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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 79 80 81 82 83 84 85 86 | import OAI.NumberTheory.DukePrimeDegree.Entropy.ConditionalNameCovers
namespace OAI
/-! # Finite entropy conditioned on a measurable observation -/
noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped BigOperators
namespace DukePrimeDegree
open PrimeDegreePackets
theorem conditionalSigmaEntropy_comap_eq_integral
{S Y α : Type*} [MeasurableSpace S] [MeasurableSpace Y]
[StandardBorelSpace Y] [Nonempty Y] [Fintype α]
(μ : Measure Y) [IsProbabilityMeasure μ]
(F : Y → S) (hF : Measurable F)
(P : Y → α) (hP : ∀ a, MeasurableSet {y | P y=a}) :
conditionalSigmaEntropy μ P (MeasurableSpace.comap F inferInstance) =
∫ s, observationEntropy (condDistrib id F μ s) P ∂μ.map F := by
have hc (a : α) :
μ[({y | P y=a}.indicator (fun _ => (1 : ℝ))) |
MeasurableSpace.comap F inferInstance] =ᵐ[μ]
fun y => (condDistrib id F μ (F y)).real {z | P z=a} := by
have H := condExp_ae_eq_integral_condDistrib_id (μ := μ) hF
((integrable_const (μ := μ) (1 : ℝ)).indicator (hP a))
have hi (s : S) : (∫ z, {y | P y=a}.indicator (fun _ => (1 : ℝ)) z
∂condDistrib id F μ s) = (condDistrib id F μ s).real {z | P z=a} := by
rw [integral_indicator_const _ (hP a)]
simp only [smul_eq_mul,mul_one]
simpa only [hi] using H
have hall := ae_all_iff.mpr hc
unfold conditionalSigmaEntropy
calc
_ = ∫ y, observationEntropy (condDistrib id F μ (F y)) P ∂μ := by
apply integral_congr_ae
filter_upwards [hall] with y hy
unfold observationEntropy
congr 1
exact funext hy
_ = _ := (integral_map hF.aemeasurable
(integrable_kernel_entropy (μ.map F) (condDistrib id F μ) P hP).aestronglyMeasurable).symm
theorem conditionalSigmaEntropy_comap_nonneg
{S Y α : Type*} [MeasurableSpace S] [MeasurableSpace Y]
[StandardBorelSpace Y] [Nonempty Y] [Fintype α]
(μ : Measure Y) [IsProbabilityMeasure μ]
(F : Y → S) (hF : Measurable F)
(P : Y → α) (hP : ∀ a, MeasurableSet {y | P y=a}) :
0 ≤ conditionalSigmaEntropy μ P (MeasurableSpace.comap F inferInstance) := by
rw [conditionalSigmaEntropy_comap_eq_integral μ F hF P hP]
exact integral_nonneg fun s => observationEntropy_nonneg (condDistrib id F μ s) P hP
theorem conditionalSigmaEntropy_comap_le_reference_bad_sets
{S Y α : Type*} [MeasurableSpace S] [MeasurableSpace Y]
[StandardBorelSpace Y] [Nonempty Y]
[MeasurableSpace α] [MeasurableSingletonClass α]
[Fintype α] [Nonempty α] [DecidableEq α]
(μ : Measure Y) [IsProbabilityMeasure μ]
(F : Y → S) (hF : Measurable F) (N M : ℕ)
(P : Y → Fin N → α) (hP : Measurable P)
(r : S → Y → Fin M) (hr : ∀ s, Measurable (r s))
(v : S → Fin M → Fin N → α) (B : Fin N → Set Y)
(hB : ∀ i, MeasurableSet (B i))
(hbad : ∀ᵐ s ∂μ.map F, ∀ i, ∀ᵐ y ∂condDistrib id F μ s,
P y i ≠ v s (r s y) i → y ∈ B i)
{δ : ℝ} (hδ : 0 < δ) (hδ1 : δ < 1) :
conditionalSigmaEntropy μ P (MeasurableSpace.comap F inferInstance) ≤
Real.log M+(N : ℝ)*(-Real.log (1-δ))+
(∑ i : Fin N, μ.real (B i)) *
(Real.log (Fintype.card α)-Real.log δ) := by
rw [conditionalSigmaEntropy_comap_eq_integral μ F hF P
(fun a => hP (measurableSet_singleton a))]
have H := integral_nameEntropy_le_reference_bad_sets (μ.map F) (condDistrib id F μ)
N M P hP r hr v B hB hbad hδ hδ1
have hm : condDistrib id F μ ∘ₘ (μ.map F)=μ := by
simpa only [Measure.map_id] using condDistrib_comp_map hF.aemeasurable aemeasurable_id
simpa only [hm] using H
end DukePrimeDegree
end
end OAI
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