| import OAI.Probability.FreeIsing.Gaussian.BinaryFusionJointLaw |
| import OAI.Probability.FreeIsing.Gaussian.LinearSignalTilt |
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| namespace OAI |
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| noncomputable section |
|
|
| open MeasureTheory |
|
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| namespace Problem367 |
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| theorem binaryFusion_density_eq_mixture {u v : ℝ} |
| (hu : |u| < 1) (hv : |v| < 1) : |
| (1 / 2 : ENNReal) * (ENNReal.ofReal (1 + u) * ENNReal.ofReal (1 + v)) + |
| (1 / 2 : ENNReal) * (ENNReal.ofReal (1 - u) * ENNReal.ofReal (1 - v)) = |
| ENNReal.ofReal (1 + u * v) := by |
| have hpu : 0 ≤ 1 + u := by have := (abs_lt.mp hu).1; linarith |
| have hpv : 0 ≤ 1 + v := by have := (abs_lt.mp hv).1; linarith |
| have hmu : 0 ≤ 1 - u := by have := (abs_lt.mp hu).2; linarith |
| have hmv : 0 ≤ 1 - v := by have := (abs_lt.mp hv).2; linarith |
| have hh : (1 / 2 : ENNReal) = ENNReal.ofReal (1 / 2 : ℝ) := by |
| norm_num [ENNReal.ofReal_div_of_pos] |
| rw [hh, ← ENNReal.ofReal_mul hpu, ← ENNReal.ofReal_mul hmu, |
| ← ENNReal.ofReal_mul (by norm_num : (0 : ℝ) ≤ 1 / 2), |
| ← ENNReal.ofReal_mul (by norm_num : (0 : ℝ) ≤ 1 / 2), |
| ← ENNReal.ofReal_add (by positivity : 0 ≤ (1 / 2 : ℝ) * ((1 + u) * (1 + v))) |
| (by positivity : 0 ≤ (1 / 2 : ℝ) * ((1 - u) * (1 - v)))] |
| congr 1 |
| ring |
|
|
| theorem linearSignalTilt_id_probability {mu : Measure ℝ} |
| [IsProbabilityMeasure mu] {a : ℝ} (ha : |a| ≤ 1) |
| (hu : ∀ᵐ u ∂mu, |u| < 1) (hsym : Measure.map (fun u : ℝ => -u) mu = mu) : |
| IsProbabilityMeasure (linearSignalTilt mu id a) := by |
| apply linearSignalTilt_probability measurable_id (hu.mono (fun _ h => h.le)) ha |
| exact integral_eq_zero_of_sign_reversal (MeasurableEquiv.neg ℝ) |
| ⟨measurable_neg, hsym⟩ (fun _ => rfl) |
|
|
| theorem binaryFusionJointLaw_eq_conditional_mixture {mu nu : Measure ℝ} |
| [IsProbabilityMeasure mu] [IsProbabilityMeasure nu] |
| (hu : ∀ᵐ u ∂mu, |u| < 1) (hv : ∀ᵐ v ∂nu, |v| < 1) : |
| (1 / 2 : ENNReal) • |
| ((linearSignalTilt mu id 1).prod (linearSignalTilt nu id 1)) + |
| (1 / 2 : ENNReal) • |
| ((linearSignalTilt mu id (-1)).prod (linearSignalTilt nu id (-1))) = |
| binaryFusionJointLaw mu nu := by |
| unfold linearSignalTilt binaryFusionJointLaw |
| simp only [id_eq, one_mul, neg_one_mul, ← sub_eq_add_neg] |
| rw [prod_withDensity (by fun_prop) (by fun_prop), |
| prod_withDensity (by fun_prop) (by fun_prop), |
| ← withDensity_smul (1 / 2) (by fun_prop), |
| ← withDensity_smul (1 / 2) (by fun_prop), |
| ← withDensity_add_left (by fun_prop)] |
| apply withDensity_congr_ae |
| filter_upwards [ae_binaryFusion_bounds hu hv] with p hp |
| exact binaryFusion_density_eq_mixture hp.1 hp.2 |
|
|
| theorem binaryFusion_conditional_product_eq_tilt {mu nu : Measure ℝ} |
| [IsProbabilityMeasure mu] [IsProbabilityMeasure nu] {a : ℝ} |
| (ha : a = 1 ∨ a = -1) |
| (hu : ∀ᵐ u ∂mu, |u| < 1) (hv : ∀ᵐ v ∂nu, |v| < 1) : |
| (linearSignalTilt mu id a).prod (linearSignalTilt nu id a) = |
| linearSignalTilt (binaryFusionJointLaw mu nu) |
| (fun p => binaryFusion p.1 p.2) a := by |
| have hab : |a| ≤ 1 := by rcases ha with rfl | rfl <;> norm_num |
| unfold linearSignalTilt binaryFusionJointLaw |
| rw [prod_withDensity (by fun_prop) (by fun_prop), ← withDensity_mul (mu.prod nu) (by fun_prop) (by unfold binaryFusion; fun_prop)] |
| apply withDensity_congr_ae |
| filter_upwards [ae_binaryFusion_bounds hu hv] with p hp |
| simp only [id_eq, Pi.mul_apply] |
| rw [← ENNReal.ofReal_mul (linearSignalTilt_density_nonneg hab hp.1.le), |
| ← ENNReal.ofReal_mul (binaryFusion_denominator_pos hp.1 hp.2).le] |
| congr 1 |
| have hd : 1 + p.1 * p.2 ≠ 0 := (binaryFusion_denominator_pos hp.1 hp.2).ne' |
| unfold binaryFusion |
| rcases ha with rfl | rfl <;> field_simp <;> ring |
| |
| end Problem367 |
| |
| end |
| |
| end OAI |
| |
