| import OAI.Probability.FreeIsing.Conditioning.CovarianceLimit |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory Filter |
| open scoped Topology |
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| namespace Problem367 |
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| variable {Ω : Type*} [mΩ : MeasurableSpace Ω] |
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| def posteriorResponse (μ : Measure Ω) (m : MeasurableSpace Ω) |
| (f g : Ω → ℝ) (ω : Ω) : ℝ := |
| posteriorCovariance (mΩ := mΩ) μ m f g ω / (1 - (μ[f | m] ω) ^ 2) |
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| lemma stronglyMeasurable_posteriorResponse (μ : Measure Ω) |
| (m : MeasurableSpace Ω) (f g : Ω → ℝ) : |
| StronglyMeasurable[m] (posteriorResponse (mΩ := mΩ) μ m f g) := |
| (stronglyMeasurable_posteriorCovariance (mΩ := mΩ) μ m f g).div |
| (stronglyMeasurable_const.sub (stronglyMeasurable_condExp.pow 2)) |
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| lemma abs_covariance_le_variance_of_binary_moment_bounds |
| {a b c : ℝ} (ha : |a| ≤ 1) |
| (h₁ : c ≤ 1 + a - b) (h₂ : c ≤ 1 - a + b) |
| (h₃ : -1 - a - b ≤ c) (h₄ : -1 + a + b ≤ c) : |
| |c - a * b| ≤ 1 - a ^ 2 := by |
| have ham : 0 ≤ 1 - a := by linarith only [(abs_le.mp ha).2] |
| have hap : 0 ≤ 1 + a := by linarith only [(abs_le.mp ha).1] |
| apply abs_le.mpr |
| constructor |
| · nlinarith [mul_nonneg ham (show 0 ≤ 1 + a + b + c by linarith), |
| mul_nonneg hap (show 0 ≤ 1 - a - b + c by linarith)] |
| · nlinarith [mul_nonneg ham (show 0 ≤ 1 + a - b - c by linarith), |
| mul_nonneg hap (show 0 ≤ 1 - a + b - c by linarith)] |
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| theorem ae_abs_posteriorCovariance_le_binary_variance |
| (μ : Measure Ω) [IsFiniteMeasure μ] |
| (m : MeasurableSpace Ω) (hm : m ≤ mΩ) |
| {f g : Ω → ℝ} (hf : Integrable f μ) (hg : Integrable g μ) |
| (hbinary : ∀ᵐ ω ∂μ, f ω = -1 ∨ f ω = 1) |
| (hgb : ∀ᵐ ω ∂μ, |g ω| ≤ 1) : |
| ∀ᵐ ω ∂μ, |posteriorCovariance (mΩ := mΩ) μ m f g ω| ≤ 1 - (μ[f | m] ω) ^ 2 := by |
| have hfb : ∀ᵐ ω ∂μ, |f ω| ≤ 1 := by |
| filter_upwards [hbinary] with ω hω |
| rcases hω with hω | hω <;> simp [hω] |
| have hfg : Integrable (fun ω => f ω * g ω) μ := by |
| apply Integrable.of_bound (hf.aestronglyMeasurable.mul hg.aestronglyMeasurable) 1 |
| filter_upwards [hfb, hgb] with ω hfw hgw |
| simpa only [Real.norm_eq_abs, Pi.mul_apply, abs_mul, one_mul] using |
| mul_le_mul hfw hgw (abs_nonneg _) (by norm_num : (0 : ℝ) ≤ 1) |
| have h₁ : (fun ω => f ω * g ω) ≤ᵐ[μ] (fun ω => 1 + f ω - g ω) := by |
| filter_upwards [hbinary, hgb] with ω hω hgω |
| rcases hω with hω | hω <;> rw [hω] <;> nlinarith [(abs_le.mp hgω).1, (abs_le.mp hgω).2] |
| have h₂ : (fun ω => f ω * g ω) ≤ᵐ[μ] (fun ω => 1 - f ω + g ω) := by |
| filter_upwards [hbinary, hgb] with ω hω hgω |
| rcases hω with hω | hω <;> rw [hω] <;> nlinarith [(abs_le.mp hgω).1, (abs_le.mp hgω).2] |
| have h₃ : (fun ω => -1 - f ω - g ω) ≤ᵐ[μ] (fun ω => f ω * g ω) := by |
| filter_upwards [hbinary, hgb] with ω hω hgω |
| rcases hω with hω | hω <;> rw [hω] <;> nlinarith [(abs_le.mp hgω).1, (abs_le.mp hgω).2] |
| have h₄ : (fun ω => -1 + f ω + g ω) ≤ᵐ[μ] (fun ω => f ω * g ω) := by |
| filter_upwards [hbinary, hgb] with ω hω hgω |
| rcases hω with hω | hω <;> rw [hω] <;> nlinarith [(abs_le.mp hgω).1, (abs_le.mp hgω).2] |
| have hc₁ := condExp_mono (m := m) hfg (((integrable_const (1 : ℝ)).add hf).sub hg) h₁ |
| have hc₂ := condExp_mono (m := m) hfg (((integrable_const (1 : ℝ)).sub hf).add hg) h₂ |
| have hc₃ := condExp_mono (m := m) (((integrable_const (-1 : ℝ)).sub hf).sub hg) hfg h₃ |
| have hc₄ := condExp_mono (m := m) (((integrable_const (-1 : ℝ)).add hf).add hg) hfg h₄ |
| filter_upwards [hc₁, hc₂, hc₃, hc₄, |
| condExp_sub ((integrable_const (1 : ℝ)).add hf) hg m, |
| condExp_add (integrable_const (1 : ℝ)) hf m, |
| condExp_add ((integrable_const (1 : ℝ)).sub hf) hg m, |
| condExp_sub (integrable_const (1 : ℝ)) hf m, |
| condExp_sub ((integrable_const (-1 : ℝ)).sub hf) hg m, |
| condExp_sub (integrable_const (-1 : ℝ)) hf m, |
| condExp_add ((integrable_const (-1 : ℝ)).add hf) hg m, |
| condExp_add (integrable_const (-1 : ℝ)) hf m, |
| ae_bdd_abs_condExp_of_ae_bdd_abs (m := m) hfb] |
| with ω hc₁ hc₂ hc₃ hc₄ he₁ he₂ he₃ he₄ he₅ he₆ he₇ he₈ hb |
| simp only [Pi.add_apply, Pi.sub_apply, condExp_const hm] at * |
| exact abs_covariance_le_variance_of_binary_moment_bounds |
| (a := μ[f | m] ω) (b := μ[g | m] ω) |
| (c := μ[fun x => f x * g x | m] ω) hb |
| (by linarith) (by linarith) (by linarith) (by linarith) |
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| theorem ae_abs_posteriorResponse_le_one |
| (μ : Measure Ω) [IsFiniteMeasure μ] |
| (m : MeasurableSpace Ω) (hm : m ≤ mΩ) |
| {f g : Ω → ℝ} (hf : Integrable f μ) (hg : Integrable g μ) |
| (hbinary : ∀ᵐ ω ∂μ, f ω = -1 ∨ f ω = 1) |
| (hgb : ∀ᵐ ω ∂μ, |g ω| ≤ 1) : |
| ∀ᵐ ω ∂μ, |posteriorResponse (mΩ := mΩ) μ m f g ω| ≤ 1 := by |
| filter_upwards [ae_abs_posteriorCovariance_le_binary_variance (mΩ := mΩ) μ m hm hf hg hbinary hgb] |
| with ω hω |
| have hnonneg : 0 ≤ 1 - (μ[f | m] ω) ^ 2 := (abs_nonneg _).trans hω |
| change |posteriorCovariance (mΩ := mΩ) μ m f g ω / (1 - (μ[f | m] ω) ^ 2)| ≤ 1 |
| by_cases hz : 1 - (μ[f | m] ω) ^ 2 = 0 |
| · simp [hz] |
| · rw [abs_div, abs_of_nonneg hnonneg] |
| exact (div_le_one (lt_of_le_of_ne hnonneg (Ne.symm hz))).mpr hω |
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| theorem tendsto_ae_posteriorResponse |
| (μ : Measure Ω) [IsFiniteMeasure μ] (F : Filtration ℕ mΩ) |
| (f g : Ω → ℝ) |
| (hinterior : ∀ᵐ ω ∂μ, μ[f | ⨆ n, F n] ω ∈ Set.Ioo (-1 : ℝ) 1) : |
| ∀ᵐ ω ∂μ, Tendsto (fun n => posteriorResponse μ (F n) f g ω) atTop |
| (𝓝 (posteriorResponse μ (⨆ n, F n) f g ω)) := by |
| filter_upwards [tendsto_ae_posteriorCovariance μ F f g, |
| tendsto_ae_condExp (μ := μ) (ℱ := F) f, hinterior] with ω hc hm hi |
| have hden : 1 - (μ[f | ⨆ n, F n] ω) ^ 2 ≠ 0 := by |
| have hsq : (μ[f | ⨆ n, F n] ω) ^ 2 < 1 := by |
| nlinarith [mul_pos (show 0 < 1 - μ[f | ⨆ n, F n] ω by linarith [hi.2]) |
| (show 0 < 1 + μ[f | ⨆ n, F n] ω by linarith [hi.1])] |
| exact ne_of_gt (sub_pos.mpr hsq) |
| exact hc.div (tendsto_const_nhds.sub (hm.pow 2)) hden |
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| theorem tendsto_integral_sq_sub_posteriorResponse |
| (μ : Measure Ω) [IsFiniteMeasure μ] (F : Filtration ℕ mΩ) |
| {f g : Ω → ℝ} (hf : Integrable f μ) (hg : Integrable g μ) |
| (hbinary : ∀ᵐ ω ∂μ, f ω = -1 ∨ f ω = 1) |
| (hgb : ∀ᵐ ω ∂μ, |g ω| ≤ 1) |
| (hinterior : ∀ᵐ ω ∂μ, μ[f | ⨆ n, F n] ω ∈ Set.Ioo (-1 : ℝ) 1) : |
| Tendsto (fun n => ∫ ω, |
| (posteriorResponse μ (F n) f g ω - |
| posteriorResponse 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_posteriorResponse μ (F n) f g).mono |
| (F.le n)).aestronglyMeasurable) |
| (((stronglyMeasurable_posteriorResponse μ (⨆ n, F n) f g).mono |
| (iSup_le fun n => F.le n)).aestronglyMeasurable) (by norm_num : (0 : ℝ) ≤ 1) |
| · intro n |
| simpa only [Real.norm_eq_abs] using |
| ae_abs_posteriorResponse_le_one μ (F n) (F.le n) hf hg hbinary hgb |
| · simpa only [Real.norm_eq_abs] using |
| ae_abs_posteriorResponse_le_one μ (⨆ n, F n) (iSup_le fun n => F.le n) |
| hf hg hbinary hgb |
| · exact tendsto_ae_posteriorResponse μ F f g hinterior |
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| end Problem367 |
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| end |
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| end OAI |
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