| 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 |
|
|
