| import OAI.Probability.FreeIsing.Gaussian.BinarySignalDensity |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal |
|
|
| namespace Problem367 |
|
|
| local instance binarySignalFusionMeasurableSingletonSpin : |
| MeasurableSingletonClass IsingSpin := ⟨fun _ => trivial⟩ |
| local instance binarySignalFusionNonemptySpin : Nonempty IsingSpin := ⟨.neg⟩ |
|
|
| theorem condDistrib_raw_signal_likelihood |
| {Ω A : Type*} [MeasurableSpace Ω] [MeasurableSpace A] |
| [StandardBorelSpace A] [Nonempty A] |
| (ρ : Measure Ω) [IsFiniteMeasure ρ] |
| (S : Ω → IsingSpin) (X : Ω → A) (u : A → ℝ) |
| (hS : Measurable S) (hX : Measurable X) (hu : Measurable u) |
| (hfair : ∀ a : IsingSpin, (ρ.map S) {a} = ENNReal.ofReal (1 / 2 : ℝ)) |
| (hpost : ρ[fun ω => spinSign (S ω) | |
| (inferInstance : MeasurableSpace A).comap X] =ᵐ[ρ] fun ω => u (X ω)) |
| (a : IsingSpin) : |
| condDistrib X S ρ a = (ρ.map X).withDensity |
| (fun x => ENNReal.ofReal (1 + spinSign a * u x)) := by |
| have hint : Integrable (fun ω => spinSign (S ω)) ρ := by |
| apply Integrable.of_bound ((measurable_of_finite spinSign).comp hS).aestronglyMeasurable 1 |
| exact Filter.Eventually.of_forall (fun ω => by |
| cases hs : S ω <;> norm_num [spinSign, hs]) |
| have hce := condExp_ae_eq_integral_condDistrib hX hS.aemeasurable |
| (measurable_of_finite spinSign).stronglyMeasurable hint |
| have hprob : ∀ᵐ x ∂ρ.map X, (condDistrib S X ρ x) {a} = |
| ENNReal.ofReal ((1 + spinSign a * u x) / 2) := by |
| apply (ae_map_iff hX.aemeasurable (measurableSet_eq_fun |
| (Kernel.measurable_coe _ (measurableSet_singleton a)) (by fun_prop))).2 |
| filter_upwards [hpost, hce] with ω hp hc |
| rw [spin_measure_singleton_eq_mean, ← hc, hp] |
| have hnonzero : (ρ.map S) {a} ≠ 0 := by rw [hfair]; norm_num |
| ext E hE |
| have hrectangle : (ρ.map (fun ω => (S ω, X ω))) ({a} ×ˢ E) = |
| ∫⁻ x in E, condDistrib S X ρ x {a} ∂ρ.map X := by |
| rw [← Measure.compProd_apply_prod hE (measurableSet_singleton a), |
| compProd_map_condDistrib hX.aemeasurable hS.aemeasurable, |
| Measure.map_apply (hS.prodMk hX) ((measurableSet_singleton a).prod hE), |
| Measure.map_apply (hX.prodMk hS) (hE.prod (measurableSet_singleton a))] |
| congr 1 |
| ext ω |
| simp only [Set.mem_preimage, Set.mem_prod, Set.mem_singleton_iff, and_comm] |
| rw [condDistrib_apply_of_ne_zero hS hX a hnonzero, hfair, hrectangle, |
| withDensity_apply _ hE, ← lintegral_const_mul _ (Kernel.measurable_coe _ |
| (measurableSet_singleton a))] |
| apply lintegral_congr_ae |
| filter_upwards [ae_restrict_of_ae hprob] with x hx |
| rw [hx, ENNReal.ofReal_div_of_pos (by norm_num : (0 : ℝ) < 2)] |
| norm_num |
| rw [show (2 : ℝ≥0∞) = ENNReal.ofReal (2 : ℝ) by norm_num, |
| ← ENNReal.ofReal_mul (by norm_num : (0 : ℝ) ≤ 2)] |
| congr 1 |
| ring |
|
|
| theorem posterior_version_calibrated |
| {Ω A : Type*} [MeasurableSpace Ω] [MeasurableSpace A] |
| (ρ : Measure Ω) [IsFiniteMeasure ρ] |
| (f : Ω → ℝ) (X : Ω → A) (u : A → ℝ) |
| (hX : Measurable X) (hu : Measurable u) |
| (hpost : ρ[f | (inferInstance : MeasurableSpace A).comap X] =ᵐ[ρ] |
| fun ω => u (X ω)) : |
| ρ[f | (inferInstance : MeasurableSpace ℝ).comap (fun ω => u (X ω))] =ᵐ[ρ] |
| fun ω => u (X ω) := by |
| have hm : MeasurableSpace.comap (fun ω => u (X ω)) (inferInstance : MeasurableSpace ℝ) ≤ |
| MeasurableSpace.comap X (inferInstance : MeasurableSpace A) := |
| (hu.comp (comap_measurable X)).comap_le |
| have htower := condExp_condExp_of_le (μ := ρ) (f := f) hm hX.comap_le |
| apply htower.symm.trans |
| apply (condExp_congr_ae hpost).trans |
| rw [condExp_of_stronglyMeasurable (hm.trans hX.comap_le) |
| (comap_measurable (fun ω => u (X ω))).stronglyMeasurable |
| (integrable_condExp.congr hpost)] |
|
|
| theorem condExp_raw_pair_eq_binaryFusion |
| {Ω A B : Type*} [MeasurableSpace Ω] [StandardBorelSpace Ω] |
| [MeasurableSpace A] [StandardBorelSpace A] [Nonempty A] |
| [MeasurableSpace B] [StandardBorelSpace B] [Nonempty B] |
| (ρ : Measure Ω) [IsProbabilityMeasure ρ] |
| (S : Ω → IsingSpin) (X : Ω → A) (Y : Ω → B) (u : A → ℝ) (v : B → ℝ) |
| (hS : Measurable S) (hX : Measurable X) (hY : Measurable Y) |
| (hu : Measurable u) (hv : Measurable v) |
| (hfair : ∀ a : IsingSpin, (ρ.map S) {a} = ENNReal.ofReal (1 / 2 : ℝ)) |
| (hpostX : ρ[fun ω => spinSign (S ω) | |
| (inferInstance : MeasurableSpace A).comap X] =ᵐ[ρ] fun ω => u (X ω)) |
| (hpostY : ρ[fun ω => spinSign (S ω) | |
| (inferInstance : MeasurableSpace B).comap Y] =ᵐ[ρ] fun ω => v (Y ω)) |
| (hXY : CondIndepFun ((inferInstance : MeasurableSpace IsingSpin).comap S) |
| hS.comap_le X Y ρ) |
| (hu1 : ∀ᵐ ω ∂ρ, |u (X ω)| < 1) (hv1 : ∀ᵐ ω ∂ρ, |v (Y ω)| < 1) : |
| ρ[fun ω => spinSign (S ω) | |
| (inferInstance : MeasurableSpace (A × B)).comap (fun ω => (X ω, Y ω))] =ᵐ[ρ] |
| fun ω => binaryFusion (u (X ω)) (v (Y ω)) := by |
| have hμ : ∀ᵐ x ∂ρ.map X, |u x| < 1 := |
| (ae_map_iff hX.aemeasurable (measurableSet_lt hu.abs measurable_const)).2 hu1 |
| have hν : ∀ᵐ y ∂ρ.map Y, |v y| < 1 := |
| (ae_map_iff hY.aemeasurable (measurableSet_lt hv.abs measurable_const)).2 hv1 |
| have hfst := (measurePreserving_fst (μ := ρ.map X) (ν := ρ.map Y)).quasiMeasurePreserving.ae hμ |
| have hsnd := (measurePreserving_snd (μ := ρ.map X) (ν := ρ.map Y)).quasiMeasurePreserving.ae hν |
| have hκ (a : IsingSpin) : |
| (condDistrib X S ρ ×ₖ condDistrib Y S ρ) a = |
| ((ρ.map X).prod (ρ.map Y)).withDensity |
| (fun z => ENNReal.ofReal ((1 + spinSign a * u z.1) * (1 + spinSign a * v z.2))) := by |
| rw [Kernel.prod_apply, |
| condDistrib_raw_signal_likelihood ρ S X u hS hX hu hfair hpostX a, |
| condDistrib_raw_signal_likelihood ρ S Y v hS hY hv hfair hpostY a, |
| prod_withDensity (by fun_prop) (by fun_prop)] |
| apply withDensity_congr_ae |
| filter_upwards [hfst] with z hz |
| have hp : 0 ≤ 1 + spinSign a * u z.1 := by |
| rcases abs_lt.mp hz with ⟨hl, hu⟩ |
| cases a <;> simp only [spinSign_neg, spinSign_pos] <;> linarith |
| exact (ENNReal.ofReal_mul hp).symm |
| have hlaw : ρ.map (fun ω => (S ω, (X ω, Y ω))) = |
| (ρ.map S) ⊗ₘ (condDistrib X S ρ ×ₖ condDistrib Y S ρ) := by |
| rw [Measure.compProd_eq_comp_prod] |
| exact (condIndepFun_iff_map_prod_eq_prod_condDistrib_prod_condDistrib hX hY hS).1 hXY |
| exact condExp_spin_eq_binaryFusion_of_likelihood ρ (ρ.map S) hfair |
| ((ρ.map X).prod (ρ.map Y)) (condDistrib X S ρ ×ₖ condDistrib Y S ρ) |
| S (fun ω => (X ω, Y ω)) hS (hX.prodMk hY) hlaw |
| (fun z => u z.1) (fun z => v z.2) (hu.comp measurable_fst) |
| (hv.comp measurable_snd) hfst hsnd hκ |
|
|
| theorem calibrated_condIndep_pair_law |
| {Ω : Type*} [MeasurableSpace Ω] [StandardBorelSpace Ω] |
| (ρ : Measure Ω) [IsProbabilityMeasure ρ] |
| (S : Ω → IsingSpin) (U V : Ω → ℝ) |
| (hS : Measurable S) (hU : Measurable U) (hV : Measurable V) |
| (hfair : ∀ a : IsingSpin, (ρ.map S) {a} = ENNReal.ofReal (1 / 2 : ℝ)) |
| (hcalU : ρ[fun ω => spinSign (S ω) | |
| (inferInstance : MeasurableSpace ℝ).comap U] =ᵐ[ρ] U) |
| (hcalV : ρ[fun ω => spinSign (S ω) | |
| (inferInstance : MeasurableSpace ℝ).comap V] =ᵐ[ρ] V) |
| (hUV : CondIndepFun ((inferInstance : MeasurableSpace IsingSpin).comap S) |
| hS.comap_le U V ρ) |
| (hU1 : ∀ᵐ ω ∂ρ, |U ω| < 1) (hV1 : ∀ᵐ ω ∂ρ, |V ω| < 1) : |
| ρ.map (fun ω => (U ω, V ω)) = binaryFusionLaw (ρ.map U) (ρ.map V) := by |
| have hμ : ∀ᵐ u ∂ρ.map U, |u| < 1 := |
| (ae_map_iff hU.aemeasurable (measurableSet_lt measurable_id.abs measurable_const)).2 hU1 |
| have hν : ∀ᵐ v ∂ρ.map V, |v| < 1 := |
| (ae_map_iff hV.aemeasurable (measurableSet_lt measurable_id.abs measurable_const)).2 hV1 |
| have hlaw : ρ.map (fun ω => (S ω, (U ω, V ω))) = |
| (ρ.map S) ⊗ₘ (condDistrib U S ρ ×ₖ condDistrib V S ρ) := by |
| rw [Measure.compProd_eq_comp_prod] |
| exact (condIndepFun_iff_map_prod_eq_prod_condDistrib_prod_condDistrib hU hV hS).1 hUV |
| rw [← Measure.snd_map_prodMk hS (hU.prodMk hV), hlaw, Measure.snd_compProd] |
| exact observation_law_eq_binaryFusionLaw (ρ.map S) hfair (ρ.map U) (ρ.map V) |
| hμ hν (condDistrib U S ρ ×ₖ condDistrib V S ρ) |
| (calibrated_signal_product_likelihood ρ S U V hS hU hV hfair hcalU hcalV) |
|
|
| theorem raw_condIndep_fusion_quadratic_loss |
| {Ω A B : Type*} [MeasurableSpace Ω] [StandardBorelSpace Ω] |
| [MeasurableSpace A] [StandardBorelSpace A] [Nonempty A] |
| [MeasurableSpace B] [StandardBorelSpace B] [Nonempty B] |
| (ρ : Measure Ω) [IsProbabilityMeasure ρ] |
| (S : Ω → IsingSpin) (X : Ω → A) (Y : Ω → B) (u : A → ℝ) (v : B → ℝ) |
| (hS : Measurable S) (hX : Measurable X) (hY : Measurable Y) |
| (hu : Measurable u) (hv : Measurable v) |
| (hfair : ∀ a : IsingSpin, (ρ.map S) {a} = ENNReal.ofReal (1 / 2 : ℝ)) |
| (hpostX : ρ[fun ω => spinSign (S ω) | |
| (inferInstance : MeasurableSpace A).comap X] =ᵐ[ρ] fun ω => u (X ω)) |
| (hpostY : ρ[fun ω => spinSign (S ω) | |
| (inferInstance : MeasurableSpace B).comap Y] =ᵐ[ρ] fun ω => v (Y ω)) |
| (hXY : CondIndepFun ((inferInstance : MeasurableSpace IsingSpin).comap S) |
| hS.comap_le X Y ρ) |
| {theta : ℝ} (hθ0 : 0 ≤ theta) (hθ1 : theta < 1) |
| (huθ : ∀ᵐ ω ∂ρ, |u (X ω)| ≤ theta) (hvθ : ∀ᵐ ω ∂ρ, |v (Y ω)| ≤ theta) |
| (hvsym : (ρ.map (fun ω => v (Y ω))).map Neg.neg = ρ.map (fun ω => v (Y ω))) : |
| (∫ ω, (ρ[fun ω => spinSign (S ω) | |
| (inferInstance : MeasurableSpace (A × B)).comap (fun ω => (X ω, Y ω))] ω)^2 ∂ρ) ≤ |
| (∫ ω, u (X ω) ^ 2 ∂ρ) + (∫ ω, v (Y ω) ^ 2 ∂ρ) - |
| 2 * (1 - theta ^ 2) * (∫ ω, u (X ω) ^ 2 ∂ρ) * (∫ ω, v (Y ω) ^ 2 ∂ρ) := by |
| let U := fun ω => u (X ω) |
| let V := fun ω => v (Y ω) |
| have hU : Measurable U := hu.comp hX |
| have hV : Measurable V := hv.comp hY |
| have hU1 : ∀ᵐ ω ∂ρ, |U ω| < 1 := huθ.mono (fun _ h => h.trans_lt hθ1) |
| have hV1 : ∀ᵐ ω ∂ρ, |V ω| < 1 := hvθ.mono (fun _ h => h.trans_lt hθ1) |
| have hμ : ∀ᵐ z ∂ρ.map U, |z| ≤ theta := |
| (ae_map_iff hU.aemeasurable (measurableSet_le measurable_id.abs measurable_const)).2 huθ |
| have hν : ∀ᵐ z ∂ρ.map V, |z| ≤ theta := |
| (ae_map_iff hV.aemeasurable (measurableSet_le measurable_id.abs measurable_const)).2 hvθ |
| have hpair := calibrated_condIndep_pair_law ρ S U V hS hU hV hfair |
| (posterior_version_calibrated ρ _ X u hX hu hpostX) |
| (posterior_version_calibrated ρ _ Y v hY hv hpostY) |
| (hXY.comp hu hv) hU1 hV1 |
| have hfull := condExp_raw_pair_eq_binaryFusion ρ S X Y u v hS hX hY hu hv |
| hfair hpostX hpostY hXY hU1 hV1 |
| have hm : Measurable (fun z : ℝ × ℝ => binaryFusion z.1 z.2 ^ 2) := by |
| unfold binaryFusion |
| fun_prop |
| have heq : (∫ ω, (ρ[fun ω => spinSign (S ω) | |
| (inferInstance : MeasurableSpace (A × B)).comap (fun ω => (X ω, Y ω))] ω)^2 ∂ρ) = |
| ∫ z, binaryFusion z.1 z.2 ^ 2 ∂binaryFusionLaw (ρ.map U) (ρ.map V) := by |
| calc |
| _ = ∫ ω, binaryFusion (U ω) (V ω) ^ 2 ∂ρ := by |
| apply integral_congr_ae |
| filter_upwards [hfull] with ω hω |
| rw [hω] |
| _ = _ := by |
| rw [← hpair] |
| exact (integral_map (hU.prodMk hV).aemeasurable hm.aestronglyMeasurable).symm |
| rw [heq] |
| have hb := integral_binaryFusion_sq_le_quadratic_loss (ρ.map U) (ρ.map V) |
| hθ0 hθ1 hμ hν hvsym |
| rw [integral_map hU.aemeasurable (Measurable.aestronglyMeasurable (by fun_prop)), |
| integral_map hV.aemeasurable (Measurable.aestronglyMeasurable (by fun_prop))] at hb |
| exact hb |
|
|
| end Problem367 |
|
|
| end |
|
|
| end OAI |
|
|
