| import OAI.Probability.FreeIsing.Conditioning.FiniteBayesCondExp |
| import OAI.Probability.FreeIsing.Gaussian.FiniteGaussianPosterior |
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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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| theorem map_planted_eq_compProd |
| {A B X : Type*} [MeasurableSpace A] [MeasurableSpace B] [MeasurableSpace X] |
| (μ : Measure A) [SFinite μ] (ν : Measure B) [SFinite ν] |
| (κ : Kernel A X) [IsSFiniteKernel κ] (F : A → B → X) |
| (hF : Measurable (fun z : A × B => F z.1 z.2)) |
| (hκ : ∀ a, ν.map (F a) = κ a) : |
| (μ.prod ν).map (fun z : A × B => (z.1, F z.1 z.2)) = μ ⊗ₘ κ := by |
| ext s hs |
| rw [Measure.map_apply (measurable_fst.prodMk hF) hs, |
| Measure.prod_apply (hs.preimage (measurable_fst.prodMk hF)), Measure.compProd_apply hs] |
| apply lintegral_congr |
| intro a |
| have hFa : Measurable (F a) := hF.comp measurable_prodMk_left |
| have hsa : MeasurableSet (Prod.mk a ⁻¹' s) := hs.preimage measurable_prodMk_left |
| rw [← hκ a, Measure.map_apply hFa hsa] |
| rfl |
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| theorem gaussianPi_map_signal {I : Type*} [Fintype I] |
| (t : ℝ≥0) (s : I → ℝ) : |
| (Measure.pi (fun _ : I => gaussianReal 0 t)).map |
| (fun x i => (t : ℝ) * s i + x i) = |
| Measure.pi (fun i => gaussianReal ((t : ℝ) * s i) t) := by |
| rw [Measure.pi_map_pi (μ := fun _ : I => gaussianReal 0 t) |
| (f := fun i (x : ℝ) => (t : ℝ) * s i + x) (fun i => by fun_prop)] |
| congr 1 |
| funext i |
| simpa using (gaussianReal_map_const_add (μ := 0) (v := t) ((t : ℝ) * s i)) |
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| theorem finiteGaussianPlanted_joint_law |
| {A I : Type*} [Fintype A] [MeasurableSpace A] [MeasurableSingletonClass A] |
| [Fintype I] (μ : Measure A) [IsProbabilityMeasure μ] |
| (t : ℝ≥0) (s : A → I → ℝ) : |
| (μ.prod (Measure.pi (fun _ : I => gaussianReal 0 t))).map |
| (fun z : A × (I → ℝ) => (z.1, fun i => (t : ℝ) * s z.1 i + z.2 i)) = |
| μ ⊗ₘ finiteGaussianObservationKernel t s := by |
| apply map_planted_eq_compProd μ (Measure.pi (fun _ : I => gaussianReal 0 t)) |
| (finiteGaussianObservationKernel t s) (fun a x i => (t : ℝ) * s a i + x i) |
| · exact Measurable.of_eval (fun index => |
| (measurable_const.mul |
| ((measurable_of_finite (fun state => s state index)).comp measurable_fst)).add |
| ((measurable_pi_apply index).comp measurable_snd)) |
| · intro a |
| exact gaussianPi_map_signal t (s a) |
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| theorem finiteGaussianPlanted_condExp |
| {A I : Type*} [Fintype A] [MeasurableSpace A] [MeasurableSingletonClass A] |
| [StandardBorelSpace A] [Nonempty A] [Fintype I] |
| (μ : Measure A) [IsProbabilityMeasure μ] |
| (t : ℝ≥0) (s : A → I → ℝ) (hs : ∀ a i, s a i ^ 2 = 1) (q : A → ℝ) : |
| let ρ := μ.prod (Measure.pi (fun _ : I => gaussianReal 0 t)) |
| let Y : A × (I → ℝ) → I → ℝ := fun z i => (t : ℝ) * s z.1 i + z.2 i |
| ρ[fun z => q z.1 | (inferInstance : MeasurableSpace (I → ℝ)).comap Y] =ᵐ[ρ] |
| fun z => (∑ a, μ.real {a} * Real.exp (∑ i, s a i * Y z i) * q a) / |
| (∑ a, μ.real {a} * Real.exp (∑ i, s a i * Y z i)) := by |
| dsimp only |
| refine FiniteBayes.condExp_state_eq_of_posterior_integral |
| (μ.prod (Measure.pi (fun _ : I => gaussianReal 0 t))) μ |
| (finiteGaussianObservationKernel t s) Prod.fst |
| (fun z i => (t : ℝ) * s z.1 i + z.2 i) measurable_fst ?_ |
| (finiteGaussianPlanted_joint_law μ t s) q |
| (fun x => (∑ a, μ.real {a} * Real.exp (∑ i, s a i * x i) * q a) / |
| (∑ a, μ.real {a} * Real.exp (∑ i, s a i * x i))) |
| (finiteGaussianObservation_posterior_mean μ t s hs q) |
| exact Measurable.of_eval (fun index => |
| (measurable_const.mul |
| ((measurable_of_finite (fun state => s state index)).comp measurable_fst)).add |
| ((measurable_pi_apply index).comp measurable_snd)) |
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| end Problem367 |
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| end |
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| end OAI |
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