponentIndex c → B) :=
fun p u => F (p.1 (v u)) (p.2 (v u))
have hG : Measurable G := by
apply Measurable.of_eval
intro u
exact hF.comp (show Measurable (fun p :
(RootedBroadcast.BranchVertex c → IsingSpin) ×
(RootedBroadcast.BranchVertex c → A) => (p.1 (v u), p.2 (v u))) from by fun_prop)
have h := hμ.condExp_root_of_noisy_branch_observable h0 h1 ν c G hG
simpa only [selectedPosterior, selectedObservationField, Function.comp_def,
G, noisyBranchArray, v] using h
theorem HasFreeIsingLaw.selectedPosterior_parent_childComponent
{d : ℕ} {theta : ℝ} {A B : Type*}
[MeasurableSpace A] [StandardBorelSpace A]
[MeasurableSpace B] [StandardBorelSpace B] [Nonempty B]
{μ : Measure (TreeVertex d → IsingSpin)} (hμ : HasFreeIsingLaw theta μ)
(h0 : -1 ≤ theta) (h1 : theta ≤ 1)
(ν : Measure A) [IsProbabilityMeasure ν]
{F : IsingSpin → A → B} (hF : Measurable (Function.uncurry F)) (c a : Fin d) :
selectedPosterior (μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν)))
F (childComponentIndexVertex c a) (RootedBroadcast.branchRoot c) =ᵐ[
μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν))]
fun z => theta * selectedPosterior
(μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν)))
F (childComponentIndexVertex c a) (prefixFlip c (RootedBroadcast.branchRoot a)) z := by
have hbase := hμ.selectedPosterior_root_component h0 h1 ν hF a
have hmp := PlantedStationarity.measurePreserving_planted_relabel hμ ν (prefixTreeAut c)
have hp := hmp.quasiMeasurePreserving.ae_eq_comp hbase
have hr := selectedPosterior_relabel_ae hμ ν hF
(componentIndexVertex (c := a)) (RootedBroadcast.root d) (prefixTreeAut c)
have hc := selectedPosterior_relabel_ae hμ ν hF
(componentIndexVertex (c := a)) (RootedBroadcast.branchRoot a) (prefixTreeAut c)
filter_upwards [hp, hr, hc] with z hp hr hc
have h := hr.symm.trans (hp.trans (congrArg (fun x : ℝ => theta * x) hc))
have hindex : (fun i : ComponentIndex a =>
(prefixTreeAut c).toEquiv.symm (componentIndexVertex i)) =
childComponentIndexVertex c a := rfl
have hchild : (prefixTreeAut c).toEquiv.symm (RootedBroadcast.branchRoot a) =
prefixFlip c (RootedBroadcast.branchRoot a) := rfl
simpa only [hindex, hchild, RootedBroadcast.prefixTreeAut_symm_root] using h
end Problem367
end
end OAI
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