| import OAI.Probability.FreeIsing.Ising.ContinuousRootResponse |
| import OAI.Probability.FreeIsing.Ising.MixedFiltering |
| import OAI.Probability.FreeIsing.Ising.ObservationPosteriorTransport |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped NNReal |
|
|
| namespace Problem367.IsingObservationInnovation |
|
|
| theorem progressive_uniformObservation {d : ℕ} |
| (hmB : ∀ t, Measurable (fun u => WienerSampler.sample u t)) |
| {H : ℝ≥0 → Paths d → ℝ} (hH : IsStronglyProgressive (filtration d) H) : |
| IsStronglyProgressive |
| (ArrayPosteriorStationarity.filtration (fun t u => WienerSampler.sample u t) hmB) |
| (fun t q => H t (uniformObservation q)) := by |
| intro t |
| have hφ : Measurable[ |
| ArrayPosteriorStationarity.filtration (fun s u => WienerSampler.sample u s) hmB t, |
| filtration d t] uniformObservation := by |
| rw [← filtration_comap_uniformObservation d t hmB] |
| exact comap_measurable _ |
| let : MeasurableSpace (IsingInnovation.State d ℝ) := |
| ArrayPosteriorStationarity.filtration (fun s u => WienerSampler.sample u s) hmB t |
| let : MeasurableSpace (Paths d) := filtration d t |
| exact (hH t).comp_measurable (measurable_id.prodMap hφ) |
|
|
| end Problem367.IsingObservationInnovation |
|
|
| namespace Problem367.IsingObservationFactor |
| open IsingObservationInnovation |
|
|
| variable {d : ℕ} {beta : ℝ} |
|
|
| theorem continuousRootCovariance_uniformObservation_progressive |
| (hmB : ∀ t, Measurable (fun u => WienerSampler.sample u t)) |
| (hb : 0 ≤ beta) (v : TreeVertex d) : |
| IsStronglyProgressive |
| (ArrayPosteriorStationarity.filtration (fun t u => WienerSampler.sample u t) hmB) |
| (fun t q => continuousRootCovariance hb v t (uniformObservation q)) := |
| progressive_uniformObservation hmB (continuousRootCovariance_progressive hb v) |
|
|
| theorem continuousRootCovariance_uniformObservation_stronglyAdapted |
| (hmB : ∀ t, Measurable (fun u => WienerSampler.sample u t)) |
| (hb : 0 ≤ beta) (v : TreeVertex d) : |
| StronglyAdapted |
| (ArrayPosteriorStationarity.filtration (fun t u => WienerSampler.sample u t) hmB) |
| (fun t q => continuousRootCovariance hb v t (uniformObservation q)) := |
| (continuousRootCovariance_uniformObservation_progressive hmB hb v).stronglyAdapted |
|
|
| theorem continuousRootCovariance_uniformObservation_abs_le_one |
| (hb : 0 ≤ beta) (v : TreeVertex d) (t : ℝ≥0) (q : IsingInnovation.State d ℝ) : |
| |continuousRootCovariance hb v t (uniformObservation q)| ≤ 1 := |
| continuousRootCovariance_abs_le_one hb v t (uniformObservation q) |
|
|
| theorem continuousRootCovariance_uniformObservation_continuous_ae |
| (hd : 3 ≤ d) {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) (v : TreeVertex d) : |
| ∀ᵐ q ∂μ.prod (labelMeasure d), |
| Continuous (fun t => continuousRootCovariance hb v t (uniformObservation q)) := |
| (measurePreserving_uniformObservation μ).quasiMeasurePreserving.ae |
| (continuousRootCovariance_continuous_ae hd hμ hb hcrit v) |
|
|
| theorem continuousRootCovariance_uniformObservation_eq_covariance_ae |
| (hd : 3 ≤ d) {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) |
| (v : TreeVertex d) (t : ℝ≥0) : |
| (fun q => continuousRootCovariance hb v t (uniformObservation q)) =ᵐ[ |
| μ.prod (labelMeasure d)] |
| IsingInnovation.covariance beta (fun s u => WienerSampler.sample u s) |
| (ConnectedHull.root d) v t := by |
| exact (measurePreserving_uniformObservation μ).quasiMeasurePreserving.ae |
| (continuousRootCovariance_eq_covariance_ae hd hμ hb hcrit v t) |
|
|
| end Problem367.IsingObservationFactor |
|
|
| end |
|
|
| end OAI |
|
|
