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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 | import Mathlib
import OAI.Probability.SKBarriers.Gaussian.GaussianAverage
import OAI.Probability.SKBarriers.Calculus.ParameterAlgebra
import OAI.Probability.SKBarriers.Calculus.ParameterProduct
import OAI.Probability.SKBarriers.Calculus.ParameterRecursion
namespace OAI
section
section
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory Filter
namespace SK.Analytic
section ParameterAverage
variable {P E : Type} [NormedAddCommGroup P] [NormedSpace ℝ P]
[NormedAddCommGroup E] [NormedSpace ℝ E]
theorem ParamRegular.exp_gaussian_pos {f : P × (E × ℝ) → ℝ}
(hf : ParamRegular f) (m : ℝ) (z : P × E) :
0 < ∫ y, Real.exp (m*f (z.1,(z.2,y))) ∂gaussianReal 0 1 := by
apply integral_exp_pos
exact (hf.2.1.exp_mul m).weakLocallyDominated.integrable_section
(Real.continuous_exp.comp (continuous_const.mul (hf.1.continuous.comp
(continuous_fst.fst.prodMk (continuous_fst.snd.prodMk continuous_snd))))) z
theorem ParamSmooth.gaussianAverage {f g : P × (E × ℝ) → ℝ}
(hf : ParamRegular f) (hg : ParamSmooth g) (m : ℝ) :
ParamSmooth (SK.Analytic.gaussianAverage m
(fun z : (P × E) × ℝ => f (z.1.1,(z.1.2,z.2)))
(fun z : (P × E) × ℝ => g (z.1.1,(z.1.2,z.2)))) := by
have hZ := (((hf.const_mul m).log_gaussian_exp).const_mul (-1)).expSmooth
have hN := ((hf.const_mul m).expSmooth.mul hg).gaussian_integral
have he : SK.Analytic.gaussianAverage m
(fun z : (P × E) × ℝ => f (z.1.1,(z.1.2,z.2)))
(fun z : (P × E) × ℝ => g (z.1.1,(z.1.2,z.2))) = fun z : P × E =>
Real.exp (-1*Real.log (∫ y, Real.exp (m*f (z.1,(z.2,y))) ∂gaussianReal 0 1))*
(∫ y, Real.exp (m*f (z.1,(z.2,y)))*g (z.1,(z.2,y)) ∂gaussianReal 0 1) := by
funext z
rw [neg_one_mul,Real.exp_neg,Real.exp_log (hf.exp_gaussian_pos m z)]
exact gaussianStepLaw_integral_inv_smul m _ z _
rw [he]
exact hZ.mul hN
end ParameterAverage
end SK.Analytic
end
end
end
end OAI
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