| import Mathlib |
| import OAI.Probability.SKGap.Gaussian.GaussianRegression |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open MeasureTheory ProbabilityTheory Matrix Real |
| open scoped BigOperators |
|
|
| lemma linear_gaussian_law {κ : Type*} [Fintype κ] [DecidableEq κ] (a : κ → ℝ) : |
| HasLaw (fun g : κ→ℝ=>∑ i,a i*g i) (gaussianReal 0 (∑ i,a i^2).toNNReal) |
| (gaussianCoordinates κ) := by |
| let A : Matrix Unit κ ℝ := fun _=>a |
| have hg := (matrix_sample_gaussian A).eval () |
| change HasGaussianLaw (fun g : κ→ℝ=>∑ i,a i*g i) (gaussianCoordinates κ) at hg |
| have hl : HasLaw _ _ _ := ⟨hg.aemeasurable,hg.map_eq_gaussianReal⟩ |
| have hm := linear_gaussian_mean a |
| have hv : Var[fun g=>∑ i,a i*g i;gaussianCoordinates κ]=∑ i,a i^2 := by |
| rw [← covariance_self hg.aemeasurable] |
| simpa only [sq] using linear_gaussian_covariance a a |
| change HasLaw (fun g : κ→ℝ=>∑ i,a i*g i) _ _ at hl |
| rw [hm,hv] at hl |
| exact hl |
|
|
| lemma linear_gaussian_exp_integrable {κ : Type*} [Fintype κ] [DecidableEq κ] (a : κ → ℝ) : |
| Integrable (fun g : κ→ℝ=>exp (∑ i,a i*g i)) (gaussianCoordinates κ) := by |
| have hl := linear_gaussian_law a |
| have hi : Integrable exp (gaussianReal 0 (∑ i,a i^2).toNNReal) := by |
| simpa using (integrable_exp_mul_gaussianReal (μ:=0) (v:=(∑ i,a i^2).toNNReal) 1) |
| rw [← hl.map_eq] at hi |
| exact (integrable_map_measure (by fun_prop) hl.aemeasurable).mp hi |
|
|
| lemma linear_gaussian_exp_integral {κ : Type*} [Fintype κ] [DecidableEq κ] (a : κ → ℝ) : |
| (∫ g : κ→ℝ,exp (∑ i,a i*g i) ∂gaussianCoordinates κ)=exp ((∑ i,a i^2)/2) := by |
| have hl := linear_gaussian_law a |
| have he := mgf_gaussianReal hl 1 |
| simpa only [mgf,one_mul,zero_mul,one_pow,mul_one,zero_add, |
| coe_toNNReal _ (Finset.sum_nonneg (fun i _=>sq_nonneg (a i)))] using he |
|
|
| end SKGap |
| end |
| end |
|
|
| end OAI |
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|
