File size: 3,518 Bytes
a44267b | 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 | import OAI.NumberTheory.PrimePackets.ConditionalSigmaEntropy
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen
namespace OAI
/-! # Conditioning on more measurable information decreases entropy
Conditional Jensen and the tower property apply directly to the actual
conditional expectations defining finite-observation entropy.
-/
noncomputable section
open MeasureTheory Set
open scoped BigOperators
namespace DukePrimeDegree
open PrimeDegreePackets
lemma integrable_negMulLog_of_unit_bounds {Y : Type*} [mY : MeasurableSpace Y]
(μ : Measure Y) [IsFiniteMeasure μ] {f : Y → ℝ}
(hf : AEStronglyMeasurable f μ) (hb : ∀ᵐ y ∂μ, f y ∈ Icc 0 1) :
Integrable (fun y => Real.negMulLog (f y)) μ := by
apply (integrable_const (1:ℝ)).mono'
(Real.continuous_negMulLog.comp_aestronglyMeasurable hf)
filter_upwards [hb] with y hy
rw [Real.norm_eq_abs,abs_of_nonneg (Real.negMulLog_nonneg hy.1 hy.2)]
have H := Real.negMulLog_le_one_sub_self hy.1
linarith [hy.1]
lemma conditionalProbability_unit_bounds {Y : Type*} [mY : MeasurableSpace Y]
(μ : Measure Y) (S : Set Y) (m : MeasurableSpace Y) :
∀ᵐ y ∂μ, (μ[S.indicator (fun _ => (1:ℝ)) | m]) y ∈ Icc 0 1 := by
have hn : 0 ≤ᵐ[μ] S.indicator (fun _ => (1:ℝ)) := by
apply ae_of_all
intro y
by_cases hy : y ∈ S <;> simp [hy]
have hu : ∀ᵐ y ∂μ, S.indicator (fun _ => (1:ℝ)) y ≤ 1 := by
apply ae_of_all
intro y
by_cases hy : y ∈ S <;> simp [hy]
filter_upwards [condExp_nonneg (m := m) hn,
condExp_le_nonneg_const (m := m) (show (0:ℝ)≤1 by norm_num) hu] with y hy hz
exact ⟨hy,hz⟩
theorem conditionalSigmaEntropy_antitone {Y α : Type*} [mY : MeasurableSpace Y] [Fintype α]
(μ : Measure Y) [IsFiniteMeasure μ] (P : Y → α)
(hP : ∀ a, MeasurableSet {y | P y=a})
{m n : MeasurableSpace Y} (hmn : m ≤ n) (hn : n ≤ mY) :
conditionalSigmaEntropy (mY := mY) μ P n ≤ conditionalSigmaEntropy (mY := mY) μ P m := by
have hm := hmn.trans hn
let g (a : α) : Y → ℝ := {y | P y=a}.indicator (fun _ => 1)
have hg a : Integrable (g a) μ := (integrable_const (1:ℝ)).indicator (hP a)
have hi (s : MeasurableSpace Y) (hs : s ≤ mY) (a : α) :
Integrable (fun y => Real.negMulLog ((μ[g a | s]) y)) μ :=
integrable_negMulLog_of_unit_bounds (mY := mY) μ
(stronglyMeasurable_condExp.mono hs).aestronglyMeasurable
(conditionalProbability_unit_bounds (mY := mY) μ {y | P y=a} s)
unfold conditionalSigmaEntropy finiteShannon
rw [integral_finsetSum _ (fun a _ => hi n hn a),
integral_finsetSum _ (fun a _ => hi m hm a)]
apply Finset.sum_le_sum
intro a _
have hJ := Real.concaveOn_negMulLog.condExp_map_le (μ := μ) (f := μ[g a | n]) hm
(Real.continuous_negMulLog.upperSemicontinuous.upperSemicontinuousOn _)
((conditionalProbability_unit_bounds (mY := mY) μ {y | P y=a} n).mono fun _ h => h.1)
isClosed_Ici integrable_condExp (hi n hn a)
have hTower := condExp_condExp_of_le (μ := μ) (f := g a) hmn hn
have hle : (μ[(fun y => Real.negMulLog ((μ[g a | n]) y)) | m]) ≤ᵐ[μ]
(fun y => Real.negMulLog ((μ[g a | m]) y)) := by
filter_upwards [hJ,hTower] with y hy ht
simpa only [Function.comp_def,ht] using hy
calc
_ = ∫ y, (μ[(fun z => Real.negMulLog ((μ[g a | n]) z)) | m]) y ∂μ :=
(integral_condExp hm).symm
_ ≤ _ := integral_mono_ae integrable_condExp (hi m hm a) hle
end DukePrimeDegree
end
end OAI
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