File size: 4,366 Bytes
f61e547 | 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 | import OAI.Probability.DirectionalWalk.ThreeChunks
namespace OAI
open MeasureTheory ProbabilityTheory Filter Preorder
open scoped ENNReal BigOperators Topology
namespace DirectionalZeroOne
open scoped Classical
section EntropyLawTransforms
variable {Ω Ξ α β γ : Type*} [Countable Ω] [Countable Ξ] [Countable α] [Countable β]
[Countable γ] [MeasurableSpace Ω] [MeasurableSpace Ξ] [MeasurableSpace α]
[MeasurableSpace β] [MeasurableSpace γ] [MeasurableSingletonClass Ω]
[MeasurableSingletonClass Ξ] [MeasurableSingletonClass α] [MeasurableSingletonClass β]
[MeasurableSingletonClass γ]
omit [Countable α] [MeasurableSingletonClass α] in
lemma informationOf_map (μ : Measure Ω) (F : Ω → Ξ) (X : Ξ → α) :
informationOf (μ.map F) X ∘ F = informationOf μ (X ∘ F) := by
funext ω
simp only [informationOf,Measure.map_map (measurable_of_countable _) (measurable_of_countable _),
Function.comp_apply]
omit [Countable α] [MeasurableSingletonClass α] in
lemma entropyOf_map (μ : Measure Ω) (F : Ω → Ξ) (X : Ξ → α) :
entropyOf (μ.map F) X = entropyOf μ (X ∘ F) := by
unfold entropyOf
rw [integral_map_of_stronglyMeasurable (measurable_of_countable _)
(measurable_of_countable _).stronglyMeasurable]
exact congrArg (fun f => ∫ ω, f ω ∂μ) (informationOf_map μ F X)
omit [Countable α] [Countable β] [MeasurableSingletonClass α] [MeasurableSingletonClass β] in
lemma conditionedEntropy_map (μ : Measure Ω) (F : Ω → Ξ) (X : Ξ → α) (Y : Ξ → β) :
conditionedEntropy (μ.map F) X Y = conditionedEntropy μ (X ∘ F) (Y ∘ F) := by
unfold conditionedEntropy conditionedInformation
rw [integral_map_of_stronglyMeasurable (measurable_of_countable _)
(measurable_of_countable _).stronglyMeasurable]
apply integral_congr_ae
filter_upwards [] with ω
change (informationOf (μ.map F) (fun ξ => (Y ξ,X ξ)) ∘ F) ω -
(informationOf (μ.map F) Y ∘ F) ω = _
rw [informationOf_map,informationOf_map]
rfl
omit [Countable α] [Countable β] in
lemma informationOf_same_fibers (μ : Measure Ω) (X : Ω → α) (Y : Ω → β)
(h : ∀ ω ξ, X ω = X ξ ↔ Y ω = Y ξ) : informationOf μ X = informationOf μ Y := by
funext ω
rw [informationOf_fiber,informationOf_fiber]
congr 3
ext ξ
exact h ξ ω
omit [Countable α] [Countable β] [Countable γ] in
lemma conditionedInformation_same_conditioner (μ : Measure Ω) (X : Ω → α) (Y : Ω → β)
(Z : Ω → γ) (h : ∀ ω ξ, Y ω = Y ξ ↔ Z ω = Z ξ) :
conditionedInformation μ X Y = conditionedInformation μ X Z := by
have hp := informationOf_same_fibers μ (fun ω => (Y ω,X ω)) (fun ω => (Z ω,X ω))
(fun ω ξ => by simp only [Prod.mk.injEq,h ω ξ])
funext ω
simp only [conditionedInformation,hp,informationOf_same_fibers μ Y Z h]
lemma conditionedEntropy_monotone_function (μ : Measure Ω) [IsProbabilityMeasure μ]
[Nonempty α] (X : Ω → α) (Y : Ω → β) (f : β → γ)
(hi : Integrable (informationOf μ X) μ) :
conditionedEntropy μ X Y ≤ conditionedEntropy μ X (f ∘ Y) := by
have hh := conditionedEntropy_le_conditionedEntropy μ X (f ∘ Y) Y hi
have he : conditionedInformation μ X (fun ω => ((f ∘ Y) ω,Y ω)) =
conditionedInformation μ X Y := by
apply conditionedInformation_same_conditioner
intro ω ξ
simp only [Function.comp_apply,Prod.mk.injEq]
exact ⟨And.right,fun h => ⟨congrArg f h,h⟩⟩
simpa only [conditionedEntropy,he] using hh
omit [Countable α] [MeasurableSingletonClass α] in
lemma entropyOf_prod_fst (μ : Measure Ω) (ν : Measure Ξ) [IsProbabilityMeasure μ]
[IsProbabilityMeasure ν] (X : Ω → α) : entropyOf (μ.prod ν) (X ∘ Prod.fst) = entropyOf μ X := by
rw [← entropyOf_map,Measure.map_fst_prod,measure_univ,one_smul]
omit [Countable α] [Countable β] [MeasurableSingletonClass α] [MeasurableSingletonClass β] in
lemma conditionedEntropy_prod_fst (μ : Measure Ω) (ν : Measure Ξ) [IsProbabilityMeasure μ]
[IsProbabilityMeasure ν] (X : Ω → α) (Y : Ω → β) :
conditionedEntropy (μ.prod ν) (X ∘ Prod.fst) (Y ∘ Prod.fst) = conditionedEntropy μ X Y := by
rw [← conditionedEntropy_map,Measure.map_fst_prod,measure_univ,one_smul]
end EntropyLawTransforms
end DirectionalZeroOne
end OAI
|