| import OAI.Probability.FreeIsing.Posterior.FiniteIsingFilteringRepresentation |
| import OAI.Probability.FreeIsing.Posterior.PlantedArrayCrossEnergy |
| import OAI.Probability.FreeIsing.Stochastic.FiniteIsingBrownianHistory |
|
|
| namespace OAI |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory Filter |
| open scoped NNReal Topology |
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| namespace Problem367.FiniteIsingFiltering |
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| variable {d : ℕ} {Ω : Type*} [MeasurableSpace Ω] |
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| theorem mean_eq_condExp_history {beta : ℝ} |
| {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) |
| (ν : TreeVertex d → Measure Ω) |
| {B : TreeVertex d → ℝ≥0 → Ω → ℝ} |
| (hB : ∀ i, IsPreBrownianReal (B i) (ν i)) |
| (hmB : ∀ i t, Measurable (B i t)) |
| (D : Finset (TreeVertex d)) (hD : ConnectedFinset D) (v : D) (t : ℝ≥0) : |
| mean beta D v B t =ᵐ[μ.prod (Measure.infinitePi ν)] |
| (μ.prod (Measure.infinitePi ν))[(fun p : State d Ω => spinSign (p.1 v)) | |
| observationFiltration B hmB D t] := by |
| rw [observationFiltration_eq_comap] |
| exact (hμ.condExp_finiteIsingBrownian_past ν hB hmB D hD |
| (fun eta => spinSign (eta v)) t).symm |
|
|
| theorem covariance_eq_condExp_history {beta : ℝ} |
| {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) |
| (ν : TreeVertex d → Measure Ω) |
| {B : TreeVertex d → ℝ≥0 → Ω → ℝ} |
| (hB : ∀ i, IsPreBrownianReal (B i) (ν i)) |
| (hmB : ∀ i t, Measurable (B i t)) |
| (D : Finset (TreeVertex d)) (hD : ConnectedFinset D) (v u : D) (t : ℝ≥0) : |
| let P := μ.prod (Measure.infinitePi ν) |
| let F := observationFiltration B hmB D |
| covariance beta D v u B t =ᵐ[P] fun p => |
| P[(fun p : State d Ω => spinSign (p.1 v) * spinSign (p.1 u)) | F t] p - |
| P[(fun p : State d Ω => spinSign (p.1 v)) | F t] p * mean beta D u B t p := by |
| have hc := hμ.finiteIsingBrownianCovariance_eq_past ν hB hmB D hD v u t |
| rw [← observationFiltration_eq_comap B hmB D t] at hc |
| filter_upwards [hc, mean_eq_condExp_history hμ ν hB hmB D hD u t] with p hp hu |
| dsimp only [posteriorCovariance] at hp |
| exact hp.trans (by rw [hu]) |
|
|
| theorem tendsto_cross_uniformSum {beta : ℝ} |
| {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) |
| (ν : TreeVertex d → Measure Ω) |
| {B : TreeVertex d → ℝ≥0 → Ω → ℝ} |
| (hB : ∀ i, IsPreBrownianReal (B i) (ν i)) |
| (hmB : ∀ i t, Measurable (B i t)) |
| (hcB : ∀ i omega, Continuous (fun t => B i t omega)) |
| (D : Finset (TreeVertex d)) (hD : ConnectedFinset D) (v u : D) (T : ℝ≥0) : |
| Tendsto (fun n : ℕ => ∫ p, mean beta D v B T p * |
| BrownianGrid.uniformSum |
| (fun t p => plantedArrayObservation B signal u t p - |
| nnrealTimePrimitive (mean beta D u B) t p) |
| (covariance beta D v u B) T n p ∂μ.prod (Measure.infinitePi ν)) |
| atTop (𝓝 (∫ r in (0 : ℝ)..(T : ℝ), |
| ∫ p, covariance beta D v u B r.toNNReal p ^ 2 ∂μ.prod (Measure.infinitePi ν))) := by |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure |
| have (index : TreeVertex d) : IsProbabilityMeasure (ν index) := |
| (hB index).isGaussianProcess.isProbabilityMeasure |
| have hs (i : TreeVertex d) : Measurable (fun sigma : TreeVertex d → IsingSpin => |
| spinSign (sigma i)) := (measurable_pi_apply i).comp measurable_signal |
| have hbound (i : TreeVertex d) : ∀ p : State d Ω, |spinSign (p.1 i)| ≤ (1 : ℝ) := by |
| intro p |
| cases p.1 i <;> norm_num [spinSign] |
| have hint : Integrable (fun p : State d Ω => spinSign (p.1 u)) |
| (μ.prod (Measure.infinitePi ν)) := |
| Integrable.of_bound ((hs u).comp measurable_fst).aestronglyMeasurable 1 |
| (Eventually.of_forall (hbound u)) |
| have hobs : StronglyAdapted (observationFiltration B hmB D) |
| (fun t p => B u t (p.2 u) + (t : ℝ) * spinSign (p.1 u)) := |
| arrayNaturalFiltration_stronglyAdapted |
| (fun i : D => plantedArrayObservation B signal i) |
| (fun i t => plantedArrayObservation_measurable hmB measurable_signal i t) u |
| have hl := FilteringCrossEnergy.tendsto_posterior_uniformSum_cross_energy |
| (preBrownian_martingale_signalArrayFiltration μ ν hB hmB (u : TreeVertex d)) |
| (observationFiltration_le_signal B hmB D) |
| (stronglyMeasurable_latent_signalArrayFiltration B hmB (hs v) 0) |
| (Eventually.of_forall (hbound v)) hint (Eventually.of_forall (hbound u)) hobs |
| (progressive_mean beta D u B hmB hcB) |
| (mean_eq_condExp_history hμ ν hB hmB D hD u) |
| (stronglyAdapted_covariance beta D v u B hmB) |
| (Eventually.of_forall (continuous_covariance beta D v u B hcB)) |
| (show (0 : ℝ) ≤ 2 by norm_num) |
| (Eventually.of_forall (fun p t => abs_covariance_le_two beta D v u B t p)) |
| (covariance_eq_condExp_history hμ ν hB hmB D hD v u) T |
| apply hl.congr |
| intro n |
| apply integral_congr_ae |
| filter_upwards [mean_eq_condExp_history hμ ν hB hmB D hD v T] with p hp |
| exact congrArg (fun x : ℝ => x * BrownianGrid.uniformSum |
| (fun t p => plantedArrayObservation B signal u t p - |
| nnrealTimePrimitive (mean beta D u B) t p) |
| (covariance beta D v u B) T n p) hp.symm |
|
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| end Problem367.FiniteIsingFiltering |
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| end |
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| end OAI |
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