| import Mathlib |
| import OAI.Probability.GaussianInformation.Kernel.SampleEquivalence |
|
|
| namespace OAI |
| open MeasureTheory ProbabilityTheory Metric Set |
| open scoped ENNReal |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal |
| open MeasureTheory ProbabilityTheory |
| namespace KernelPotential.Bayes |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal |
| open EntropySupport Coercivity Sphere |
|
|
| noncomputable def mass [MeasurableSpace α] (ν : Measure α) (r : α → ℝ) : ℝ := ∫ s,r s ∂ν |
| noncomputable def post [MeasurableSpace α] (ν : Measure α) (r : α → ℝ) : Measure α := |
| ν.withDensity (fun s => ENNReal.ofReal (r s/mass ν r)) |
|
|
| lemma integrable_rule [MeasurableSpace α] (ν : Measure α) [IsFiniteMeasure ν] |
| (r : α → ℝ) (hr : Measurable r) (hr0 : ∀ s,0 ≤ r s) (hr1 : ∀ s,r s ≤ 1) : Integrable r ν := |
| integrable_bounded_measurable ν r hr (fun s => by rw [abs_of_nonneg (hr0 s)]; exact hr1 s) |
|
|
| lemma mass_bounds [MeasurableSpace α] (ν : Measure α) [IsProbabilityMeasure ν] |
| (r : α → ℝ) (hr : Measurable r) (hε : 0 < ε) |
| (hrε : ∀ s,ε ≤ r s) (hr1 : ∀ s,r s ≤ 1) : ε ≤ mass ν r ∧ mass ν r ≤ 1 := by |
| have hi := integrable_rule ν r hr (fun s => hε.le.trans (hrε s)) hr1 |
| constructor |
| · simpa [mass] using integral_mono (integrable_const ε) hi hrε |
| · simpa [mass] using integral_mono hi (integrable_const 1) hr1 |
| |
| lemma post_probability [MeasurableSpace α] (ν : Measure α) [IsFiniteMeasure ν] |
| (r : α → ℝ) (hr : Measurable r) (hr0 : ∀ s,0 ≤ r s) (hr1 : ∀ s,r s ≤ 1) |
| (ha : 0 < mass ν r) : IsProbabilityMeasure (post ν r) := by |
| have hi := integrable_rule ν r hr hr0 hr1 |
| constructor |
| rw [post,withDensity_apply _ MeasurableSet.univ,Measure.restrict_univ, |
| ← ofReal_integral_eq_lintegral_ofReal (hi.div_const _) (ae_of_all _ (fun s => div_nonneg (hr0 s) ha.le)), |
| integral_div] |
| change ENNReal.ofReal (mass ν r/mass ν r)=1 |
| simp [ha.ne'] |
| |
| lemma post_integral [MeasurableSpace α] (ν : Measure α) |
| (r : α → ℝ) (hr : Measurable r) (hr0 : ∀ s,0 ≤ r s) (ha : 0 < mass ν r) |
| (g : α → ℝ) : (∫ s,g s ∂post ν r) = (∫ s,r s*g s ∂ν)/mass ν r := by |
| rw [post,integral_withDensity_eq_integral_toReal_smul (hr.div_const _).ennreal_ofReal |
| (ae_of_all _ (fun _ => ENNReal.ofReal_lt_top))] |
| simp only [ENNReal.toReal_ofReal (div_nonneg (hr0 _) ha.le),smul_eq_mul] |
| simp_rw [div_mul_eq_mul_div] |
| exact integral_div _ _ |
|
|
| lemma post_le [MeasurableSpace α] (ν : Measure α) [SFinite ν] |
| (r : α → ℝ) (hr1 : ∀ s,r s ≤ 1) (ha : 0 < mass ν r) : |
| post ν r ≤ ENNReal.ofReal (1/mass ν r) • ν := |
| density_bounded ν _ (fun s => div_le_div_of_nonneg_right (hr1 s) ha.le) |
|
|
| lemma post_bounded (ν : Measure (Ambient d)) [IsProbabilityMeasure ν] |
| (hH : 0 < H) (hν : ν ≤ ENNReal.ofReal H • sphereProbability d) |
| (r : Ambient d → ℝ) (hr1 : ∀ s,r s ≤ 1) (ha : 0 < mass ν r) : |
| post ν r ≤ ENNReal.ofReal (H/mass ν r) • sphereProbability d := by |
| calc |
| _ ≤ ENNReal.ofReal (1/mass ν r) • ν := post_le ν r hr1 ha |
| _ ≤ ENNReal.ofReal (1/mass ν r) • (ENNReal.ofReal H • sphereProbability d) := smul_le_smul_left _ hν |
| _ = _ := by rw [smul_smul,← ENNReal.ofReal_mul' hH.le]; congr 2; ring |
| |
| lemma post_density {α : Type*} [MeasurableSpace α] (μ ν : Measure α) |
| (f r : α → ℝ) (hf : Measurable f) (hf0 : ∀ s,0 ≤ f s) (hr : Measurable r) |
| (hν : ν=μ.withDensity (fun s => ENNReal.ofReal (f s))) : |
| post ν r = μ.withDensity (fun s => ENNReal.ofReal (f s*r s/mass ν r)) := by |
| unfold post |
| nth_rw 1 [hν] |
| rw [← withDensity_mul _ hf.ennreal_ofReal (hr.div_const _).ennreal_ofReal] |
| congr 1 |
| ext s |
| simp only [Pi.mul_apply,← ENNReal.ofReal_mul (hf0 s),mul_div_assoc] |
| |
| noncomputable def clippedJ {d : ℕ} (ν : Measure (Ambient d)) (s : Ambient d) : ℝ := |
| max ((1/2:ℝ)^(2*(d/16))) (J ν s).toReal |
| |
| lemma clippedJ_measurable {d : ℕ} (ν : Measure (Ambient d)) [SFinite ν] : |
| Measurable (clippedJ ν) := measurable_const.max (measurable_J ν).ennreal_toReal |
| lemma clippedJ_pos {d : ℕ} (ν : Measure (Ambient d)) (s : Ambient d) : 0 < clippedJ ν s := |
| lt_of_lt_of_le (by positivity) (le_max_left _ _) |
| lemma clippedJ_eq {d : ℕ} (hd : 16 ≤ d) (ν : Measure (Ambient d)) [IsProbabilityMeasure ν] |
| {H : ℝ} (hH : 0 < H) (hν : ν ≤ ENNReal.ofReal H • sphereProbability d) |
| (s : Ambient d) (hs : ‖s‖=1) : clippedJ ν s=(J ν s).toReal := |
| max_eq_right (J_real_bounds hd ν hH hν s hs).1 |
| |
| lemma integrable_log_clippedJ {d : ℕ} (hd : 16 ≤ d) (ν : Measure (Ambient d)) [IsProbabilityMeasure ν] |
| (hν : BoundedDensity ν) (μ : Measure (Ambient d)) [IsFiniteMeasure μ] : |
| Integrable (fun s => Real.log (clippedJ ν s)) μ := by |
| obtain ⟨H,hH,hνH⟩ := hν |
| apply integrable_log_of_bounds μ _ (clippedJ_measurable ν) |
| (b := max ((1/2:ℝ)^(2*(d/16))) (H*(1+9*3^d*2^((d-1)+2*(d/16)+1)))) |
| (by positivity : (0:ℝ) < (1/2:ℝ)^(2*(d/16))) |
| filter_upwards [] with s |
| refine ⟨le_max_left _ _, max_le_max le_rfl ?_⟩ |
| exact (ENNReal.toReal_mono ENNReal.ofReal_ne_top (bounded_J hd ν hH hνH s)).trans_eq |
| (ENNReal.toReal_ofReal (by positivity)) |
| |
| noncomputable def referenceWeight {d : ℕ} (ν : Measure (Ambient d)) (r : Ambient d → ℝ) |
| (s : Ambient d) : ℝ := r s/(mass ν r*clippedJ (post ν r) s) |
| |
| lemma xlog_posterior_identity {f r a j : ℝ} (hr : 0 < r) (ha : 0 < a) (hj : 0 < j) : |
| a*(xlog (f*r/a)-(f*r/a)*Real.log j) = |
| r*xlog f+f*r*Real.log (r/(a*j)) := by |
| rw [show f*r/a=f*(r/a) by ring,xlog_mul] |
| simp only [xlog] |
| rw [Real.log_div hr.ne' ha.ne',Real.log_div hr.ne' (mul_ne_zero ha.ne' hj.ne'), |
| Real.log_mul ha.ne' hj.ne'] |
| field_simp |
| ring |
|
|
| lemma weighted_Phi_identity (hd : 16 ≤ d) |
| (ν : Measure (Ambient d)) [IsProbabilityMeasure ν] |
| (f : Ambient d → ℝ) (hf : Measurable f) (hf0 : ∀ s,0 ≤ f s) |
| (hH : 0 < H) (hfH : ∀ s,f s ≤ H) |
| (hν : ν=(sphereProbability d).withDensity (fun s => ENNReal.ofReal (f s))) |
| (r : Ambient d → ℝ) (hr : Measurable r) (hε : 0 < ε) |
| (hrε : ∀ s,ε ≤ r s) (hr1 : ∀ s,r s ≤ 1) : |
| mass ν r*Phi (post ν r) = |
| (∫ s,r s*xlog (f s) ∂sphereProbability d)+ |
| ∫ s,r s*Real.log (referenceWeight ν r s) ∂ν := by |
| have hr0 (s : Ambient d) : 0 ≤ r s := hε.le.trans (hrε s) |
| have hrpos (s : Ambient d) : 0 < r s := hε.trans_le (hrε s) |
| have ha := hε.trans_le (mass_bounds ν r hr hε hrε hr1).1 |
| have hνb : ν ≤ ENNReal.ofReal H • sphereProbability d := by rw [hν]; exact density_bounded _ _ hfH |
| have hp := post_bounded ν hH hνb r hr1 ha |
| have := post_probability ν r hr hr0 hr1 ha |
| have hpb : BoundedDensity (post ν r) := ⟨H/mass ν r,div_pos hH ha,hp⟩ |
| let F : Ambient d → ℝ := fun s => f s*r s/mass ν r |
| have hF : Measurable F := (hf.mul hr).div_const _ |
| have hF0 (s : Ambient d) : 0 ≤ F s := div_nonneg (mul_nonneg (hf0 s) (hr0 s)) ha.le |
| have hFH (s : Ambient d) : F s ≤ H/mass ν r := by |
| apply div_le_div_of_nonneg_right _ ha.le |
| exact (mul_le_mul_of_nonneg_left (hr1 s) (hf0 s)).trans (by simpa using hfH s) |
| have hpost : post ν r=(sphereProbability d).withDensity (fun s => ENNReal.ofReal (F s)) := |
| post_density _ ν f r hf hf0 hr hν |
| have hlogr : Integrable (fun s => Real.log (r s)) (sphereProbability d) := |
| integrable_log_of_bounds _ _ hr hε (ae_of_all _ (fun s => ⟨hrε s,hr1 s⟩)) |
| have hlogj := integrable_log_clippedJ hd (post ν r) hpb (sphereProbability d) |
| have hlogx : Integrable (fun s => Real.log (referenceWeight ν r s)) (sphereProbability d) := by |
| have he : (fun s => Real.log (referenceWeight ν r s)) = |
| (fun s => Real.log (r s)-Real.log (mass ν r)-Real.log (clippedJ (post ν r) s)) := by |
| ext s |
| rw [referenceWeight,Real.log_div (hrpos s).ne' (mul_ne_zero ha.ne' (clippedJ_pos _ _).ne'), |
| Real.log_mul ha.ne' (clippedJ_pos _ _).ne'] |
| ring |
| rw [he] |
| exact (hlogr.sub (integrable_const _)).sub hlogj |
| have hix := integrable_xlog_bounded (sphereProbability d) f hf hf0 hfH |
| have hleft := hix.bdd_mul hr.aestronglyMeasurable (ae_of_all _ (fun s => by |
| rw [Real.norm_eq_abs,abs_of_nonneg (hr0 s)]; exact hr1 s)) |
| have hxright : Integrable (fun s => f s*(r s*Real.log (referenceWeight ν r s))) (sphereProbability d) := |
| (hlogx.bdd_mul hr.aestronglyMeasurable (ae_of_all _ (fun s => by |
| rw [Real.norm_eq_abs,abs_of_nonneg (hr0 s)]; exact hr1 s))).bdd_mul hf.aestronglyMeasurable |
| (ae_of_all _ (fun s => by rw [Real.norm_eq_abs,abs_of_nonneg (hf0 s)]; exact hfH s)) |
| rw [Phi_density_formula hd F hF hF0 hFH (post ν r) hpb hpost,← integral_const_mul] |
| have hνint : (∫ s,r s*Real.log (referenceWeight ν r s) ∂ν) = |
| ∫ s,f s*(r s*Real.log (referenceWeight ν r s)) ∂sphereProbability d := by |
| rw [hν,integral_withDensity_eq_integral_toReal_smul hf.ennreal_ofReal |
| (ae_of_all _ (fun _ => ENNReal.ofReal_lt_top))] |
| simp only [ENNReal.toReal_ofReal (hf0 _),smul_eq_mul] |
| rw [hνint,← integral_add hleft hxright] |
| apply integral_congr_ae |
| filter_upwards [ae_unit_norm (by omega : 0 < d)] with s hs |
| rw [referenceWeight,clippedJ_eq hd (post ν r) (div_pos hH ha) hp s hs] |
| simpa only [F,mul_assoc] using xlog_posterior_identity (f := f s) (hrpos s) ha |
| (lt_of_lt_of_le (by positivity : (0:ℝ)<(1/2:ℝ)^(2*(d/16))) |
| (J_real_bounds hd (post ν r) (div_pos hH ha) hp s hs).1) |
| |
| end KernelPotential.Bayes |
| |
| end OAI |
| |
