| import OAI.Probability.FreeIsing.Spins.ObservationLaw |
| import OAI.Probability.FreeIsing.Spins.PosteriorCertification |
| import OAI.Probability.FreeIsing.Spins.PosteriorFusion |
|
|
| namespace OAI |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory |
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| namespace Problem367 |
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| 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)) |
|
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| 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 |
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|
| 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 |
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|
| 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 |
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| end Problem367 |
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| end |
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| end OAI |
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