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b797a5b | 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 | 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
namespace PrimeDegreePackets
variable {Y α β : Type*} [MeasurableSpace Y]
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
end PrimeDegreePackets
end
end OAI
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