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b317718 | 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 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 | import OAI.Probability.FreeIsing.Spins.ObservationLaw
import OAI.Probability.FreeIsing.Spins.PosteriorCertification
import OAI.Probability.FreeIsing.Spins.PosteriorFusion
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
namespace Problem367
def observationFusionJointLaw {A B : Type*} [MeasurableSpace A] [MeasurableSpace B]
(mu : Measure A) (nu : Measure B) (q : A → ℝ) (r : B → ℝ) : Measure (A × B) :=
(mu.prod nu).withDensity (fun p => ENNReal.ofReal (1 + q p.1 * r p.2))
private theorem ae_observation_pair_bounds
{A B : Type*} [MeasurableSpace A] [MeasurableSpace B]
{mu : Measure A} {nu : Measure B} [IsProbabilityMeasure mu] [IsProbabilityMeasure nu]
{q : A → ℝ} {r : B → ℝ}
(hq : ∀ᵐ x ∂mu, |q x| < 1) (hr : ∀ᵐ y ∂nu, |r y| < 1) :
∀ᵐ p ∂mu.prod nu, |q p.1| < 1 ∧ |r p.2| < 1 := by
have hq' : ∀ᵐ p ∂mu.prod nu, |q p.1| < 1 :=
measurePreserving_fst.quasiMeasurePreserving.tendsto_ae hq
have hr' : ∀ᵐ p ∂mu.prod nu, |r p.2| < 1 :=
measurePreserving_snd.quasiMeasurePreserving.tendsto_ae hr
exact hq'.and hr'
theorem observationFusionJointLaw_eq_mixture
{A B : Type*} [MeasurableSpace A] [MeasurableSpace B]
{mu : Measure A} {nu : Measure B} [IsProbabilityMeasure mu] [IsProbabilityMeasure nu]
{q : A → ℝ} {r : B → ℝ} (hq : Measurable q) (hr : Measurable r)
(hqb : ∀ᵐ x ∂mu, |q x| < 1) (hrb : ∀ᵐ y ∂nu, |r y| < 1) :
(1 / 2 : ENNReal) • ((linearSignalTilt mu q 1).prod (linearSignalTilt nu r 1)) +
(1 / 2 : ENNReal) • ((linearSignalTilt mu q (-1)).prod (linearSignalTilt nu r (-1))) =
observationFusionJointLaw mu nu q r := by
unfold linearSignalTilt observationFusionJointLaw
simp only [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_observation_pair_bounds hqb hrb] with p hp
exact binaryFusion_density_eq_mixture hp.1 hp.2
theorem observationFusion_positive_product_eq_tilt
{A B : Type*} [MeasurableSpace A] [MeasurableSpace B]
{mu : Measure A} {nu : Measure B} [IsProbabilityMeasure mu] [IsProbabilityMeasure nu]
{q : A → ℝ} {r : B → ℝ} (hq : Measurable q) (hr : Measurable r)
(hqb : ∀ᵐ x ∂mu, |q x| < 1) (hrb : ∀ᵐ y ∂nu, |r y| < 1) :
(linearSignalTilt mu q 1).prod (linearSignalTilt nu r 1) =
linearSignalTilt (observationFusionJointLaw mu nu q r)
(fun p => binaryFusion (q p.1) (r p.2)) 1 := by
unfold linearSignalTilt observationFusionJointLaw
rw [prod_withDensity (by fun_prop) (by fun_prop),
← withDensity_mul _ (by fun_prop) (by unfold binaryFusion; fun_prop)]
apply withDensity_congr_ae
filter_upwards [ae_observation_pair_bounds hqb hrb] with p hp
simp only [one_mul, Pi.mul_apply]
have hpq : 0 ≤ 1 + q p.1 := by have := (abs_lt.mp hp.1).1; linarith
have hden := binaryFusion_denominator_pos hp.1 hp.2
rw [← ENNReal.ofReal_mul hpq, ← ENNReal.ofReal_mul hden.le]
congr 1
unfold binaryFusion
field_simp
ring
theorem condExp_spin_given_observation_pair_eq_binaryFusion
{Ω A B : Type*} {mΩ : MeasurableSpace Ω} {mA : MeasurableSpace A}
{mB : MeasurableSpace B} {rho : Measure Ω} [IsProbabilityMeasure rho]
{S : Ω → IsingSpin} (hS : Measurable S)
{X : Ω → A} {Y : Ω → B} (hX : Measurable X) (hY : Measurable Y)
{q : A → ℝ} {r : B → ℝ} (hq : Measurable q) (hr : Measurable r)
(hpostX : rho[fun x => spinSign (S x) | mA.comap X] =ᵐ[rho] fun x => q (X x))
(hpostY : rho[fun x => spinSign (S x) | mB.comap Y] =ᵐ[rho] fun x => r (Y x))
(hqb : ∀ᵐ x ∂rho, |q (X x)| < 1) (hrb : ∀ᵐ x ∂rho, |r (Y x)| < 1)
(hfair : ∀ a : IsingSpin, rho {x | S x = a} = (1 / 2 : ENNReal))
(hInd : ∀ a : IsingSpin, IndepFun X Y (cond rho {x | S x = a})) :
rho[fun x => spinSign (S x) |
(inferInstance : MeasurableSpace (A × B)).comap (fun x => (X x, Y x))] =ᵐ[rho]
fun x => binaryFusion (q (X x)) (r (Y x)) := by
let mu := Measure.map X rho
let nu := Measure.map Y rho
have hqm : ∀ᵐ x ∂mu, |q x| < 1 :=
(ae_map_iff hX.aemeasurable (measurableSet_lt hq.abs measurable_const)).2 hqb
have hrm : ∀ᵐ y ∂nu, |r y| < 1 :=
(ae_map_iff hY.aemeasurable (measurableSet_lt hr.abs measurable_const)).2 hrb
have hpair (a : IsingSpin) : Measure.map (fun x => (X x, Y x)) (cond rho {x | S x = a}) =
(linearSignalTilt mu q (spinSign a)).prod (linearSignalTilt nu r (spinSign a)) := by
let : IsProbabilityMeasure (cond rho {x | S x = a}) :=
cond_isProbabilityMeasure (by rw [hfair]; norm_num)
rw [(hInd a).map_prod_eq_prod_map_map hX.aemeasurable hY.aemeasurable,
map_cond_spin_observation_eq_linearSignalTilt hS hX hq hpostX a (hfair a),
map_cond_spin_observation_eq_linearSignalTilt hS hY hr hpostY a (hfair a)]
have hjoint : Measure.map (fun x => (X x, Y x)) rho =
observationFusionJointLaw mu nu q r := by
calc
_ = Measure.map (fun x => (X x, Y x))
((1 / 2 : ENNReal) • cond rho {x | S x = .pos} +
(1 / 2 : ENNReal) • cond rho {x | S x = .neg}) := by
rw [← measure_eq_fair_spin_mixture hS hfair]
_ = _ := by
rw [Measure.map_add _ _ (hX.prodMk hY),
Measure.map_smul _ (hX.prodMk hY).aemeasurable,
Measure.map_smul _ (hX.prodMk hY).aemeasurable,
hpair .pos, hpair .neg]
exact observationFusionJointLaw_eq_mixture hq hr hqm hrm
apply condExp_spin_eq_of_conditional_linearTilt hS (hX.prodMk hY)
(show Measurable (fun p : A × B => binaryFusion (q p.1) (r p.2)) by
unfold binaryFusion
fun_prop)
· rw [hjoint]
have hb : ∀ᵐ p ∂observationFusionJointLaw mu nu q r, |q p.1| < 1 ∧ |r p.2| < 1 :=
(withDensity_absolutelyContinuous (mu.prod nu)
(fun p => ENNReal.ofReal (1 + q p.1 * r p.2))).ae_le
(ae_observation_pair_bounds hqm hrm)
filter_upwards [hb] with p hp
exact (abs_binaryFusion_lt_one hp.1 hp.2).le
· exact hfair .pos
· rw [hpair .pos, hjoint]
exact observationFusion_positive_product_eq_tilt hq hr hqm hrm
end Problem367
end
end OAI
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