| import Mathlib |
| import OAI.Probability.GaussianInformation.Kernel.PositiveRules |
|
|
| namespace OAI |
| open MeasureTheory ProbabilityTheory Metric Set |
| open scoped ENNReal |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal |
| open MeasureTheory ProbabilityTheory |
| open MeasureTheory |
| open scoped ENNReal |
|
|
| namespace KernelPotential.Noise |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal Topology |
| open Projection Projection.Support GaussianTools ReferenceGaussian |
|
|
| lemma noise_norm_le {k : ℕ} {η : ℝ} (hη : 0 ≤ η) : |
| ∀ᵐ e ∂noiseLaw k η, ‖e‖ ≤ η := by |
| filter_upwards [noise_support k η] with e he |
| apply (pi_norm_le_iff_of_nonneg hη).mpr |
| intro i |
| simpa [closedBox,Real.norm_eq_abs] using he i |
|
|
| lemma tendsto_noise_average {k : ℕ} (φ : (Fin k → ℝ) → ℝ) (hφ : Continuous φ) |
| (hφ0 : ∀ e,0 ≤ φ e) (hφ1 : ∀ e,φ e ≤ 1) (y : Fin k → ℝ) |
| (η : ℕ → ℝ) (hη : ∀ n,0 < η n) (hlim : Tendsto η atTop (𝓝 0)) : |
| Tendsto (fun n => ∫ e,φ (y+e) ∂noiseLaw k (η n)) atTop (𝓝 (φ y)) := by |
| have hi (n) : Integrable (fun e => φ (y+e)) (noiseLaw k (η n)) := by |
| have := noise_probability k (hη n) |
| refine Integrable.of_bound (hφ.comp (continuous_const.add continuous_id)).measurable.aestronglyMeasurable 1 ?_ |
| filter_upwards [] with e |
| rw [Real.norm_eq_abs,abs_of_nonneg (hφ0 _)] |
| exact hφ1 _ |
| apply Metric.tendsto_nhds.mpr |
| intro ε hε |
| obtain ⟨δ,hδ,hh⟩ := Metric.continuousAt_iff.mp (hφ.continuousAt (x := y)) (ε/2) (half_pos hε) |
| filter_upwards [hlim.eventually (eventually_lt_nhds hδ)] with n hn |
| have := noise_probability k (hη n) |
| have hb : ∀ᵐ e ∂noiseLaw k (η n),‖φ (y+e)-φ y‖ ≤ ε/2 := by |
| filter_upwards [noise_norm_le (hη n).le] with e he |
| have hdist : dist (y+e) y < δ := by |
| simpa only [dist_eq_norm,add_sub_cancel_left] using he.trans_lt hn |
| simpa only [dist_eq_norm] using (hh hdist).le |
| have hb' := norm_integral_le_of_norm_le_const hb |
| rw [integral_sub (hi n) (integrable_const _)] at hb' |
| simp only [integral_const,probReal_univ,one_smul,mul_one] at hb' |
| rw [dist_eq_norm] |
| exact hb'.trans_lt (half_lt_self hε) |
|
|
| end KernelPotential.Noise |
| namespace KernelPotential.NoisyDrift.PositiveRule |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal Topology |
| open Projection Projection.Support GaussianTools ReferenceGaussian Noise Bayes DriftTools.FiniteReference |
| variable {k d N : ℕ} (rule : PositiveRule k d N) |
| noncomputable def exactResponse (w : Fin N) (s : Ambient d) : ℝ := |
| ∫ X,rule.prob (X,projectVector X s) w ∂gaussianRows k d |
| lemma measurable_exactResponse (w : Fin N) : Measurable (rule.exactResponse w) := by |
| have : IsProbabilityMeasure (gaussianRows k d) := by unfold gaussianRows; infer_instance |
| have hm : Measurable (fun t : Ambient d × Mat k d => rule.prob (t.2,projectVector t.2 t.1) w) := by |
| apply (rule.measurable w).comp |
| exact measurable_snd.prodMk ((continuous_projectVector.comp (continuous_snd.prodMk continuous_fst)).measurable) |
| exact hm.stronglyMeasurable.integral_prod_right.measurable |
| lemma exactResponse_bounds (w : Fin N) (s : Ambient d) : |
| rule.lower ≤ rule.exactResponse w s ∧ rule.exactResponse w s ≤ 1 := by |
| have : IsProbabilityMeasure (gaussianRows k d) := by unfold gaussianRows; infer_instance |
| have hi : Integrable (fun X => rule.prob (X,projectVector X s) w) (gaussianRows k d) := by |
| refine Integrable.of_bound ((rule.measurable w).comp (measurable_id.prodMk ((continuous_projectVector.comp (continuous_id.prodMk continuous_const)).measurable))).aestronglyMeasurable 1 ?_ |
| filter_upwards [] with X |
| rw [Real.norm_eq_abs,abs_of_nonneg (rule.nonneg _ _)] |
| exact rule.le_one _ _ |
| constructor |
| · have h := integral_mono (integrable_const rule.lower) hi (fun X => rule.lower_prob _ w) |
| simpa [exactResponse] using h |
| · have h := integral_mono hi (integrable_const (1:ℝ)) (fun X => rule.le_one _ w) |
| simpa [exactResponse] using h |
| lemma tendsto_response (hc : ∀ w,Continuous (fun t => rule.prob t w)) (w : Fin N) (s : Ambient d) |
| (η : ℕ → ℝ) (hη : ∀ n,0 < η n) (hlim : Tendsto η atTop (𝓝 0)) : |
| Tendsto (fun n => rule.response (η n) w s) atTop (𝓝 (rule.exactResponse w s)) := by |
| have : IsProbabilityMeasure (gaussianRows k d) := by unfold gaussianRows; infer_instance |
| have he (n) : rule.response (η n) w s = ∫ X,∫ e,rule.prob (X,projectVector X s+e) w ∂noiseLaw k (η n) ∂gaussianRows k d := |
| likelihood_first (hη n) _ (rule.measurable w) (fun t => rule.nonneg t w) (fun t => rule.le_one t w) s |
| simp_rw [he] |
| apply tendsto_integral_of_dominated_convergence (fun _ => (1:ℝ)) |
| (fun n => (noiseAverage_measurable (hη n) _ (rule.measurable w) s).aestronglyMeasurable) |
| (integrable_const 1) |
| · intro n |
| have := noise_probability k (hη n) |
| filter_upwards [] with X |
| have hb := norm_integral_le_of_norm_le_const (μ := noiseLaw k (η n)) (f := fun e => rule.prob (X,projectVector X s+e) w) |
| (ae_of_all _ (fun e => by rw [Real.norm_eq_abs,abs_of_nonneg (rule.nonneg _ _)]; exact rule.le_one _ _)) |
| simpa using hb |
| · filter_upwards [] with X |
| exact tendsto_noise_average _ ((hc w).comp (continuous_const.prodMk continuous_id)) |
| (fun e => rule.nonneg _ _) (fun e => rule.le_one _ _) _ η hη hlim |
| end KernelPotential.NoisyDrift.PositiveRule |
|
|
| end OAI |
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