| import Mathlib |
| import OAI.Probability.GaussianInformation.Kernel.BayesContinuity |
|
|
| 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.NoisyDrift |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal Topology |
| open Projection Projection.Support GaussianTools ReferenceGaussian Noise Bayes DriftTools.FiniteReference EntropySupport Coercivity |
|
|
| lemma positive_continuous_bound {d N : ℕ} (hd : 16 ≤ d) (hN : 0 < N) |
| (ν : Measure (Ambient d)) [IsProbabilityMeasure ν] {H : ℝ} (hH : 1 ≤ H) |
| (hν : ν ≤ ENNReal.ofReal H • sphereProbability d) |
| (rule : PositiveRule (d/16) d N) (hc : ∀ w,Continuous (fun t => rule.prob t w)) : |
| (∑ w,mass ν (rule.exactResponse w)*Phi (post ν (rule.exactResponse w))) ≤ |
| Phi ν+95*(d:ℝ)+6*Real.log H/(d:ℝ)+2*Real.log (N:ℝ)/(d:ℝ)+Real.log (N:ℝ)/(d/16:ℕ) := by |
| let R := Scale.radius (DriftCutoff.level d H) |
| have hR : 0 < R := Scale.radius_pos _ |
| let η : ℕ → ℝ := fun n => R/((n:ℝ)+1) |
| have hη (n) : 0 < η n := div_pos hR (by positivity) |
| have hηR (n) : η n ≤ R := div_le_self hR.le (by have : (0:ℝ) ≤ n := Nat.cast_nonneg n; linarith) |
| have ht : Tendsto η atTop (𝓝 0) := by |
| have hh := tendsto_const_nhds.mul (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)) (a := R) |
| simpa only [mul_one_div,mul_zero] using hh |
| have hconv (w : Fin N) : Tendsto (fun n => mass ν (rule.response (η n) w)*Phi (post ν (rule.response (η n) w))) atTop |
| (𝓝 (mass ν (rule.exactResponse w)*Phi (post ν (rule.exactResponse w)))) := by |
| apply tendsto_weighted_Phi_positive hd ν (lt_of_lt_of_le zero_lt_one hH) hν |
| (fun n => rule.response (η n) w) (rule.exactResponse w) |
| (fun n => rule.measurable_response _ w) (rule.measurable_exactResponse w) rule.lower_pos |
| (fun n s => (rule.response_bounds (hη n) w s).1) |
| (fun s => (rule.exactResponse_bounds w s).1) |
| (fun n s => (rule.response_bounds (hη n) w s).2) |
| (fun s => (rule.exactResponse_bounds w s).2) |
| exact ae_of_all _ (fun s => rule.tendsto_response hc w s η hη ht) |
| exact le_of_tendsto_of_tendsto' (tendsto_finsetSum _ (fun w _ => hconv w)) tendsto_const_nhds |
| (fun n => positive_noisy_bound hd hN ν hH hν rule (hη n) (hηR n)) |
| |
| end KernelPotential.NoisyDrift |
| |
| end OAI |
| |
