| 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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|
