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b224933 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 | import OAI.MeasureTheory.SelfSimilar.WordAverage
namespace OAI
universe uα uβ uX
namespace EntropyRateDimension.MeasureEntropy
open MeasureTheory
open scoped BigOperators ENNReal
theorem finite_entropy_integral {X : Type uX} {β : Type uβ} [MeasurableSpace X]
(μ : Measure X) [IsProbabilityMeasure μ] (s : Finset β) (p : X → β → ℝ)
(hm : ∀ b ∈ s, Measurable (fun y => p y b))
(hp : ∀ b ∈ s, ∀ y, 0 ≤ p y b ∧ p y b ≤ 1) :
(∫ y, ∑ b ∈ s, Real.negMulLog (p y b) ∂μ) ≤
∑ b ∈ s, Real.negMulLog (∫ y, p y b ∂μ) := by
have hi (b) (hb : b ∈ s) : Integrable (fun y => p y b) μ :=
(integrable_const (1 : ℝ)).mono' (hm b hb).aestronglyMeasurable
(Filter.Eventually.of_forall (fun y => by rw [Real.norm_eq_abs,abs_of_nonneg (hp b hb y).1]; exact (hp b hb y).2))
have hni (b) (hb : b ∈ s) : Integrable (fun y => Real.negMulLog (p y b)) μ := by
apply (integrable_const (1 : ℝ)).mono'
(Real.continuous_negMulLog.measurable.comp (hm b hb)).aestronglyMeasurable
filter_upwards [] with y
simp only [Function.comp_apply]
rw [Real.norm_eq_abs,abs_of_nonneg (Real.negMulLog_nonneg (hp b hb y).1 (hp b hb y).2)]
exact (Real.negMulLog_le_one_sub_self (hp b hb y).1).trans (sub_le_self _ (hp b hb y).1)
rw [integral_finsetSum s hni]
apply Finset.sum_le_sum
intro b hb
exact Real.concaveOn_negMulLog.le_map_integral Real.continuous_negMulLog.continuousOn
isClosed_Ici (Filter.Eventually.of_forall (fun y => (hp b hb y).1)) (hi b hb) (hni b hb)
theorem random_observation_entropy {X : Type uX} {α : Type uα} [MeasurableSpace X] [Fintype α]
(μ : Measure X) [IsProbabilityMeasure μ] (ν : Measure ℝ) [IsProbabilityMeasure ν]
(p : α → ℝ) (hp : ∀ i, 0 ≤ p i) (hp1 : ∑ i, p i=1)
(F : α → X → ℝ) (hF : ∀ i, Measurable (F i))
(hm : ν=∑ i, ENNReal.ofReal (p i) • μ.map (F i))
(f : ℝ → ℤ) (hf : Measurable f) (s : Finset ℤ)
(hs : ∀ i, ∀ᵐ y ∂μ, f (F i y) ∈ s) :
(∫ y, FiniteEntropy.observableEntropy Finset.univ p (fun i => f (F i y)) ∂μ) ≤ entropy ν f := by
classical
let K : X → ℤ → ℝ := fun y b => FiniteEntropy.mass Finset.univ p (fun i => f (F i y)) b
have hk (y : X) (b : ℤ) : K y b=∑ i, if f (F i y)=b then p i else 0 := by
simp only [K,FiniteEntropy.mass,Finset.sum_filter]
have hmK (b : ℤ) : Measurable (fun y => K y b) := by
simp_rw [hk]
exact Finset.measurable_sum _ (fun i _ => measurable_const.ite
(measurableSet_eq_fun (hf.comp (hF i)) measurable_const) measurable_const)
have hpK (y : X) (b : ℤ) : 0 ≤ K y b ∧ K y b ≤ 1 := by
refine ⟨FiniteEntropy.mass_nonneg _ _ _ (fun i _ => hp i) _,?_⟩
change (∑ i ∈ Finset.univ.filter (fun i => f (F i y)=b), p i) ≤ 1
rw [← hp1]
exact Finset.sum_le_sum_of_subset_of_nonneg (Finset.filter_subset _ _) (fun i _ _ => hp i)
have hmi (i : α) : IsProbabilityMeasure (μ.map (F i)) :=
(Measure.isProbabilityMeasure_map_iff (hF i).aemeasurable).2 inferInstance
let := hmi
have hsupport : ∀ᵐ x ∂ν, f x ∈ s := by
rw [hm]
rw [ae_iff]
simp only [Measure.coe_finsetSum,Finset.sum_apply,Measure.smul_apply]
apply Finset.sum_eq_zero
intro i _
have hh : ∀ᵐ x ∂μ.map (F i), f x ∈ s :=
(ae_map_iff (hF i).aemeasurable (hf (s.finite_toSet.measurableSet))).mpr (hs i)
rw [ae_iff] at hh
rw [hh,smul_zero]
have hmasses (b : ℤ) : (∫ y, K y b ∂μ)=mass ν f b := by
rw [mass_mixture ν (fun i => μ.map (F i)) p hp hm f]
simp_rw [hk]
have hint (i : α) : Integrable (fun y => if f (F i y)=b then p i else 0) μ := by
have hh : MeasurableSet {y | f (F i y)=b} :=
measurableSet_eq_fun (hf.comp (hF i)) (measurable_const (a := b))
have h := (integrable_const (p i) : Integrable (fun _ : X => p i) μ).indicator hh
exact h.congr (Filter.Eventually.of_forall (fun y => by
rw [Set.indicator_apply]
rfl))
rw [integral_finsetSum Finset.univ (fun i _ => hint i)]
apply Finset.sum_congr rfl
intro i _
rw [mass_map μ _ (hF i) f hf]
have hind : (fun y => if f (F i y)=b then p i else 0) =
({y | f (F i y)=b} : Set X).indicator (fun _ => p i) := by
funext y
rw [Set.indicator_apply]
rfl
rw [hind,integral_indicator_const (p i) (show MeasurableSet {y | f (F i y)=b} from
measurableSet_eq_fun (hf.comp (hF i)) (measurable_const (a := b)))]
simp only [smul_eq_mul,mass,Function.comp_apply,mul_comm]
have he : (fun y => FiniteEntropy.observableEntropy Finset.univ p (fun i => f (F i y))) =ᵐ[μ]
fun y => ∑ b ∈ s, Real.negMulLog (K y b) := by
filter_upwards [ae_all_iff.mpr hs] with y hy
exact FiniteEntropy.observableEntropy_on_superset Finset.univ s p _ (fun i _ => hy i)
rw [integral_congr_ae he,entropy_eq_sum ν f s hsupport]
simpa only [hmasses] using finite_entropy_integral μ s K (fun b _ => hmK b) (fun b _ y => hpK y b)
theorem random_observation_integrable {X : Type uX} {α : Type uα} [MeasurableSpace X] [Fintype α]
(μ : Measure X) [IsProbabilityMeasure μ] (p : α → ℝ)
(hp : ∀ i, 0 ≤ p i) (hp1 : ∑ i, p i=1)
(f : α → X → ℤ) (hf : ∀ i, Measurable (f i)) (s : Finset ℤ)
(hs : ∀ i, ∀ᵐ y ∂μ, f i y ∈ s) :
Integrable (fun y => FiniteEntropy.observableEntropy Finset.univ p (fun i => f i y)) μ := by
classical
let K : X → ℤ → ℝ := fun y b => FiniteEntropy.mass Finset.univ p (fun i => f i y) b
have hk (y : X) (b : ℤ) : K y b=∑ i, if f i y=b then p i else 0 := by
simp only [K,FiniteEntropy.mass,Finset.sum_filter]
have hmK (b : ℤ) : Measurable (fun y => K y b) := by
simp_rw [hk]
exact Finset.measurable_sum _ (fun i _ => measurable_const.ite
(measurableSet_eq_fun (hf i) measurable_const) measurable_const)
have hpK (y : X) (b : ℤ) : 0 ≤ K y b ∧ K y b ≤ 1 := by
refine ⟨FiniteEntropy.mass_nonneg _ _ _ (fun i _ => hp i) _,?_⟩
change (∑ i ∈ Finset.univ.filter (fun i => f i y=b), p i) ≤ 1
rw [← hp1]
exact Finset.sum_le_sum_of_subset_of_nonneg (Finset.filter_subset _ _) (fun i _ _ => hp i)
have hi (b : ℤ) : Integrable (fun y => Real.negMulLog (K y b)) μ := by
apply (integrable_const (1 : ℝ)).mono'
(Real.continuous_negMulLog.measurable.comp (hmK b)).aestronglyMeasurable
filter_upwards [] with y
simp only [Function.comp_apply]
rw [Real.norm_eq_abs,abs_of_nonneg (Real.negMulLog_nonneg (hpK y b).1 (hpK y b).2)]
exact (Real.negMulLog_le_one_sub_self (hpK y b).1).trans (sub_le_self _ (hpK y b).1)
apply (integrable_finsetSum s (fun b _ => hi b)).congr
filter_upwards [ae_all_iff.mpr hs] with y hy
exact (FiniteEntropy.observableEntropy_on_superset Finset.univ s p _ (fun i _ => hy i)).symm
end EntropyRateDimension.MeasureEntropy
end OAI
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