| import OAI.Probability.FreeIsing.Trees.RootComponentAttenuation |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory |
| open scoped NNReal |
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| namespace Problem367.ComponentSignalAttenuation |
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| open PlantedSymmetry |
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| theorem condExp_attenuation_le_critical_relabel_root_others |
| {d : ℕ} {theta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw theta μ) (hθ0 : 0 < theta) (hθ1 : theta < 1) |
| (c : Fin d) (hcritical : ((d - 1 : ℕ) : ℝ) * theta ^ 2 = 1) |
| {t : ℝ≥0} (ht : 0 < t) (g : TreeAut d) |
| {B : Type*} [mB : MeasurableSpace B] [StandardBorelSpace B] [Nonempty B] |
| (G : NoisyRootOtherData ℝ c → B) (hG : Measurable G) |
| (shift : B → ℝ) (hs : Measurable shift) : |
| let P := μ.prod (gaussianNoiseLaw (TreeVertex d) t) |
| let X := (G ∘ noisyRootOtherArray c) ∘ Prod.map (relabel g) (relabel g) |
| let Q := selectedPosterior P (fun s z => (t : ℝ) * spinSign s + z) |
| (fun i => g.toEquiv.symm (componentIndexVertex (c := c) i)) |
| (g.toEquiv.symm (RootedBroadcast.branchRoot c)) |
| ∀ᵐ w ∂P, |
| (P[fun v => responseAttenuation theta |
| (shift (X v) + statisticMessage theta (Q v)) ^ 2 | mB.comap X]) w ≤ |
| 1 - lossCoefficient theta * min (Real.sqrt (t : ℝ)) 1 := by |
| let := hμ.isProbabilityMeasure |
| let P := μ.prod (gaussianNoiseLaw (TreeVertex d) t) |
| let T : ((TreeVertex d → IsingSpin) × (TreeVertex d → ℝ)) → |
| ((TreeVertex d → IsingSpin) × (TreeVertex d → ℝ)) := |
| Prod.map (relabel g) (relabel g) |
| let X := G ∘ noisyRootOtherArray (A := ℝ) c |
| let f := fun v => responseAttenuation theta |
| (shift (X v) + statisticMessage theta (posteriorMean μ c t v)) ^ 2 |
| have hX : Measurable X := hG.comp (measurable_noisyRootOtherArray c) |
| have hθsq : theta ^ 2 ≤ 1 := by nlinarith |
| have hfm : Measurable f := |
| ((continuous_responseAttenuation hθsq).measurable.comp |
| ((hs.comp hX).add ((measurable_statisticMessage theta).comp |
| (measurable_posteriorMean μ c t)))).pow_const 2 |
| have hfi : Integrable f P := by |
| apply (integrable_const (1 : ℝ)).mono' hfm.aestronglyMeasurable |
| filter_upwards [] with v |
| rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)] |
| have hpos := responseAttenuation_pos hθsq |
| (shift (X v) + statisticMessage theta (posteriorMean μ c t v)) |
| have hle := responseAttenuation_le_one hθsq |
| (shift (X v) + statisticMessage theta (posteriorMean μ c t v)) |
| dsimp only [f] |
| nlinarith |
| have hmp : MeasurePreserving T P P := |
| PlantedStationarity.measurePreserving_planted_relabel hμ (gaussianReal 0 t) g |
| have hCE := condExp_comp_measurePreserving P P T hmp (mB.comap X) hX.comap_le hfi |
| have hbase := hmp.quasiMeasurePreserving.ae |
| (condExp_attenuation_le_critical_root_others hμ hθ0 hθ1 c hcritical ht G hG shift hs) |
| have hchannel : Measurable (Function.uncurry |
| (fun s z => (t : ℝ) * spinSign s + z)) := |
| ((PlantedInformation.measurable_spinSign.comp measurable_fst).const_mul |
| (t : ℝ)).add measurable_snd |
| have hpost := selectedPosterior_relabel_ae hμ (gaussianReal 0 t) hchannel |
| (componentIndexVertex (c := c)) (RootedBroadcast.branchRoot c) g |
| have hfc : f ∘ T =ᵐ[P] fun v => responseAttenuation theta |
| (shift ((X ∘ T) v) + statisticMessage theta |
| (selectedPosterior P (fun s z => (t : ℝ) * spinSign s + z) |
| (fun i => g.toEquiv.symm (componentIndexVertex (c := c) i)) |
| (g.toEquiv.symm (RootedBroadcast.branchRoot c)) v)) ^ 2 := by |
| filter_upwards [hpost] with v hv |
| exact congrArg (fun q => responseAttenuation theta |
| (shift ((X ∘ T) v) + statisticMessage theta q) ^ 2) hv |
| have hcongr := condExp_congr_ae (m := mB.comap (X ∘ T)) hfc |
| rw [MeasurableSpace.comap_comp] at hCE |
| filter_upwards [hCE, hcongr, hbase] with w hw hc hb |
| exact hc.symm.trans_le (hw.trans_le hb) |
| |
| end Problem367.ComponentSignalAttenuation |
| |
| end |
| |
| end OAI |
| |
