File size: 3,118 Bytes
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 | import OAI.NumberTheory.PrimePackets.ConditionalSigmaEntropy
namespace OAI
/-! # Conditional information in disintegration coordinates -/
noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped Topology
namespace PrimeDegreePackets
variable {S T α : Type*} [MeasurableSpace S] [MeasurableSpace T]
[StandardBorelSpace T] [Nonempty T] [Fintype α]
omit [StandardBorelSpace T] [Nonempty T] in
lemma measurable_fiber_observationEntropy (κ : Kernel S T) [IsMarkovKernel κ]
(P : S × T → α) (hP : ∀ a, MeasurableSet {z | P z = a}) :
Measurable (fun s => observationEntropy (κ s) (fun t => P (s,t))) := by
apply continuous_finiteShannon.measurable.comp
apply measurable_pi_iff.mpr
intro a
exact (Kernel.measurable_kernel_prodMk_left (κ := κ) (hP a)).ennreal_toReal
omit [Fintype α] in
lemma condExp_fst_observation_probability
(μ : Measure (S × T)) [IsFiniteMeasure μ]
(P : S × T → α) (hP : ∀ a, MeasurableSet {z | P z = a}) (a : α) :
μ[({z | P z = a}.indicator (fun _ => (1 : ℝ))) |
MeasurableSpace.comap Prod.fst (inferInstance : MeasurableSpace S)] =ᵐ[μ]
fun z => (μ.condKernel z.1).real {t | P (z.1,t) = a} := by
let f : S × T → ℝ := {z | P z = a}.indicator (fun _ => 1)
have hid : (fun z : S × T => (z.1,z.2)) = id := rfl
have hf : Integrable f (μ.map (fun z : S × T => (z.1,z.2))) := by
rw [hid, Measure.map_id]
exact (integrable_const (1 : ℝ)).indicator (hP a)
have H := condExp_prod_ae_eq_integral_condDistrib'
(μ := μ) (X := (Prod.fst : S × T → S)) (Y := (Prod.snd : S × T → T))
measurable_fst measurable_snd.aemeasurable hf
have hκ : condDistrib (Prod.snd : S × T → T) Prod.fst μ = μ.condKernel := by
simp only [condDistrib, hid, Measure.map_id]
rw [hκ] at H
have hI (s : S) : (∫ t, f (s,t) ∂μ.condKernel s) =
(μ.condKernel s).real {t | P (s,t) = a} := by
change (∫ t, {t | P (s,t) = a}.indicator (fun _ => (1 : ℝ)) t ∂μ.condKernel s) = _
have hs : MeasurableSet {t | P (s,t) = a} := (hP a).preimage measurable_prodMk_left
rw [integral_indicator_const _ hs]
simp only [smul_eq_mul, mul_one]
simpa only [Prod.eta, hI] using H
theorem conditionalSigmaEntropy_fst
(μ : Measure (S × T)) [IsFiniteMeasure μ]
(P : S × T → α) (hP : ∀ a, MeasurableSet {z | P z = a}) :
conditionalSigmaEntropy μ P
(MeasurableSpace.comap Prod.fst (inferInstance : MeasurableSpace S)) =
∫ s, observationEntropy (μ.condKernel s) (fun t => P (s,t)) ∂μ.fst := by
have H := ae_all_iff.mpr (fun a => condExp_fst_observation_probability μ P hP a)
unfold conditionalSigmaEntropy
calc
_ = ∫ z, observationEntropy (μ.condKernel z.1) (fun t => P (z.1,t)) ∂μ := by
apply integral_congr_ae
filter_upwards [H] with z hz
unfold observationEntropy
congr 1
exact funext hz
_ = _ := by
exact (integral_map measurable_fst.aemeasurable
(measurable_fiber_observationEntropy μ.condKernel P hP).aestronglyMeasurable).symm
end PrimeDegreePackets
end
end OAI
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