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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 | import OAI.Probability.FreeIsing.Conditioning.NoisyBranchConditionalLaw
import OAI.Probability.FreeIsing.Posterior.ComponentObservationRegression
import OAI.Probability.FreeIsing.Posterior.StatisticTilt
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
namespace Problem367
local instance componentPosteriorTiltMeasurableSingletonSpin : MeasurableSingletonClass IsingSpin :=
⟨fun _ => trivial⟩
local instance componentPosteriorTiltNonemptySpin : Nonempty IsingSpin := ⟨.neg⟩
theorem HasFreeIsingLaw.condDistrib_componentPosterior_root
{d : ℕ} {theta : ℝ} {A B : Type*}
[MeasurableSpace A] [StandardBorelSpace A]
[MeasurableSpace B] [StandardBorelSpace B] [Nonempty 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) (a : IsingSpin) :
let P := μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν))
let q := selectedPosterior P F (componentIndexVertex (c := c))
(RootedBroadcast.branchRoot c)
condDistrib q (fun z => z.1 (RootedBroadcast.root d)) P a =
linearSignalTilt (P.map q) id (spinSign a * theta) := by
let := hμ.isProbabilityMeasure
dsimp only
apply condDistrib_statistic_eq_tilt_of_linear_posterior
(selectedObservationField_le hF (componentIndexVertex (c := c)))
((measurable_pi_apply (RootedBroadcast.root d)).comp measurable_fst)
stronglyMeasurable_condExp.measurable integrable_condExp
· exact hμ.selectedPosterior_root_component h0 h1 ν hF c
· exact hμ.planted_spin_event_mass (fun _ => ν) _
theorem HasFreeIsingLaw.map_cond_componentPosterior_root
{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) (a : IsingSpin) :
let P := μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν))
let q := selectedPosterior P F (componentIndexVertex (c := c))
(RootedBroadcast.branchRoot c)
(P[|{z | z.1 (RootedBroadcast.root d) = a}]).map q =
linearSignalTilt (P.map q) id (spinSign a * theta) := by
let := hμ.isProbabilityMeasure
have h := hμ.condDistrib_componentPosterior_root h0 h1 ν hF c a
dsimp only at h ⊢
rw [condDistrib_eq_map_cond_fiber
(μ := μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν)))
(S := fun z => z.1 (RootedBroadcast.root d))
(Y := selectedPosterior (μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν)))
F (componentIndexVertex (c := c)) (RootedBroadcast.branchRoot c))
((measurable_pi_apply (RootedBroadcast.root d)).comp measurable_fst)
(measurable_selectedPosterior _ hF _ _) a (by
change (μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν)))
{z | z.1 (RootedBroadcast.root d) = a} ≠ 0
rw [hμ.planted_spin_event_mass]
norm_num)] at h
exact h
theorem HasFreeIsingLaw.condDistrib_childComponentPosterior_parent
{d : ℕ} {theta : ℝ} {A B : Type*}
[MeasurableSpace A] [StandardBorelSpace A]
[MeasurableSpace B] [StandardBorelSpace B] [Nonempty 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 e : Fin d) (a : IsingSpin) :
let P := μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν))
let q := selectedPosterior P F (childComponentIndexVertex c e)
(prefixFlip c (RootedBroadcast.branchRoot e))
condDistrib q (fun z => z.1 (RootedBroadcast.branchRoot c)) P a =
linearSignalTilt (P.map q) id (spinSign a * theta) := by
let := hμ.isProbabilityMeasure
dsimp only
apply condDistrib_statistic_eq_tilt_of_linear_posterior
(selectedObservationField_le hF (childComponentIndexVertex c e))
((measurable_pi_apply (RootedBroadcast.branchRoot c)).comp measurable_fst)
stronglyMeasurable_condExp.measurable integrable_condExp
· exact hμ.selectedPosterior_parent_childComponent h0 h1 ν hF c e
· exact hμ.planted_spin_event_mass (fun _ => ν) _
end Problem367
end
end OAI
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