| import OAI.Probability.FreeIsing.Gaussian.SideSignalLoss | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory | |
| namespace Problem367 | |
| def statisticMessage (theta q : ℝ) : ℝ := Real.artanh (theta * q) | |
| theorem measurable_statisticMessage (theta : ℝ) : Measurable (statisticMessage theta) := by | |
| unfold statisticMessage Real.artanh | |
| fun_prop | |
| theorem statisticMessage_argument_mem {theta q : ℝ} | |
| (hθ0 : 0 ≤ theta) (hθ1 : theta < 1) (hq : |q| ≤ 1) : | |
| theta * q ∈ Set.Ioo (-1 : ℝ) 1 := by | |
| apply abs_lt.mp | |
| rw [abs_mul, abs_of_nonneg hθ0] | |
| exact (mul_le_mul_of_nonneg_left hq hθ0).trans_lt (by simpa using hθ1) | |
| theorem statisticMessage_abs_le {theta q : ℝ} | |
| (hθ0 : 0 ≤ theta) (hθ1 : theta < 1) (hq : |q| ≤ 1) : | |
| |statisticMessage theta q| ≤ Real.artanh theta := by | |
| have hm := statisticMessage_argument_mem hθ0 hθ1 hq | |
| have hupper : Real.artanh (theta * q) ≤ Real.artanh theta := by | |
| apply Real.artanh_le_artanh hm.1 hθ1 | |
| simpa using mul_le_mul_of_nonneg_left (abs_le.mp hq).2 hθ0 | |
| have hneg : Real.artanh (-(theta * q)) ≤ Real.artanh theta := by | |
| apply Real.artanh_le_artanh (by linarith [hm.2]) hθ1 | |
| nlinarith [(abs_le.mp hq).1] | |
| rw [artanh_neg_of_mem hm] at hneg | |
| exact abs_le.mpr ⟨by simpa only [statisticMessage, neg_neg] using (neg_le_neg hneg), hupper⟩ | |
| theorem statisticMessage_abs_ge {theta q : ℝ} | |
| (hθ0 : 0 ≤ theta) (hθ1 : theta < 1) (hq : |q| ≤ 1) : | |
| theta * |q| ≤ |statisticMessage theta q| := by | |
| have h := abs_tanh_le_abs_response (Real.artanh (theta * q)) | |
| rw [Real.tanh_artanh (statisticMessage_argument_mem hθ0 hθ1 hq), | |
| abs_mul, abs_of_nonneg hθ0] at h | |
| exact h | |
| section Probability | |
| variable {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ] | |
| theorem variance_statisticMessage_linearSignalTilt | |
| {theta eta : ℝ} (hθ0 : 0 ≤ theta) (hθ1 : theta < 1) (hη : |eta| ≤ 1) | |
| {q : Ω → ℝ} (hq : Measurable q) (hqb : ∀ᵐ w ∂μ, |q w| ≤ 1) | |
| (T : Ω ≃ᵐ Ω) (hT : MeasurePreserving T μ μ) | |
| (hqodd : ∀ᵐ w ∂μ, q (T w) = -q w) : | |
| (1 - theta ^ 2) * theta ^ 2 * (∫ w, q w ^ 2 ∂μ) ≤ | |
| variance (fun w => statisticMessage theta (q w)) | |
| (linearSignalTilt μ q (eta * theta)) := by | |
| have hθsq : theta ^ 2 ≤ 1 := by nlinarith | |
| have ha : |eta * theta| ≤ 1 := by | |
| rw [abs_mul, abs_of_nonneg hθ0] | |
| exact (mul_le_mul_of_nonneg_right hη hθ0).trans (by simpa using hθ1.le) | |
| have hg : MemLp (fun w => statisticMessage theta (q w)) 2 μ := | |
| MemLp.of_bound ((measurable_statisticMessage theta).comp hq).aestronglyMeasurable | |
| (Real.artanh theta) (by | |
| filter_upwards [hqb] with w hw | |
| simpa only [Real.norm_eq_abs] using statisticMessage_abs_le hθ0 hθ1 hw) | |
| apply variance_linearSignalTilt_ge_signal_moment hq hqb ha hg T hT hqodd | |
| · filter_upwards [hqodd, hqb] with w hw hb | |
| simp only [statisticMessage, hw, mul_neg] | |
| exact artanh_neg_of_mem (statisticMessage_argument_mem hθ0 hθ1 hb) | |
| · have hηsq := (sq_le_one_iff_abs_le_one eta).mpr hη | |
| nlinarith [mul_le_mul_of_nonneg_right hηsq (sq_nonneg theta)] | |
| · exact hθsq | |
| · filter_upwards [hqb] with w hw | |
| have hs := (sq_le_sq₀ (mul_nonneg hθ0 (abs_nonneg _)) (abs_nonneg _)).mpr | |
| (statisticMessage_abs_ge hθ0 hθ1 hw) | |
| simpa only [mul_pow, sq_abs] using hs | |
| theorem integral_tilted_statistic_attenuation_le | |
| {theta eta : ℝ} (hθ0 : 0 ≤ theta) (hθ1 : theta < 1) (hη : |eta| ≤ 1) | |
| {q : Ω → ℝ} (hq : Measurable q) (hqb : ∀ᵐ w ∂μ, |q w| ≤ 1) | |
| (T : Ω ≃ᵐ Ω) (hT : MeasurePreserving T μ μ) | |
| (hqodd : ∀ᵐ w ∂μ, q (T w) = -q w) (z : ℝ) : | |
| (∫ w, responseAttenuation theta (z + statisticMessage theta (q w)) ^ 2 | |
| ∂linearSignalTilt μ q (eta * theta)) ≤ | |
| 1 - sideSignalLossCoefficient theta * (∫ w, q w ^ 2 ∂μ) := by | |
| have hθsq : theta ^ 2 < 1 := by nlinarith | |
| have ha : |eta * theta| ≤ 1 := by | |
| rw [abs_mul, abs_of_nonneg hθ0] | |
| exact (mul_le_mul_of_nonneg_right hη hθ0).trans (by simpa using hθ1.le) | |
| let := linearSignalTilt_probability hq hqb ha | |
| (integral_eq_zero_of_ae_sign_reversal T hT hqodd) | |
| have hb : ∀ᵐ w ∂linearSignalTilt μ q (eta * theta), | |
| |statisticMessage theta (q w)| ≤ Real.artanh theta := by | |
| apply (withDensity_absolutelyContinuous _ _).ae_le | |
| exact hqb.mono (fun w hw => statisticMessage_abs_le hθ0 hθ1 hw) | |
| have hloss := integral_shifted_attenuation_loss_explicit | |
| (μ := linearSignalTilt μ q (eta * theta)) hθsq.le | |
| ((measurable_statisticMessage theta).comp hq) hb z | |
| have hvariance := variance_statisticMessage_linearSignalTilt hθ0 hθ1 hη hq hqb T hT hqodd | |
| have hmul := mul_le_mul_of_nonneg_left hvariance | |
| (shiftedLossCoefficient_pos (beta := Real.artanh theta) hθsq).le | |
| simp only [Function.comp_def] at hloss | |
| unfold sideSignalLossCoefficient | |
| nlinarith [hloss] | |
| end Probability | |
| end Problem367 | |
| end | |
| end OAI | |