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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
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