| 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 |
|
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| end OAI |
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