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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 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 | import OAI.NumberTheory.DukePrimeDegree.Entropy.ObservationInformation
namespace OAI
/-! # Conditional entropy as the mean pointwise information
The finite observed variable has a regular conditional distribution even
when the conditioning sigma algebra is arbitrary. Lifting its underlying
space keeps the ambient and conditioning measurable structures distinct.
-/
noncomputable section
open MeasureTheory ProbabilityTheory Set
namespace DukePrimeDegree
open PrimeDegreePackets
def conditionalInformation {Y α : Type*} [MeasurableSpace Y]
(μ : Measure Y) (P : Y → α) (m : MeasurableSpace Y) (y : Y) : ℝ :=
-Real.log ((μ[({z | P z=P y}.indicator (fun _ => (1:ℝ))) | m]) y)
theorem conditionalInformation_integrable_integral {Y : Type*} (m : MeasurableSpace Y)
[mY : MeasurableSpace Y]
(μ : Measure Y) [IsFiniteMeasure μ] {N : ℕ} [NeZero N]
(P : Y → Fin N) (hP : Measurable P) (hm : m ≤ mY) :
Integrable (conditionalInformation μ P m) μ ∧
(∫ y, conditionalInformation μ P m y ∂μ) = conditionalSigmaEntropy μ P m := by
let : MeasurableSpace (ULift.{0} Y) := m.comap ULift.down
let X : Y → ULift.{0} Y := ULift.up
have hmX : MeasurableSpace.comap X (inferInstance : MeasurableSpace (ULift.{0} Y)) = m := by
change (m.comap ULift.down).comap ULift.up = m
rw [MeasurableSpace.comap_comp]
exact MeasurableSpace.comap_id
have hX : Measurable X := by
apply Measurable.of_comap_le
simpa only [hmX] using hm
let σ := μ.map (fun y => (X y,P y))
let κ := σ.condKernel
have hκ : condDistrib P X μ = κ := by rw [condDistrib]
have heq (a : Fin N) :
μ[({z | P z=a}.indicator (fun _ => (1:ℝ))) | m] =ᵐ[μ]
fun y => (κ (X y)).real {a} := by
have H := condExp_ae_eq_integral_condDistrib (μ := μ) (X := X) (Y := P)
hX hP.aemeasurable
((stronglyMeasurable_const.indicator (measurableSet_singleton a)) :
StronglyMeasurable (({a} : Set (Fin N)).indicator (fun _ : Fin N => (1:ℝ))))
((integrable_const (1:ℝ)).indicator (hP (measurableSet_singleton a)))
rw [hκ,hmX] at H
have hleft : (fun y => ({a} : Set (Fin N)).indicator (fun _ => (1:ℝ)) (P y)) =
{z | P z=a}.indicator (fun _ => (1:ℝ)) := by
funext y
simp only [Set.indicator_apply,Set.mem_singleton_iff,Set.mem_ofPred_eq]
simpa only [hleft,integral_indicator_const (1:ℝ) (measurableSet_singleton a),
smul_eq_mul,mul_one] using H
let I : ULift.{0} Y × Fin N → ℝ :=
fun z => observationInformation (κ z.1) (id : Fin N → Fin N) z.2
have hI : Measurable I := measurable_kernel_observationInformation κ Prod.snd
(fun a => measurable_snd (measurableSet_singleton a))
have hIp : conditionalInformation μ P m =ᵐ[μ] fun y => I (X y,P y) := by
filter_upwards [ae_all_iff.mpr heq] with y hy
unfold conditionalInformation I observationInformation
rw [hy (P y)]
rfl
have hIint : Integrable I σ := by
have H := integrable_kernel_observationInformation σ.fst κ Prod.snd
(fun a => measurable_snd (measurableSet_singleton a))
change Integrable I (σ.fst ⊗ₘ σ.condKernel) at H
rwa [Measure.disintegrate] at H
have hIf : Integrable (fun y => I (X y,P y)) μ :=
hIint.comp_measurable (hX.prodMk hP)
refine ⟨hIf.congr hIp.symm,?_⟩
rw [integral_congr_ae hIp]
have HI := conditionalEntropy_fst_information σ Prod.snd
(fun a => measurable_snd (measurableSet_singleton a))
rw [conditionalSigmaEntropy_fst σ Prod.snd
(fun a => measurable_snd (measurableSet_singleton a))] at HI
have HE : conditionalSigmaEntropy μ P m =
∫ y, observationEntropy (κ (X y)) (id : Fin N → Fin N) ∂μ := by
unfold conditionalSigmaEntropy observationEntropy
apply integral_congr_ae
filter_upwards [ae_all_iff.mpr heq] with y hy
congr 1
funext a
simpa only [id_eq,Set.ofPred_eq_eq_singleton] using hy a
rw [HE]
calc
_ = ∫ z, I z ∂σ :=
(integral_map (hX.prodMk hP).aemeasurable hI.aestronglyMeasurable).symm
_ = ∫ s, observationEntropy (κ s) (id : Fin N → Fin N) ∂σ.fst := HI.symm
_ = ∫ z, observationEntropy (κ z.1) (id : Fin N → Fin N) ∂σ :=
integral_map measurable_fst.aemeasurable
(measurable_fiber_observationEntropy κ Prod.snd
(fun a => measurable_snd (measurableSet_singleton a))).aestronglyMeasurable
_ = _ := integral_map (hX.prodMk hP).aemeasurable
((measurable_fiber_observationEntropy κ Prod.snd
(fun a => measurable_snd (measurableSet_singleton a))).comp measurable_fst).aestronglyMeasurable
end DukePrimeDegree
end
end OAI
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