| import OAI.Probability.PosteriorReplicas.Regression.GaussianDomination |
| import OAI.Probability.PosteriorReplicas.Regression.ReplicaKernel |
|
|
| namespace OAI |
|
|
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal NNReal RealInnerProductSpace |
| noncomputable section |
| universe u |
| namespace PosteriorRegression |
| open scoped Topology |
| open InnerProductSpace |
| section GaussianKernel |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
|
|
| lemma gaussian_rows_kernel_bound {n : ℕ} (v : Fin n → E) (hv : LinearIndependent ℝ v) |
| (φ : ℝ → ℝ≥0∞) (hφ : Measurable φ) (hφ1 : ∫⁻ z, φ z = 1) |
| (m : ℕ) (y : Fin m → ℝ) : |
| ∫⁻ A : Fin m → E, (∏ j, φ (y j)) * ∏ i, ∏ j, φ (y j - ⟪v i, A j⟫) |
| ∂Measure.pi (fun _ => stdGaussian E) ≤ |
| ENNReal.ofReal ((∏ i, ‖gramSchmidt ℝ v i‖)⁻¹ ^ m) * ∏ j, φ (y j) := by |
| let H : (Fin m → Fin n → ℝ) → ℝ≥0∞ := |
| fun z => ∏ j, φ (y j) * ∏ i, φ (y j - z j i) |
| have hH : Measurable H := by dsimp [H]; fun_prop |
| have heq (A : Fin m → E) : |
| (∏ j, φ (y j)) * ∏ i, ∏ j, φ (y j - ⟪v i, A j⟫) = |
| H (fun j i => ⟪v i, A j⟫) := by |
| dsimp [H] |
| rw [Finset.prod_mul_distrib, Finset.prod_comm] |
| simp_rw [heq] |
| rw [← lintegral_map hH (by fun_prop)] |
| calc |
| _ ≤ ∫⁻ z, H z ∂(ENNReal.ofReal ((∏ i, ‖gramSchmidt ℝ v i‖)⁻¹ ^ m) • |
| (volume : Measure (Fin m → Fin n → ℝ))) := |
| lintegral_mono' (gaussian_rows_pairings_le v hv m) (fun _ => le_rfl) |
| _ = _ := by |
| rw [lintegral_smul_measure] |
| change _ * (∫⁻ z : Fin m → Fin n → ℝ, ∏ j, φ (y j) * ∏ i, φ (y j - z j i)) = _ |
| rw [lintegral_product_differences φ hφ hφ1] |
|
|
| lemma gaussian_replica_kernel_bound {n : ℕ} (p : E) (v : Fin n → E) |
| (hv : LinearIndependent ℝ (fun i => v i - p)) |
| (φ : ℝ → ℝ≥0∞) (hφ : Measurable φ) (hφ1 : ∫⁻ z, φ z = 1) (m : ℕ) : |
| ∫⁻ A : Fin m → E, ∫⁻ y : Fin m → ℝ, |
| (∏ j, φ (y j - ⟪p, A j⟫)) * ∏ i, ∏ j, φ (y j - ⟪v i, A j⟫) |
| ∂volume ∂Measure.pi (fun _ => stdGaussian E) ≤ |
| ENNReal.ofReal ((∏ i, ‖gramSchmidt ℝ (fun i => v i - p) i‖)⁻¹ ^ m) := by |
| have ht (A : Fin m → E) : |
| (∫⁻ y : Fin m → ℝ, (∏ j, φ (y j - ⟪p, A j⟫)) * |
| ∏ i, ∏ j, φ (y j - ⟪v i, A j⟫)) = |
| ∫⁻ y : Fin m → ℝ, (∏ j, φ (y j)) * ∏ i, ∏ j, φ (y j - ⟪v i - p, A j⟫) := by |
| have he (y : Fin m → ℝ) (i : Fin n) (j : Fin m) : |
| y j - ⟪p, A j⟫ - ⟪v i - p, A j⟫ = y j - ⟪v i, A j⟫ := by |
| rw [inner_sub_left] |
| ring |
| simpa only [Pi.sub_apply, he] using |
| lintegral_sub_right_eq_self |
| (fun y : Fin m → ℝ => (∏ j, φ (y j)) * ∏ i, ∏ j, φ (y j - ⟪v i - p, A j⟫)) |
| (fun j => ⟪p, A j⟫) |
| simp_rw [ht] |
| rw [lintegral_lintegral_swap (by fun_prop)] |
| calc |
| _ ≤ ∫⁻ y : Fin m → ℝ, |
| ENNReal.ofReal ((∏ i, ‖gramSchmidt ℝ (fun i => v i - p) i‖)⁻¹ ^ m) * |
| ∏ j, φ (y j) := lintegral_mono fun y => |
| gaussian_rows_kernel_bound (fun i => v i - p) hv φ hφ hφ1 m y |
| _ = _ := by |
| rw [lintegral_const_mul _ (by fun_prop), volume_pi, |
| lintegral_fin_prod volume m (fun _ => φ) (fun _ => hφ)] |
| simp [hφ1] |
| end GaussianKernel |
|
|
| end PosteriorRegression |
| end |
|
|
| end OAI |
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