| import OAI.Probability.FreeIsing.Posterior.LatentObservationIndependence |
| import OAI.Probability.FreeIsing.Trees.BroadcastBranchLabels |
| import OAI.Probability.FreeIsing.Trees.BroadcastJointLaw |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory |
|
|
| namespace Problem367.RootedBroadcast |
|
|
| local instance : MeasurableSingletonClass IsingSpin := ⟨fun _ => trivial⟩ |
|
|
| local instance (d : ℕ) : IsProbabilityMeasure (labelMeasure d) := by |
| unfold labelMeasure |
| infer_instance |
|
|
| theorem map_factor_disjoint_blocks {d : ℕ} (theta : ℝ) (v : TreeVertex d) |
| (S T : Set (NonrootVertex d)) (hST : Disjoint S T) : |
| (labelMeasure d).map (fun U => (factor theta U v, |
| (fun i : S => U i.val.val), (fun i : T => U i.val.val))) = |
| (uniform01.map fairSpin).prod |
| ((Measure.infinitePi (fun _ : S => uniform01)).prod |
| (Measure.infinitePi (fun _ : T => uniform01))) := by |
| let H : (NonrootVertex d → ℝ) → (S → ℝ) × (T → ℝ) := |
| fun Y => ((fun i => Y i.val), (fun i => Y i.val)) |
| have hH : Measurable H := by fun_prop |
| have hJ : Measurable (fun U : TreeVertex d → ℝ => |
| (factor theta U v, fun w : NonrootVertex d => U w.val)) := |
| ((measurable_pi_apply v).comp (measurable_factor d theta)).prodMk |
| (Measurable.of_eval (fun vertex => measurable_pi_apply vertex.val)) |
| have hmap : (Measure.infinitePi (fun _ : NonrootVertex d => uniform01)).map H = |
| (Measure.infinitePi (fun _ : S => uniform01)).prod |
| (Measure.infinitePi (fun _ : T => uniform01)) := |
| map_disjoint_subarrays (fun _ : NonrootVertex d => uniform01) S T hST |
| calc |
| _ = ((labelMeasure d).map (fun U => |
| (factor theta U v, fun w : NonrootVertex d => U w.val))).map (Prod.map id H) := by |
| rw [Measure.map_map (measurable_id.prodMap hH) hJ] |
| rfl |
| _ = _ := by |
| rw [map_factor_rest] |
| rw [← Measure.map_prod_map _ _ measurable_id hH] |
| rw [Measure.map_id, hmap] |
|
|
| theorem condIndepFun_factor_disjoint_blocks {d : ℕ} (theta : ℝ) (v : TreeVertex d) |
| (S T : Set (NonrootVertex d)) (hST : Disjoint S T) |
| {A B : Type*} [MeasurableSpace A] [MeasurableSpace B] |
| (F : IsingSpin × (S → ℝ) → A) (G : IsingSpin × (T → ℝ) → B) |
| (hF : Measurable F) (hG : Measurable G) : |
| CondIndepFun (MeasurableSpace.comap (fun U => factor theta U v) inferInstance) |
| (((measurable_pi_apply v).comp (measurable_factor d theta)).comap_le) |
| (fun U => F (factor theta U v, fun i : S => U i.val.val)) |
| (fun U => G (factor theta U v, fun i : T => U i.val.val)) |
| (labelMeasure d) := by |
| let : IsProbabilityMeasure (labelMeasure d) := by |
| unfold labelMeasure |
| infer_instance |
| exact condIndepFun_of_product_jointLaw (labelMeasure d) (uniform01.map fairSpin) |
| (Measure.infinitePi (fun _ : S => uniform01)) |
| (Measure.infinitePi (fun _ : T => uniform01)) |
| ((measurable_pi_apply v).comp (measurable_factor d theta)) |
| (by fun_prop) (by fun_prop) (map_factor_disjoint_blocks theta v S T hST) |
| F G hF hG |
|
|
| end Problem367.RootedBroadcast |
|
|
| end |
|
|
| end OAI |
|
|
