openAI-math-dataset / lean /OAI /Probability /FreeIsing /Posterior /ContinuousPosteriorInterior.lean
| import OAI.Probability.FreeIsing.Gibbs.PosteriorSymmetry | |
| import OAI.Probability.FreeIsing.Posterior.InteriorLocalization | |
| import OAI.Probability.FreeIsing.Posterior.ProgressiveIsingPosterior | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped NNReal Topology | |
| namespace Problem367 | |
| theorem HasFreeIsingLaw.continuous_posterior_all_time_field_bound | |
| {d : ℕ} {beta : ℝ} {Ω : Type*} [MeasurableSpace Ω] | |
| {μ : Measure (TreeVertex d → IsingSpin)} | |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) | |
| (ν : TreeVertex d → Measure Ω) | |
| {B : TreeVertex d → ℝ≥0 → Ω → ℝ} | |
| (hB : ∀ v, IsBrownianReal (B v) (ν v)) | |
| (hmB : ∀ v t, Measurable (B v t)) | |
| {m : TreeVertex d → ℝ≥0 → | |
| ((TreeVertex d → IsingSpin) × (TreeVertex d → Ω)) → ℝ} | |
| (hcm : ∀ v, ∀ᵐ p ∂μ.prod (Measure.infinitePi ν), Continuous (fun t => m v t p)) | |
| (hversion : ∀ v t, m v t =ᵐ[μ.prod (Measure.infinitePi ν)] | |
| (fun p => infiniteGibbsSpinMean beta v | |
| (fun i => (t : ℝ) * spinSign (p.1 i) + B i t (p.2 i)))) : | |
| ∀ᵐ p ∂μ.prod (Measure.infinitePi ν), ∀ v, | |
| (∀ t, m v t p ∈ Set.Ioo (-1 : ℝ) 1 ∧ | |
| |Real.artanh (m v t p) - ((t : ℝ) * spinSign (p.1 v) + B v t (p.2 v))| ≤ | |
| (d : ℝ) * beta) ∧ | |
| Continuous (fun t => Real.artanh (m v t p)) := by | |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure | |
| have (vertex : TreeVertex d) : IsProbabilityMeasure (ν vertex) := | |
| (hB vertex).isGaussianProcess.isProbabilityMeasure | |
| apply ae_all_iff.mpr | |
| intro v | |
| have hX : ∀ᵐ p ∂μ.prod (Measure.infinitePi ν), | |
| Continuous (fun t : ℝ≥0 => (t : ℝ) * spinSign (p.1 v) + B v t (p.2 v)) := by | |
| have hnoise := (planted_noise_coordinate_measurePreserving μ ν v).quasiMeasurePreserving.ae | |
| (hB v).cont | |
| filter_upwards [hnoise] with p hp | |
| exact (continuous_subtype_val.mul continuous_const).add hp | |
| have hb : ∀ t, ∀ᵐ p ∂μ.prod (Measure.infinitePi ν), | |
| m v t p ∈ Set.Ioo (-1 : ℝ) 1 ∧ | |
| |Real.artanh (m v t p) - ((t : ℝ) * spinSign (p.1 v) + B v t (p.2 v))| ≤ | |
| (d : ℝ) * beta := by | |
| intro t | |
| filter_upwards [hversion v t, | |
| hμ.infiniteGibbsSpinMean_brownian_field_bound hbeta ν | |
| (fun i => (hB i).toIsPreBrownianReal) hmB t v] with p hp hb | |
| rw [hp] | |
| exact hb | |
| filter_upwards [hcm v, | |
| PosteriorInteriorLocalization.ae_all_field_bound_of_continuous (hcm v) hX hb] | |
| with p hp hb | |
| exact ⟨hb, PosteriorInteriorLocalization.continuous_artanh_of_interior hp | |
| (fun t => (hb t).1)⟩ | |
| theorem HasFreeIsingLaw.continuous_posterior_compact_localization | |
| {d : ℕ} {beta : ℝ} {Ω : Type*} [MeasurableSpace Ω] | |
| {μ : Measure (TreeVertex d → IsingSpin)} | |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) | |
| (ν : TreeVertex d → Measure Ω) | |
| {B : TreeVertex d → ℝ≥0 → Ω → ℝ} | |
| (hB : ∀ v, IsBrownianReal (B v) (ν v)) | |
| (hmB : ∀ v t, Measurable (B v t)) | |
| {m : TreeVertex d → ℝ≥0 → | |
| ((TreeVertex d → IsingSpin) × (TreeVertex d → Ω)) → ℝ} | |
| (hcm : ∀ v, ∀ᵐ p ∂μ.prod (Measure.infinitePi ν), Continuous (fun t => m v t p)) | |
| (hversion : ∀ v t, m v t =ᵐ[μ.prod (Measure.infinitePi ν)] | |
| (fun p => infiniteGibbsSpinMean beta v | |
| (fun i => (t : ℝ) * spinSign (p.1 i) + B i t (p.2 i)))) : | |
| ∀ᵐ p ∂μ.prod (Measure.infinitePi ν), ∀ v, ∀ T : ℝ≥0, ∀ᶠ n in atTop, | |
| ∀ t ≤ T, |m v t p| < PosteriorInteriorLocalization.radius n := by | |
| filter_upwards [ae_all_iff.mpr hcm, | |
| hμ.continuous_posterior_all_time_field_bound hbeta ν hB hmB hcm hversion] | |
| with p hc hb | |
| intro v T | |
| have h := PosteriorInteriorLocalization.eventually_uniform_abs_lt_radius | |
| (s := Set.Icc (0 : ℝ≥0) T) isCompact_Icc (hc v).continuousOn | |
| (fun t _ => (hb v).1 t |>.1) | |
| filter_upwards [h] with n hn | |
| exact fun t ht => hn t ⟨bot_le, ht⟩ | |
| theorem HasFreeIsingLaw.posterior_version_initial_zero | |
| {d : ℕ} {beta : ℝ} {Ω : Type*} [MeasurableSpace Ω] | |
| {μ : Measure (TreeVertex d → IsingSpin)} | |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) | |
| (ν : TreeVertex d → Measure Ω) | |
| {B : TreeVertex d → ℝ≥0 → Ω → ℝ} | |
| (hB : ∀ v, IsPreBrownianReal (B v) (ν v)) | |
| {m : TreeVertex d → ℝ≥0 → | |
| ((TreeVertex d → IsingSpin) × (TreeVertex d → Ω)) → ℝ} | |
| (hversion : ∀ v, m v 0 =ᵐ[μ.prod (Measure.infinitePi ν)] | |
| (fun p => infiniteGibbsSpinMean beta v (fun i => B i 0 (p.2 i)))) : | |
| ∀ᵐ p ∂μ.prod (Measure.infinitePi ν), ∀ v, | |
| m v 0 p = 0 ∧ Real.artanh (m v 0 p) = 0 := by | |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure | |
| filter_upwards [ae_all_iff.mpr hversion, | |
| infiniteGibbsSpinMean_brownian_zero beta μ ν hB] with p hp hz | |
| intro v | |
| have hzero : m v 0 p = 0 := by | |
| rw [hp v] | |
| simpa only [zero_mul, zero_add] using hz v | |
| exact ⟨hzero, by rw [hzero, Real.artanh_zero]⟩ | |
| theorem isStronglyProgressive_artanh | |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] | |
| (F : Filtration ℝ≥0 mΩ) {m : ℝ≥0 → Ω → ℝ} | |
| (hm : IsStronglyProgressive F m) : | |
| IsStronglyProgressive F (fun t ω => Real.artanh (m t ω)) := by | |
| have ha : Measurable Real.artanh := by | |
| unfold Real.artanh | |
| fun_prop | |
| intro t | |
| let : MeasurableSpace Ω := F t | |
| exact (ha.comp (hm t).measurable).stronglyMeasurable | |
| end Problem367 | |
| end | |
| end OAI | |