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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 | import OAI.NumberTheory.DukePrimeDegree.Entropy.ConditionalMapEntropy
import OAI.NumberTheory.DukePrimeDegree.Entropy.BilateralConditionalSupport
import OAI.NumberTheory.DukePrimeDegree.Entropy.PredecessorSymbols
namespace OAI
/-! # Finite recoding within the fibers of a countable observation -/
noncomputable section
open MeasureTheory ProbabilityTheory Set
namespace DukePrimeDegree
open PrimeDegreePackets
lemma observationEntropy_eq_of_ae_eq {Y α : Type*} [MeasurableSpace Y]
[Fintype α] (ν : Measure Y) (P R : Y → α) (h : P =ᵐ[ν] R) :
observationEntropy ν P=observationEntropy ν R := by
unfold observationEntropy
congr 1
funext a
apply congrArg ENNReal.toReal
apply measure_congr
filter_upwards [h] with y hy
simp only [hy]
theorem conditionalSigmaEntropy_le_fiber_recoding
{Y S α β : Type*} [MeasurableSpace Y] [StandardBorelSpace Y] [Nonempty Y]
[MeasurableSpace S] [MeasurableEq S]
[MeasurableSpace α] [MeasurableSingletonClass α] [Fintype α]
[MeasurableSpace β] [MeasurableSingletonClass β] [Fintype β]
(μ : Measure Y) [IsProbabilityMeasure μ]
(F : Y → S) (hF : Measurable F)
(P : Y → α) (hP : Measurable P) (B : Y → β) (hB : Measurable B)
(f : S → β → α) (hf : ∀ x, P x=f (F x) (B x)) :
conditionalSigmaEntropy μ P (MeasurableSpace.comap F inferInstance) ≤
conditionalSigmaEntropy μ B (MeasurableSpace.comap F inferInstance) := by
rw [conditionalSigmaEntropy_comap_eq_integral μ F hF P
(fun a => hP (measurableSet_singleton a)),
conditionalSigmaEntropy_comap_eq_integral μ F hF B
(fun a => hB (measurableSet_singleton a))]
apply integral_mono_ae
(integrable_kernel_entropy (μ.map F) (condDistrib id F μ) P
(fun a => hP (measurableSet_singleton a)))
(integrable_kernel_entropy (μ.map F) (condDistrib id F μ) B
(fun a => hB (measurableSet_singleton a)))
filter_upwards [ae_condDistrib_self_fiber μ F hF] with s hs
have he : P =ᵐ[condDistrib id F μ s] (f s ∘ B) := by
filter_upwards [hs] with y hy
simpa only [Function.comp_apply,hy] using hf y
rw [observationEntropy_eq_of_ae_eq _ P (f s ∘ B) he]
apply observationEntropy_le_of_refines _ _ B
(fun a => ((measurable_of_finite (f s)).comp hB) (measurableSet_singleton a))
(fun a => hB (measurableSet_singleton a))
intro x y hxy
exact congrArg (f s) hxy
theorem predecessor_clip_conditionalEntropy_le
{Y : Type*} [MeasurableSpace Y] [StandardBorelSpace Y] [Nonempty Y]
(μ : Measure Y) [IsProbabilityMeasure μ]
(T : Y → Y) (hT : Measurable T) (Q : Y → ℕ) (hQ : Measurable Q)
{M : ℕ} (B : Y → Fin M) (hB : Measurable B)
(decode : ℕ → Fin M → ℕ) (hdecode : ∀ x, decode (Q (T x)) (B x)=Q x)
(L : ℕ) :
conditionalSigmaEntropy μ (heightClip Q L)
(MeasurableSpace.comap (Q ∘ T) inferInstance) ≤
conditionalSigmaEntropy μ B (MeasurableSpace.comap (Q ∘ T) inferInstance) := by
apply conditionalSigmaEntropy_le_fiber_recoding μ (Q ∘ T) (hQ.comp hT)
(heightClip Q L) (heightClip_measurable hQ L) B hB
(fun q b => heightClip id L (decode q b))
intro x
change heightClip id L (Q x)=heightClip id L (decode (Q (T x)) (B x))
rw [hdecode]
end DukePrimeDegree
end
end OAI
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