| import OAI.Probability.GaussianPropeller.Translation |
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| namespace OAI |
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| universe uι |
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| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal |
| open scoped RealInnerProductSpace |
| open scoped RealInnerProductSpace |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal RealInnerProductSpace |
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| open Filter |
| open scoped Topology |
| namespace GaussianPropeller.Stein |
| open GaussianPropeller.Translation |
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| variable {ι : Type uι} [Fintype ι] |
| local notation "E" => EuclideanSpace ℝ ι |
| local notation "γ" => stdGaussian E |
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| lemma integrable_bounded {H : E → ℝ} (hH : AEStronglyMeasurable H γ) |
| {B : ℝ} (hB : ∀ x, ‖H x‖ ≤ B) : Integrable H γ := |
| Integrable.mono' (integrable_const B) hH (ae_of_all _ hB) |
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| lemma translation_expectation (v : E) {H : E → ℝ} (hH : Continuous H) : |
| ∫ x, H (x+v) ∂γ = ∫ x, Real.exp (⟪v,x⟫-‖v‖^2/2)*H x ∂γ := by |
| rw [← integral_map (by fun_prop) hH.aestronglyMeasurable, stdGaussian_map_add] |
| rw [integral_withDensity_eq_integral_toReal_smul (by fun_prop) |
| (ae_of_all _ (fun x => ENNReal.ofReal_lt_top))] |
| simp only [ENNReal.toReal_ofReal (le_of_lt (Real.exp_pos _)), smul_eq_mul] |
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| lemma weighted_mgf_derivative {H : E → ℝ} (hH : Continuous H) |
| {B : ℝ} (hB : ∀ x, ‖H x‖ ≤ B) (v : E) : |
| HasDerivAt (fun s : ℝ => ∫ x, H x*Real.exp (s*⟪v,x⟫) ∂γ) |
| (∫ x, H x*⟪v,x⟫ ∂γ) 0 := by |
| have hB0 : 0 ≤ B := (norm_nonneg (H 0)).trans (hB 0) |
| have hi : Integrable (fun x : E => |⟪v,x⟫| *Real.exp |⟪v,x⟫|) γ := by |
| simpa using (integrable_pow_abs_mul_exp_add_of_integrable_exp_mul |
| (X := fun x : E => ⟪v,x⟫) (v := 0) (t := 2) (x := 1) |
| (by simpa using integrable_exp_inner v 2) |
| (by simpa using integrable_exp_inner v (-2)) (by norm_num) (by norm_num) 1) |
| have hd := hasDerivAt_integral_of_dominated_loc_of_deriv_le |
| (μ := γ) (x₀ := 0) (F := fun s : ℝ => fun x : E => H x*Real.exp (s*⟪v,x⟫)) |
| (F' := fun s x => H x*(Real.exp (s*⟪v,x⟫)*⟪v,x⟫)) |
| (s := Ioo (-1) 1) (bound := fun x => B*(|⟪v,x⟫| *Real.exp |⟪v,x⟫|)) |
| (by exact Ioo_mem_nhds (by norm_num) (by norm_num)) |
| (Filter.Eventually.of_forall (fun s => by fun_prop)) |
| (by simpa using integrable_bounded hH.aestronglyMeasurable hB) |
| (by fun_prop) ?_ (hi.const_mul B) ?_ |
| · simpa [mul_comm, mul_left_comm, mul_assoc] using hd.2 |
| · filter_upwards [] with x s hs |
| simp only [norm_mul, Real.norm_eq_abs, abs_of_pos (Real.exp_pos _)] |
| have hs1 : |s| ≤ 1 := (abs_lt.mpr hs).le |
| have he : Real.exp (s*⟪v,x⟫) ≤ Real.exp |⟪v,x⟫| := by |
| apply Real.exp_le_exp.mpr |
| exact (le_abs_self _).trans (by rw [abs_mul]; nlinarith [abs_nonneg ⟪v,x⟫]) |
| calc |
| |H x| *(Real.exp (s*⟪v,x⟫)*|⟪v,x⟫|) ≤ B*(Real.exp |⟪v,x⟫| *|⟪v,x⟫|) := |
| mul_le_mul (hB x) (mul_le_mul_of_nonneg_right he (abs_nonneg _)) |
| (by positivity) hB0 |
| _ = _ := by ring |
| · filter_upwards [] with x s _ |
| simpa only [id_eq, one_mul] using (((hasDerivAt_id s).mul_const ⟪v,x⟫).exp).const_mul (H x) |
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| lemma integration_by_parts {H : E → ℝ} {D : E → E →L[ℝ] ℝ} |
| (hD : ∀ x, HasFDerivAt H (D x) x) (hcD : Continuous D) |
| {B C : ℝ} (hB : ∀ x, ‖H x‖ ≤ B) (hC : ∀ x, ‖D x‖ ≤ C) (v : E) : |
| ∫ x, D x v ∂γ = ∫ x, H x*⟪v,x⟫ ∂γ := by |
| have hc : Continuous H := continuous_iff_continuousAt.mpr (fun x => (hD x).continuousAt) |
| have hd := hasDerivAt_integral_of_dominated_loc_of_deriv_le |
| (μ := γ) (x₀ := 0) (F := fun s : ℝ => fun x : E => H (x+s•v)) |
| (F' := fun s x => D (x+s•v) v) (s := univ) |
| (bound := fun _ => C*‖v‖) (Filter.univ_mem) |
| (Filter.Eventually.of_forall (fun s => by fun_prop)) |
| (by simpa using integrable_bounded hc.aestronglyMeasurable hB) |
| (by fun_prop) (ae_of_all _ (fun x s _ => |
| (D (x+s•v)).le_opNorm v |>.trans (mul_le_mul_of_nonneg_right (hC _) (norm_nonneg _)))) |
| (integrable_const _) (ae_of_all _ (fun x s _ => by |
| simpa only [Function.comp_def, one_smul, id_eq] using (hD (x+s•v)).comp_hasDerivAt s |
| ((hasDerivAt_id s).smul_const v |>.const_add x))) |
| have he : HasDerivAt (fun s : ℝ => Real.exp (-s^2*‖v‖^2/2)) 0 0 := by |
| convert (((hasDerivAt_pow 2 (0:ℝ)).neg.mul_const (‖v‖^2)).div_const 2).exp using 1 |
| simp |
| have hr := he.mul (weighted_mgf_derivative hc hB v) |
| simp only [zero_pow (by norm_num : 2 ≠ 0), neg_zero, zero_mul, zero_div, |
| Real.exp_zero, one_mul, zero_add] at hr |
| have heq : (fun s : ℝ => Real.exp (-s^2*‖v‖^2/2)* |
| (∫ x, H x*Real.exp (s*⟪v,x⟫) ∂γ)) = (fun s : ℝ => ∫ x, H (x+s•v) ∂γ) := by |
| funext s |
| rw [translation_expectation _ hc, ← integral_const_mul] |
| apply integral_congr_ae |
| filter_upwards [] with x |
| simp only [inner_smul_left, norm_smul, Real.norm_eq_abs, mul_pow, sq_abs, |
| starRingEnd_apply, star_trivial] |
| rw [mul_left_comm, ← Real.exp_add] |
| rw [show -s^2*‖v‖^2/2+s*⟪v,x⟫ = s*⟪v,x⟫-s^2*‖v‖^2/2 by ring] |
| ring |
| change HasDerivAt (fun s : ℝ => Real.exp (-s^2*‖v‖^2/2)* |
| (∫ x, H x*Real.exp (s*⟪v,x⟫) ∂γ)) _ 0 at hr |
| rw [heq] at hr |
| simpa using hd.2.unique hr |
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| end GaussianPropeller.Stein |
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| end OAI |
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