| import OAI.Probability.FreeIsing.Conditioning.FiniteBayesCondExp |
| import OAI.Probability.FreeIsing.Gaussian.Likelihood |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory |
| open scoped NNReal ENNReal ProbabilityTheory |
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| namespace Problem367 |
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| def finiteGaussianObservationKernel |
| {A I : Type*} [Fintype A] [MeasurableSpace A] [MeasurableSingletonClass A] |
| [Fintype I] (t : ℝ≥0) (s : A → I → ℝ) : Kernel A (I → ℝ) where |
| toFun a := Measure.pi (fun i => gaussianReal ((t : ℝ) * s a i) t) |
| measurable' := measurable_of_finite _ |
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| instance finiteGaussianObservationKernel_isMarkov |
| {A I : Type*} [Fintype A] [MeasurableSpace A] [MeasurableSingletonClass A] |
| [Fintype I] (t : ℝ≥0) (s : A → I → ℝ) : |
| IsMarkovKernel (finiteGaussianObservationKernel t s) where |
| isProbabilityMeasure a := by |
| change IsProbabilityMeasure (Measure.pi (fun i => gaussianReal ((t : ℝ) * s a i) t)) |
| infer_instance |
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| theorem finiteGaussianObservation_posterior_mean |
| {A I : Type*} [Fintype A] [MeasurableSpace A] [MeasurableSingletonClass A] |
| [StandardBorelSpace A] [Nonempty A] [Fintype I] |
| (mu : Measure A) [IsProbabilityMeasure mu] |
| (t : ℝ≥0) (s : A → I → ℝ) (hs : ∀ a i, s a i ^ 2 = 1) |
| (q : A → ℝ) : |
| ∀ᵐ x ∂finiteGaussianObservationKernel t s ∘ₘ mu, |
| (∫ a, q a ∂(posterior (finiteGaussianObservationKernel t s) mu) x) = |
| (∑ a, mu.real {a} * Real.exp (∑ i, s a i * x i) * q a) / |
| (∑ a, mu.real {a} * Real.exp (∑ i, s a i * x i)) := by |
| have hlikelihood (a : A) : |
| finiteGaussianObservationKernel t s a = |
| (Measure.pi (fun _ : I => gaussianReal 0 t)).withDensity |
| (fun x => ENNReal.ofReal |
| (Real.exp (-((t : ℝ) * Fintype.card I / 2)) * |
| Real.exp (∑ i, s a i * x i))) := by |
| change Measure.pi (fun i => gaussianReal ((t : ℝ) * s a i) t) = _ |
| rw [gaussianPi_spin_withDensity t (s a) (hs a)] |
| congr 1 |
| funext x |
| rw [sub_eq_add_neg, Real.exp_add, mul_comm] |
| exact FiniteBayes.posterior_integral_exp mu (finiteGaussianObservationKernel t s) |
| (Measure.pi (fun _ : I => gaussianReal 0 t)) |
| (fun a x => ∑ i, s a i * x i) |
| (fun a => Finset.measurable_sum _ |
| (fun i _ => measurable_const.mul (measurable_pi_apply i))) |
| (fun _ => Real.exp (-((t : ℝ) * Fintype.card I / 2))) measurable_const |
| (fun _ => Real.exp_pos _) hlikelihood q |
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| theorem finiteGaussianObservation_condExp |
| {A I : Type*} [Fintype A] [MeasurableSpace A] [MeasurableSingletonClass A] |
| [StandardBorelSpace A] [Nonempty A] [Fintype I] |
| (mu : Measure A) [IsProbabilityMeasure mu] |
| (t : ℝ≥0) (s : A → I → ℝ) (hs : ∀ a i, s a i ^ 2 = 1) |
| (q : A → ℝ) : |
| (mu ⊗ₘ finiteGaussianObservationKernel t s)[fun z : A × (I → ℝ) => q z.1 | |
| (inferInstance : MeasurableSpace (I → ℝ)).comap Prod.snd] |
| =ᵐ[mu ⊗ₘ finiteGaussianObservationKernel t s] fun z => |
| (∑ a, mu.real {a} * Real.exp (∑ i, s a i * z.2 i) * q a) / |
| (∑ a, mu.real {a} * Real.exp (∑ i, s a i * z.2 i)) := by |
| apply (FiniteBayes.condExp_state_eq_posterior_integral mu |
| (finiteGaussianObservationKernel t s) q).trans |
| have hp := finiteGaussianObservation_posterior_mean mu t s hs q |
| rw [← Measure.snd_compProd mu (finiteGaussianObservationKernel t s)] at hp |
| exact ae_of_ae_map measurable_snd.aemeasurable hp |
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| end Problem367 |
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| end |
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| end OAI |
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