| import Mathlib |
| import OAI.Probability.GaussianInformation.Kernel.NoisyMoments |
|
|
| 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.DriftTools.FiniteReference |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal |
| variable {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] |
| lemma kernel_average_measurable (Λ : Measure β) [SigmaFinite Λ] |
| (K : β → α → ℝ≥0∞) (hK : Measurable (Function.uncurry K)) |
| (hKfin : ∀ z s,K z s ≠ ∞) (g : β → ℝ) (hg : Measurable g) : |
| Measurable (fun s => ∫ z,g z ∂Λ.withDensity (fun z => K z s)) := by |
| have hm : Measurable (fun t : α × β => (K t.2 t.1).toReal*g t.2) := by fun_prop |
| have hh := (hm.stronglyMeasurable.integral_prod_right' (ν := Λ)).measurable |
| convert hh using 1 |
| ext s |
| have hKs : Measurable (fun z => K z s) := hK.comp (measurable_id.prodMk measurable_const) |
| rw [integral_withDensity_eq_integral_toReal_smul hKs |
| (ae_of_all _ (fun z => lt_top_iff_ne_top.mpr (hKfin z s)))] |
| simp only [smul_eq_mul] |
|
|
| lemma kernel_average_integrable (ν : Measure α) [IsFiniteMeasure ν] (Λ : Measure β) [SigmaFinite Λ] |
| (K : β → α → ℝ≥0∞) (hK : Measurable (Function.uncurry K)) |
| (hKfin : ∀ z s,K z s ≠ ∞) (hK1 : ∀ s,∫⁻ z,K z s ∂Λ=1) |
| (g : β → ℝ) (hg : Measurable g) {M : ℝ} (hgM : ∀ z,‖g z‖ ≤ M) : |
| Integrable (fun s => ∫ z,g z ∂Λ.withDensity (fun z => K z s)) ν := by |
| refine Integrable.of_bound (kernel_average_measurable Λ K hK hKfin g hg).aestronglyMeasurable M ?_ |
| filter_upwards [] with s |
| have : IsProbabilityMeasure (Λ.withDensity (fun z => K z s)) := by |
| constructor |
| rw [withDensity_apply _ MeasurableSet.univ,Measure.restrict_univ,hK1] |
| simpa using norm_integral_le_of_norm_le_const (μ := Λ.withDensity (fun z => K z s)) |
| (ae_of_all _ hgM) |
| end KernelPotential.DriftTools.FiniteReference |
|
|
| namespace KernelPotential.NoisyDrift |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal |
| open Projection Projection.Support GaussianTools ReferenceGaussian Noise Bayes DriftTools.FiniteReference EntropySupport Coercivity Sphere |
| noncomputable def postCharge {d N : ℕ} (ν : Measure (Ambient d)) |
| (rule : PositiveRule (d/16) d N) (η : ℝ) (z : Observation (d/16) d) (w : Fin N) : ℝ := |
| Real.log (1+(projectedReference (post ν (rule.response η w)) η z.1.1 z.1.2 z.2.1 z.2.2).toReal) |
|
|
| lemma measurable_postCharge {d N : ℕ} (hd : 16 ≤ d) (ν : Measure (Ambient d)) [IsProbabilityMeasure ν] |
| {H : ℝ} (hν : ν ≤ ENNReal.ofReal H • sphereProbability d) |
| (rule : PositiveRule (d/16) d N) {η : ℝ} (hη : 0 < η) : |
| Measurable (Function.uncurry (postCharge ν rule η)) := by |
| apply measurable_from_prod_countable_left |
| intro w |
| have := rule.post_prob hη ν w |
| have hnorm : ∀ᵐ s ∂ν,‖s‖=1 := (Measure.absolutelyContinuous_of_le_smul hν).ae_le (ae_unit_norm (by omega : 0 < d)) |
| have := Reweight.rho_finite (post ν (rule.response η w)) (withDensity_absolutelyContinuous _ _ |>.ae_le hnorm) |
| exact (measurable_const.add (projectedReference_measurable (post ν (rule.response η w)) η).ennreal_toReal).log |
|
|
| lemma postCharge_uniform_bound {d N : ℕ} (hd : 16 ≤ d) (ν : Measure (Ambient d)) |
| [IsProbabilityMeasure ν] {H : ℝ} (hH : 0 < H) (hν : ν ≤ ENNReal.ofReal H • sphereProbability d) |
| (rule : PositiveRule (d/16) d N) {η : ℝ} (hη : 0 < η) : |
| ∃ M : ℝ,0 ≤ M ∧ ∀ z w,0 ≤ postCharge ν rule η z w ∧ postCharge ν rule η z w ≤ M := by |
| classical |
| have hh (w : Fin N) : ∃ M : ℝ,0 ≤ M ∧ ∀ z,0 ≤ postCharge ν rule η z w ∧ postCharge ν rule η z w ≤ M := by |
| have := rule.post_prob hη ν w |
| have hb := post_bounded ν hH hν (rule.response η w) (fun s => (rule.response_bounds hη w s).2) (rule.mass_pos hη ν w) |
| have hnorm : ∀ᵐ s ∂post ν (rule.response η w),‖s‖=1 := |
| (Measure.absolutelyContinuous_of_le_smul hb).ae_le (ae_unit_norm (by omega : 0 < d)) |
| exact projected_log_bounded (post ν (rule.response η w)) hnorm hη |
| choose M hM hb using hh |
| refine ⟨∑ w,M w,Finset.sum_nonneg (fun w _ => hM w),fun z w => ⟨(hb w z).1,(hb w z).2.trans ?_⟩⟩ |
| exact Finset.single_le_sum (fun v _ => hM v) (Finset.mem_univ w) |
|
|
| lemma postCharge_pointwise {d N : ℕ} (hd : 16 ≤ d) (ν : Measure (Ambient d)) |
| [IsProbabilityMeasure ν] {H : ℝ} (hH : 1 ≤ H) (hν : ν ≤ ENNReal.ofReal H • sphereProbability d) |
| (rule : PositiveRule (d/16) d N) {η : ℝ} (hη : 0 < η) (w : Fin N) |
| {s : Ambient d} (hs : ‖s‖=1) : |
| (∫ z : Observation (d/16) d,rule.prob (z.1.1,z.2.1) w*postCharge ν rule η z w |
| ∂(observationBase (d/16) d).withDensity (fun z => observationKernel η z s)) ≤ |
| -xlog (rule.response η w s)/(d/16:ℕ)+rule.response η w s* |
| (10*(d:ℝ)+Real.log ((DriftCutoff.level d (H/mass ν (rule.response η w)):ℝ)+2)) := by |
| have hHp : 0 < H := lt_of_lt_of_le zero_lt_one hH |
| have ha := rule.mass_pos hη ν w |
| have hHdiv : 1 ≤ H/mass ν (rule.response η w) := |
| (le_div_iff₀ ha).mpr (by simpa using (rule.mass_le_one hη ν w).trans hH) |
| have := rule.post_prob hη ν w |
| have hpost := post_bounded ν hHp hν _ (fun s => (rule.response_bounds hη w s).2) ha |
| have hc := noisy_weighted_projection hd (post ν (rule.response η w)) (div_pos hHp ha) hpost s hs |
| (DriftCutoff.level d (H/mass ν (rule.response η w))) (DriftCutoff.moment_cut d hd hHdiv) |
| hη (fun t => rule.prob t w) (rule.measurable w) (fun t => rule.nonneg t w) (fun t => rule.le_one t w) |
| simpa only [← rule.response_average hη,xlog,postCharge,neg_mul] using hc |
|
|
| lemma integrable_postCharge_average {d N : ℕ} (hd : 16 ≤ d) (ν : Measure (Ambient d)) |
| [IsProbabilityMeasure ν] {H : ℝ} (hHp : 0 < H) (hν : ν ≤ ENNReal.ofReal H • sphereProbability d) |
| (rule : PositiveRule (d/16) d N) {η : ℝ} (hη : 0 < η) (w : Fin N) : |
| Integrable (fun s => ∫ z : Observation (d/16) d,rule.prob (z.1.1,z.2.1) w*postCharge ν rule η z w |
| ∂(observationBase (d/16) d).withDensity (fun z => observationKernel η z s)) ν := by |
| obtain ⟨M,hM,hMb⟩ := postCharge_uniform_bound hd ν hHp hν rule hη |
| have hm1 : Measurable (fun z : Observation (d/16) d => rule.prob (z.1.1,z.2.1) w) := |
| (rule.measurable w).comp (show Measurable (fun z : Observation (d/16) d => (z.1.1,z.2.1)) from measurable_fst.fst.prodMk measurable_snd.fst) |
| have hm2 : Measurable (fun z : Observation (d/16) d => postCharge ν rule η z w) := |
| (measurable_postCharge hd ν hν rule hη).comp (f := fun z : Observation (d/16) d => (z,w)) (show Measurable (fun z : Observation (d/16) d => (z,w)) from measurable_id.prodMk measurable_const) |
| have hg : Measurable (fun z : Observation (d/16) d => rule.prob (z.1.1,z.2.1) w*postCharge ν rule η z w) := hm1.mul hm2 |
| have hgb (z : Observation (d/16) d) : ‖rule.prob (z.1.1,z.2.1) w*postCharge ν rule η z w‖ ≤ M := by |
| rw [Real.norm_eq_abs,abs_of_nonneg (mul_nonneg (rule.nonneg _ w) (hMb z w).1)] |
| exact (mul_le_mul_of_nonneg_right (rule.le_one _ w) (hMb z w).1).trans (by simpa using (hMb z w).2) |
| exact kernel_average_integrable ν (observationBase (d/16) d) (observationKernel η) |
| (observationKernel_measurable η) (observationKernel_ne_top hη) (observationKernel_probability hη) _ hg hgb |
|
|
| lemma integrated_postCharge {d N : ℕ} (hd : 16 ≤ d) (ν : Measure (Ambient d)) |
| [IsProbabilityMeasure ν] {H : ℝ} (hH : 1 ≤ H) (hν : ν ≤ ENNReal.ofReal H • sphereProbability d) |
| (rule : PositiveRule (d/16) d N) {η : ℝ} (hη : 0 < η) (w : Fin N) : |
| (∫ s,∫ z : Observation (d/16) d,rule.prob (z.1.1,z.2.1) w*postCharge ν rule η z w |
| ∂(observationBase (d/16) d).withDensity (fun z => observationKernel η z s) ∂ν) ≤ |
| -(∫ s,xlog (rule.response η w s) ∂ν)/(d/16:ℕ)+ |
| mass ν (rule.response η w)*(10*(d:ℝ)+Real.log ((DriftCutoff.level d (H/mass ν (rule.response η w)):ℝ)+2)) := by |
| have hHp : 0 < H := lt_of_lt_of_le zero_lt_one hH |
| have ha := rule.mass_pos hη ν w |
| have ha1 := rule.mass_le_one hη ν w |
| have hHdiv : 1 ≤ H/mass ν (rule.response η w) := (le_div_iff₀ ha).mpr (by simpa using ha1.trans hH) |
| have := rule.post_prob hη ν w |
| have hpost := post_bounded ν hHp hν (rule.response η w) (fun s => (rule.response_bounds hη w s).2) ha |
| have hnorm : ∀ᵐ s ∂ν,‖s‖=1 := |
| (Measure.absolutelyContinuous_of_le_smul hν).ae_le (ae_unit_norm (by omega : 0 < d)) |
| have hi := integrable_postCharge_average hd ν hHp hν rule hη w |
| have hir := integrable_rule ν _ (rule.measurable_response η w) |
| (fun s => rule.lower_pos.le.trans (rule.response_bounds hη w s).1) (fun s => (rule.response_bounds hη w s).2) |
| have hix := integrable_xlog_bounded ν _ (rule.measurable_response η w) |
| (fun s => rule.lower_pos.le.trans (rule.response_bounds hη w s).1) (fun s => (rule.response_bounds hη w s).2) |
| let L := DriftCutoff.level d (H/mass ν (rule.response η w)) |
| have hh := integral_mono_ae hi ((hix.neg.div_const (d/16:ℕ)).add (hir.mul_const (10*(d:ℝ)+Real.log ((L:ℝ)+2)))) (by |
| filter_upwards [hnorm] with s hs |
| exact postCharge_pointwise hd ν hH hν rule hη w hs) |
| have hi1 : Integrable (fun s => -xlog (rule.response η w s)/(d/16:ℕ)) ν := hix.neg.div_const _ |
| have hi2 : Integrable (fun s => rule.response η w s*(10*(d:ℝ)+Real.log ((L:ℝ)+2))) ν := hir.mul_const _ |
| change _ ≤ ∫ s, -xlog (rule.response η w s)/(d/16:ℕ)+rule.response η w s*(10*(d:ℝ)+Real.log ((L:ℝ)+2)) ∂ν at hh |
| rw [integral_add hi1 hi2,integral_div,integral_neg,integral_mul_const] at hh |
| exact hh |
| end KernelPotential.NoisyDrift |
|
|
| end OAI |
|
|
