| import Mathlib |
| import OAI.Probability.SKBarriers.Hierarchy.HierarchyPathLaw |
| import OAI.Probability.SKBarriers.Gaussian.ExponentialAverage |
|
|
| namespace OAI |
|
|
| section |
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| section |
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter |
| namespace SK.Analytic |
| attribute [local instance 2000] parameterNormedGroup parameterNormedSpace |
| theorem hierarchyPath_integral_succ_exp (n : ℕ) (m : Fin (n+1) → ℝ) |
| (f : ParameterSpace (n+1) → ℝ) (hf : BoundedDerivs f) (x : ℝ) |
| (g : ParameterSpace (n+1) → ℝ) (hg : Continuous g) (hb : HasExpGrowth g) : |
| (∫ z, g z ∂hierarchyPathLaw (n+1) m f x) = |
| ∫ z, gaussianAverage (m (Fin.last n)) f g z |
| ∂hierarchyPathLaw n (fun i => m i.castSucc) (gaussianStep (m (Fin.last n)) f) x := by |
| rw [hierarchyPathLaw_integral (n+1) m f hf, |
| hierarchyPathLaw_integral n _ _ (hf.gaussianStep _),fiberGaussian, |
| integral_prod _ (hierarchyPath_exp_integrable (n+1) m f hf x g hg hb)] |
| apply integral_congr_ae |
| filter_upwards [] with z |
| simp only [hierarchyPathWeight] |
| have he (y : ℝ) : Real.exp (m (Fin.last n)*(f (z,y)-gaussianStep (m (Fin.last n)) f z)) * |
| hierarchyPathWeight n (fun i => m i.castSucc) (gaussianStep (m (Fin.last n)) f) z*g (z,y) = |
| hierarchyPathWeight n (fun i => m i.castSucc) (gaussianStep (m (Fin.last n)) f) z * |
| (Real.exp (m (Fin.last n)*(f (z,y)-gaussianStep (m (Fin.last n)) f z))*g (z,y)) := by ring |
| simp_rw [he] |
| rw [integral_const_mul] |
| congr 1 |
| simp_rw [gaussianTransition_density hf] |
| rw [gaussianAverage,gaussianStepLaw_integral_inv_smul] |
| simp only [smul_eq_mul,div_eq_mul_inv,mul_assoc,← integral_const_mul] |
| apply integral_congr_ae |
| filter_upwards [] with y |
| ring |
|
|
| theorem hierarchyAverage_eq_integral_exp (n : ℕ) (m : Fin n → ℝ) |
| (f : ParameterSpace n → ℝ) (hf : BoundedDerivs f) (g : ParameterSpace n → ℝ) |
| (hg : Continuous g) (hb : HasExpGrowth g) (x : ℝ) : |
| hierarchyAverage n m f g x = ∫ z, g z ∂hierarchyPathLaw n m f x := by |
| induction n with |
| | zero => |
| rw [hierarchyPathLaw_integral 0 m f hf] |
| simp only [hierarchyAverage,hierarchyPathWeight,one_mul,fiberGaussian,integral_dirac] |
| | succ n ih => |
| rw [hierarchyPath_integral_succ_exp n m f hf x g hg hb] |
| exact ih (fun i => m i.castSucc) _ (hf.gaussianStep _) _ |
| (gaussianAverage_continuous_of_expGrowth hf _ hb hg) (gaussianAverage_expGrowth hf _ hb) |
|
|
| end SK.Analytic |
|
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| end |
| end |
|
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| end |
|
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| end OAI |
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|
