| 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 |
|
|
