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import OAI.Probability.Ballisticity.Entropy.EntropyVariational
namespace OAI
section
open MeasureTheory ProbabilityTheory InformationTheory TopologicalSpace
open scoped ENNReal Classical Topology BoundedContinuousFunction
namespace DirectionalTransience.Entropy
lemma exp_sub_le_exp_mul {x y : ℝ} (h : x≤y) :
Real.exp y-Real.exp x ≤ Real.exp y*(y-x) := by
rcases h.eq_or_lt with rfl | _
· simp
· have hh := mul_le_mul_of_nonneg_left (Real.add_one_le_exp (x-y)) (Real.exp_pos y).le
rw [← Real.exp_add,add_sub_cancel] at hh
nlinarith
lemma abs_exp_sub_le {x y B : ℝ} (hx : x≤B) (hy : y≤B) :
|Real.exp x-Real.exp y| ≤ Real.exp B*|x-y| := by
rcases le_total x y with h|h
· rw [abs_of_nonpos (sub_nonpos.mpr (Real.exp_le_exp.mpr h)),
abs_of_nonpos (sub_nonpos.mpr h),neg_sub,neg_sub]
exact (exp_sub_le_exp_mul h).trans
(mul_le_mul_of_nonneg_right (Real.exp_le_exp.mpr hy) (sub_nonneg.mpr h))
· rw [abs_of_nonneg (sub_nonneg.mpr (Real.exp_le_exp.mpr h)),abs_of_nonneg (sub_nonneg.mpr h)]
exact (exp_sub_le_exp_mul h).trans
(mul_le_mul_of_nonneg_right (Real.exp_le_exp.mpr hx) (sub_nonneg.mpr h))
lemma abs_sub_min_le {x y B : ℝ} (hx : x≤B) : |x-min y B| ≤ |x-y| := by
by_cases hy : y≤B
· simp [min_eq_left hy]
· have hy : B≤y := le_of_not_ge hy
rw [min_eq_right hy,abs_of_nonpos (sub_nonpos.mpr hx),
abs_of_nonpos (sub_nonpos.mpr (hx.trans hy))]
linarith
section Approximation
variable {X : Type*} [TopologicalSpace X] [MetrizableSpace X] [MeasurableSpace X] [BorelSpace X]
lemma continuous_exp_test_approx (μ ν : Measure X) [IsProbabilityMeasure μ]
[IsProbabilityMeasure ν] (F : X → ℝ) (hF : Measurable F)
(hB : ∃ B : ℝ, ∀ x, |F x|≤B) {ε : ℝ} (hε : 0<ε) :
∃ g : X →ᵇ ℝ,
|(∫ x, F x ∂μ)-(∫ x, g x ∂μ)| ≤ ε ∧
|(∫ x, Real.exp (F x) ∂ν)-(∫ x, Real.exp (g x) ∂ν)| ≤ ε := by
obtain ⟨B,hB⟩ := hB
let δ := ε/(Real.exp B+1)
have hδ : 0<δ := div_pos hε (by positivity)
have hi : Integrable F (μ+ν) := integrable_of_bounded hF ⟨B,hB⟩
obtain ⟨g,hg,hgi⟩ := hi.exists_boundedContinuous_integral_sub_le hδ
let k : X →ᵇ ℝ := min g (BoundedContinuousFunction.const X B)
have hk : ∀ x, k x≤B := fun x => min_le_right (g x) B
have hkB : ∃ b : ℝ, ∀ x, |k x|≤b :=
⟨‖k‖,fun x => by simpa only [Real.norm_eq_abs] using k.norm_coe_le_norm x⟩
have hμF : Integrable F μ := integrable_of_bounded hF ⟨B,hB⟩
have hνF : Integrable F ν := integrable_of_bounded hF ⟨B,hB⟩
have herr (x : X) : |F x-k x|≤|F x-g x| := abs_sub_min_le ((le_abs_self _).trans (hB x))
have hgμ : (∫ x, |F x-g x| ∂μ)≤δ := by
have h1 := integral_add_measure ((hμF.sub (g.integrable μ)).norm)
((hνF.sub (g.integrable ν)).norm)
simp only [Real.norm_eq_abs, Pi.sub_apply] at h1 hg
rw [h1] at hg
have hn : 0 ≤ ∫ x, |F x-g x| ∂ν := integral_nonneg (fun x => abs_nonneg (F x-g x))
linarith
have hgν : (∫ x, |F x-g x| ∂ν)≤δ := by
have h1 := integral_add_measure ((hμF.sub (g.integrable μ)).norm)
((hνF.sub (g.integrable ν)).norm)
simp only [Real.norm_eq_abs, Pi.sub_apply] at h1 hg
rw [h1] at hg
have hn : 0 ≤ ∫ x, |F x-g x| ∂μ := integral_nonneg (fun x => abs_nonneg (F x-g x))
linarith
have hkμ : (∫ x, |F x-k x| ∂μ)≤δ :=
(integral_mono (hμF.sub (k.integrable μ)).abs (hμF.sub (g.integrable μ)).abs herr).trans hgμ
have hkν : (∫ x, |F x-k x| ∂ν)≤δ :=
(integral_mono (hνF.sub (k.integrable ν)).abs (hνF.sub (g.integrable ν)).abs herr).trans hgν
have hδle : δ≤ε := by
apply (div_le_iff₀ (show 0<Real.exp B+1 by positivity)).mpr
nlinarith [Real.exp_pos B]
refine ⟨k,?_,?_⟩
· rw [← integral_sub hμF (k.integrable μ)]
exact (abs_integral_le_integral_abs).trans (hkμ.trans hδle)
· have heF : Integrable (fun x => Real.exp (F x)) ν := exp_integrable_of_bounded hF ⟨B,hB⟩
have hek : Integrable (fun x => Real.exp (k x)) ν := exp_integrable_of_bounded k.continuous.measurable hkB
rw [← integral_sub heF hek]
calc
_ ≤ ∫ x, |Real.exp (F x)-Real.exp (k x)| ∂ν := abs_integral_le_integral_abs
_ ≤ ∫ x, Real.exp B*|F x-k x| ∂ν := integral_mono (heF.sub hek).abs
(((hνF.sub (k.integrable ν)).abs).const_mul _) (fun x =>
abs_exp_sub_le ((le_abs_self _).trans (hB x)) (hk x))
_ = Real.exp B*(∫ x, |F x-k x| ∂ν) := integral_const_mul _ _
_ ≤ Real.exp B*δ := mul_le_mul_of_nonneg_left hkν (Real.exp_pos B).le
_ ≤ ε := by
dsimp only [δ]
rw [← mul_div_assoc]
apply (div_le_iff₀ (show 0<Real.exp B+1 by positivity)).mpr
nlinarith
theorem expBudget_of_continuous (μ ν : Measure X) [IsProbabilityMeasure μ]
[IsProbabilityMeasure ν] {C : ℝ}
(h : ∀ g : X →ᵇ ℝ, (∫ x, g x ∂μ)-(∫ x, Real.exp (g x) ∂ν)+1≤C) :
ExpBudget μ ν C := by
intro F hF hB
apply le_of_forall_pos_le_add
intro ε hε
obtain ⟨g,hg,hge⟩ := continuous_exp_test_approx μ ν F hF hB (half_pos hε)
have h1 := (abs_le.mp hg).2
have h2 := (abs_le.mp hge).1
have h3 := h g
linarith
end Approximation
end DirectionalTransience.Entropy
end
section
open MeasureTheory ProbabilityTheory InformationTheory TopologicalSpace Filter
open scoped ENNReal Topology BoundedContinuousFunction
namespace DirectionalTransience.Entropy
variable {X I : Type*} [TopologicalSpace X] [CompactSpace X] [MetrizableSpace X]
[MeasurableSpace X] [BorelSpace X]
lemma kl_le_of_weak_limit {l : Filter I} [l.NeBot]
(μs νs : I → ProbabilityMeasure X) (μ ν : ProbabilityMeasure X)
(hμ : Tendsto μs l (𝓝 μ)) (hν : Tendsto νs l (𝓝 ν))
{C : ℝ} (hC : 0≤C)
(hbudget : ∀ᶠ i in l, klDiv (μs i : Measure X) (νs i : Measure X) ≤ ENNReal.ofReal C) :
klDiv (μ : Measure X) (ν : Measure X) ≤ ENNReal.ofReal C := by
apply kl_le_of_expBudget
apply expBudget_of_continuous
intro f
let ef : X →ᵇ ℝ := BoundedContinuousFunction.mkOfCompact ⟨fun x => Real.exp (f x), Real.continuous_exp.comp f.continuous⟩
have h1 := ProbabilityMeasure.tendsto_iff_forall_integral_tendsto.mp hμ f
have h2 := ProbabilityMeasure.tendsto_iff_forall_integral_tendsto.mp hν ef
apply le_of_tendsto ((h1.sub h2).add_const 1)
filter_upwards [hbudget] with i hi
exact expBudget_of_kl_le (μs i : Measure X) (νs i : Measure X) hC hi
f f.continuous.measurable ⟨‖f‖,fun x => by simpa only [Real.norm_eq_abs] using f.norm_coe_le_norm x⟩
end DirectionalTransience.Entropy
end
end OAI
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