| import OAI.Probability.GaussianPropeller.CellBounds |
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| namespace OAI |
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| universe uE uι |
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| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal |
| open scoped RealInnerProductSpace |
| open scoped RealInnerProductSpace |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal RealInnerProductSpace |
| open Filter |
| open scoped Topology |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped Topology |
| open scoped RealInnerProductSpace |
| open Set Filter |
| open scoped Topology RealInnerProductSpace |
| open scoped NNReal |
| open Set Filter |
| open scoped Topology RealInnerProductSpace NNReal |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped Topology RealInnerProductSpace |
| open MeasureTheory Set Filter |
| open scoped Topology BigOperators |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped RealInnerProductSpace Topology |
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| namespace GaussianPropeller.Analytic |
| section FamilyLaw |
| variable {E : Type uE} {ι : Type uι} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] [Fintype ι] |
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| lemma covariance_inner_stdGaussian (u v : E) : |
| cov[fun x => ⟪u,x⟫, fun x => ⟪v,x⟫; stdGaussian E] = ⟪u,v⟫ := by |
| rw [← covarianceBilin_apply_eq_cov IsGaussian.memLp_two_id, |
| covarianceBilin_stdGaussian, innerSL_apply_apply] |
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| lemma hasGaussianLaw_innerFamily (v : ι → E) : |
| HasGaussianLaw (fun x : E => fun i => ⟪v i,x⟫) (stdGaussian E) := by |
| exact (IsGaussian.hasGaussianLaw_id (μ := stdGaussian E)).map_fun |
| (ContinuousLinearMap.pi (fun i => InnerProductSpace.toDual ℝ E (v i))) |
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| lemma indep_innerFamily (v : ι → E) (hv : Pairwise (fun i j => ⟪v i,v j⟫ = 0)) : |
| iIndepFun (fun i x => ⟪v i,x⟫) (stdGaussian E) := by |
| exact (hasGaussianLaw_innerFamily v).iIndepFun_of_covariance_eq_zero |
| (fun i j hij => (covariance_inner_stdGaussian _ _).trans (hv hij)) |
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| lemma map_inner_stdGaussian (v : E) : |
| (stdGaussian E).map (fun x => ⟪v,x⟫) = gaussianReal 0 (‖v‖^2).toNNReal := by |
| change (stdGaussian E).map (InnerProductSpace.toDual ℝ E v) = _ |
| rw [IsGaussian.map_eq_gaussianReal, integral_strongDual_stdGaussian, |
| variance_dual_stdGaussian, LinearIsometryEquiv.norm_map] |
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| lemma map_orthonormal_stdGaussian (v : ι → E) (hv : Orthonormal ℝ v) : |
| (stdGaussian E).map (fun x => fun i => ⟪v i,x⟫) = |
| Measure.pi (fun _ : ι => gaussianReal 0 1) := by |
| have hi := indep_innerFamily v (fun _ _ hij => hv.2 hij) |
| rw [hi.map_fun_eq_pi_map (fun _ => by fun_prop)] |
| congr 1 |
| funext i |
| rw [map_inner_stdGaussian, hv.1 i] |
| norm_num |
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| end FamilyLaw |
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| end GaussianPropeller.Analytic |
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| namespace GaussianPropeller.Analytic |
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| open scoped ENNReal |
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| section TripleLaw |
| variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
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| lemma map_orthonormal_triple_stdGaussian (v : Fin 3 → E) (hv : Orthonormal ℝ v) : |
| (stdGaussian E).map (fun x => (⟪v 0,x⟫, ⟪v 1,x⟫, ⟪v 2,x⟫)) = |
| (gaussianReal 0 1).prod ((gaussianReal 0 1).prod (gaussianReal 0 1)) := by |
| have hi := indep_innerFamily v (fun _ _ hij => hv.2 hij) |
| have hm (i : Fin 3) : (stdGaussian E).map (fun x => ⟪v i,x⟫) = gaussianReal 0 1 := by |
| rw [map_inner_stdGaussian, hv.1 i] |
| norm_num |
| have hp := hi.indepFun_prodMk (fun _ => by fun_prop) 1 2 0 (by decide) (by decide) |
| rw [hp.symm.map_prod_eq_prod_map_map (by fun_prop) (by fun_prop), hm, |
| (hi.indepFun (by decide : (1:Fin 3) ≠ 2)).map_prod_eq_prod_map_map |
| (by fun_prop) (by fun_prop), hm, hm] |
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| lemma lintegral_orthonormal_triple (v : Fin 3 → E) (hv : Orthonormal ℝ v) |
| (f : ℝ × ℝ × ℝ → ℝ≥0∞) (hf : Measurable f) : |
| ∫⁻ x, f (⟪v 0,x⟫,⟪v 1,x⟫,⟪v 2,x⟫) ∂stdGaussian E = |
| ∫⁻ s : ℝ, (∫⁻ u : ℝ, (∫⁻ w : ℝ, f (s,u,w) ∂gaussianReal 0 1) |
| ∂gaussianReal 0 1) ∂gaussianReal 0 1 := by |
| rw [← lintegral_map hf (by fun_prop), map_orthonormal_triple_stdGaussian v hv, |
| lintegral_prod _ hf.aemeasurable] |
| apply lintegral_congr |
| intro s |
| exact lintegral_prod _ (hf.comp (measurable_const.prodMk measurable_id)).aemeasurable |
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| end TripleLaw |
| end GaussianPropeller.Analytic |
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| end OAI |
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