| import OAI.Probability.FreeIsing.Estimates.LocalizedArtanhFormula |
| import OAI.Probability.FreeIsing.Ising.CanonicalContinuousFields |
| import OAI.Probability.FreeIsing.Ising.CopyEnergyIntegrability |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped NNReal Topology |
|
|
| namespace Problem367.IsingObservationFactor |
| open IsingObservationInnovation IsingCopyEnergy PosteriorCompensatorStopping |
|
|
| private lemma continuous_real_tanh : Continuous Real.tanh := |
| lipschitzWith_tanh_response.continuous |
|
|
| def continuousPosterior {d : ℕ} (beta : ℝ) (v : TreeVertex d) |
| (t : ℝ≥0) (x : Paths d) : ℝ := Real.tanh (continuousField beta v t x) |
|
|
| lemma continuousPosterior_progressive {d : ℕ} (beta : ℝ) (v : TreeVertex d) : |
| IsStronglyProgressive (filtration d) (continuousPosterior beta v) := by |
| intro T |
| exact continuous_real_tanh.comp_stronglyMeasurable (continuousField_progressive beta v T) |
|
|
| lemma continuousPosterior_interior {d : ℕ} (beta : ℝ) (v : TreeVertex d) |
| (t : ℝ≥0) (x : Paths d) : continuousPosterior beta v t x ∈ Ioo (-1 : ℝ) 1 := |
| ⟨Real.neg_one_lt_tanh _, Real.tanh_lt_one _⟩ |
|
|
| lemma artanh_continuousPosterior {d : ℕ} (beta : ℝ) (v : TreeVertex d) |
| (t : ℝ≥0) (x : Paths d) : |
| Real.artanh (continuousPosterior beta v t x) = continuousField beta v t x := |
| Real.artanh_tanh _ |
|
|
| theorem continuousPosterior_modification |
| {d : ℕ} (hd : 3 ≤ d) {beta : ℝ} {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 x ∂observationLaw μ, Continuous (fun t => continuousPosterior beta v t x)) ∧ |
| (∀ t, continuousPosterior beta v t =ᵐ[observationLaw μ] posterior beta v t) := by |
| obtain ⟨hc, he⟩ := continuousField_modification hd hμ hb hcrit v |
| refine ⟨hc.mono (fun x hx => continuous_real_tanh.comp hx), fun t => ?_⟩ |
| filter_upwards [he t, tanh_field_eq_posterior_ae hμ hb v t] with x hx ht |
| exact (congrArg Real.tanh hx).trans ht |
|
|
| lemma copyProjection_measurePreserving |
| {d : ℕ} (beta : ℝ) (μ : Measure (TreeVertex d → IsingSpin)) |
| [IsProbabilityMeasure μ] (j : Bool) : |
| MeasurePreserving (projection (d := d) j) (copies beta μ) (observationLaw μ) := by |
| have : IsProbabilityMeasure ((observationLaw μ).map (innovationMap beta)) := inferInstance |
| obtain ⟨hl, hr⟩ := IsingConditionalCopyBounds.measurePreserving_observation_copies |
| (beta := beta) (μ := μ) (rho := (observationLaw μ).map (innovationMap beta)) |
| (eta := kernel beta μ) kernel_graphLaw |
| cases j |
| · exact hr |
| · exact hl |
|
|
| def copyContinuousField {d : ℕ} (beta : ℝ) (v : TreeVertex d) (j : Bool) |
| (t : ℝ≥0) (q : Joint d) : ℝ := continuousField beta v t (projection j q) |
|
|
| def copyContinuousPosterior {d : ℕ} (beta : ℝ) (v : TreeVertex d) (j : Bool) |
| (t : ℝ≥0) (q : Joint d) : ℝ := continuousPosterior beta v t (projection j q) |
|
|
| lemma copyContinuousField_progressive {d : ℕ} (beta : ℝ) (v : TreeVertex d) (j : Bool) : |
| IsStronglyProgressive ContinuousCopyBrownian.joinedFiltration |
| (copyContinuousField beta v j) := by |
| cases j |
| · exact ProgressiveVersionPushforward.progressive_version |
| (isStronglyProgressive_artanh _ (isStronglyProgressive_infiniteGibbsSpinMean _ beta v _ |
| IsingConditionalCopyBounds.progressive_rightObservation)) |
| · exact ProgressiveVersionPushforward.progressive_version |
| (isStronglyProgressive_artanh _ (isStronglyProgressive_infiniteGibbsSpinMean _ beta v _ |
| IsingConditionalCopyBounds.progressive_leftObservation)) |
|
|
| lemma copyContinuousPosterior_progressive {d : ℕ} (beta : ℝ) (v : TreeVertex d) (j : Bool) : |
| IsStronglyProgressive ContinuousCopyBrownian.joinedFiltration |
| (copyContinuousPosterior beta v j) := by |
| intro T |
| exact continuous_real_tanh.comp_stronglyMeasurable (copyContinuousField_progressive beta v j T) |
|
|
| theorem copyContinuousPosterior_modification |
| {d : ℕ} (hd : 3 ≤ d) {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| [IsProbabilityMeasure μ] |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) (v : TreeVertex d) (j : Bool) : |
| (∀ᵐ q ∂copies beta μ, Continuous (fun t => copyContinuousPosterior beta v j t q)) ∧ |
| (∀ t, copyContinuousPosterior beta v j t =ᵐ[copies beta μ] |
| (fun q => posterior beta v t (projection j q))) := by |
| obtain ⟨hc, he⟩ := continuousPosterior_modification hd hμ hb hcrit v |
| have hp := (copyProjection_measurePreserving beta μ j).quasiMeasurePreserving |
| exact ⟨hp.ae hc, fun t => hp.ae (he t)⟩ |
|
|
| theorem copyContinuousPosterior_compact |
| {d : ℕ} (hd : 3 ≤ d) {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| [IsProbabilityMeasure μ] |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) (v : TreeVertex d) (j : Bool) : |
| ∀ᵐ q ∂copies beta μ, ∀ T : ℝ≥0, |
| ∃ r < (1 : ℝ), ∀ t ≤ T, |copyContinuousPosterior beta v j t q| ≤ r := by |
| filter_upwards [(copyContinuousPosterior_modification hd hμ hb hcrit v j).1] with q hc |
| intro T |
| obtain ⟨r, hr0, hr1, hr⟩ := PosteriorInteriorLocalization.exists_uniform_radius_of_compact |
| (s := Icc (0 : ℝ≥0) T) isCompact_Icc hc.continuousOn |
| (fun t _ => continuousPosterior_interior beta v t (projection j q)) |
| exact ⟨r, hr1, fun t ht => hr t ⟨zero_le, ht⟩⟩ |
|
|
| def copyResponseEnergy {d : ℕ} (beta : ℝ) (v : TreeVertex d) (j : Bool) |
| (t : ℝ≥0) (q : Joint d) : ℝ := |
| infiniteGibbsResponseRow beta v (fun i => projection j q i t) |
|
|
| lemma copyResponseEnergy_progressive {d : ℕ} (beta : ℝ) (v : TreeVertex d) (j : Bool) : |
| IsStronglyProgressive ContinuousCopyBrownian.joinedFiltration |
| (copyResponseEnergy beta v j) := by |
| cases j |
| · exact isStronglyProgressive_infiniteGibbsResponseRow _ beta v _ |
| IsingConditionalCopyBounds.progressive_rightObservation |
| · exact isStronglyProgressive_infiniteGibbsResponseRow _ beta v _ |
| IsingConditionalCopyBounds.progressive_leftObservation |
|
|
| theorem copyResponseEnergy_integrable |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| [IsProbabilityMeasure μ] |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (v : TreeVertex d) (j : Bool) (T : ℝ≥0) : |
| Integrable (fun p : ℝ × Joint d => copyResponseEnergy beta v j p.1.toNNReal p.2) |
| ((volume.restrict (Ioc (0 : ℝ) T)).prod (copies beta μ)) := |
| (copy_row_energy_integrable hμ hb v j T.coe_nonneg).2.1 |
|
|
| def commonResponseCompensator {d : ℕ} (beta : ℝ) (v : TreeVertex d) : |
| ℝ≥0 → Joint d → ℝ := |
| nnrealTimePrimitive (fun t q => |
| copyResponseEnergy beta v true t q + copyResponseEnergy beta v false t q) |
|
|
| lemma commonResponseCompensator_progressive {d : ℕ} (beta : ℝ) (v : TreeVertex d) : |
| IsStronglyProgressive ContinuousCopyBrownian.joinedFiltration |
| (commonResponseCompensator beta v) := |
| isStronglyProgressive_nnrealTimePrimitive |
| ((copyResponseEnergy_progressive beta v true).add (copyResponseEnergy_progressive beta v false)) |
|
|
| theorem commonResponseCompensator_continuous |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| [IsProbabilityMeasure μ] |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) (v : TreeVertex d) : |
| ∀ᵐ q ∂copies beta μ, Continuous (fun t => commonResponseCompensator beta v t q) := by |
| apply ae_continuous_nnrealTimePrimitive_of_joint |
| intro n |
| have hi := (copyResponseEnergy_integrable hμ hb v true (n : ℝ≥0)).add |
| (copyResponseEnergy_integrable hμ hb v false (n : ℝ≥0)) |
| convert hi using 1; rfl |
|
|
| def commonPosteriorExit {d : ℕ} (beta : ℝ) (v : TreeVertex d) |
| (n : ℕ) (q : Joint d) : WithTop ℝ≥0 := |
| min (jointExit (copyContinuousPosterior beta v true) (commonResponseCompensator beta v) n q) |
| (jointExit (copyContinuousPosterior beta v false) (commonResponseCompensator beta v) n q) |
|
|
| theorem isLocalizingSequence_commonPosteriorExit |
| {d : ℕ} (hd : 3 ≤ d) {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| [IsProbabilityMeasure μ] |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) (v : TreeVertex d) : |
| IsLocalizingSequence ContinuousCopyBrownian.joinedFiltration.rightCont |
| (commonPosteriorExit beta v) (copies beta μ) := by |
| have h (j : Bool) := isLocalizingSequence_jointExit_rightCont |
| (copyContinuousPosterior_progressive beta v j).stronglyAdapted.adapted |
| (commonResponseCompensator_progressive beta v).stronglyAdapted.adapted |
| (copyContinuousPosterior_compact hd hμ hb hcrit v j) |
| (commonResponseCompensator_continuous hμ hb v) |
| exact (h true).min (h false) |
|
|
| theorem ae_eventually_le_commonPosteriorExit |
| {d : ℕ} (hd : 3 ≤ d) {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| [IsProbabilityMeasure μ] |
| (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) : |
| ∀ᵐ q ∂copies beta μ, ∀ᶠ n in atTop, |
| (T : WithTop ℝ≥0) ≤ commonPosteriorExit beta v n q := by |
| filter_upwards [copyContinuousPosterior_compact hd hμ hb hcrit v true, |
| copyContinuousPosterior_compact hd hμ hb hcrit v false, |
| commonResponseCompensator_continuous hμ hb v] with q hl hr hA |
| filter_upwards [eventually_le_jointExit (hl T) hA, |
| eventually_le_jointExit (hr T) hA] with n hn hn' |
| exact le_min hn hn' |
|
|
| end Problem367.IsingObservationFactor |
|
|
| end |
|
|
| end OAI |
|
|
