| import OAI.Probability.FreeIsing.Energy.TiltedVariance | |
| import OAI.Probability.FreeIsing.Gaussian.LinearSignalTilt | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory | |
| namespace Problem367 | |
| variable {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} | |
| lemma memLp_two_linearSignalTilt [IsProbabilityMeasure μ] | |
| {q g : Ω → ℝ} {a : ℝ} (hq : Measurable q) | |
| (hqb : ∀ᵐ x ∂μ, |q x| ≤ 1) (ha : |a| ≤ 1) (hg : MemLp g 2 μ) : | |
| MemLp g 2 (linearSignalTilt μ q a) := by | |
| have hgi := integrable_linearSignalTilt hq hqb ha (hg.integrable one_le_two) | |
| exact (memLp_two_iff_integrable_sq hgi.aestronglyMeasurable).mpr | |
| (integrable_linearSignalTilt hq hqb ha hg.integrable_sq) | |
| lemma variance_linearSignalTilt_ge [IsProbabilityMeasure μ] | |
| {q g : Ω → ℝ} {a : ℝ} (hq : Measurable q) | |
| (hqb : ∀ᵐ x ∂μ, |q x| ≤ 1) (ha : |a| ≤ 1) (hg : MemLp g 2 μ) | |
| (T : Ω ≃ᵐ Ω) (hT : MeasurePreserving T μ μ) | |
| (hqodd : ∀ᵐ x ∂μ, q (T x) = -q x) | |
| (hgodd : ∀ᵐ x ∂μ, g (T x) = -g x) : | |
| (1 - a ^ 2) * (∫ x, g x ^ 2 ∂μ) ≤ variance g (linearSignalTilt μ q a) := by | |
| let := linearSignalTilt_probability hq hqb ha | |
| (integral_eq_zero_of_ae_sign_reversal T hT hqodd) | |
| apply variance_ge_of_binary_tilt_moments hg | |
| (memLp_two_linearSignalTilt hq hqb ha hg) hq.aestronglyMeasurable hqb | |
| · rw [integral_linearSignalTilt hq hqb ha (hg.integrable one_le_two), | |
| integral_eq_zero_of_ae_sign_reversal T hT hgodd, zero_add] | |
| congr 1 | |
| apply integral_congr_ae | |
| filter_upwards [] with x | |
| ring | |
| · apply integral_linearSignalTilt_ae_even hq hqb ha hg.integrable_sq T hT hqodd | |
| filter_upwards [hgodd] with x hx | |
| rw [hx, neg_sq] | |
| lemma variance_linearSignalTilt_ge_signal_moment [IsProbabilityMeasure μ] | |
| {q g : Ω → ℝ} {a theta : ℝ} (hq : Measurable q) | |
| (hqb : ∀ᵐ x ∂μ, |q x| ≤ 1) (ha : |a| ≤ 1) (hg : MemLp g 2 μ) | |
| (T : Ω ≃ᵐ Ω) (hT : MeasurePreserving T μ μ) | |
| (hqodd : ∀ᵐ x ∂μ, q (T x) = -q x) | |
| (hgodd : ∀ᵐ x ∂μ, g (T x) = -g x) | |
| (haTheta : a ^ 2 ≤ theta ^ 2) (hTheta : theta ^ 2 ≤ 1) | |
| (hdom : ∀ᵐ x ∂μ, theta ^ 2 * q x ^ 2 ≤ g x ^ 2) : | |
| (1 - theta ^ 2) * theta ^ 2 * (∫ x, q x ^ 2 ∂μ) ≤ | |
| variance g (linearSignalTilt μ q a) := by | |
| have hqLp : MemLp q 2 μ := | |
| MemLp.of_bound hq.aestronglyMeasurable 1 (by simpa only [Real.norm_eq_abs] using hqb) | |
| have henergy : theta ^ 2 * (∫ x, q x ^ 2 ∂μ) ≤ ∫ x, g x ^ 2 ∂μ := by | |
| rw [← integral_const_mul] | |
| exact integral_mono_ae (hqLp.integrable_sq.const_mul _) hg.integrable_sq hdom | |
| have hnonneg : 0 ≤ ∫ x, g x ^ 2 ∂μ := integral_nonneg fun x => sq_nonneg _ | |
| calc | |
| (1 - theta ^ 2) * theta ^ 2 * (∫ x, q x ^ 2 ∂μ) = | |
| (1 - theta ^ 2) * (theta ^ 2 * (∫ x, q x ^ 2 ∂μ)) := by ring | |
| _ ≤ (1 - theta ^ 2) * (∫ x, g x ^ 2 ∂μ) := | |
| mul_le_mul_of_nonneg_left henergy (sub_nonneg.mpr hTheta) | |
| _ ≤ (1 - a ^ 2) * (∫ x, g x ^ 2 ∂μ) := | |
| mul_le_mul_of_nonneg_right (sub_le_sub_left haTheta 1) hnonneg | |
| _ ≤ _ := variance_linearSignalTilt_ge hq hqb ha hg T hT hqodd hgodd | |
| end Problem367 | |
| end | |
| end OAI | |