| import OAI.Probability.FreeIsing.Spins.ObservationFusion | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory | |
| namespace Problem367 | |
| section Observations | |
| variable {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})) | |
| include hS hX hY hq hr hpostX hpostY hqb hrb hfair hInd | |
| theorem map_observation_posterior_pair_eq_binaryFusionJointLaw : | |
| Measure.map (fun x => (q (X x), r (Y x))) rho = | |
| binaryFusionJointLaw (Measure.map (fun x => q (X x)) rho) | |
| (Measure.map (fun x => r (Y x)) rho) := by | |
| have hL : ∀ᵐ x ∂rho, |rho[fun y => spinSign (S y) | mA.comap X] x| < 1 := by | |
| filter_upwards [hpostX, hqb] with x hx hb | |
| simpa only [hx] using hb | |
| have hR : ∀ᵐ x ∂rho, |rho[fun y => spinSign (S y) | mB.comap Y] x| < 1 := by | |
| filter_upwards [hpostY, hrb] with x hx hb | |
| simpa only [hx] using hb | |
| have hInd' (a : IsingSpin) : | |
| IndepFun (rho[fun y => spinSign (S y) | mA.comap X]) | |
| (rho[fun y => spinSign (S y) | mB.comap Y]) (cond rho {x | S x = a}) := | |
| ((hInd a).comp hq hr).congr | |
| (cond_absolutelyContinuous.ae_le hpostX.symm) | |
| (cond_absolutelyContinuous.ae_le hpostY.symm) | |
| have hjoint := map_posterior_pair_eq_binaryFusionJointLaw | |
| hX.comap_le hY.comap_le hS hfair hInd' hL hR | |
| have hp : (fun x => (rho[fun y => spinSign (S y) | mA.comap X] x, | |
| rho[fun y => spinSign (S y) | mB.comap Y] x)) =ᵐ[rho] | |
| (fun x => (q (X x), r (Y x))) := by | |
| filter_upwards [hpostX, hpostY] with x hx hy | |
| rw [hx, hy] | |
| rw [Measure.map_congr hp, Measure.map_congr hpostX, Measure.map_congr hpostY] at hjoint | |
| exact hjoint | |
| theorem condExp_observation_pair_map_eq_binaryFusion : | |
| Measure.map (rho[fun x => spinSign (S x) | | |
| (inferInstance : MeasurableSpace (A × B)).comap (fun x => (X x, Y x))]) rho = | |
| Measure.map (fun p : ℝ × ℝ => binaryFusion p.1 p.2) | |
| (binaryFusionJointLaw (Measure.map (fun x => q (X x)) rho) | |
| (Measure.map (fun x => r (Y x)) rho)) := by | |
| have hCE := condExp_spin_given_observation_pair_eq_binaryFusion | |
| hS hX hY hq hr hpostX hpostY hqb hrb hfair hInd | |
| rw [Measure.map_congr hCE, | |
| ← map_observation_posterior_pair_eq_binaryFusionJointLaw | |
| hS hX hY hq hr hpostX hpostY hqb hrb hfair hInd, | |
| Measure.map_map (show Measurable (fun p : ℝ × ℝ => binaryFusion p.1 p.2) by | |
| unfold binaryFusion; fun_prop) (show Measurable (fun x => (q (X x), r (Y x))) from | |
| (hq.comp hX).prodMk (hr.comp hY))] | |
| rfl | |
| theorem condExp_observation_pair_secondMoment | |
| (hsym : Measure.map (fun u : ℝ => -u) | |
| (Measure.map (fun x => r (Y x)) rho) = Measure.map (fun x => r (Y x)) rho) : | |
| (∫ x, (rho[fun x => spinSign (S x) | | |
| (inferInstance : MeasurableSpace (A × B)).comap (fun x => (X x, Y x))] x) ^ 2 ∂rho) = | |
| (∫ x, q (X x) ^ 2 ∂rho) + (∫ x, r (Y x) ^ 2 ∂rho) - | |
| ∫ p : ℝ × ℝ, binaryFusionLoss p.1 p.2 ∂ | |
| (Measure.map (fun x => q (X x)) rho).prod (Measure.map (fun x => r (Y x)) rho) := by | |
| have hCE := condExp_spin_given_observation_pair_eq_binaryFusion | |
| hS hX hY hq hr hpostX hpostY hqb hrb hfair hInd | |
| have hiCE := integral_congr_ae (hCE.fun_comp (fun x : ℝ => x ^ 2)) | |
| simp only [Function.comp_apply] at hiCE | |
| rw [hiCE] | |
| have hpair : Measurable (fun x => (q (X x), r (Y x))) := | |
| (hq.comp hX).prodMk (hr.comp hY) | |
| have hmap := integral_map (μ := rho) hpair.aemeasurable | |
| (show AEStronglyMeasurable (fun p : ℝ × ℝ => binaryFusion p.1 p.2 ^ 2) _ by | |
| apply Measurable.aestronglyMeasurable; unfold binaryFusion; fun_prop) | |
| rw [map_observation_posterior_pair_eq_binaryFusionJointLaw | |
| hS hX hY hq hr hpostX hpostY hqb hrb hfair hInd] at hmap | |
| rw [← hmap] | |
| have hqm : ∀ᵐ u ∂Measure.map (fun x => q (X x)) rho, |u| < 1 := | |
| (ae_map_iff (hq.comp hX).aemeasurable | |
| (measurableSet_lt measurable_abs measurable_const)).2 hqb | |
| have hrm : ∀ᵐ u ∂Measure.map (fun x => r (Y x)) rho, |u| < 1 := | |
| (ae_map_iff (hr.comp hY).aemeasurable | |
| (measurableSet_lt measurable_abs measurable_const)).2 hrb | |
| rw [binaryFusionJointLaw_secondMoment hqm hrm hsym, | |
| integral_map (μ := rho) (φ := fun x => q (X x)) (f := fun u : ℝ => u ^ 2) | |
| (hq.comp hX).aemeasurable (by fun_prop), | |
| integral_map (μ := rho) (φ := fun x => r (Y x)) (f := fun u : ℝ => u ^ 2) | |
| (hr.comp hY).aemeasurable (by fun_prop)] | |
| theorem condExp_observation_pair_secondMoment_le_add | |
| (hsym : Measure.map (fun u : ℝ => -u) | |
| (Measure.map (fun x => r (Y x)) rho) = Measure.map (fun x => r (Y x)) rho) : | |
| (∫ x, (rho[fun x => spinSign (S x) | | |
| (inferInstance : MeasurableSpace (A × B)).comap (fun x => (X x, Y x))] x) ^ 2 ∂rho) ≤ | |
| (∫ x, q (X x) ^ 2 ∂rho) + (∫ x, r (Y x) ^ 2 ∂rho) := by | |
| rw [condExp_observation_pair_secondMoment | |
| hS hX hY hq hr hpostX hpostY hqb hrb hfair hInd hsym] | |
| apply sub_le_self | |
| apply integral_binaryFusionLoss_nonneg | |
| · exact (ae_map_iff (hq.comp hX).aemeasurable | |
| (measurableSet_lt measurable_abs measurable_const)).2 hqb | |
| · exact (ae_map_iff (hr.comp hY).aemeasurable | |
| (measurableSet_lt measurable_abs measurable_const)).2 hrb | |
| theorem condExp_observation_pair_abs_lt_one : | |
| ∀ᵐ x ∂rho, |rho[fun x => spinSign (S x) | | |
| (inferInstance : MeasurableSpace (A × B)).comap (fun x => (X x, Y x))] x| < 1 := by | |
| have hCE := condExp_spin_given_observation_pair_eq_binaryFusion | |
| hS hX hY hq hr hpostX hpostY hqb hrb hfair hInd | |
| filter_upwards [hCE, hqb, hrb] with x hx hqx hrx | |
| rw [hx] | |
| exact abs_binaryFusion_lt_one hqx hrx | |
| theorem condExp_observation_pair_neg_invariant | |
| (hsymq : Measure.map (fun u : ℝ => -u) | |
| (Measure.map (fun x => q (X x)) rho) = Measure.map (fun x => q (X x)) rho) | |
| (hsymr : Measure.map (fun u : ℝ => -u) | |
| (Measure.map (fun x => r (Y x)) rho) = Measure.map (fun x => r (Y x)) rho) : | |
| Measure.map (fun u : ℝ => -u) | |
| (Measure.map (rho[fun x => spinSign (S x) | | |
| (inferInstance : MeasurableSpace (A × B)).comap (fun x => (X x, Y x))]) rho) = | |
| Measure.map (rho[fun x => spinSign (S x) | | |
| (inferInstance : MeasurableSpace (A × B)).comap (fun x => (X x, Y x))]) rho := by | |
| rw [condExp_observation_pair_map_eq_binaryFusion | |
| hS hX hY hq hr hpostX hpostY hqb hrb hfair hInd] | |
| exact binaryFusion_map_neg_invariant hsymq hsymr | |
| omit hqb hrb in | |
| theorem condExp_observation_pair_secondMoment_le | |
| {theta : ℝ} | |
| (hqtheta : ∀ᵐ x ∂rho, |q (X x)| ≤ theta) | |
| (hrtheta : ∀ᵐ x ∂rho, |r (Y x)| ≤ theta) (htheta : theta < 1) | |
| (hsym : Measure.map (fun u : ℝ => -u) | |
| (Measure.map (fun x => r (Y x)) rho) = Measure.map (fun x => r (Y x)) rho) : | |
| (∫ x, (rho[fun x => spinSign (S x) | | |
| (inferInstance : MeasurableSpace (A × B)).comap (fun x => (X x, Y x))] x) ^ 2 ∂rho) ≤ | |
| (∫ x, q (X x) ^ 2 ∂rho) + (∫ x, r (Y x) ^ 2 ∂rho) - | |
| 2 * (1 - theta ^ 2) * (∫ x, q (X x) ^ 2 ∂rho) * (∫ x, r (Y x) ^ 2 ∂rho) := by | |
| have hq1 := hqtheta.mono (fun _ hx => lt_of_le_of_lt hx htheta) | |
| have hr1 := hrtheta.mono (fun _ hx => lt_of_le_of_lt hx htheta) | |
| rw [condExp_observation_pair_secondMoment | |
| hS hX hY hq hr hpostX hpostY hq1 hr1 hfair hInd hsym] | |
| have hqm : ∀ᵐ u ∂Measure.map (fun x => q (X x)) rho, |u| ≤ theta := | |
| (ae_map_iff (hq.comp hX).aemeasurable | |
| (measurableSet_le measurable_abs measurable_const)).2 hqtheta | |
| have hrm : ∀ᵐ u ∂Measure.map (fun x => r (Y x)) rho, |u| ≤ theta := | |
| (ae_map_iff (hr.comp hY).aemeasurable | |
| (measurableSet_le measurable_abs measurable_const)).2 hrtheta | |
| have hloss := integral_binaryFusionLoss_lower_bound hqm hrm htheta | |
| rw [integral_map (μ := rho) (φ := fun x => q (X x)) (f := fun u : ℝ => u ^ 2) | |
| (hq.comp hX).aemeasurable (by fun_prop), | |
| integral_map (μ := rho) (φ := fun x => r (Y x)) (f := fun u : ℝ => u ^ 2) | |
| (hr.comp hY).aemeasurable (by fun_prop)] at hloss | |
| exact sub_le_sub_left hloss _ | |
| end Observations | |
| end Problem367 | |
| end | |
| end OAI | |