File size: 3,718 Bytes
f61e547 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 | import OAI.Probability.FreeIsing.Gaussian.BinaryFusionJointLaw
import OAI.Probability.FreeIsing.Gaussian.LinearSignalTilt
namespace OAI
noncomputable section
open MeasureTheory
namespace Problem367
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
|