| import Mathlib |
| import OAI.Probability.GaussianInformation.Kernel.MassContinuity |
|
|
| 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 |
| namespace LocalRule |
| lemma positive_mass {k d N : ℕ} (rule : LocalRule k d N) (hN : 0 < N) |
| (ν : Measure (Ambient d)) [IsProbabilityMeasure ν] (ε : ℝ) (hε : 0 < ε) (hε1 : ε ≤ 1) (w : Fin N) : |
| mass ν ((rule.positive hN ε hε hε1).exactResponse w)=(1-ε)*mass ν (rule.response w)+ε/(N:ℝ) := by |
| simp_rw [mass,rule.positive_response] |
| rw [integral_add ((integrable_rule ν _ (rule.measurable_response w) |
| (fun s => (rule.response_bounds w s).1) (fun s => (rule.response_bounds w s).2)).const_mul _) (integrable_const _),integral_const_mul] |
| simp |
| lemma positive_post_zero {k d N : ℕ} (rule : LocalRule k d N) (hN : 0 < N) |
| (ν : Measure (Ambient d)) [IsProbabilityMeasure ν] (w : Fin N) (ha : mass ν (rule.response w)=0) |
| (ε : ℝ) (hε : 0 < ε) (hε1 : ε ≤ 1) : post ν ((rule.positive hN ε hε hε1).exactResponse w)=ν := by |
| have hr : rule.response w=ᵐ[ν] 0 := (integral_eq_zero_iff_of_nonneg |
| (fun s => (rule.response_bounds w s).1) (integrable_rule ν _ (rule.measurable_response w) |
| (fun s => (rule.response_bounds w s).1) (fun s => (rule.response_bounds w s).2))).mp ha |
| have hp : 0 < ε/(N:ℝ) := div_pos hε (by exact_mod_cast hN) |
| rw [post,rule.positive_mass hN ν ε hε hε1 w,ha,mul_zero,zero_add] |
| calc |
| _ = ν.withDensity (fun _ => (1:ℝ≥0∞)) := withDensity_congr_ae (by |
| filter_upwards [hr] with s hs |
| rw [rule.positive_response,hs,Pi.zero_apply,mul_zero,zero_add,div_self hp.ne'] |
| simp) |
| _ = ν := by simp |
| end LocalRule |
| |
| lemma borel_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 : LocalRule (d/16) d N) : |
| (∑ w,mass ν (rule.response w)*Phi (post ν (rule.response 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 ε : ℕ → ℝ := fun n => (1/2:ℝ)/((n:ℝ)+1) |
| have hε (n) : 0 < ε n := div_pos (by norm_num) (by positivity) |
| have hε2 (n) : ε n ≤ (1/2:ℝ) := div_le_self (by norm_num) (by have : (0:ℝ) ≤ n := Nat.cast_nonneg n; linarith) |
| have hε1 (n) : ε n ≤ 1 := (hε2 n).trans (by norm_num) |
| have ht : Tendsto ε atTop (𝓝 0) := by |
| have hh := tendsto_const_nhds.mul (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)) (a := (1/2:ℝ)) |
| simpa only [mul_one_div,mul_zero] using hh |
| let R := fun n => rule.positive hN (ε n) (hε n) (hε1 n) |
| have hconv (w : Fin N) : Tendsto (fun n => mass ν ((R n).exactResponse w)*Phi (post ν ((R n).exactResponse w))) atTop |
| (𝓝 (mass ν (rule.response w)*Phi (post ν (rule.response w)))) := by |
| have hg0 (s) := (rule.response_bounds w s).1 |
| have hg1 (s) := (rule.response_bounds w s).2 |
| have ha0 : 0 ≤ mass ν (rule.response w) := integral_nonneg hg0 |
| by_cases haz : mass ν (rule.response w)=0 |
| · have he (n) : post ν ((R n).exactResponse w)=ν := rule.positive_post_zero hN ν w haz (ε n) (hε n) (hε1 n) |
| simp_rw [he,show ∀ n,mass ν ((R n).exactResponse w)=(1-ε n)*mass ν (rule.response w)+ε n/(N:ℝ) from |
| fun _ => rule.positive_mass hN ν _ _ _ w,haz,mul_zero,zero_add,zero_mul] |
| simpa using (ht.div_const (N:ℝ)).mul_const (Phi ν) |
| · have hap : 0 < mass ν (rule.response w) := lt_of_le_of_ne ha0 (Ne.symm haz) |
| apply tendsto_weighted_Phi_mass_positive hd ν (lt_of_lt_of_le zero_lt_one hH) hν |
| (fun n => (R n).exactResponse w) (rule.response w) |
| (fun n => (R n).measurable_exactResponse w) (rule.measurable_response w) |
| (fun n s => (R n).lower_pos.le.trans ((R n).exactResponse_bounds w s).1) hg0 (half_pos hap) |
| (fun n => ?_) (by linarith) |
| (fun n s => ((R n).exactResponse_bounds w s).2) hg1 |
| · apply ae_of_all |
| intro s |
| simp_rw [show ∀ n,(R n).exactResponse w s=(1-ε n)*rule.response w s+ε n/(N:ℝ) from |
| fun _ => rule.positive_response hN _ _ _ w s] |
| simpa using (((tendsto_const_nhds (x := (1:ℝ))).sub ht).mul_const (rule.response w s)).add (ht.div_const (N:ℝ)) |
| · change _ ≤ mass ν ((rule.positive hN _ _ _).exactResponse w) |
| rw [rule.positive_mass] |
| have := hε2 n |
| have hp : 0 ≤ ε n/(N:ℝ) := (div_pos (hε n) (by exact_mod_cast hN)).le |
| nlinarith |
| exact le_of_tendsto_of_tendsto' (tendsto_finsetSum _ (fun w _ => hconv w)) tendsto_const_nhds |
| (fun n => positive_borel_bound hd hN ν hH hν (R n)) |
| end KernelPotential.NoisyDrift |
|
|
| end OAI |
|
|
