File size: 5,424 Bytes
f61e547 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 | import OAI.Probability.FreeIsing.Conditioning.FiniteBayesExp
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped ENNReal ProbabilityTheory
namespace Problem367.FiniteBayes
variable {S X : Type*} [Fintype S] [MeasurableSpace S] [MeasurableSingletonClass S]
[StandardBorelSpace S] [Nonempty S] [MeasurableSpace X]
theorem condExp_state_eq_posterior_integral (π : Measure S) [IsFiniteMeasure π]
(κ : Kernel S X) [IsFiniteKernel κ] (A : S → ℝ) :
(π ⊗ₘ κ)[fun z : S × X => A z.1 | (inferInstance : MeasurableSpace X).comap Prod.snd]
=ᵐ[π ⊗ₘ κ] fun z => ∫ s, A s ∂(κ†π) z.2 := by
have hint : Integrable (fun z : S × X => A z.1) (π ⊗ₘ κ) := by
apply (integrable_const (∑ s, |A s|)).mono'
· exact ((measurable_of_finite A).comp measurable_fst).aestronglyMeasurable
· filter_upwards [] with z
rw [Real.norm_eq_abs]
exact Finset.single_le_sum (fun s _ => abs_nonneg (A s)) (Finset.mem_univ z.1)
have hcd : condDistrib (Prod.fst : S × X → S) Prod.snd (π ⊗ₘ κ) = κ†π := by
rw [condDistrib, posterior]
rfl
have hc := condExp_ae_eq_integral_condDistrib
(μ := π ⊗ₘ κ) (X := Prod.snd) (Y := Prod.fst)
measurable_snd measurable_fst.aemeasurable (measurable_of_finite A).stronglyMeasurable hint
rwa [hcd] at hc
theorem condExp_state_eq_real_likelihood (π : Measure S) [IsProbabilityMeasure π]
(ν : Measure X) [SigmaFinite ν] (κ : Kernel S X) [IsFiniteKernel κ]
{L : S → X → ℝ} (hL : ∀ s, Measurable (L s)) (hLpos : ∀ s x, 0 < L s x)
(hκ : ∀ s, κ s = ν.withDensity (fun x => ENNReal.ofReal (L s x))) (A : S → ℝ) :
(π ⊗ₘ κ)[fun z : S × X => A z.1 | (inferInstance : MeasurableSpace X).comap Prod.snd]
=ᵐ[π ⊗ₘ κ] fun z =>
(∑ s, π.real {s} * L s z.2 * A s) / (∑ s, π.real {s} * L s z.2) := by
apply (condExp_state_eq_posterior_integral π κ A).trans
have hp := posterior_integral π κ ν L hL hLpos hκ A
rw [← Measure.snd_compProd π κ] at hp
exact ae_of_ae_map measurable_snd.aemeasurable hp
theorem condExp_state_eq_exponential_likelihood (π : Measure S) [IsProbabilityMeasure π]
(ν : Measure X) [SigmaFinite ν] (κ : Kernel S X) [IsFiniteKernel κ]
(E : S → X → ℝ) (hE : ∀ s, Measurable (E s))
(c : X → ℝ) (hc : Measurable c) (hcpos : ∀ x, 0 < c x)
(hκ : ∀ s, κ s = ν.withDensity (fun x => ENNReal.ofReal (c x * Real.exp (E s x))))
(A : S → ℝ) :
(π ⊗ₘ κ)[fun z : S × X => A z.1 | (inferInstance : MeasurableSpace X).comap Prod.snd]
=ᵐ[π ⊗ₘ κ] fun z =>
(∑ s, π.real {s} * Real.exp (E s z.2) * A s) /
(∑ s, π.real {s} * Real.exp (E s z.2)) := by
apply (condExp_state_eq_posterior_integral π κ A).trans
have hp := posterior_integral_exp π κ ν E hE c hc hcpos hκ A
rw [← Measure.snd_compProd π κ] at hp
exact ae_of_ae_map measurable_snd.aemeasurable hp
section OriginalSpace
variable {Ω : Type*} [MeasurableSpace Ω]
omit [Fintype S] [MeasurableSingletonClass S] in
theorem condDistrib_eq_posterior_of_joint_law
(μ : Measure Ω) [IsFiniteMeasure μ] (π : Measure S) [IsFiniteMeasure π]
(κ : Kernel S X) [IsFiniteKernel κ] (Z : Ω → S) (Y : Ω → X)
(hZ : Measurable Z) (hY : Measurable Y)
(hlaw : μ.map (fun ω => (Z ω, Y ω)) = π ⊗ₘ κ) :
condDistrib Z Y μ = κ†π := by
rw [condDistrib, posterior]
congr 1
rw [← hlaw, Measure.map_map measurable_swap (hZ.prodMk hY)]
rfl
theorem condExp_state_eq_posterior_integral_of_joint_law
(μ : Measure Ω) [IsFiniteMeasure μ] (π : Measure S) [IsFiniteMeasure π]
(κ : Kernel S X) [IsFiniteKernel κ] (Z : Ω → S) (Y : Ω → X)
(hZ : Measurable Z) (hY : Measurable Y)
(hlaw : μ.map (fun ω => (Z ω, Y ω)) = π ⊗ₘ κ) (A : S → ℝ) :
μ[fun ω => A (Z ω) | (inferInstance : MeasurableSpace X).comap Y]
=ᵐ[μ] fun ω => ∫ s, A s ∂(κ†π) (Y ω) := by
have hint : Integrable (fun ω => A (Z ω)) μ := by
apply (integrable_const (∑ s, |A s|)).mono'
· exact ((measurable_of_finite A).comp hZ).aestronglyMeasurable
· filter_upwards [] with ω
rw [Real.norm_eq_abs]
exact Finset.single_le_sum (fun s _ => abs_nonneg (A s)) (Finset.mem_univ (Z ω))
have hc := condExp_ae_eq_integral_condDistrib hY hZ.aemeasurable
(measurable_of_finite A).stronglyMeasurable hint
rwa [condDistrib_eq_posterior_of_joint_law μ π κ Z Y hZ hY hlaw] at hc
theorem condExp_state_eq_of_posterior_integral
(μ : Measure Ω) [IsFiniteMeasure μ] (π : Measure S) [IsFiniteMeasure π]
(κ : Kernel S X) [IsFiniteKernel κ] (Z : Ω → S) (Y : Ω → X)
(hZ : Measurable Z) (hY : Measurable Y)
(hlaw : μ.map (fun ω => (Z ω, Y ω)) = π ⊗ₘ κ)
(A : S → ℝ) (g : X → ℝ)
(hpost : (fun x => ∫ s, A s ∂(κ†π) x) =ᵐ[κ ∘ₘ π] g) :
μ[fun ω => A (Z ω) | (inferInstance : MeasurableSpace X).comap Y]
=ᵐ[μ] fun ω => g (Y ω) := by
apply (condExp_state_eq_posterior_integral_of_joint_law μ π κ Z Y hZ hY hlaw A).trans
have hYlaw : μ.map Y = κ ∘ₘ π := by
rw [← Measure.snd_map_prodMk hZ hY, hlaw, Measure.snd_compProd]
rw [← hYlaw] at hpost
exact ae_of_ae_map hY.aemeasurable hpost
end OriginalSpace
end Problem367.FiniteBayes
end
end OAI
|