| import OAI.NumberTheory.PrimePackets.ObservationRefinement |
|
|
| namespace OAI |
|
|
| /-! |
| # Entropy-rate comparison from one-step conditional entropy |
|
|
| The finite-block estimate here is the approximation step needed when |
| replacing finite observations by successively finer geometric partitions. |
| -/ |
|
|
| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped BigOperators Topology |
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| namespace PrimeDegreePackets |
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|
| variable {Y α β : Type*} [MeasurableSpace Y] |
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|
| lemma observationEntropy_unique [Fintype α] [Unique α] |
| (μ : Measure Y) [IsProbabilityMeasure μ] (P : Y → α) : |
| observationEntropy μ P = 0 := by |
| have hP : P = fun _ => default := funext (fun y => Subsingleton.elim _ _) |
| rw [hP] |
| simp [observationEntropy, finiteShannon] |
|
|
| omit [MeasurableSpace Y] in |
| lemma orbitName_one (T : Y → Y) (P : Y → α) : |
| orbitName T P 1 = fun y (_ : Fin 1) => P y := by |
| funext y j |
| have hj : j = 0 := Subsingleton.elim _ _ |
| subst j |
| rfl |
|
|
| lemma conditionalOrbitEntropy_one [Fintype α] [Fintype β] |
| (μ : Measure Y) (T : Y → Y) (P : Y → α) (Q : Y → β) : |
| conditionalObservationEntropy μ (orbitName T P 1) (orbitName T Q 1) = |
| conditionalObservationEntropy μ P Q := by |
| rw [orbitName_one, orbitName_one] |
| exact conditionalObservationEntropy_injective μ P Q |
| (fun a (_ : Fin 1) => a) (fun b (_ : Fin 1) => b) |
| (fun _ _ h => congrFun h 0) (fun _ _ h => congrFun h 0) |
|
|
| theorem conditionalOrbitEntropy_le [Fintype α] [Fintype β] |
| (μ : Measure Y) [IsProbabilityMeasure μ] {T : Y → Y} |
| (hT : MeasurePreserving T μ μ) (P : Y → α) (Q : Y → β) |
| (hP : ∀ a, MeasurableSet {y | P y = a}) |
| (hQ : ∀ b, MeasurableSet {y | Q y = b}) (N : ℕ) : |
| conditionalObservationEntropy μ (orbitName T P N) (orbitName T Q N) ≤ |
| (N : ℝ) * conditionalObservationEntropy μ P Q := by |
| induction N with |
| | zero => |
| let : Unique ((Fin 0 → β) × (Fin 0 → α)) := |
| uniqueOfSubsingleton (default, default) |
| unfold conditionalObservationEntropy |
| rw [observationEntropy_unique, observationEntropy_unique] |
| simp |
| | succ N ih => |
| have hPN := orbitName_measurable_fiber hT.measurable hP N |
| have hQN := orbitName_measurable_fiber hT.measurable hQ N |
| have hP1 := orbitName_measurable_fiber hT.measurable hP 1 |
| have hQ1 := orbitName_measurable_fiber hT.measurable hQ 1 |
| have heq : |
| conditionalObservationEntropy μ (orbitName T P (N+1)) (orbitName T Q (N+1)) = |
| conditionalObservationEntropy μ |
| (fun y => (orbitName T P N y, orbitName T P 1 (T^[N] y))) |
| (fun y => (orbitName T Q N y, orbitName T Q 1 (T^[N] y))) := by |
| rw [orbitName_add T P N 1, orbitName_add T Q N 1] |
| exact conditionalObservationEntropy_injective μ _ _ |
| (Fin.appendEquiv N 1) (Fin.appendEquiv N 1) |
| (Fin.appendEquiv N 1).injective (Fin.appendEquiv N 1).injective |
| rw [heq] |
| have H := conditionalObservationEntropy_pair_pair_le μ |
| (orbitName T P N) (fun y => orbitName T P 1 (T^[N] y)) |
| (orbitName T Q N) (fun y => orbitName T Q 1 (T^[N] y)) |
| hPN (fun a => (hP1 a).preimage (hT.measurable.iterate N)) |
| hQN (fun b => (hQ1 b).preimage (hT.measurable.iterate N)) |
| rw [conditionalObservationEntropy_comp_preserving μ _ _ hP1 hQ1 (hT.iterate N), |
| conditionalOrbitEntropy_one] at H |
| refine H.trans ?_ |
| have H' := add_le_add ih (le_refl (conditionalObservationEntropy μ P Q)) |
| simpa only [Nat.cast_add, Nat.cast_one, add_mul, one_mul] using H' |
|
|
| /-- A finite observation approximates entropy rates up to its one-step |
| conditional entropy. This is the standard finite-partition approximation bound. -/ |
| theorem observationEntropyRate_le_add_conditional [Fintype α] [Fintype β] |
| (μ : Measure Y) [IsProbabilityMeasure μ] {T : Y → Y} |
| (hT : MeasurePreserving T μ μ) (P : Y → α) (Q : Y → β) |
| (hP : ∀ a, MeasurableSet {y | P y = a}) |
| (hQ : ∀ b, MeasurableSet {y | Q y = b}) : |
| observationEntropyRate μ T P ≤ |
| observationEntropyRate μ T Q + conditionalObservationEntropy μ P Q := by |
| have H : ∀ᶠ N : ℕ in atTop, |
| observationEntropy μ (orbitName T P N) / (N : ℝ) ≤ |
| observationEntropy μ (orbitName T Q N) / (N : ℝ) + |
| conditionalObservationEntropy μ P Q := by |
| filter_upwards [eventually_ge_atTop 1] with N hN |
| have hn : (N : ℝ) ≠ 0 := by exact_mod_cast (show N ≠ 0 by omega) |
| have H₁ := observationEntropy_le_add_conditional μ |
| (orbitName T P N) (orbitName T Q N) |
| (orbitName_measurable_fiber hT.measurable hP N) |
| (orbitName_measurable_fiber hT.measurable hQ N) |
| have H₂ := conditionalOrbitEntropy_le μ hT P Q hP hQ N |
| calc |
| _ ≤ (observationEntropy μ (orbitName T Q N) + |
| (N : ℝ) * conditionalObservationEntropy μ P Q) / (N : ℝ) := |
| div_le_div_of_nonneg_right (H₁.trans (add_le_add le_rfl H₂)) (Nat.cast_nonneg N) |
| _ = _ := by rw [add_div]; congr 1; field_simp |
| exact le_of_tendsto_of_tendsto (tendsto_observationEntropyRate μ hT P hP) |
| ((tendsto_observationEntropyRate μ hT Q hQ).add tendsto_const_nhds) H |
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|
| end PrimeDegreePackets |
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| end |
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| end OAI |
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