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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 | import OAI.Probability.FreeIsing.Posterior.ChildObservationRegression
import OAI.Probability.FreeIsing.Trees.BranchFlipGeometry
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
namespace Problem367
abbrev NoisyRootOtherData {d : ℕ} (A : Type*) (c : Fin d) :=
(IsingSpin × A) × (∀ b : {b : Fin d // b ≠ c},
(RootedBroadcast.BranchVertex b.val → IsingSpin) ×
(RootedBroadcast.BranchVertex b.val → A))
private theorem exists_otherBranch_of_nonroot {d : ℕ} (c : Fin d)
(w : {w : TreeVertex d // ¬ [c] <+: w.val}) (hw : w.val.val ≠ []) :
∃ b : {b : Fin d // b ≠ c}, [b.val] <+: w.val.val := by
rcases w with ⟨⟨w, hwred⟩, hwc⟩
cases w with
| nil => exact False.elim (hw rfl)
| cons b t =>
have hbc : b ≠ c := by
intro h
subst b
exact hwc (by simp)
exact ⟨⟨b, hbc⟩, by simp⟩
def noisyRootOtherDecode {d : ℕ} {A : Type*} (c : Fin d)
(p : NoisyRootOtherData A c)
(w : {w : TreeVertex d // ¬ [c] <+: w.val}) : IsingSpin × A :=
if hw : w.val.val = [] then p.1 else
let b := Classical.choose (exists_otherBranch_of_nonroot c w hw)
let v : RootedBroadcast.BranchVertex b.val :=
⟨w.val, Classical.choose_spec (exists_otherBranch_of_nonroot c w hw)⟩
((p.2 b).1 v, (p.2 b).2 v)
theorem measurable_noisyRootOtherDecode {d : ℕ} {A : Type*} [MeasurableSpace A]
(c : Fin d) (w : {w : TreeVertex d // ¬ [c] <+: w.val}) :
Measurable (fun p : NoisyRootOtherData A c => noisyRootOtherDecode c p w) := by
unfold noisyRootOtherDecode
split_ifs <;> dsimp <;> fun_prop
theorem noisyRootOtherDecode_array {d : ℕ} {A : Type*} (c : Fin d)
(z : (TreeVertex d → IsingSpin) × (TreeVertex d → A))
(w : {w : TreeVertex d // ¬ [c] <+: w.val}) :
noisyRootOtherDecode c (noisyRootOtherArray c z) w = (z.1 w.val, z.2 w.val) := by
unfold noisyRootOtherDecode
split_ifs with hw
· have hroot : w.val = RootedBroadcast.root d := Subtype.ext hw
simp only [noisyRootOtherArray, hroot]
· rfl
def decodeFlippedBranch {d : ℕ} {A : Type*} (c : Fin d)
(p : NoisyRootOtherData A c) :
(RootedBroadcast.BranchVertex c → IsingSpin) × (RootedBroadcast.BranchVertex c → A) :=
((fun v => (noisyRootOtherDecode c p (RootedBroadcast.flipBranchToComplement c v)).1),
fun v => (noisyRootOtherDecode c p (RootedBroadcast.flipBranchToComplement c v)).2)
theorem measurable_decodeFlippedBranch {d : ℕ} {A : Type*} [MeasurableSpace A]
(c : Fin d) : Measurable (decodeFlippedBranch (A := A) c) :=
(Measurable.of_eval (fun v =>
(measurable_noisyRootOtherDecode c (RootedBroadcast.flipBranchToComplement c v)).fst)).prodMk
(Measurable.of_eval (fun v =>
(measurable_noisyRootOtherDecode c (RootedBroadcast.flipBranchToComplement c v)).snd))
theorem decodeFlippedBranch_relabel {d : ℕ} {A : Type*} (c : Fin d)
(z : (TreeVertex d → IsingSpin) × (TreeVertex d → A)) :
decodeFlippedBranch c (noisyRootOtherArray c
(Prod.map (relabel (prefixTreeAut c)) (relabel (prefixTreeAut c)) z)) =
noisyBranchArray c z := by
apply Prod.ext
· funext v
change (noisyRootOtherDecode c _ (RootedBroadcast.flipBranchToComplement c v)).1 = _
rw [noisyRootOtherDecode_array]
change z.1 (prefixFlip c (prefixFlip c v.val)) = z.1 v.val
rw [prefixFlip_twice]
· funext v
change (noisyRootOtherDecode c _ (RootedBroadcast.flipBranchToComplement c v)).2 = _
rw [noisyRootOtherDecode_array]
change z.2 (prefixFlip c (prefixFlip c v.val)) = z.2 v.val
rw [prefixFlip_twice]
theorem HasFreeIsingLaw.condExp_root_of_noisy_branch_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] (c : Fin d)
(G : (RootedBroadcast.BranchVertex c → IsingSpin) ×
(RootedBroadcast.BranchVertex c → A) → C) (hG : Measurable G) :
(μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν)))[
(fun z : (TreeVertex d → IsingSpin) × (TreeVertex d → A) =>
spinSign (z.1 (RootedBroadcast.root d))) |
mC.comap (G ∘ noisyBranchArray c)] =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 (RootedBroadcast.branchRoot c))) |
mC.comap (G ∘ noisyBranchArray c)] z := by
have h := hμ.condExp_relabel_spin_of_noisy_root_others_observable h0 h1 ν
(prefixTreeAut c) c (G ∘ decodeFlippedBranch c)
(hG.comp (measurable_decodeFlippedBranch c))
have hcomp : (G ∘ decodeFlippedBranch c) ∘ noisyRootOtherArray c ∘
Prod.map (relabel (prefixTreeAut c)) (relabel (prefixTreeAut c)) =
G ∘ noisyBranchArray c := by
funext z
exact congrArg G (decodeFlippedBranch_relabel c z)
have hparent : (prefixTreeAut c).toEquiv.symm ⟨[c], List.IsChain.singleton c⟩ =
RootedBroadcast.root d := RootedBroadcast.prefixTreeAut_symm_branchRoot c
simpa only [hcomp, hparent, RootedBroadcast.prefixTreeAut_symm_root] using h
end Problem367
end
end OAI
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