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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 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 | import OAI.Probability.FreeIsing.Estimates.BalancedLatentMean
import OAI.Probability.FreeIsing.Trees.NoisyCutLevelIndependence
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped ENNReal
namespace Problem367.RootedBroadcast
local instance cutLevelConditionalMeanMeasurableSingletonIsingSpin :
MeasurableSingletonClass IsingSpin := ⟨fun _ => trivial⟩
local instance cutLevelConditionalMeanIsProbabilityLabelMeasure (d : ℕ) :
IsProbabilityMeasure (labelMeasure d) := by
unfold labelMeasure
infer_instance
theorem cutSpins_coordinate_fair {d : ℕ} {theta : ℝ}
(h0 : -1 ≤ theta) (h1 : theta ≤ 1) (n : ℕ) (c : TreeShell d n) (a : IsingSpin) :
((labelMeasure d).map (cutSpins theta n)) {s | s c = a} = (1 / 2 : ℝ≥0∞) := by
rw [Measure.map_apply (measurable_cutSpins d theta n)
(measurableSet_eq_fun (measurable_pi_apply c) measurable_const)]
have h := (freeIsingMeasure_hasFreeIsingLaw d h0 h1).marginal c.1 a
rw [freeIsingMeasure, Measure.map_apply (measurable_factor d theta)
(measurableSet_eq_fun (measurable_pi_apply c.1) measurable_const)] at h
change (labelMeasure d) {U | factor theta U c.1 = a} = _
change (labelMeasure d) {U | factor theta U c.1 = a} = _ at h
simpa using h
theorem condExp_noisy_factor_cone_le_two_integral {d : ℕ} {theta : ℝ} {A : Type*}
[MeasurableSpace A] [StandardBorelSpace A]
(h0 : -1 ≤ theta) (h1 : theta ≤ 1) (n : ℕ)
(ν : TreeVertex d → Measure A) [∀ v, IsProbabilityMeasure (ν v)]
(c : TreeShell d n)
(F : (DescendantVertex c.1 → IsingSpin) × (DescendantVertex c.1 → A) → ℝ)
(hF : Measurable F) (hbound : ∀ p, 0 ≤ F p ∧ F p ≤ 1) :
∀ᵐ z ∂(labelMeasure d).prod (Measure.infinitePi ν),
(((labelMeasure d).prod (Measure.infinitePi ν))[
(fun z : (TreeVertex d → ℝ) × (TreeVertex d → A) =>
F ((fun v => factor theta z.1 v.1), fun v => z.2 v.1)) |
(inferInstance : MeasurableSpace (TreeShell d n → IsingSpin)).comap
(fun z => cutSpins theta n z.1)]) z ≤
2 * ∫ w : (TreeVertex d → ℝ) × (TreeVertex d → A),
F ((fun v => factor theta w.1 v.1), fun v => w.2 v.1)
∂(labelMeasure d).prod (Measure.infinitePi ν) := by
let S := TreeShell d n → IsingSpin
let R := (e : TreeShell d n) →
(StrictDescendantVertex e.1 → ℝ) × (DescendantVertex e.1 → A)
let rho := (labelMeasure d).map (cutSpins theta n)
let muR := Measure.infinitePi (fun e : TreeShell d n =>
(Measure.infinitePi (fun _ : StrictDescendantVertex e.1 => uniform01)).prod
(Measure.infinitePi (fun v : DescendantVertex e.1 => ν v.1)))
let e : (TreeVertex d → ℝ) × (TreeVertex d → A) → S × R :=
fun z => (cutSpins theta n z.1,
fun c => (cutLabels z.1 c, fun v => z.2 v.1))
have he : MeasurePreserving e ((labelMeasure d).prod (Measure.infinitePi ν))
(rho.prod muR) :=
⟨((measurable_cutSpins d theta n).comp measurable_fst).prodMk
(by unfold cutLabels; fun_prop), map_cutSpins_cutLabels_noises theta n ν⟩
let G : IsingSpin × R → ℝ := fun p =>
F (coneFactor theta c.1 (p.1, (p.2 c).1), (p.2 c).2)
have hG : Measurable G :=
hF.comp (((measurable_coneFactor theta c.1).comp
(measurable_fst.prodMk
(measurable_fst.comp ((measurable_pi_apply c).comp measurable_snd)))).prodMk
(measurable_snd.comp ((measurable_pi_apply c).comp measurable_snd)))
have h := condExp_binary_channel_comp_le_two_integral
((labelMeasure d).prod (Measure.infinitePi ν)) rho muR e he
(fun s : S => s c) (measurable_pi_apply c)
(cutSpins_coordinate_fair h0 h1 n c) G hG (fun p => hbound _)
have hfun : (fun z => G ((e z).1 c, (e z).2)) =
(fun z : (TreeVertex d → ℝ) × (TreeVertex d → A) =>
F ((fun v => factor theta z.1 v.1), fun v => z.2 v.1)) := by
funext z
apply congrArg F
apply Prod.ext
· exact (factor_cone_representation theta c.1 z.1).symm
· rfl
rw [hfun] at h
exact h
end Problem367.RootedBroadcast
namespace Problem367
local instance cutLevelConditionalMeanFreeIsingMeasurableSingletonIsingSpin :
MeasurableSingletonClass IsingSpin := ⟨fun _ => trivial⟩
theorem freeIsingMeasure_condExp_cutLevel_observable_le_two_integral
{d : ℕ} {theta : ℝ} {A : Type*}
[MeasurableSpace A] [StandardBorelSpace A]
(h0 : -1 ≤ theta) (h1 : theta ≤ 1) (n : ℕ)
(ν : TreeVertex d → Measure A) [∀ v, IsProbabilityMeasure (ν v)]
(c : TreeShell d n)
(F : (RootedBroadcast.DescendantVertex c.1 → IsingSpin) ×
(RootedBroadcast.DescendantVertex c.1 → A) → ℝ)
(hF : Measurable F) (hbound : ∀ p, 0 ≤ F p ∧ F p ≤ 1) :
∀ᵐ z ∂(freeIsingMeasure d theta).prod (Measure.infinitePi ν),
(((freeIsingMeasure d theta).prod (Measure.infinitePi ν))[
(fun z : (TreeVertex d → IsingSpin) × (TreeVertex d → A) =>
F ((fun v => z.1 v.1), fun v => z.2 v.1)) |
(inferInstance : MeasurableSpace (TreeShell d n → IsingSpin)).comap
(fun z => fun e : TreeShell d n => z.1 e.1)]) z ≤
2 * ∫ w : (TreeVertex d → IsingSpin) × (TreeVertex d → A),
F ((fun v => w.1 v.1), fun v => w.2 v.1)
∂(freeIsingMeasure d theta).prod (Measure.infinitePi ν) := by
let : IsProbabilityMeasure (labelMeasure d) := by
unfold labelMeasure
infer_instance
let P := (freeIsingMeasure d theta).prod (Measure.infinitePi ν)
let Q := (labelMeasure d).prod (Measure.infinitePi ν)
let e : (TreeVertex d → ℝ) × (TreeVertex d → A) →
(TreeVertex d → IsingSpin) × (TreeVertex d → A) :=
Prod.map (RootedBroadcast.factor theta) id
let H : (TreeVertex d → IsingSpin) × (TreeVertex d → A) → ℝ :=
fun z => F ((fun v => z.1 v.1), fun v => z.2 v.1)
let S : (TreeVertex d → IsingSpin) × (TreeVertex d → A) →
TreeShell d n → IsingSpin := fun z c => z.1 c.1
have hS : Measurable S := by fun_prop
have hH : Measurable H := hF.comp (by fun_prop)
have hInt : Integrable H P := by
apply Integrable.of_bound hH.aestronglyMeasurable 1
exact Filter.Eventually.of_forall (fun z => by
rw [Real.norm_eq_abs, abs_of_nonneg (hbound _).1]
exact (hbound _).2)
have hf : MeasurePreserving (RootedBroadcast.factor theta)
(labelMeasure d) (freeIsingMeasure d theta) :=
⟨RootedBroadcast.measurable_factor d theta, rfl⟩
have he : MeasurePreserving e Q P := hf.prod (MeasurePreserving.id (Measure.infinitePi ν))
have hpull := condExp_comp_measurePreserving Q P e he
((inferInstance : MeasurableSpace (TreeShell d n → IsingSpin)).comap S)
hS.comap_le hInt
have hMean : (∫ x, H (e x) ∂Q) = ∫ z, H z ∂P := by
rw [← he.map_eq]
exact (integral_map he.measurable.aemeasurable hH.aestronglyMeasurable).symm
have hsource := RootedBroadcast.condExp_noisy_factor_cone_le_two_integral
h0 h1 n ν c F hF hbound
change ∀ᵐ z ∂P, (P[H | MeasurableSpace.comap S inferInstance]) z ≤
2 * ∫ w, H w ∂P
have hgoal : ∀ᵐ z ∂Q.map e,
(P[H | MeasurableSpace.comap S inferInstance]) z ≤ 2 * ∫ w, H w ∂P := by
apply (ae_map_iff he.measurable.aemeasurable
(measurableSet_le (stronglyMeasurable_condExp.mono hS.comap_le).measurable
measurable_const)).mpr
filter_upwards [hpull, hsource] with z hz hzbound
have hmeanbound : (Q[H ∘ e | (MeasurableSpace.comap S inferInstance).comap e]) z ≤
2 * ∫ x, H (e x) ∂Q := by
simp only [MeasurableSpace.comap_comp, Function.comp_def, S, H, e,
Prod.map_fst, Prod.map_snd, id_eq]
exact hzbound
rw [hz, hMean] at hmeanbound
exact hmeanbound
simpa only [he.map_eq] using hgoal
theorem HasFreeIsingLaw.condExp_cutLevel_observable_le_two_integral
{d : ℕ} {theta : ℝ} {A : Type*}
[MeasurableSpace A] [StandardBorelSpace A]
{μ : Measure (TreeVertex d → IsingSpin)} [IsFiniteMeasure μ]
(hμ : HasFreeIsingLaw theta μ) (h0 : -1 ≤ theta) (h1 : theta ≤ 1) (n : ℕ)
(ν : TreeVertex d → Measure A) [∀ v, IsProbabilityMeasure (ν v)]
(c : TreeShell d n)
(F : (RootedBroadcast.DescendantVertex c.1 → IsingSpin) ×
(RootedBroadcast.DescendantVertex c.1 → A) → ℝ)
(hF : Measurable F) (hbound : ∀ p, 0 ≤ F p ∧ F p ≤ 1) :
∀ᵐ z ∂μ.prod (Measure.infinitePi ν),
((μ.prod (Measure.infinitePi ν))[
(fun z : (TreeVertex d → IsingSpin) × (TreeVertex d → A) =>
F ((fun v => z.1 v.1), fun v => z.2 v.1)) |
(inferInstance : MeasurableSpace (TreeShell d n → IsingSpin)).comap
(fun z => fun e : TreeShell d n => z.1 e.1)]) z ≤
2 * ∫ w : (TreeVertex d → IsingSpin) × (TreeVertex d → A),
F ((fun v => w.1 v.1), fun v => w.2 v.1) ∂μ.prod (Measure.infinitePi ν) := by
have heq := hμ.eq_freeIsingMeasure h0 h1
subst μ
exact freeIsingMeasure_condExp_cutLevel_observable_le_two_integral h0 h1 n ν c F hF hbound
end Problem367
end
end OAI
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