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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 | 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
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