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f61e547 | 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 | import OAI.Probability.FreeIsing.Conditioning.FamilyConditionalPullback
import OAI.Probability.FreeIsing.Trees.BroadcastBranchFamily
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 iCondIndepFun_factor_branches (d : ℕ) (theta : ℝ) :
iCondIndepFun
((inferInstance : MeasurableSpace IsingSpin).comap
(fun U : TreeVertex d → ℝ => fairSpin (U (root d))))
(measurable_fairSpin.comp (measurable_pi_apply (root d))).comap_le
(fun c U => fun v : BranchVertex c => factor theta U v.val)
(labelMeasure d) := by
let e : (TreeVertex d → ℝ) → IsingSpin × (∀ c : Fin d, BranchVertex c → ℝ) :=
fun U => (fairSpin (U (root d)), fun c v => U v.val)
let ν : Measure (IsingSpin × (∀ c : Fin d, BranchVertex c → ℝ)) :=
(uniform01.map fairSpin).prod (Measure.infinitePi (fun c : Fin d =>
Measure.infinitePi (fun _ : BranchVertex c => uniform01)))
have he : MeasurePreserving e (labelMeasure d) ν :=
⟨(measurable_fairSpin.comp (measurable_pi_apply (root d))).prodMk
(by fun_prop), map_root_branch_family d⟩
have hbase := iCondIndepFun_branch_product d theta
have hpull := iCondIndepFun_comp_measurePreserving (labelMeasure d) ν e he
((inferInstance : MeasurableSpace IsingSpin).comap Prod.fst)
measurable_fst.comap_le
(fun c : Fin d => show Measurable
(fun z : IsingSpin × (∀ c : Fin d, BranchVertex c → ℝ) =>
branchFactor theta c (z.1, z.2 c)) from
(measurable_branchFactor theta c).comp
(measurable_fst.prodMk ((measurable_pi_apply c).comp measurable_snd))) hbase
have hfun : (fun c : Fin d =>
(fun z : IsingSpin × (∀ c : Fin d, BranchVertex c → ℝ) =>
branchFactor theta c (z.1, z.2 c)) ∘ e) =
(fun c U => fun v : BranchVertex c => factor theta U v.val) := by
funext c U
exact (factor_branch_representation theta c U).symm
rw [hfun] at hpull
simpa only [MeasurableSpace.comap_comp, Function.comp_def, e] using hpull
end Problem367.RootedBroadcast
end
end OAI
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