| import OAI.Probability.FreeIsing.Gaussian.BinaryFusionPosterior |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory ProbabilityTheory |
|
|
| namespace Problem367 |
|
|
| def binaryFusionLaw (μ ν : Measure ℝ) : Measure (ℝ × ℝ) := |
| (μ.prod ν).withDensity (fun z => ENNReal.ofReal (1 + z.1 * z.2)) |
|
|
| private theorem ae_binaryFusion_support (μ ν : Measure ℝ) |
| [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] |
| (hμ : ∀ᵐ u ∂μ, |u| < 1) (hν : ∀ᵐ v ∂ν, |v| < 1) : |
| ∀ᵐ z ∂μ.prod ν, |z.1| < 1 ∧ |z.2| < 1 := by |
| have hfst := (measurePreserving_fst (μ := μ) (ν := ν)).quasiMeasurePreserving.ae hμ |
| have hsnd := (measurePreserving_snd (μ := μ) (ν := ν)).quasiMeasurePreserving.ae hν |
| filter_upwards [hfst, hsnd] with z hu hv |
| exact ⟨hu, hv⟩ |
|
|
| theorem binaryFusionLoss_le_sum_squares {u v : ℝ} |
| (hu : |u| < 1) (hv : |v| < 1) : |
| binaryFusionLoss u v ≤ u ^ 2 + v ^ 2 := by |
| have hp := binaryFusion_denominator_pos hu hv |
| have hm := binaryFusion_denominator_pos hu (by simpa using hv : |-v| < 1) |
| have heq := binaryFusion_symmetric_square hu hv |
| have hnonneg : 0 ≤ ((u + v) ^ 2 / (1 + u * v) + |
| (u - v) ^ 2 / (1 - u * v)) / 2 := by |
| have hm' : 0 < 1 - u * v := by simpa using hm |
| positivity |
| linarith |
|
|
| private theorem integrable_binaryFusion_integrand (μ ν : Measure ℝ) |
| [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] |
| (hμ : ∀ᵐ u ∂μ, |u| < 1) (hν : ∀ᵐ v ∂ν, |v| < 1) : |
| Integrable (fun z : ℝ × ℝ => (z.1 + z.2)^2 / (1 + z.1 * z.2)) (μ.prod ν) := by |
| apply Integrable.of_bound (Measurable.aestronglyMeasurable (by fun_prop)) 2 |
| filter_upwards [ae_binaryFusion_support μ ν hμ hν] with z hz |
| have hd := binaryFusion_denominator_pos hz.1 hz.2 |
| rw [Real.norm_eq_abs, abs_of_nonneg (by positivity : |
| 0 ≤ (z.1 + z.2)^2 / (1 + z.1 * z.2))] |
| exact binaryFusion_integrand_le_two hz.1 hz.2 |
|
|
| theorem integrable_binaryFusionLoss_product (μ ν : Measure ℝ) |
| [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] |
| (hμ : ∀ᵐ u ∂μ, |u| < 1) (hν : ∀ᵐ v ∂ν, |v| < 1) : |
| Integrable (fun z : ℝ × ℝ => binaryFusionLoss z.1 z.2) (μ.prod ν) := by |
| apply Integrable.of_bound (Measurable.aestronglyMeasurable (by unfold binaryFusionLoss; fun_prop)) 2 |
| filter_upwards [ae_binaryFusion_support μ ν hμ hν] with z hz |
| rw [Real.norm_eq_abs, abs_of_nonneg (binaryFusionLoss_nonneg hz.1 hz.2)] |
| apply (binaryFusionLoss_le_sum_squares hz.1 hz.2).trans |
| have hu := (sq_lt_one_iff_abs_lt_one z.1).2 hz.1 |
| have hv := (sq_lt_one_iff_abs_lt_one z.2).2 hz.2 |
| linarith |
|
|
| theorem integral_binaryFusion_sq_eq_raw (μ ν : Measure ℝ) |
| [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] |
| (hμ : ∀ᵐ u ∂μ, |u| < 1) (hν : ∀ᵐ v ∂ν, |v| < 1) : |
| (∫ z, binaryFusion z.1 z.2 ^ 2 ∂binaryFusionLaw μ ν) = |
| ∫ z : ℝ × ℝ, (z.1 + z.2)^2 / (1 + z.1 * z.2) ∂μ.prod ν := by |
| unfold binaryFusionLaw |
| rw [integral_withDensity_eq_integral_toReal_smul |
| (by fun_prop) (Filter.Eventually.of_forall (fun _ => ENNReal.ofReal_lt_top))] |
| apply integral_congr_ae |
| filter_upwards [ae_binaryFusion_support μ ν hμ hν] with z hz |
| have hd := binaryFusion_denominator_pos hz.1 hz.2 |
| rw [ENNReal.toReal_ofReal hd.le, smul_eq_mul] |
| unfold binaryFusion |
| field_simp |
|
|
| theorem integral_binaryFusion_sq (μ ν : Measure ℝ) |
| [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] |
| (hμ : ∀ᵐ u ∂μ, |u| < 1) (hν : ∀ᵐ v ∂ν, |v| < 1) |
| (hνsym : ν.map Neg.neg = ν) : |
| (∫ z, binaryFusion z.1 z.2 ^ 2 ∂binaryFusionLaw μ ν) = |
| (∫ u, u ^ 2 ∂μ) + (∫ v, v ^ 2 ∂ν) - |
| ∫ z : ℝ × ℝ, binaryFusionLoss z.1 z.2 ∂μ.prod ν := by |
| let F : ℝ × ℝ → ℝ := fun z => (z.1 + z.2)^2 / (1 + z.1 * z.2) |
| let e : (ℝ × ℝ) ≃ᵐ (ℝ × ℝ) := |
| (MeasurableEquiv.refl ℝ).prodCongr (MeasurableEquiv.neg ℝ) |
| have hp : MeasurePreserving e (μ.prod ν) (μ.prod ν) := |
| (MeasurePreserving.id μ).prod ⟨measurable_neg, hνsym⟩ |
| have hsym : (∫ z : ℝ × ℝ, (z.1 - z.2)^2 / (1 - z.1 * z.2) ∂μ.prod ν) = |
| ∫ z, F z ∂μ.prod ν := by |
| have hh := hp.integral_comp' F |
| change (∫ z : ℝ × ℝ, (z.1 + -z.2)^2 / (1 + z.1 * -z.2) ∂μ.prod ν) = _ at hh |
| simpa only [mul_neg, sub_eq_add_neg] using hh |
| have hF : Integrable F (μ.prod ν) := integrable_binaryFusion_integrand μ ν hμ hν |
| have hFm : Integrable (fun z : ℝ × ℝ => |
| (z.1 - z.2)^2 / (1 - z.1 * z.2)) (μ.prod ν) := by |
| have hh := (hp.integrable_comp hF.aestronglyMeasurable).2 hF |
| change Integrable (fun z : ℝ × ℝ => (z.1 + -z.2)^2 / |
| (1 + z.1 * -z.2)) (μ.prod ν) at hh |
| simpa only [mul_neg, sub_eq_add_neg] using hh |
| have hU : Integrable (fun z : ℝ × ℝ => z.1^2) (μ.prod ν) := by |
| apply Integrable.of_bound (Measurable.aestronglyMeasurable (by fun_prop)) 1 |
| filter_upwards [ae_binaryFusion_support μ ν hμ hν] with z hz |
| rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)] |
| exact ((sq_lt_one_iff_abs_lt_one z.1).2 hz.1).le |
| have hV : Integrable (fun z : ℝ × ℝ => z.2^2) (μ.prod ν) := by |
| apply Integrable.of_bound (Measurable.aestronglyMeasurable (by fun_prop)) 1 |
| filter_upwards [ae_binaryFusion_support μ ν hμ hν] with z hz |
| rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)] |
| exact ((sq_lt_one_iff_abs_lt_one z.2).2 hz.2).le |
| rw [integral_binaryFusion_sq_eq_raw μ ν hμ hν] |
| change (∫ z, F z ∂μ.prod ν) = _ |
| calc |
| (∫ z, F z ∂μ.prod ν) = |
| ((∫ z, F z ∂μ.prod ν) + |
| ∫ z : ℝ × ℝ, (z.1 - z.2)^2 / (1 - z.1 * z.2) ∂μ.prod ν) / 2 := by |
| rw [hsym] |
| ring |
| _ = ∫ z : ℝ × ℝ, |
| (F z + (z.1 - z.2)^2 / (1 - z.1 * z.2)) / 2 ∂μ.prod ν := by |
| rw [integral_div, integral_add hF hFm] |
| _ = ∫ z : ℝ × ℝ, z.1^2 + z.2^2 - binaryFusionLoss z.1 z.2 ∂μ.prod ν := by |
| apply integral_congr_ae |
| filter_upwards [ae_binaryFusion_support μ ν hμ hν] with z hz |
| exact binaryFusion_symmetric_square hz.1 hz.2 |
| _ = _ := by |
| rw [integral_sub (f := fun z : ℝ × ℝ => z.1^2 + z.2^2) |
| (hU.add hV) (integrable_binaryFusionLoss_product μ ν hμ hν), |
| integral_add hU hV, integral_fun_fst (fun u : ℝ => u^2), |
| integral_fun_snd (fun v : ℝ => v^2)] |
| simp |
|
|
| theorem integral_binaryFusion_sq_le_add (μ ν : Measure ℝ) |
| [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] |
| (hμ : ∀ᵐ u ∂μ, |u| < 1) (hν : ∀ᵐ v ∂ν, |v| < 1) |
| (hνsym : ν.map Neg.neg = ν) : |
| (∫ z, binaryFusion z.1 z.2 ^ 2 ∂binaryFusionLaw μ ν) ≤ |
| (∫ u, u ^ 2 ∂μ) + (∫ v, v ^ 2 ∂ν) := by |
| rw [integral_binaryFusion_sq μ ν hμ hν hνsym] |
| apply sub_le_self |
| apply integral_nonneg_of_ae |
| filter_upwards [ae_binaryFusion_support μ ν hμ hν] with z hz |
| exact binaryFusionLoss_nonneg hz.1 hz.2 |
|
|
| theorem integral_binaryFusion_sq_le_quadratic_loss (μ ν : Measure ℝ) |
| [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] {theta : ℝ} |
| (hθ0 : 0 ≤ theta) (hθ1 : theta < 1) |
| (hμ : ∀ᵐ u ∂μ, |u| ≤ theta) (hν : ∀ᵐ v ∂ν, |v| ≤ theta) |
| (hνsym : ν.map Neg.neg = ν) : |
| (∫ z, binaryFusion z.1 z.2 ^ 2 ∂binaryFusionLaw μ ν) ≤ |
| (∫ u, u ^ 2 ∂μ) + (∫ v, v ^ 2 ∂ν) - |
| 2 * (1 - theta ^ 2) * (∫ u, u ^ 2 ∂μ) * (∫ v, v ^ 2 ∂ν) := by |
| have hμ1 : ∀ᵐ u ∂μ, |u| < 1 := hμ.mono (fun _ h => h.trans_lt hθ1) |
| have hν1 : ∀ᵐ v ∂ν, |v| < 1 := hν.mono (fun _ h => h.trans_lt hθ1) |
| have hθsq : 0 ≤ 1 - theta ^ 2 := by |
| have := (sq_lt_one_iff₀ hθ0).2 hθ1 |
| linarith |
| have hfst := (measurePreserving_fst (μ := μ) (ν := ν)).quasiMeasurePreserving.ae hμ |
| have hsnd := (measurePreserving_snd (μ := μ) (ν := ν)).quasiMeasurePreserving.ae hν |
| have hloss : 2 * (1 - theta ^ 2) * (∫ u, u ^ 2 ∂μ) * (∫ v, v ^ 2 ∂ν) ≤ |
| ∫ z : ℝ × ℝ, binaryFusionLoss z.1 z.2 ∂μ.prod ν := by |
| calc |
| _ = ∫ z : ℝ × ℝ, 2 * (1 - theta ^ 2) * (z.1 ^ 2 * z.2 ^ 2) ∂μ.prod ν := by |
| rw [integral_const_mul, integral_prod_mul (fun u : ℝ => u^2) (fun v : ℝ => v^2)] |
| ring |
| _ ≤ _ := by |
| refine integral_mono_of_nonneg ?_ (integrable_binaryFusionLoss_product μ ν hμ1 hν1) ?_ |
| · exact Filter.Eventually.of_forall (fun z => by positivity) |
| · filter_upwards [hfst, hsnd] with z hu hv |
| simpa only [mul_assoc] using binaryFusionLoss_lower_bound hu hv hθ1 |
| rw [integral_binaryFusion_sq μ ν hμ1 hν1 hνsym] |
| linarith |
|
|
| local instance : MeasurableSingletonClass IsingSpin := ⟨fun _ => trivial⟩ |
| local instance : Nonempty IsingSpin := ⟨.neg⟩ |
|
|
| theorem observation_law_eq_binaryFusionLaw |
| (π : Measure IsingSpin) (hπ : ∀ a, π {a} = ENNReal.ofReal (1 / 2 : ℝ)) |
| (μ ν : Measure ℝ) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] |
| (hμ : ∀ᵐ u ∂μ, |u| < 1) (hν : ∀ᵐ v ∂ν, |v| < 1) |
| (κ : Kernel IsingSpin (ℝ × ℝ)) |
| (hκ : ∀ a, κ a = (μ.prod ν).withDensity |
| (fun z => ENNReal.ofReal ((1 + spinSign a * z.1) * (1 + spinSign a * z.2)))) : |
| κ ∘ₘ π = binaryFusionLaw μ ν := by |
| rw [finite_observation_law_density π (μ.prod ν) κ _ (fun _ => by fun_prop) hκ] |
| unfold binaryFusionLaw |
| apply withDensity_congr_ae |
| filter_upwards [ae_binaryFusion_support μ ν hμ hν] with z hz |
| simp only [hπ] |
| exact binaryFusion_likelihood_sum hz.1.le hz.2.le |
|
|
| theorem integral_posterior_spinMean_sq_le_quadratic_loss |
| (π : Measure IsingSpin) [IsProbabilityMeasure π] |
| (hπ : ∀ a, π {a} = ENNReal.ofReal (1 / 2 : ℝ)) |
| (μ ν : Measure ℝ) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] |
| {theta : ℝ} (hθ0 : 0 ≤ theta) (hθ1 : theta < 1) |
| (hμ : ∀ᵐ u ∂μ, |u| ≤ theta) (hν : ∀ᵐ v ∂ν, |v| ≤ theta) |
| (hνsym : ν.map Neg.neg = ν) |
| (κ : Kernel IsingSpin (ℝ × ℝ)) [IsFiniteKernel κ] |
| (hκ : ∀ a, κ a = (μ.prod ν).withDensity |
| (fun z => ENNReal.ofReal ((1 + spinSign a * z.1) * (1 + spinSign a * z.2)))) : |
| (∫ z, (∫ a, spinSign a ∂(posterior κ π) z) ^ 2 ∂κ ∘ₘ π) ≤ |
| (∫ u, u ^ 2 ∂μ) + (∫ v, v ^ 2 ∂ν) - |
| 2 * (1 - theta ^ 2) * (∫ u, u ^ 2 ∂μ) * (∫ v, v ^ 2 ∂ν) := by |
| have hμ1 : ∀ᵐ u ∂μ, |u| < 1 := hμ.mono (fun _ h => h.trans_lt hθ1) |
| have hν1 : ∀ᵐ v ∂ν, |v| < 1 := hν.mono (fun _ h => h.trans_lt hθ1) |
| have hsup := ae_binaryFusion_support μ ν hμ1 hν1 |
| have hpost := posterior_spinMean_eq_binaryFusion π hπ (μ.prod ν) κ |
| Prod.fst Prod.snd measurable_fst measurable_snd |
| (hsup.mono (fun _ h => h.1)) (hsup.mono (fun _ h => h.2)) hκ |
| calc |
| _ = ∫ z, binaryFusion z.1 z.2 ^ 2 ∂κ ∘ₘ π := by |
| apply integral_congr_ae |
| filter_upwards [hpost] with z hz |
| rw [hz] |
| _ = ∫ z, binaryFusion z.1 z.2 ^ 2 ∂binaryFusionLaw μ ν := by |
| rw [observation_law_eq_binaryFusionLaw π hπ μ ν hμ1 hν1 κ hκ] |
| _ ≤ _ := integral_binaryFusion_sq_le_quadratic_loss μ ν hθ0 hθ1 hμ hν hνsym |
|
|
| theorem integral_condExp_spin_sq_eq_binaryFusion |
| {Ω : Type*} [MeasurableSpace Ω] (ρ : Measure Ω) [IsFiniteMeasure ρ] |
| (π : Measure IsingSpin) [IsProbabilityMeasure π] |
| (hπ : ∀ a, π {a} = ENNReal.ofReal (1 / 2 : ℝ)) |
| (μ ν : Measure ℝ) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] |
| (hμ : ∀ᵐ u ∂μ, |u| < 1) (hν : ∀ᵐ v ∂ν, |v| < 1) |
| (hνsym : ν.map Neg.neg = ν) |
| (κ : Kernel IsingSpin (ℝ × ℝ)) [IsFiniteKernel κ] |
| (hκ : ∀ a, κ a = (μ.prod ν).withDensity |
| (fun z => ENNReal.ofReal ((1 + spinSign a * z.1) * (1 + spinSign a * z.2)))) |
| (S : Ω → IsingSpin) (Y : Ω → ℝ × ℝ) (hS : Measurable S) (hY : Measurable Y) |
| (hlaw : ρ.map (fun ω => (S ω, Y ω)) = π ⊗ₘ κ) : |
| (∫ ω, (ρ[fun ω => spinSign (S ω) | |
| (inferInstance : MeasurableSpace (ℝ × ℝ)).comap Y] ω) ^ 2 ∂ρ) = |
| (∫ u, u ^ 2 ∂μ) + (∫ v, v ^ 2 ∂ν) - |
| ∫ z : ℝ × ℝ, binaryFusionLoss z.1 z.2 ∂μ.prod ν := by |
| have hsup := ae_binaryFusion_support μ ν hμ hν |
| have hpost := condExp_spin_eq_binaryFusion_of_likelihood ρ π hπ (μ.prod ν) κ |
| S Y hS hY hlaw Prod.fst Prod.snd measurable_fst measurable_snd |
| (hsup.mono (fun _ h => h.1)) (hsup.mono (fun _ h => h.2)) hκ |
| have hYlaw : ρ.map Y = binaryFusionLaw μ ν := by |
| rw [← Measure.snd_map_prodMk hS hY, hlaw, Measure.snd_compProd] |
| exact observation_law_eq_binaryFusionLaw π hπ μ ν hμ hν κ hκ |
| have hm : Measurable (fun z : ℝ × ℝ => binaryFusion z.1 z.2 ^ 2) := by |
| unfold binaryFusion |
| fun_prop |
| calc |
| _ = ∫ ω, binaryFusion (Y ω).1 (Y ω).2 ^ 2 ∂ρ := by |
| apply integral_congr_ae |
| filter_upwards [hpost] with ω hω |
| rw [hω] |
| _ = ∫ z, binaryFusion z.1 z.2 ^ 2 ∂binaryFusionLaw μ ν := by |
| rw [← hYlaw] |
| exact (integral_map hY.aemeasurable hm.aestronglyMeasurable).symm |
| _ = _ := integral_binaryFusion_sq μ ν hμ hν hνsym |
|
|
| theorem integral_condExp_spin_sq_le_quadratic_loss |
| {Ω : Type*} [MeasurableSpace Ω] (ρ : Measure Ω) [IsFiniteMeasure ρ] |
| (π : Measure IsingSpin) [IsProbabilityMeasure π] |
| (hπ : ∀ a, π {a} = ENNReal.ofReal (1 / 2 : ℝ)) |
| (μ ν : Measure ℝ) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] |
| {theta : ℝ} (hθ0 : 0 ≤ theta) (hθ1 : theta < 1) |
| (hμ : ∀ᵐ u ∂μ, |u| ≤ theta) (hν : ∀ᵐ v ∂ν, |v| ≤ theta) |
| (hνsym : ν.map Neg.neg = ν) |
| (κ : Kernel IsingSpin (ℝ × ℝ)) [IsFiniteKernel κ] |
| (hκ : ∀ a, κ a = (μ.prod ν).withDensity |
| (fun z => ENNReal.ofReal ((1 + spinSign a * z.1) * (1 + spinSign a * z.2)))) |
| (S : Ω → IsingSpin) (Y : Ω → ℝ × ℝ) (hS : Measurable S) (hY : Measurable Y) |
| (hlaw : ρ.map (fun ω => (S ω, Y ω)) = π ⊗ₘ κ) : |
| (∫ ω, (ρ[fun ω => spinSign (S ω) | |
| (inferInstance : MeasurableSpace (ℝ × ℝ)).comap Y] ω) ^ 2 ∂ρ) ≤ |
| (∫ u, u ^ 2 ∂μ) + (∫ v, v ^ 2 ∂ν) - |
| 2 * (1 - theta ^ 2) * (∫ u, u ^ 2 ∂μ) * (∫ v, v ^ 2 ∂ν) := by |
| have hμ1 : ∀ᵐ u ∂μ, |u| < 1 := hμ.mono (fun _ h => h.trans_lt hθ1) |
| have hν1 : ∀ᵐ v ∂ν, |v| < 1 := hν.mono (fun _ h => h.trans_lt hθ1) |
| rw [integral_condExp_spin_sq_eq_binaryFusion ρ π hπ μ ν hμ1 hν1 hνsym |
| κ hκ S Y hS hY hlaw, |
| ← integral_binaryFusion_sq μ ν hμ1 hν1 hνsym] |
| exact integral_binaryFusion_sq_le_quadratic_loss μ ν hθ0 hθ1 hμ hν hνsym |
|
|
| end Problem367 |
|
|
| end |
|
|
| end OAI |
|
|
