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180bc03 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 | import Mathlib
import OAI.Probability.Ballisticity.Entropy.EntropyVariational
namespace OAI
section
open MeasureTheory ProbabilityTheory InformationTheory
open scoped ENNReal Classical BigOperators
namespace DirectionalTransience.Entropy
variable {X I : Type*} [MeasurableSpace X]
lemma restricted_exp_test (μ ν : Measure X) [IsProbabilityMeasure μ]
[IsProbabilityMeasure ν] (hkl : klDiv μ ν ≠ ∞)
(E : Set X) (hE : MeasurableSet E)
(F : X → ℝ) (hF : Measurable F) (hB : ∃ B : ℝ, ∀ x, |F x| ≤ B) :
(∫ x, F x ∂μ.restrict E) - (∫ x, Real.exp (F x) ∂ν.restrict E) + ν.real E ≤
(klDiv μ ν).toReal := by
let G : X → ℝ := E.indicator F
have hG : Measurable G := hF.indicator hE
have hGB : ∃ B : ℝ, ∀ x, |G x| ≤ B := by
obtain ⟨B,hB⟩ := hB
refine ⟨max B 0,fun x => ?_⟩
by_cases hx : x ∈ E
· simpa [G,hx] using (hB x).trans (le_max_left B 0)
· simp [G,hx]
have h := entropy_exp_test_le μ ν hkl G (integrable_of_bounded hG hGB)
(exp_integrable_of_bounded hG hGB)
have heq : (fun x => Real.exp (G x)) =
fun x => E.indicator (fun x => Real.exp (F x)) x + Eᶜ.indicator (fun _ => (1:ℝ)) x := by
funext x
by_cases hx : x ∈ E <;> simp [G,hx]
have hei := (exp_integrable_of_bounded (μ:=ν) hF hB).indicator hE
have hci := (integrable_const (μ:=ν) (1:ℝ)).indicator hE.compl
rw [heq, integral_add hei hci, integral_indicator hE,
integral_indicator_const 1 hE.compl, smul_eq_mul, mul_one,
probReal_compl_eq_one_sub hE] at h
rw [integral_indicator hE] at h
linarith
noncomputable def activeSum (s : Finset I) (μ : I → Measure X) (E : I → Set X) : Measure X :=
∑ i ∈ s, (μ i).restrict (E i)
instance activeSum_finite (s : Finset I) (μ : I → Measure X)
[∀ i, IsFiniteMeasure (μ i)] (E : I → Set X) : IsFiniteMeasure (activeSum s μ E) := by
unfold activeSum
infer_instance
lemma activeSum_integral (s : Finset I) (μ : I → Measure X)
[∀ i, IsFiniteMeasure (μ i)] (E : I → Set X)
(F : X → ℝ) (hF : Measurable F) (hB : ∃ B : ℝ, ∀ x, |F x| ≤ B) :
(∫ x, F x ∂activeSum s μ E) = ∑ i ∈ s, ∫ x, F x ∂(μ i).restrict (E i) := by
unfold activeSum
exact integral_finsetSum_measure (fun i _ => integrable_of_bounded hF hB)
lemma activeSum_mass (s : Finset I) (μ : I → Measure X)
[∀ i, IsFiniteMeasure (μ i)] (E : I → Set X) :
(activeSum s μ E).real Set.univ = ∑ i ∈ s, (μ i).real (E i) := by
have h := activeSum_integral s μ E (fun _ => (1:ℝ)) measurable_const ⟨1,fun _ => by norm_num⟩
simpa only [integral_const,smul_eq_mul,mul_one,measureReal_restrict_apply_univ] using h
theorem activeSum_exp_test (s : Finset I) (μ ν : I → Measure X)
[∀ i, IsProbabilityMeasure (μ i)] [∀ i, IsProbabilityMeasure (ν i)]
(hkl : ∀ i ∈ s, klDiv (μ i) (ν i) ≠ ∞)
(E : I → Set X) (hE : ∀ i ∈ s, MeasurableSet (E i))
(F : X → ℝ) (hF : Measurable F) (hB : ∃ B : ℝ, ∀ x, |F x| ≤ B) :
(∫ x, F x ∂activeSum s μ E) - (∫ x, Real.exp (F x) ∂activeSum s ν E) +
(activeSum s ν E).real Set.univ ≤ ∑ i ∈ s, (klDiv (μ i) (ν i)).toReal := by
have heB : ∃ B : ℝ, ∀ x, |Real.exp (F x)| ≤ B := by
obtain ⟨B,hB⟩ := hB
exact ⟨Real.exp B,fun x => by
rw [abs_of_pos (Real.exp_pos _)]
exact Real.exp_le_exp.mpr ((le_abs_self _).trans (hB x))⟩
rw [activeSum_integral s μ E F hF hB,activeSum_integral s ν E _ hF.exp heB,
activeSum_mass,← Finset.sum_sub_distrib,← Finset.sum_add_distrib]
exact Finset.sum_le_sum fun i hi => restricted_exp_test (μ i) (ν i) (hkl i hi)
(E i) (hE i hi) F hF hB
theorem normalized_activeSum_kl_le [Nonempty X]
(s : Finset I) (μ ν : I → Measure X)
[∀ i, IsProbabilityMeasure (μ i)] [∀ i, IsProbabilityMeasure (ν i)]
(hkl : ∀ i ∈ s, klDiv (μ i) (ν i) ≠ ∞)
(E : I → Set X) (hE : ∀ i ∈ s, MeasurableSet (E i))
(hequal : ∀ i ∈ s, μ i (E i) = ν i (E i))
(hpos : 0 < (activeSum s μ E).real Set.univ) :
let μs : FiniteMeasure X := ⟨activeSum s μ E,inferInstance⟩
let νs : FiniteMeasure X := ⟨activeSum s ν E,inferInstance⟩
klDiv (μs.normalize : Measure X) (νs.normalize : Measure X) ≤
ENNReal.ofReal ((∑ i ∈ s, (klDiv (μ i) (ν i)).toReal) /
(activeSum s μ E).real Set.univ) := by
let μs : FiniteMeasure X := ⟨activeSum s μ E,inferInstance⟩
let νs : FiniteMeasure X := ⟨activeSum s ν E,inferInstance⟩
have hm : (activeSum s ν E).real Set.univ = (activeSum s μ E).real Set.univ := by
rw [activeSum_mass,activeSum_mass]
apply Finset.sum_congr rfl
intro i hi
exact congrArg ENNReal.toReal (hequal i hi).symm
have hμne : μs ≠ 0 := by
intro h
have hz : (activeSum s μ E).real Set.univ = 0 := by
change (μs : Measure X).real Set.univ = 0
rw [h]; simp
linarith
have hνne : νs ≠ 0 := by
intro h
have hz : (activeSum s ν E).real Set.univ = 0 := by
change (νs : Measure X).real Set.univ = 0
rw [h]; simp
linarith
apply kl_le_of_expBudget
intro F hF hB
have h := activeSum_exp_test s μ ν hkl E hE F hF hB
rw [hm] at h
rw [← μs.average_eq_integral_normalize hμne,← νs.average_eq_integral_normalize hνne,
average,average,integral_smul_measure,integral_smul_measure]
change ((activeSum s μ E) Set.univ)⁻¹.toReal * _ -
((activeSum s ν E) Set.univ)⁻¹.toReal * _ + 1 ≤ _
rw [ENNReal.toReal_inv,ENNReal.toReal_inv]
change ((activeSum s μ E).real Set.univ)⁻¹ * _ -
((activeSum s ν E).real Set.univ)⁻¹ * _ + 1 ≤ _
rw [hm]
apply (le_div_iff₀ hpos).mpr
field_simp
exact h
end DirectionalTransience.Entropy
end
end OAI
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