| import OAI.Probability.FreeIsing.Conditioning.SideSignalLoss | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory | |
| namespace Problem367 | |
| lemma condExp_cavity_attenuation_le_of_condIndep | |
| {Ω S B : Type*} {mΩ : MeasurableSpace Ω} {mS : MeasurableSpace S} | |
| {mB : MeasurableSpace B} [StandardBorelSpace Ω] [StandardBorelSpace B] [Nonempty B] | |
| {μ : Measure Ω} [IsProbabilityMeasure μ] | |
| {theta : ℝ} (hθ0 : 0 ≤ theta) (hθ1 : theta < 1) | |
| {ρ : Measure ℝ} [IsProbabilityMeasure ρ] | |
| (hρsym : MeasurePreserving Neg.neg ρ ρ) | |
| {latent : Ω → S} {X : Ω → B} {Y : Ω → ℝ} {shift : B → ℝ} {eta : S → ℝ} | |
| (hS : Measurable latent) (hX : Measurable X) (hY : Measurable Y) | |
| (hs : Measurable shift) (hInd : X ⟂ᵢ[latent, hS; μ] Y) | |
| (hη : ∀ᵐ w ∂μ, |eta (latent w)| ≤ 1) | |
| (hLaw : ∀ᵐ w ∂μ, condDistrib Y latent μ (latent w) = | |
| linearSignalTilt ρ Real.tanh (eta (latent w) * theta)) : | |
| ∀ᵐ w ∂μ, | |
| (μ[fun v => responseAttenuation theta | |
| (shift (X v) + cavityMessage theta (Y v)) ^ 2 | mB.comap X]) w ≤ | |
| 1 - sideSignalLossCoefficient theta * (∫ y, Real.tanh y ^ 2 ∂ρ) := by | |
| have hPair := hS.prodMk hX | |
| have hIndLaw := | |
| (condIndepFun_iff_condDistrib_prod_ae_eq_prodMkRight hY hX hS).mp hInd | |
| have hIndAE := ae_of_ae_map hPair.aemeasurable hIndLaw | |
| have hLawPair : ∀ᵐ w ∂μ, | |
| condDistrib Y (fun v => (latent v, X v)) μ (latent w, X w) = | |
| linearSignalTilt ρ Real.tanh (eta (latent w) * theta) := by | |
| filter_upwards [hIndAE, hLaw] with w hw hLaww | |
| exact hw.trans hLaww | |
| have hFine := condExp_cavity_attenuation_le_of_tiltLaw | |
| (X := fun w => (latent w, X w)) (Y := Y) | |
| (shift := fun p : S × B => shift p.2) (eta := fun p : S × B => eta p.1) | |
| hθ0 hθ1 hρsym hPair hY (hs.comp measurable_snd) hη hLawPair | |
| have hcoarse : mB.comap X ≤ | |
| (inferInstance : MeasurableSpace (S × B)).comap (fun w => (latent w, X w)) := by | |
| have h := MeasurableSpace.comap_mono (g := fun w => (latent w, X w)) | |
| (measurable_snd : Measurable (Prod.snd : S × B → B)).comap_le | |
| simpa only [MeasurableSpace.comap_comp, Function.comp_def] using h | |
| let q : ℝ := 1 - sideSignalLossCoefficient theta * (∫ y, Real.tanh y ^ 2 ∂ρ) | |
| have hmono := condExp_mono (m := mB.comap X) | |
| integrable_condExp (integrable_const q) hFine | |
| rw [condExp_const hX.comap_le q] at hmono | |
| have htower := condExp_condExp_of_le (μ := μ) | |
| (f := fun v => responseAttenuation theta | |
| (shift (X v) + cavityMessage theta (Y v)) ^ 2) hcoarse hPair.comap_le | |
| filter_upwards [hmono, htower] with w hw htw | |
| rw [htw] at hw | |
| exact hw | |
| end Problem367 | |
| end | |
| end OAI | |