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b317718 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 | import Mathlib
import OAI.Probability.GaussianInformation.Kernel.BorelDrift
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.Bayes
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal Topology
open EntropySupport Coercivity Sphere DriftTools Continuity
lemma tendsto_weighted_Phi_mass_positive {d : ℕ} (hd : 16 ≤ d) (ν : Measure (Ambient d))
[IsProbabilityMeasure ν] {H : ℝ} (hH : 0 < H) (hν : ν ≤ ENNReal.ofReal H • sphereProbability d)
(r : ℕ → Ambient d → ℝ) (g : Ambient d → ℝ) (hr : ∀ n,Measurable (r n)) (hg : Measurable g)
(hr0 : ∀ n s,0 ≤ r n s) (hg0 : ∀ s,0 ≤ g s)
{ε : ℝ} (hε : 0 < ε) (ha : ∀ n,ε ≤ mass ν (r n)) (hb : ε ≤ mass ν g)
(hr1 : ∀ n s,r n s ≤ 1) (hg1 : ∀ s,g s ≤ 1)
(hlim : ∀ᵐ s ∂ν,Tendsto (fun n => r n s) atTop (𝓝 (g s))) :
Tendsto (fun n => mass ν (r n)*Phi (post ν (r n))) atTop (𝓝 (mass ν g*Phi (post ν g))) := by
have hap (n) := hε.trans_le (ha n)
have hbp := hε.trans_le hb
have hmass := tendsto_mass ν r g hr hr0 hr1 hlim
obtain ⟨f,hf,hf0,hfH,hνf⟩ := bounded_density_representation (sphereProbability d) ν hH hν
let F := fun n s => f s*r n s/mass ν (r n)
let G := fun s => f s*g s/mass ν g
have hF (n) : Measurable (F n) := (hf.mul (hr n)).div_const _
have hG : Measurable G := (hf.mul hg).div_const _
have hF0 (n s) : 0 ≤ F n s := div_nonneg (mul_nonneg (hf0 s) (hr0 n s)) (hap n).le
have hG0 (s) : 0 ≤ G s := div_nonneg (mul_nonneg (hf0 s) (hg0 s)) hbp.le
have hFb (n s) : F n s ≤ H/ε := by
apply (div_le_div_of_nonneg_left (mul_nonneg (hf0 s) (hr0 n s)) hε (ha n)).trans
exact div_le_div_of_nonneg_right ((mul_le_mul_of_nonneg_left (hr1 n s) (hf0 s)).trans (by simpa using hfH s)) hε.le
have hGb (s) : G s ≤ H/ε := by
apply (div_le_div_of_nonneg_left (mul_nonneg (hf0 s) (hg0 s)) hε hb).trans
exact div_le_div_of_nonneg_right ((mul_le_mul_of_nonneg_left (hg1 s) (hf0 s)).trans (by simpa using hfH s)) hε.le
have hpost (n) : post ν (r n)=densityLaw (F n) := post_density _ _ f _ hf hf0 (hr n) hνf
have hpostg : post ν g=densityLaw G := post_density _ _ f _ hf hf0 hg hνf
have hfp (n) : IsProbabilityMeasure (densityLaw (F n)) := by
rw [← hpost n]; exact post_probability ν _ (hr n) (hr0 n) (hr1 n) (hap n)
have hgp : IsProbabilityMeasure (densityLaw G) := by
rw [← hpostg]; exact post_probability ν _ hg hg0 hg1 hbp
have hlimf : ∀ᵐ s ∂sphereProbability d,Tendsto (fun n => F n s) atTop (𝓝 (G s)) := by
rw [hνf,ae_withDensity_iff hf.ennreal_ofReal] at hlim
filter_upwards [hlim] with s hs
by_cases hfz : f s=0
· simp only [F,G,hfz,zero_mul,zero_div]; exact tendsto_const_nhds
· exact (tendsto_const_nhds.mul (hs (by simpa using (lt_of_le_of_ne (hf0 s) (Ne.symm hfz))))).div hmass hbp.ne'
have hp := tendsto_Phi hd F G hF hG hF0 hG0 (div_pos hH hε) hFb hGb hfp hgp hlimf
simp_rw [← hpost,← hpostg] at hp
exact hmass.mul hp
end KernelPotential.Bayes
end OAI
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