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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 | import OAI.Probability.FreeIsing.Posterior.ComponentPosteriorStationarity
import OAI.Probability.FreeIsing.Trees.NoisyBranchTransport
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
namespace Problem367
theorem HasFreeIsingLaw.iCondIndepFun_child_component_observations
{d : ℕ} {theta : ℝ} {A B : Type*}
[MeasurableSpace A] [StandardBorelSpace A] [MeasurableSpace B]
{μ : Measure (TreeVertex d → IsingSpin)} [IsFiniteMeasure μ] (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) :
iCondIndepFun
((inferInstance : MeasurableSpace IsingSpin).comap
(fun p : (TreeVertex d → IsingSpin) × (TreeVertex d → A) =>
p.1 ((⟨[c], List.IsChain.singleton c⟩ : TreeVertex d))))
((measurable_pi_apply ((⟨[c], List.IsChain.singleton c⟩ : TreeVertex d))).comp measurable_fst).comap_le
(fun a : Fin d => fun p : (TreeVertex d → IsingSpin) × (TreeVertex d → A) =>
fun u : ComponentIndex a => F (p.1 (childComponentIndexVertex c a u))
(p.2 (childComponentIndexVertex c a u)))
(μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν))) := by
let := hμ.isProbabilityMeasure
let v (a : Fin d) (u : ComponentIndex a) : RootedBroadcast.BranchVertex a :=
⟨componentIndexVertex u, by simp [componentIndexVertex, componentVertex]⟩
let G (a : Fin d) : (RootedBroadcast.BranchVertex a → IsingSpin) ×
(RootedBroadcast.BranchVertex a → A) → (ComponentIndex a → B) :=
fun p u => F (p.1 (v a u)) (p.2 (v a u))
have hG (a : Fin d) : Measurable (G a) := by
apply Measurable.of_eval
intro u
exact hF.comp (show Measurable (fun p :
(RootedBroadcast.BranchVertex a → IsingSpin) ×
(RootedBroadcast.BranchVertex a → A) => (p.1 (v a u), p.2 (v a u))) by fun_prop)
have h := (hμ.iCondIndepFun_relabel_noisy_branches h0 h1 ν (prefixTreeAut c)).comp G hG
exact h
theorem HasFreeIsingLaw.iCondIndepFun_forward_child_observations
{d : ℕ} {theta : ℝ} {A B : Type*}
[MeasurableSpace A] [StandardBorelSpace A] [MeasurableSpace B]
{μ : Measure (TreeVertex d → IsingSpin)} [IsFiniteMeasure μ] (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) :
iCondIndepFun
((inferInstance : MeasurableSpace IsingSpin).comap
(fun p : (TreeVertex d → IsingSpin) × (TreeVertex d → A) =>
p.1 ((⟨[c], List.IsChain.singleton c⟩ : TreeVertex d))))
((measurable_pi_apply ((⟨[c], List.IsChain.singleton c⟩ : TreeVertex d))).comp measurable_fst).comap_le
(fun a : {a : Fin d // c ≠ a} =>
fun p : (TreeVertex d → IsingSpin) × (TreeVertex d → A) =>
fun u : ComponentIndex a.1 => F (p.1 (childComponentIndexVertex c a.1 u))
(p.2 (childComponentIndexVertex c a.1 u)))
(μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν))) := by
exact Kernel.iIndepFun.precomp Subtype.val_injective
(hμ.iCondIndepFun_child_component_observations h0 h1 ν hF c)
theorem HasFreeIsingLaw.condIndepFun_own_forward_child_observations
{d : ℕ} {theta : ℝ} {A B : Type*}
[MeasurableSpace A] [StandardBorelSpace A] [MeasurableSpace B]
{μ : Measure (TreeVertex d → IsingSpin)} [IsFiniteMeasure μ] (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) :
CondIndepFun
((inferInstance : MeasurableSpace IsingSpin).comap
(fun p : (TreeVertex d → IsingSpin) × (TreeVertex d → A) =>
p.1 ((⟨[c], List.IsChain.singleton c⟩ : TreeVertex d))))
((measurable_pi_apply ((⟨[c], List.IsChain.singleton c⟩ : TreeVertex d))).comp measurable_fst).comap_le
(fun p : (TreeVertex d → IsingSpin) × (TreeVertex d → A) =>
F (p.1 ((⟨[c], List.IsChain.singleton c⟩ : TreeVertex d))) (p.2 ((⟨[c], List.IsChain.singleton c⟩ : TreeVertex d))))
(fun p : (TreeVertex d → IsingSpin) × (TreeVertex d → A) =>
fun a : {a : Fin d // c ≠ a} => fun u : ComponentIndex a.1 =>
F (p.1 (childComponentIndexVertex c a.1 u))
(p.2 (childComponentIndexVertex c a.1 u)))
(μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν))) := by
let := hμ.isProbabilityMeasure
let v (a : Fin d) (u : ComponentIndex a) : RootedBroadcast.BranchVertex a :=
⟨componentIndexVertex u, by simp [componentIndexVertex, componentVertex]⟩
let G : (∀ a : Fin d, (RootedBroadcast.BranchVertex a → IsingSpin) ×
(RootedBroadcast.BranchVertex a → A)) →
(∀ a : {a : Fin d // c ≠ a}, ComponentIndex a.1 → B) :=
fun p a u => F ((p a.1).1 (v a.1 u)) ((p a.1).2 (v a.1 u))
have hG : Measurable G := by
apply Measurable.of_eval
intro a
apply Measurable.of_eval
intro u
exact hF.comp (show Measurable (fun p :
(∀ a : Fin d, (RootedBroadcast.BranchVertex a → IsingSpin) ×
(RootedBroadcast.BranchVertex a → A)) =>
((p a.1).1 (v a.1 u), (p a.1).2 (v a.1 u))) by fun_prop)
have h := hμ.condIndepFun_relabel_root_branch_observables h0 h1 ν (prefixTreeAut c)
(Function.uncurry F) G hF hG
exact h
end Problem367
end
end OAI
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