File size: 4,798 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 99 100 | import OAI.Probability.FreeIsing.Conditioning.NoisyBranchConditionalLaw
import OAI.Probability.FreeIsing.Estimates.IndependentOwnFamily
import OAI.Probability.FreeIsing.Posterior.ComponentObservationIndependence
import OAI.Probability.FreeIsing.Trees.BranchFlipGeometry
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
namespace Problem367
def componentOwnChildFields {d : ℕ} {A : Type*} [MeasurableSpace A]
(ρ : Measure ((TreeVertex d → IsingSpin) × (TreeVertex d → A)))
(F : IsingSpin → A → ℝ) (c : Fin d) :
Option {a : Fin d // c ≠ a} → ((TreeVertex d → IsingSpin) × (TreeVertex d → A)) → ℝ
| none => fun p => F (p.1 (RootedBroadcast.branchRoot c))
(p.2 (RootedBroadcast.branchRoot c))
| some a => fun p => Real.artanh (selectedPosterior ρ F
(childComponentIndexVertex c a.1)
(prefixFlip c (RootedBroadcast.branchRoot a.1)) p)
private theorem measurable_artanh : Measurable Real.artanh := by
unfold Real.artanh
fun_prop
theorem measurable_componentOwnChildFields {d : ℕ} {A : Type*} [MeasurableSpace A]
(ρ : Measure ((TreeVertex d → IsingSpin) × (TreeVertex d → A)))
{F : IsingSpin → A → ℝ} (hF : Measurable (Function.uncurry F))
(c : Fin d) (i : Option {a : Fin d // c ≠ a}) :
Measurable (componentOwnChildFields ρ F c i) := by
cases i with
| none =>
exact hF.comp (show Measurable (fun p :
(TreeVertex d → IsingSpin) × (TreeVertex d → A) =>
(p.1 (RootedBroadcast.branchRoot c), p.2 (RootedBroadcast.branchRoot c))) by fun_prop)
| some a => exact measurable_artanh.comp (measurable_selectedPosterior ρ hF _ _)
theorem HasFreeIsingLaw.iIndepFun_component_fields_cond_spin
{d : ℕ} {θ : ℝ} {A : Type*}
[MeasurableSpace A] [StandardBorelSpace A]
{μ : Measure (TreeVertex d → IsingSpin)} (hμ : HasFreeIsingLaw θ μ)
(h0 : -1 ≤ θ) (h1 : θ ≤ 1)
(ν : Measure A) [IsProbabilityMeasure ν]
{F : IsingSpin → A → ℝ} (hF : Measurable (Function.uncurry F))
(c : Fin d) (s : IsingSpin) :
let ρ := μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν))
iIndepFun (componentOwnChildFields ρ F c)
ρ[|{p | p.1 (RootedBroadcast.branchRoot c) = s}] := by
let := hμ.isProbabilityMeasure
let Ω := (TreeVertex d → IsingSpin) × (TreeVertex d → A)
let ρ := μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν))
let I := {a : Fin d // c ≠ a}
let S : Ω → IsingSpin := fun p => p.1 (RootedBroadcast.branchRoot c)
let X (a : I) (p : Ω) : ComponentIndex a.1 → ℝ :=
fun u => F (p.1 (childComponentIndexVertex c a.1 u))
(p.2 (childComponentIndexVertex c a.1 u))
let Y : Ω → ℝ := componentOwnChildFields ρ F c none
let Z (a : I) : Ω → ℝ := componentOwnChildFields ρ F c (some a)
have hS : Measurable S := (measurable_pi_apply _).comp measurable_fst
have hX (a : I) : Measurable (X a) :=
Measurable.of_eval fun u => hF.comp (show Measurable (fun p : Ω =>
(p.1 (childComponentIndexVertex c a.1 u),
p.2 (childComponentIndexVertex c a.1 u))) from
((measurable_pi_apply _).comp measurable_fst).prodMk
((measurable_pi_apply _).comp measurable_snd))
have hY : Measurable Y := measurable_componentOwnChildFields ρ hF c none
have hZ (a : I) : Measurable (Z a) :=
measurable_componentOwnChildFields ρ hF c (some a)
have hmass : ρ (S ⁻¹' {s}) ≠ 0 := by
change ρ {p | p.1 (RootedBroadcast.branchRoot c) = s} ≠ 0
rw [hμ.planted_spin_event_mass]
norm_num
let : IsProbabilityMeasure ρ[|S ⁻¹' {s}] := cond_isProbabilityMeasure hmass
have hchildren : iIndepFun X ρ[|S ⁻¹' {s}] :=
iIndepFun_cond_fiber_of_iCondIndepFun hS hX
(hμ.iCondIndepFun_forward_child_observations h0 h1 ν hF c) s hmass
have hown : IndepFun Y (fun p a => X a p) ρ[|S ⁻¹' {s}] :=
indepFun_cond_fiber_of_condIndepFun hS hY (Measurable.of_eval hX)
(hμ.condIndepFun_own_forward_child_observations h0 h1 ν hF c) s
have hsmall (a : I) :
Measurable[MeasurableSpace.comap (X a) inferInstance, inferInstance] (Z a) :=
measurable_artanh.comp stronglyMeasurable_condExp.measurable
have hZchildren : iIndepFun Z ρ[|S ⁻¹' {s}] :=
iIndep_of_iIndep_of_le hchildren (fun a => (hsmall a).comap_le)
have hZown : IndepFun Y (fun p a => Z a p) ρ[|S ⁻¹' {s}] := by
apply indep_of_indep_of_le hown le_rfl
rw [MeasurableSpace.comap_process_pi, MeasurableSpace.comap_process_pi]
exact iSup_mono fun a => (hsmall a).comap_le
change iIndepFun (componentOwnChildFields ρ F c) ρ[|S ⁻¹' {s}]
exact iIndepFun_option_of_indep_tuple (μ := ρ[|S ⁻¹' {s}])
(measurable_componentOwnChildFields ρ hF c) hZchildren hZown
end Problem367
end
end OAI
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