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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 | import OAI.Probability.FreeIsing.Conditioning.ComponentInformation
import OAI.Probability.FreeIsing.Gaussian.StatisticSideSignalLoss
import OAI.Probability.FreeIsing.Posterior.Symmetry
namespace OAI
noncomputable section
open MeasureTheory
open scoped NNReal
namespace Problem367.ComponentSignalAttenuation
open PlantedSymmetry
def posteriorMean {d : ℕ} (μ : Measure (TreeVertex d → IsingSpin))
(c : Fin d) (t : ℝ≥0) :
((TreeVertex d → IsingSpin) × (TreeVertex d → ℝ)) → ℝ :=
(μ.prod (gaussianNoiseLaw (TreeVertex d) t))[
fun p => spinSign (p.1 ⟨[c], List.IsChain.singleton c⟩) |
PlantedInformation.observationSigma (componentIndexVertex (c := c)) t]
theorem measurable_posteriorMean {d : ℕ} (μ : Measure (TreeVertex d → IsingSpin))
(c : Fin d) (t : ℝ≥0) : Measurable (posteriorMean μ c t) :=
stronglyMeasurable_condExp.measurable.mono
(PlantedInformation.observationSigma_le _ _) le_rfl
theorem posteriorMean_abs_le_one {d : ℕ} (μ : Measure (TreeVertex d → IsingSpin))
(c : Fin d) (t : ℝ≥0) :
∀ᵐ p ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t), |posteriorMean μ c t p| ≤ 1 := by
apply ae_bdd_abs_condExp_of_ae_bdd_abs
filter_upwards [] with p
cases h : p.1 (⟨[c], List.IsChain.singleton c⟩ : TreeVertex d) <;>
norm_num [spinSign, h]
theorem posteriorMean_flip_ae {d : ℕ} {theta : ℝ}
{μ : Measure (TreeVertex d → IsingSpin)} (hμ : HasFreeIsingLaw theta μ)
(c : Fin d) (t : ℝ≥0) :
(posteriorMean μ c t) ∘ plantedFlipEquiv (TreeVertex d) =ᵐ[
μ.prod (gaussianNoiseLaw (TreeVertex d) t)] -posteriorMean μ c t := by
let := hμ.isProbabilityMeasure
let m := PlantedInformation.observationSigma (componentIndexVertex (c := c)) t
have hsmall : Measurable[m, m] (plantedFlipEquiv (TreeVertex d)) := by
apply measurable_comap_iff.mpr
have heq : (fun p i => observation t p (componentIndexVertex (c := c) i)) ∘
plantedFlipEquiv (TreeVertex d) =
fun p i => -observation t p (componentIndexVertex (c := c) i) := by
funext p i
exact congrFun (observation_flip t p) (componentIndexVertex i)
rw [heq]
exact (comap_measurable
(fun p i => observation t p (componentIndexVertex (c := c) i))).neg
have hnat := condExp_comp_measurePreserving_equiv
(PlantedInformation.observationSigma_le (componentIndexVertex (c := c)) t)
(plantedFlipEquiv (TreeVertex d))
(product_measurePreserving_flip hμ.flip_invariant
(gaussianNoiseLaw_map_neg (TreeVertex d) t)) hsmall hsmall
(integrable_plantedSpin (μ.prod (gaussianNoiseLaw (TreeVertex d) t))
⟨[c], List.IsChain.singleton c⟩)
have hodd : (fun p : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ) =>
spinSign (p.1 ⟨[c], List.IsChain.singleton c⟩)) ∘ plantedFlipEquiv (TreeVertex d) =
-(fun p : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ) =>
spinSign (p.1 ⟨[c], List.IsChain.singleton c⟩)) := by
funext p
cases h : p.1 (⟨[c], List.IsChain.singleton c⟩ : TreeVertex d) <;>
simp [plantedFlipEquiv, flipConfig, IsingSpin.flip, spinSign, h]
rw [hodd] at hnat
exact hnat.trans (condExp_neg _ _)
def posteriorLaw {d : ℕ} (μ : Measure (TreeVertex d → IsingSpin))
(c : Fin d) (t : ℝ≥0) : Measure ℝ :=
(μ.prod (gaussianNoiseLaw (TreeVertex d) t)).map (posteriorMean μ c t)
theorem posteriorLaw_probability {d : ℕ} {theta : ℝ}
{μ : Measure (TreeVertex d → IsingSpin)} (hμ : HasFreeIsingLaw theta μ)
(c : Fin d) (t : ℝ≥0) : IsProbabilityMeasure (posteriorLaw μ c t) := by
let := hμ.isProbabilityMeasure
unfold posteriorLaw
infer_instance
theorem posteriorLaw_abs_le_one {d : ℕ} (μ : Measure (TreeVertex d → IsingSpin))
(c : Fin d) (t : ℝ≥0) : ∀ᵐ q ∂posteriorLaw μ c t, |q| ≤ 1 := by
exact (ae_map_iff (measurable_posteriorMean μ c t).aemeasurable
(measurableSet_le measurable_id.abs measurable_const)).mpr
(posteriorMean_abs_le_one μ c t)
theorem posteriorLaw_neg_invariant {d : ℕ} {theta : ℝ}
{μ : Measure (TreeVertex d → IsingSpin)} (hμ : HasFreeIsingLaw theta μ)
(c : Fin d) (t : ℝ≥0) :
MeasurePreserving Neg.neg (posteriorLaw μ c t) (posteriorLaw μ c t) := by
let := hμ.isProbabilityMeasure
refine ⟨measurable_neg, ?_⟩
exact FactorTransport.map_invariant_of_ae_equivariant
(product_measurePreserving_flip hμ.flip_invariant
(gaussianNoiseLaw_map_neg (TreeVertex d) t)) measurable_neg
(measurable_posteriorMean μ c t) (posteriorMean_flip_ae hμ c t)
theorem posteriorLaw_information_lower {d : ℕ} {theta : ℝ}
{μ : Measure (TreeVertex d → IsingSpin)} (hμ : HasFreeIsingLaw theta μ)
(hθ0 : -1 ≤ theta) (hθ1 : theta ≤ 1) (c : Fin d)
(hcritical : ((d - 1 : ℕ) : ℝ) * theta ^ 2 = 1)
{t : ℝ≥0} (ht : 0 < t) :
min (Real.sqrt (t : ℝ)) 1 / 5 ≤ ∫ q, q ^ 2 ∂posteriorLaw μ c t := by
have hmap := integral_map (μ := μ.prod (gaussianNoiseLaw (TreeVertex d) t))
(measurable_posteriorMean μ c t).aemeasurable
(measurable_id.pow_const 2).aestronglyMeasurable
simp only [id_eq] at hmap
rw [posteriorLaw, hmap]
exact hμ.component_critical_information_lower hθ0 hθ1 c hcritical ht
def lossCoefficient (theta : ℝ) : ℝ := min (sideSignalLossCoefficient theta / 5) (1 / 2)
theorem lossCoefficient_pos {theta : ℝ} (hθ0 : 0 < theta) (hθ1 : theta < 1) :
0 < lossCoefficient theta :=
lt_min (div_pos (sideSignalLossCoefficient_pos hθ0 hθ1) (by norm_num)) (by norm_num)
theorem lossCoefficient_le_half (theta : ℝ) : lossCoefficient theta ≤ 1 / 2 :=
min_le_right _ _
theorem integral_tilted_posteriorLaw_attenuation_le {d : ℕ} {theta eta : ℝ}
{μ : Measure (TreeVertex d → IsingSpin)} (hμ : HasFreeIsingLaw theta μ)
(hθ0 : 0 < theta) (hθ1 : theta < 1) (hη : |eta| ≤ 1) (c : Fin d)
(hcritical : ((d - 1 : ℕ) : ℝ) * theta ^ 2 = 1)
{t : ℝ≥0} (ht : 0 < t) (z : ℝ) :
(∫ q, responseAttenuation theta (z + statisticMessage theta q) ^ 2
∂linearSignalTilt (posteriorLaw μ c t) id (eta * theta)) ≤
1 - lossCoefficient theta * min (Real.sqrt (t : ℝ)) 1 := by
let := posteriorLaw_probability hμ c t
have h := integral_tilted_statistic_attenuation_le hθ0.le hθ1 hη
measurable_id (posteriorLaw_abs_le_one μ c t) (MeasurableEquiv.neg ℝ)
(posteriorLaw_neg_invariant hμ c t) (Filter.Eventually.of_forall (fun _ => rfl)) z
have hm := posteriorLaw_information_lower hμ (by linarith) hθ1.le c hcritical ht
have hc := sideSignalLossCoefficient_pos hθ0 hθ1
have hω : 0 ≤ min (Real.sqrt (t : ℝ)) 1 := le_min (Real.sqrt_nonneg _) zero_le_one
have hco := mul_le_mul_of_nonneg_right
(show lossCoefficient theta ≤ sideSignalLossCoefficient theta / 5 from min_le_left _ _) hω
have hmul := mul_le_mul_of_nonneg_left hm hc.le
simp only [id_eq] at h
nlinarith
end Problem367.ComponentSignalAttenuation
end
end OAI
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