| import OAI.Probability.FreeIsing.Posterior.Limit | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter | |
| open scoped Topology | |
| namespace Problem367 | |
| variable {Ω : Type*} [mΩ : MeasurableSpace Ω] | |
| def posteriorCovariance (μ : Measure Ω) (m : MeasurableSpace Ω) | |
| (f g : Ω → ℝ) : Ω → ℝ := | |
| fun ω => μ[(fun x => f x * g x) | m] ω - μ[f | m] ω * μ[g | m] ω | |
| lemma stronglyMeasurable_posteriorCovariance (μ : Measure Ω) | |
| (m : MeasurableSpace Ω) (f g : Ω → ℝ) : | |
| StronglyMeasurable[m] (posteriorCovariance (mΩ := mΩ) μ m f g) := | |
| stronglyMeasurable_condExp.sub | |
| (stronglyMeasurable_condExp.mul stronglyMeasurable_condExp) | |
| theorem tendsto_ae_posteriorCovariance (μ : Measure Ω) [IsFiniteMeasure μ] | |
| (F : Filtration ℕ mΩ) (f g : Ω → ℝ) : | |
| ∀ᵐ ω ∂μ, Tendsto (fun n => posteriorCovariance μ (F n) f g ω) atTop | |
| (𝓝 (posteriorCovariance μ (⨆ n, F n) f g ω)) := by | |
| exact tendsto_conditional_covariance_ae μ F f g | |
| theorem ae_norm_posteriorCovariance_le (μ : Measure Ω) | |
| (m : MeasurableSpace Ω) {f g : Ω → ℝ} {R S : ℝ} | |
| (hR : 0 ≤ R) (_hS : 0 ≤ S) | |
| (hf : ∀ᵐ ω ∂μ, ‖f ω‖ ≤ R) (hg : ∀ᵐ ω ∂μ, ‖g ω‖ ≤ S) : | |
| ∀ᵐ ω ∂μ, ‖posteriorCovariance (mΩ := mΩ) μ m f g ω‖ ≤ 2 * R * S := by | |
| have hfg : ∀ᵐ ω ∂μ, ‖f ω * g ω‖ ≤ R * S := by | |
| filter_upwards [hf, hg] with ω hfω hgω | |
| rw [norm_mul] | |
| exact mul_le_mul hfω hgω (norm_nonneg _) hR | |
| filter_upwards [ae_bdd_norm_condExp_of_ae_bdd_norm (m := m) hf, | |
| ae_bdd_norm_condExp_of_ae_bdd_norm (m := m) hg, | |
| ae_bdd_norm_condExp_of_ae_bdd_norm (m := m) hfg] with ω hfω hgω hfgω | |
| calc | |
| ‖posteriorCovariance (mΩ := mΩ) μ m f g ω‖ ≤ | |
| ‖μ[(fun x => f x * g x) | m] ω‖ + ‖μ[f | m] ω * μ[g | m] ω‖ := | |
| norm_sub_le _ _ | |
| _ ≤ R * S + R * S := add_le_add hfgω (by | |
| rw [norm_mul] | |
| exact mul_le_mul hfω hgω (norm_nonneg _) hR) | |
| _ = 2 * R * S := by ring | |
| theorem tendsto_integral_sq_sub_of_bounded_ae | |
| (μ : Measure Ω) [IsFiniteMeasure μ] {X : ℕ → Ω → ℝ} {Y : Ω → ℝ} | |
| (hX : ∀ n, AEStronglyMeasurable (X n) μ) (hY : AEStronglyMeasurable Y μ) | |
| {R : ℝ} (hR : 0 ≤ R) | |
| (hbX : ∀ n, ∀ᵐ ω ∂μ, ‖X n ω‖ ≤ R) | |
| (hbY : ∀ᵐ ω ∂μ, ‖Y ω‖ ≤ R) | |
| (hlim : ∀ᵐ ω ∂μ, Tendsto (fun n => X n ω) atTop (𝓝 (Y ω))) : | |
| Tendsto (fun n => ∫ ω, (X n ω - Y ω) ^ 2 ∂μ) atTop (𝓝 0) := by | |
| have hbound (n : ℕ) : ∀ᵐ ω ∂μ, ‖(X n ω - Y ω) ^ 2‖ ≤ (2 * R) ^ 2 := by | |
| filter_upwards [hbX n, hbY] with ω hx hy | |
| have hdiff : ‖X n ω - Y ω‖ ≤ 2 * R := by | |
| calc | |
| ‖X n ω - Y ω‖ ≤ ‖X n ω‖ + ‖Y ω‖ := norm_sub_le _ _ | |
| _ ≤ 2 * R := by linarith | |
| rw [norm_pow] | |
| nlinarith [norm_nonneg (X n ω - Y ω)] | |
| have ht := tendsto_integral_of_dominated_convergence (μ := μ) | |
| (fun _ : Ω => (2 * R) ^ 2) | |
| (fun n => ((hX n).sub hY).pow 2) (integrable_const _) hbound | |
| (hlim.mono (fun ω hω => by | |
| simpa using (hω.sub tendsto_const_nhds).pow 2)) | |
| simpa using ht | |
| theorem tendsto_integral_sq_sub_posteriorCovariance | |
| (μ : Measure Ω) [IsFiniteMeasure μ] (F : Filtration ℕ mΩ) | |
| {f g : Ω → ℝ} {R S : ℝ} (hR : 0 ≤ R) (hS : 0 ≤ S) | |
| (hf : ∀ᵐ ω ∂μ, ‖f ω‖ ≤ R) (hg : ∀ᵐ ω ∂μ, ‖g ω‖ ≤ S) : | |
| Tendsto (fun n => ∫ ω, | |
| (posteriorCovariance μ (F n) f g ω - | |
| posteriorCovariance μ (⨆ n, F n) f g ω) ^ 2 ∂μ) atTop (𝓝 0) := by | |
| apply tendsto_integral_sq_sub_of_bounded_ae μ | |
| (fun n => ((stronglyMeasurable_posteriorCovariance μ (F n) f g).mono | |
| (F.le n)).aestronglyMeasurable) | |
| (((stronglyMeasurable_posteriorCovariance μ (⨆ n, F n) f g).mono | |
| (iSup_le fun n => F.le n)).aestronglyMeasurable) | |
| (show 0 ≤ 2 * R * S by positivity) | |
| · exact fun n => ae_norm_posteriorCovariance_le μ (F n) hR hS hf hg | |
| · exact ae_norm_posteriorCovariance_le μ (⨆ n, F n) hR hS hf hg | |
| · exact tendsto_ae_posteriorCovariance μ F f g | |
| theorem tendsto_integral_prod_sq_sub_of_bounded_ae | |
| {T : Type*} [MeasurableSpace T] | |
| (ν : Measure T) (μ : Measure Ω) [IsFiniteMeasure ν] [IsFiniteMeasure μ] | |
| {X : ℕ → T × Ω → ℝ} {Y : T × Ω → ℝ} | |
| (hX : ∀ n, Measurable (X n)) (hY : Measurable Y) | |
| {R : ℝ} (hR : 0 ≤ R) | |
| (hbX : ∀ n, ∀ᵐ t ∂ν, ∀ᵐ ω ∂μ, ‖X n (t, ω)‖ ≤ R) | |
| (hbY : ∀ᵐ t ∂ν, ∀ᵐ ω ∂μ, ‖Y (t, ω)‖ ≤ R) | |
| (hlim : ∀ᵐ t ∂ν, ∀ᵐ ω ∂μ, | |
| Tendsto (fun n => X n (t, ω)) atTop (𝓝 (Y (t, ω)))) : | |
| Tendsto (fun n => ∫ p, (X n p - Y p) ^ 2 ∂ν.prod μ) atTop (𝓝 0) := by | |
| apply tendsto_integral_sq_sub_of_bounded_ae (ν.prod μ) | |
| (fun n => (hX n).aestronglyMeasurable) hY.aestronglyMeasurable hR | |
| · intro n | |
| exact (Measure.ae_prod_iff_ae_ae (measurableSet_le (hX n).norm measurable_const)).mpr (hbX n) | |
| · exact (Measure.ae_prod_iff_ae_ae (measurableSet_le hY.norm measurable_const)).mpr hbY | |
| · exact (Measure.ae_prod_iff_ae_ae (measurableSet_tendsto_fun hX hY)).mpr hlim | |
| theorem tendsto_integral_prod_sq_sub_posteriorCovariance_versions | |
| {T : Type*} [MeasurableSpace T] | |
| (ν : Measure T) (μ : Measure Ω) [IsFiniteMeasure ν] [IsFiniteMeasure μ] | |
| (F : T → Filtration ℕ mΩ) {f g : Ω → ℝ} {R S : ℝ} | |
| (hR : 0 ≤ R) (hS : 0 ≤ S) | |
| (hf : ∀ᵐ ω ∂μ, ‖f ω‖ ≤ R) (hg : ∀ᵐ ω ∂μ, ‖g ω‖ ≤ S) | |
| {X : ℕ → T × Ω → ℝ} {Y : T × Ω → ℝ} | |
| (hX : ∀ n, Measurable (X n)) (hY : Measurable Y) | |
| (hXversion : ∀ n, ∀ᵐ t ∂ν, | |
| (fun ω => X n (t, ω)) =ᵐ[μ] posteriorCovariance μ (F t n) f g) | |
| (hYversion : ∀ᵐ t ∂ν, | |
| (fun ω => Y (t, ω)) =ᵐ[μ] posteriorCovariance μ (⨆ n, F t n) f g) : | |
| Tendsto (fun n => ∫ p, (X n p - Y p) ^ 2 ∂ν.prod μ) atTop (𝓝 0) := by | |
| apply tendsto_integral_prod_sq_sub_of_bounded_ae ν μ hX hY | |
| (show 0 ≤ 2 * R * S by positivity) | |
| · intro n | |
| filter_upwards [hXversion n] with t ht | |
| filter_upwards [ht, ae_norm_posteriorCovariance_le μ (F t n) hR hS hf hg] | |
| with ω heq hbound | |
| rw [heq] | |
| exact hbound | |
| · filter_upwards [hYversion] with t ht | |
| filter_upwards [ht, ae_norm_posteriorCovariance_le μ (⨆ n, F t n) hR hS hf hg] | |
| with ω heq hbound | |
| rw [heq] | |
| exact hbound | |
| · filter_upwards [ae_all_iff.mpr hXversion, hYversion] with t hXt hYt | |
| filter_upwards [ae_all_iff.mpr hXt, hYt, tendsto_ae_posteriorCovariance μ (F t) f g] | |
| with ω hXω hYω hlim | |
| simpa only [hXω, hYω] using hlim | |
| def observationExhaustionFiltration {ι : Type*} {E : ι → Type*} | |
| [∀ i, MeasurableSpace (E i)] (X : Ω → ∀ i, E i) | |
| (hX : ∀ i, Measurable (fun ω => X ω i)) | |
| (s : ℕ → Set ι) (hs : Monotone s) : Filtration ℕ mΩ where | |
| seq n := ⨆ i, ⨆ (_ : i ∈ s n), MeasurableSpace.comap (fun ω => X ω i) inferInstance | |
| mono' _ _ hnk := iSup₂_le fun i hi => | |
| le_iSup_of_le i (le_iSup_of_le (hs hnk hi) le_rfl) | |
| le' _ := iSup₂_le fun i _ => (hX i).comap_le | |
| theorem observationExhaustionFiltration_eq_comap {ι : Type*} {E : ι → Type*} | |
| [∀ i, MeasurableSpace (E i)] (X : Ω → ∀ i, E i) | |
| (hX : ∀ i, Measurable (fun ω => X ω i)) | |
| (s : ℕ → Set ι) (hs : Monotone s) (n : ℕ) : | |
| observationExhaustionFiltration X hX s hs n = | |
| MeasurableSpace.comap (fun ω (i : s n) => X ω i) inferInstance := by | |
| simp only [observationExhaustionFiltration, MeasurableSpace.comap_process_pi, | |
| iSup_subtype'] | |
| theorem observationExhaustionFiltration_iSup {ι : Type*} {E : ι → Type*} | |
| [∀ i, MeasurableSpace (E i)] (X : Ω → ∀ i, E i) | |
| (hX : ∀ i, Measurable (fun ω => X ω i)) | |
| (s : ℕ → Set ι) (hs : Monotone s) (hex : ∀ i, ∃ n, i ∈ s n) : | |
| (⨆ n, observationExhaustionFiltration X hX s hs n) = | |
| MeasurableSpace.comap X inferInstance := by | |
| rw [MeasurableSpace.comap_process_pi (fun i ω => X ω i)] | |
| apply le_antisymm | |
| · exact iSup_le fun n => iSup₂_le fun i _ => le_iSup_of_le i le_rfl | |
| · apply iSup_le | |
| intro i | |
| obtain ⟨n, hn⟩ := hex i | |
| exact le_iSup_of_le n (le_iSup_of_le i (le_iSup_of_le hn le_rfl)) | |
| theorem tendsto_integral_sq_sub_observationCovariance | |
| {ι : Type*} {E : ι → Type*} [∀ i, MeasurableSpace (E i)] | |
| (μ : Measure Ω) [IsFiniteMeasure μ] | |
| (X : Ω → ∀ i, E i) (hX : ∀ i, Measurable (fun ω => X ω i)) | |
| (s : ℕ → Set ι) (hs : Monotone s) (hex : ∀ i, ∃ n, i ∈ s n) | |
| {f g : Ω → ℝ} {R S : ℝ} (hR : 0 ≤ R) (hS : 0 ≤ S) | |
| (hf : ∀ᵐ ω ∂μ, ‖f ω‖ ≤ R) (hg : ∀ᵐ ω ∂μ, ‖g ω‖ ≤ S) : | |
| Tendsto (fun n => ∫ ω, | |
| (posteriorCovariance μ | |
| (MeasurableSpace.comap (fun ω (i : s n) => X ω i) inferInstance) f g ω - | |
| posteriorCovariance μ (MeasurableSpace.comap X inferInstance) f g ω) ^ 2 ∂μ) | |
| atTop (𝓝 0) := by | |
| have h := tendsto_integral_sq_sub_posteriorCovariance μ | |
| (observationExhaustionFiltration X hX s hs) hR hS hf hg | |
| rw [observationExhaustionFiltration_iSup X hX s hs hex] at h | |
| simpa only [observationExhaustionFiltration_eq_comap] using h | |
| end Problem367 | |
| end | |
| end OAI | |