File size: 3,063 Bytes
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 | 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
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