| 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 |
| |
