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135052a | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 | import Mathlib
import OAI.Probability.SKBarriers.Hierarchy.HierarchyPathLaw
import OAI.Probability.SKBarriers.Gaussian.ExponentialAverage
namespace OAI
section
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
end
end
end
end OAI
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