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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 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 | import Mathlib
import OAI.Probability.GaussianInformation.Kernel.JointRules
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
open scoped ENNReal
open Projection Projection.Support GaussianTools ReferenceGaussian Noise Bayes DriftTools.FiniteReference EntropySupport Coercivity
lemma reference_log_measurable {d N : ℕ} (ν : Measure (Ambient d)) [IsProbabilityMeasure ν]
(rule : PositiveRule (d/16) d N) {η : ℝ} (hη : 0 < η) :
Measurable (fun a : Ambient d × Fin N => Real.log (referenceWeight ν (rule.response η a.2) a.1)) := by
apply measurable_from_prod_countable_left
intro w
exact (referenceWeight_measurable 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) (rule.mass_pos hη ν w)).log
lemma reference_log_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,∀ s w,‖Real.log (referenceWeight ν (rule.response η w) s)‖ ≤ M := by
classical
have hb (w : Fin N) := referenceWeight_log_bounded hd ν hH hν _ (rule.measurable_response η w)
rule.lower_pos (fun s => (rule.response_bounds hη w s).1) (fun s => (rule.response_bounds hη w s).2)
choose M hM hb using hb
refine ⟨∑ w,M w,fun s w => (hb w s).trans ?_⟩
exact Finset.single_le_sum (fun v _ => hM v) (Finset.mem_univ w)
lemma joint_Phi_identity {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) :
(∑ w,mass ν (rule.response η w)*Phi (post ν (rule.response η w)))=
(InformationTheory.klDiv ν (sphereProbability d)).toReal+
∫ t : Ambient d × Observation (d/16) d × Fin N,
Real.log (referenceWeight ν (rule.response η t.2.2) t.1)
∂joint ν (observationBase (d/16) d) (observationKernel η) rule.kernel := by
obtain ⟨f,hf,hf0,hfH,hνf⟩ := bounded_density_representation (sphereProbability d) ν hH hν
have : IsProbabilityMeasure ((sphereProbability d).withDensity (fun s => ENNReal.ofReal (f s))) := hνf ▸ inferInstance
have hD : (InformationTheory.klDiv ν (sphereProbability d)).toReal=∫ s,xlog (f s) ∂sphereProbability d := by
rw [hνf,klDiv_withDensity_bounded _ f hf hf0 hfH,
ENNReal.toReal_ofReal (density_entropy_nonneg _ f hf hf0 hfH)]
obtain ⟨M,hM⟩ := reference_log_uniform_bound hd ν hH hν rule hη
have hj := joint_integral_signal ν (observationBase (d/16) d) (observationKernel η) rule.kernel
(observationKernel_measurable η) rule.measurable_kernel (observationKernel_ne_top hη)
(observationKernel_probability hη) rule.sum_kernel
(fun s w => Real.log (referenceWeight ν (rule.response η w) s)) (reference_log_measurable ν rule hη) hM
rw [hj]
change (∑ w,mass ν (rule.response η w)*Phi (post ν (rule.response η w)))=
(InformationTheory.klDiv ν (sphereProbability d)).toReal+
∑ w,∫ s,rule.response η w s*Real.log (referenceWeight ν (rule.response η w) s) ∂ν
simp_rw [weighted_Phi_identity hd ν f hf hf0 hH hfH hνf _ (rule.measurable_response η _)
rule.lower_pos (fun s => (rule.response_bounds hη _ s).1) (fun s => (rule.response_bounds hη _ s).2)]
rw [Finset.sum_add_distrib,hD]
congr 1
have hix := integrable_xlog_bounded (sphereProbability d) f hf hf0 hfH
rw [← integral_finsetSum _ (fun w _ => hix.bdd_mul (rule.measurable_response η w).aestronglyMeasurable
(ae_of_all _ (fun s => by
rw [Real.norm_eq_abs,abs_of_nonneg (rule.lower_pos.le.trans (rule.response_bounds hη w s).1)]
exact (rule.response_bounds hη w s).2)))]
apply integral_congr_ae
filter_upwards [] with s
rw [← Finset.sum_mul,rule.response_sum hη,one_mul]
end KernelPotential.NoisyDrift
namespace KernelPotential.NoisyDrift
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal
open Projection Projection.Support GaussianTools ReferenceGaussian Noise Bayes DriftTools.FiniteReference EntropySupport Coercivity Sphere
lemma referenceWeight_pos {d N : ℕ} (ν : Measure (Ambient d)) [IsProbabilityMeasure ν]
(rule : PositiveRule (d/16) d N) {η : ℝ} (hη : 0 < η) (w) (s) :
0 < referenceWeight ν (rule.response η w) s := by
have := rule.post_prob hη ν w
exact div_pos (rule.lower_pos.trans_le (rule.response_bounds hη w s).1)
(mul_pos (rule.mass_pos hη ν w) (clippedJ_pos _ s))
lemma positive_noisy_comparison {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) (hηL : η ≤ Scale.radius (DriftCutoff.level d H)) :
(∑ w,mass ν (rule.response η w)*Phi (post ν (rule.response η w))) ≤
Phi ν+10*(d:ℝ)+2*Real.log ((DriftCutoff.level d H:ℝ)+2)+
∫ t : Ambient d × Observation (d/16) d × Fin N,postCharge ν rule η t.2.1 t.2.2
∂joint ν (observationBase (d/16) d) (observationKernel η) rule.kernel := by
have hHp : 0 < H := lt_of_lt_of_le zero_lt_one hH
let L := DriftCutoff.level d H
have hL : 0 < L := DriftCutoff.level_pos d H
let q : Observation (d/16) d → ℝ := fun z =>
PairReference.density ν (fun j : Fin L => Scale.radius j) z.1.1 z.1.2 z.2.1 z.2.2
have hq : Measurable q := PairReference.density_measurable ν _
have hqp (z) : 0 < q z := PairReference.density_pos hL ν _ (fun j => Scale.radius_pos j) _ _ _ _
have hq1 : ∫⁻ z,ENNReal.ofReal (q z) ∂observationBase (d/16) d=1 := by
unfold observationBase
rw [lintegral_prod _ hq.ennreal_ofReal.aemeasurable]
simp_rw [q,PairReference.density_probability hL ν _ (fun j => Scale.radius_pos j)]
have : IsProbabilityMeasure (gaussianRows (d/16) d) := by unfold gaussianRows; infer_instance
simp
let x : Fin N → Ambient d → ℝ≥0∞ := fun w s => ENNReal.ofReal (referenceWeight ν (rule.response η w) s)
have hx : Measurable (Function.uncurry x) := by
apply measurable_from_prod_countable_right
intro w
exact (referenceWeight_measurable 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) (rule.mass_pos hη ν w)).ennreal_ofReal
have hxe (w s) : (x w s).toReal=referenceWeight ν (rule.response η w) s :=
ENNReal.toReal_ofReal (referenceWeight_pos ν rule hη w s).le
have hCe (z : Observation (d/16) d) (w : Fin N) : C ν (observationKernel η) x z w=
projectedReference (post ν (rule.response η w)) η z.1.1 z.1.2 z.2.1 z.2.2 :=
C_eq_projectedReference hd ν hHp hν _ (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) (rule.mass_pos hη ν w) η z
have hCf (z : Observation (d/16) d) (w : Fin N) : C ν (observationKernel η) x z w ≠ ∞ := by
rw [hCe]
have := rule.post_prob hη ν w
have hp := post_bounded ν hHp hν _ (fun s => (rule.response_bounds hη w s).2) (rule.mass_pos hη ν w)
exact projectedReference_finite _ ((Measure.absolutelyContinuous_of_le_smul hp).ae_le
(ae_unit_norm (by omega : 0 < d))) hη z
have := joint_probability ν (observationBase (d/16) d) (observationKernel η) rule.kernel
(observationKernel_measurable η) rule.measurable_kernel (observationKernel_probability hη) rule.sum_kernel
obtain ⟨Mx,hMx⟩ := reference_log_uniform_bound hd ν hHp hν rule hη
have hix : Integrable (fun t : Ambient d × Observation (d/16) d × Fin N => Real.log (x t.2.2 t.1).toReal)
(joint ν (observationBase (d/16) d) (observationKernel η) rule.kernel) := by
simp_rw [hxe]
refine Integrable.of_bound ((reference_log_measurable ν rule hη).comp (f := fun t : Ambient d × Observation (d/16) d × Fin N => (t.1,t.2.2))
(measurable_fst.prodMk measurable_snd.snd)).aestronglyMeasurable Mx ?_
exact ae_of_all _ (fun _ => hMx _ _)
obtain ⟨Mq,_hMq,hMqb⟩ := postCharge_uniform_bound hd ν hHp hν rule hη
have hiC : Integrable (fun t : Ambient d × Observation (d/16) d × Fin N =>
Real.log (1+(C ν (observationKernel η) x t.2.1 t.2.2).toReal))
(joint ν (observationBase (d/16) d) (observationKernel η) rule.kernel) := by
simp_rw [hCe]
refine Integrable.of_bound ((measurable_postCharge hd ν hν rule hη).comp (f := fun t : Ambient d × Observation (d/16) d × Fin N => t.2) measurable_snd).aestronglyMeasurable Mq ?_
exact ae_of_all _ (fun t => by change ‖postCharge ν rule η t.2.1 t.2.2‖ ≤ Mq; rw [Real.norm_eq_abs,abs_of_nonneg (hMqb _ _).1]; exact (hMqb _ _).2)
obtain ⟨hiq,hqlo⟩ := NoisyReference.joint_reference_log hd ν hHp hν rule.kernel rule.measurable_kernel
rule.sum_kernel L hL (DriftCutoff.reference_cut d hd hH) hη hηL
have hqto (z) : (ENNReal.ofReal (q z)).toReal=q z := ENNReal.toReal_ofReal (hqp z).le
have hc := log_comparison ν (observationBase (d/16) d) (observationKernel η) rule.kernel
(observationKernel_measurable η) rule.measurable_kernel (observationKernel_probability hη) rule.sum_kernel
x hx (fun w s => ENNReal.ofReal_pos.mpr (referenceWeight_pos ν rule hη w s))
(fun _ _ => ENNReal.ofReal_ne_top) (fun z => ENNReal.ofReal (q z)) hq.ennreal_ofReal
(fun z => ENNReal.ofReal_pos.mpr (hqp z)) (fun _ => ENNReal.ofReal_ne_top) hq1 hCf hix
(by simpa only [hqto] using hiq) hiC
simp_rw [hxe,hCe,hqto] at hc
rw [joint_Phi_identity hd ν hHp hν rule hη]
change (InformationTheory.klDiv ν (sphereProbability d)).toReal+_ ≤ _
unfold Phi
change B ν-10*(d:ℝ)-2*Real.log ((L:ℝ)+2) ≤ ∫ t,Real.log (q t.2.1)
∂joint ν (observationBase (d/16) d) (observationKernel η) rule.kernel at hqlo
change _ ≤ (∫ t,postCharge ν rule η t.2.1 t.2.2 ∂joint ν (observationBase (d/16) d) (observationKernel η) rule.kernel)-
∫ t,Real.log (q t.2.1) ∂joint ν (observationBase (d/16) d) (observationKernel η) rule.kernel at hc
linarith
end KernelPotential.NoisyDrift
end OAI
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