| import Mathlib |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open scoped BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped Topology Interval |
| noncomputable section |
| open MeasureTheory Set |
| open scoped Interval |
| noncomputable section |
| open MeasureTheory Set Filter ProbabilityTheory |
| open scoped Topology |
| noncomputable section |
| open MeasureTheory Set Filter ProbabilityTheory |
| open scoped Topology NNReal |
| namespace SK.Analytic |
|
|
| theorem hasDerivAt_standardGaussianPDF (x : ℝ) : |
| HasDerivAt (gaussianPDFReal 0 1) (-x * gaussianPDFReal 0 1 x) x := by |
| have h := (((hasDerivAt_id x).pow 2).neg.div_const 2).exp.const_mul |
| ((Real.sqrt (2*Real.pi))⁻¹) |
| convert! h using 1 |
| · funext y |
| simp only [gaussianPDFReal,NNReal.coe_one,sub_zero,mul_one, |
| Pi.neg_apply,Pi.pow_apply,id_eq] |
| · simp only [Pi.neg_apply,Pi.pow_apply,id_eq] |
| simp only [gaussianPDFReal,NNReal.coe_one,sub_zero,mul_one] |
| norm_num |
| ring |
|
|
| theorem integrable_standardGaussian_iff (f : ℝ → ℝ) : |
| Integrable f (gaussianReal 0 1) ↔ Integrable (fun x => f x*gaussianPDFReal 0 1 x) := by |
| rw [gaussianReal_of_var_ne_zero _ (by norm_num : (1 : ℝ≥0) ≠ 0)] |
| rw [integrable_withDensity_iff_integrable_smul' (measurable_gaussianPDF _ _) |
| (ae_of_all _ fun _ => gaussianPDF_lt_top)] |
| simp only [gaussianPDF,ENNReal.toReal_ofReal (gaussianPDFReal_nonneg _ _ _),smul_eq_mul,mul_comm] |
| |
| theorem gaussian_integral_mul_eq_integral_deriv (f f' : ℝ → ℝ) |
| (hd : ∀ x, HasDerivAt f (f' x) x) |
| (hi : Integrable f (gaussianReal 0 1)) |
| (hi' : Integrable f' (gaussianReal 0 1)) |
| (hix : Integrable (fun x => x*f x) (gaussianReal 0 1)) : |
| (∫ x, x*f x ∂gaussianReal 0 1) = ∫ x, f' x ∂gaussianReal 0 1 := by |
| have hi₀ := (integrable_standardGaussian_iff f).mp hi |
| have hi₁ := (integrable_standardGaussian_iff f').mp hi' |
| have hi₂ := (integrable_standardGaussian_iff (fun x => x*f x)).mp hix |
| have hx : Integrable (fun x => f x*(-x*gaussianPDFReal 0 1 x)) := by |
| convert hi₂.neg using 1 |
| funext x |
| change _ = -(x*f x*gaussianPDFReal 0 1 x) |
| ring |
| have h := integral_mul_deriv_eq_deriv_mul_of_integrable |
| (fun x _ => hd x) (fun x _ => hasDerivAt_standardGaussianPDF x) hx hi₁ hi₀ |
| have he : (fun x => f x*(-x*gaussianPDFReal 0 1 x)) = |
| (fun x => -(gaussianPDFReal 0 1 x*(x*f x))) := by funext x; ring |
| rw [he,integral_neg] at h |
| have h' := neg_injective h |
| rw [integral_gaussianReal_eq_integral_smul (by norm_num : (1 : ℝ≥0) ≠ 0), |
| integral_gaussianReal_eq_integral_smul (by norm_num : (1 : ℝ≥0) ≠ 0)] |
| simpa only [smul_eq_mul,mul_comm] using h' |
|
|
| theorem integral_tilted_real_eq_div {Ω : Type} [MeasurableSpace Ω] |
| (μ : Measure Ω) (V g : Ω → ℝ) : |
| (∫ x, g x ∂μ.tilted V) = |
| (∫ x, g x*Real.exp (V x) ∂μ) / ∫ x, Real.exp (V x) ∂μ := by |
| rw [integral_tilted] |
| simp only [smul_eq_mul] |
| calc |
| (∫ x, (Real.exp (V x)/(∫ x, Real.exp (V x) ∂μ))*g x ∂μ) = |
| ∫ x, (g x*Real.exp (V x))/(∫ x, Real.exp (V x) ∂μ) ∂μ := by |
| apply integral_congr_ae |
| filter_upwards [] with x |
| ring |
| _ = _ := integral_div _ _ |
|
|
| theorem gaussian_tilted_stein (V V' g g' : ℝ → ℝ) |
| (hV : ∀ x, HasDerivAt V (V' x) x) (hg : ∀ x, HasDerivAt g (g' x) x) |
| (hi : Integrable (fun x => g x*Real.exp (V x)) (gaussianReal 0 1)) |
| (hi' : Integrable (fun x => g' x*Real.exp (V x)) (gaussianReal 0 1)) |
| (hiV : Integrable (fun x => g x*V' x*Real.exp (V x)) (gaussianReal 0 1)) |
| (hix : Integrable (fun x => x*(g x*Real.exp (V x))) (gaussianReal 0 1)) : |
| (∫ x, x*g x ∂(gaussianReal 0 1).tilted V) = |
| (∫ x, g' x ∂(gaussianReal 0 1).tilted V) + |
| ∫ x, g x*V' x ∂(gaussianReal 0 1).tilted V := by |
| have hd (x : ℝ) : HasDerivAt (fun x => g x*Real.exp (V x)) |
| (g' x*Real.exp (V x)+g x*V' x*Real.exp (V x)) x := by |
| convert! (hg x).mul (hV x).exp using 1 |
| ring |
| have h := gaussian_integral_mul_eq_integral_deriv _ _ hd hi (hi'.add hiV) hix |
| simp only [integral_tilted_real_eq_div] |
| rw [← add_div] |
| congr 1 |
| rw [← integral_add hi' hiV] |
| convert! h using 1 |
| apply integral_congr_ae |
| filter_upwards [] with x |
| ring |
| |
| end SK.Analytic |
| |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| |
| end OAI |
| |
