| import OAI.Probability.FreeIsing.Conditioning.FiniteBayesExp |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal ProbabilityTheory |
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| namespace Problem367.FiniteBayes |
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| variable {S X : Type*} [Fintype S] [MeasurableSpace S] [MeasurableSingletonClass S] |
| [StandardBorelSpace S] [Nonempty S] [MeasurableSpace X] |
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| 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 |
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| 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 |
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| 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 |
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| section OriginalSpace |
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| variable {Ω : Type*} [MeasurableSpace Ω] |
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| omit [Fintype S] [MeasurableSingletonClass S] in |
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| 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 |
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| 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 |
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| 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 |
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| end OriginalSpace |
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| end Problem367.FiniteBayes |
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| end |
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| end OAI |
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