File size: 4,368 Bytes
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 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.DiscreteConditioning
import OAI.Probability.FreeIsing.Trees.NoisyBranchIndependence
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
namespace Problem367
local instance noisyBranchConditionalLawMeasurableSingletonIsingSpin :
MeasurableSingletonClass IsingSpin := ⟨fun _ => trivial⟩
theorem HasFreeIsingLaw.planted_spin_event_mass
{d : ℕ} {theta : ℝ} {A : Type*} [MeasurableSpace A]
{μ : Measure (TreeVertex d → IsingSpin)} (hμ : HasFreeIsingLaw theta μ)
(ν : TreeVertex d → Measure A) [∀ v, IsProbabilityMeasure (ν v)]
(v : TreeVertex d) (s : IsingSpin) :
(μ.prod (Measure.infinitePi ν))
{z : (TreeVertex d → IsingSpin) × (TreeVertex d → A) | z.1 v = s} = 1 / 2 := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
have hset :
{z : (TreeVertex d → IsingSpin) × (TreeVertex d → A) | z.1 v = s} =
{x : TreeVertex d → IsingSpin | x v = s} ×ˢ Set.univ := by
ext z
simp
rw [hset, Measure.prod_prod, measure_univ, mul_one, hμ.marginal]
norm_num [ENNReal.ofReal_div_of_pos]
def plantedBranchObservations {d : ℕ} {A T : Type*}
(a : T → ℝ) (B : TreeVertex d → T → A → ℝ) (c : Fin d)
(z : (TreeVertex d → IsingSpin) × (TreeVertex d → A))
(v : RootedBroadcast.BranchVertex c) (t : T) : ℝ :=
a t * (z.1 v.val).toReal + B v.val t (z.2 v.val)
theorem measurable_plantedBranchObservations {d : ℕ} {A T : Type*}
[MeasurableSpace A] (a : T → ℝ) (B : TreeVertex d → T → A → ℝ)
(hB : ∀ v t, Measurable (B v t)) (c : Fin d) :
Measurable (plantedBranchObservations a B c) := by
apply Measurable.of_eval
intro v
apply Measurable.of_eval
intro t
exact ((measurable_spinToReal.comp ((measurable_pi_apply v.val).comp
measurable_fst)).const_mul (a t)).add
((hB v.val t).comp ((measurable_pi_apply v.val).comp measurable_snd))
theorem HasFreeIsingLaw.iIndepFun_planted_branches_cond_root
{d : ℕ} {theta : ℝ} {A T : Type*}
[MeasurableSpace A] [StandardBorelSpace A]
{μ : Measure (TreeVertex d → IsingSpin)} (hμ : HasFreeIsingLaw theta μ)
(h0 : -1 ≤ theta) (h1 : theta ≤ 1)
(ν : TreeVertex d → Measure A) [∀ v, IsProbabilityMeasure (ν v)]
(a : T → ℝ) (B : TreeVertex d → T → A → ℝ)
(hB : ∀ v t, Measurable (B v t)) (s : IsingSpin) :
iIndepFun (plantedBranchObservations a B)
(μ.prod (Measure.infinitePi ν))[|
{z : (TreeVertex d → IsingSpin) × (TreeVertex d → A) |
z.1 (RootedBroadcast.root d) = s}] := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
have hroot : Measurable (fun z : (TreeVertex d → IsingSpin) × (TreeVertex d → A) =>
z.1 (RootedBroadcast.root d)) :=
(measurable_pi_apply (RootedBroadcast.root d)).comp measurable_fst
apply iIndepFun_cond_fiber_of_iCondIndepFun hroot
(measurable_plantedBranchObservations a B hB)
(hμ.iCondIndepFun_planted_branch_observations h0 h1 ν a B hB) s
change (μ.prod (Measure.infinitePi ν))
{z : (TreeVertex d → IsingSpin) × (TreeVertex d → A) |
z.1 (RootedBroadcast.root d) = s} ≠ 0
rw [hμ.planted_spin_event_mass ν]
norm_num
theorem HasFreeIsingLaw.map_planted_branches_cond_root
{d : ℕ} {theta : ℝ} {A T : Type*}
[MeasurableSpace A] [StandardBorelSpace A]
{μ : Measure (TreeVertex d → IsingSpin)} (hμ : HasFreeIsingLaw theta μ)
(h0 : -1 ≤ theta) (h1 : theta ≤ 1)
(ν : TreeVertex d → Measure A) [∀ v, IsProbabilityMeasure (ν v)]
(a : T → ℝ) (B : TreeVertex d → T → A → ℝ)
(hB : ∀ v t, Measurable (B v t)) (s : IsingSpin) :
((μ.prod (Measure.infinitePi ν))[|
{z : (TreeVertex d → IsingSpin) × (TreeVertex d → A) |
z.1 (RootedBroadcast.root d) = s}]).map
(fun z c => plantedBranchObservations a B c z) =
Measure.infinitePi (fun c : Fin d =>
((μ.prod (Measure.infinitePi ν))[|
{z : (TreeVertex d → IsingSpin) × (TreeVertex d → A) |
z.1 (RootedBroadcast.root d) = s}]).map (plantedBranchObservations a B c)) := by
exact (hμ.iIndepFun_planted_branches_cond_root h0 h1 ν a B hB s).map_fun_eq_infinitePi_map
(measurable_plantedBranchObservations a B hB)
end Problem367
end
end OAI
|