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b317718 | 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 | import Mathlib
import OAI.Probability.GaussianInformation.Kernel.EntropyDrift
namespace OAI
open MeasureTheory ProbabilityTheory Metric Set
open scoped ENNReal
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal
namespace KernelPotential.DriftTools
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal
lemma fresh_weighted_log_moment {α β : Type*} [MeasurableSpace α] [MeasurableSpace β]
(μ : Measure α) (ν : Measure β) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
(u : α → ℝ) (m : α × β → ℝ≥0∞)
(hu : Measurable u) (hm : Measurable m)
(hu0 : ∀ a,0 ≤ u a) (hu1 : ∀ a,u a ≤ 1)
{B G c : ℝ} (hB : 0 ≤ B) (hG : 0 ≤ G) (hc : 0 ≤ c)
(hmB : ∀ a b,m (a,b) ≤ ENNReal.ofReal B) (k : ℕ) (hk : 0 < k)
(hmoment : (∫⁻ a,(∫⁻ b,m (a,b) ∂ν)^k ∂μ) ≤ ENNReal.ofReal G^k) :
(∫ a,u a*(∫ b,Real.log (1+c*(m (a,b)).toReal) ∂ν) ∂μ) ≤
-(∫ a,u a ∂μ)*Real.log (∫ a,u a ∂μ)/(k:ℝ)+
(∫ a,u a ∂μ)*Real.log (4*(1+c*G)) := by
let g : α → ℝ := fun a => (∫⁻ b,m (a,b) ∂ν).toReal
have hgE : Measurable (fun a => ∫⁻ b,m (a,b) ∂ν) := Measurable.lintegral_prod_right hm
have hg : Measurable g := hgE.ennreal_toReal
have hmt (a : α) (b : β) : m (a,b) ≠ ∞ := ne_top_of_le_ne_top ENNReal.ofReal_ne_top (hmB a b)
have hgBE (a : α) : (∫⁻ b,m (a,b) ∂ν) ≤ ENNReal.ofReal B := by
calc
_ ≤ ∫⁻ _b,ENNReal.ofReal B ∂ν := lintegral_mono (hmB a)
_ = _ := by simp
have hgt (a : α) : (∫⁻ b,m (a,b) ∂ν) ≠ ∞ := ne_top_of_le_ne_top ENNReal.ofReal_ne_top (hgBE a)
have hgB (a : α) : g a ≤ B := by
exact (ENNReal.toReal_mono ENNReal.ofReal_ne_top (hgBE a)).trans_eq (ENNReal.toReal_ofReal hB)
have hmBr (a : α) (b : β) : (m (a,b)).toReal ≤ B :=
(ENNReal.toReal_mono ENNReal.ofReal_ne_top (hmB a b)).trans_eq (ENNReal.toReal_ofReal hB)
have hfi (a : α) : Integrable (fun b => (m (a,b)).toReal) ν :=
integrable_of_bounded ν _ ((hm.comp (measurable_const.prodMk measurable_id)).ennreal_toReal.aestronglyMeasurable)
(Filter.Eventually.of_forall (fun b => by rw [Real.norm_eq_abs,abs_of_nonneg ENNReal.toReal_nonneg]; exact hmBr a b))
have hgeq (a : α) : (∫ b,(m (a,b)).toReal ∂ν)=g a :=
integral_toReal (hm.comp (measurable_const.prodMk measurable_id)).aemeasurable
(Filter.Eventually.of_forall (fun b => (hmt a b).lt_top))
have hgmoment : (∫ a,(c*g a)^k ∂μ) ≤ (c*G)^k := by
have he : (∫ a,(g a)^k ∂μ) = (∫⁻ a,(∫⁻ b,m (a,b) ∂ν)^k ∂μ).toReal := by
dsimp only [g]
simp_rw [← ENNReal.toReal_pow]
exact integral_toReal (hgE.pow_const k).aemeasurable
(Filter.Eventually.of_forall (fun a => (ENNReal.pow_ne_top (hgt a)).lt_top))
have hb : (∫ a,(g a)^k ∂μ) ≤ G^k := by
rw [he]
have h := ENNReal.toReal_mono (ENNReal.pow_ne_top ENNReal.ofReal_ne_top) hmoment
simpa only [ENNReal.toReal_pow,ENNReal.toReal_ofReal hG] using h
simp_rw [mul_pow]
rw [integral_const_mul]
exact mul_le_mul_of_nonneg_left hb (pow_nonneg hc k)
have hcg : Measurable (fun a => c*g a) := measurable_const.mul hg
have hpoint (a : α) : (∫ b,Real.log (1+c*(m (a,b)).toReal) ∂ν) ≤ Real.log (1+c*g a) := by
have hvpos (b : β) : 0 < 1+c*(m (a,b)).toReal := by positivity
have hvlog : Integrable (fun b => Real.log (1+c*(m (a,b)).toReal)) ν := by
apply integrable_of_bounded ν _
((measurable_const.add (measurable_const.mul
(hm.comp (measurable_const.prodMk measurable_id)).ennreal_toReal)).log.aestronglyMeasurable)
(C := Real.log (1+c*B))
filter_upwards [] with b
dsimp only [Pi.add_apply, Pi.mul_apply, Function.comp_apply, id_eq]
rw [Real.norm_eq_abs,abs_of_nonneg (Real.log_nonneg (by linarith [mul_nonneg hc (ENNReal.toReal_nonneg (a := m (a,b))) ]))]
exact Real.log_le_log (hvpos b) (by nlinarith [hmBr a b])
apply integral_log_le_log_of_integral_le ν _
(Filter.Eventually.of_forall hvpos) ((integrable_const 1).add ((hfi a).const_mul c))
hvlog (by dsimp [g]; positivity)
rw [integral_add (integrable_const 1) ((hfi a).const_mul c),integral_const_mul,hgeq]
simp
have hlogM : ∀ a b, ‖Real.log (1+c*(m (a,b)).toReal)‖ ≤ Real.log (1+c*B) := by
intro a b
rw [Real.norm_eq_abs,abs_of_nonneg (Real.log_nonneg (by linarith [mul_nonneg hc (ENNReal.toReal_nonneg (a := m (a,b))) ]))]
exact Real.log_le_log (by positivity) (by nlinarith [hmBr a b])
have hlogint : Integrable (fun t : α × β => Real.log (1+c*(m t).toReal)) (μ.prod ν) :=
integrable_of_bounded _ _
((measurable_const.add (measurable_const.mul hm.ennreal_toReal)).log.aestronglyMeasurable)
(Filter.Eventually.of_forall (fun t => hlogM t.1 t.2))
have hlefti : Integrable (fun a => u a*(∫ b,Real.log (1+c*(m (a,b)).toReal) ∂ν)) μ :=
hlogint.integral_prod_left.bdd_mul hu.aestronglyMeasurable
(Filter.Eventually.of_forall (fun a => by rw [Real.norm_eq_abs,abs_of_nonneg (hu0 a)]; exact hu1 a))
have hrighti : Integrable (fun a => u a*Real.log (1+c*g a)) μ := by
have hi : Integrable (fun a => Real.log (1+c*g a)) μ := by
apply integrable_of_bounded μ _ (measurable_const.add hcg).log.aestronglyMeasurable
(C := Real.log (1+c*B))
filter_upwards [] with a
have hg0 : 0 ≤ g a := ENNReal.toReal_nonneg
dsimp only [Pi.add_apply]
rw [Real.norm_eq_abs,abs_of_nonneg (Real.log_nonneg (by linarith [mul_nonneg hc hg0]))]
exact Real.log_le_log (by positivity) (by nlinarith [hgB a])
exact hi.bdd_mul hu.aestronglyMeasurable
(Filter.Eventually.of_forall (fun a => by rw [Real.norm_eq_abs,abs_of_nonneg (hu0 a)]; exact hu1 a))
calc
_ ≤ ∫ a,u a*Real.log (1+c*g a) ∂μ := integral_mono hlefti hrighti
(fun a => mul_le_mul_of_nonneg_left (hpoint a) (hu0 a))
_ ≤ _ := integral_weighted_log_moment μ u (fun a => c*g a) hu hcg hu0 hu1
(fun _ => mul_nonneg hc ENNReal.toReal_nonneg) (M := c*B)
(fun a => mul_le_mul_of_nonneg_left (hgB a) hc) (mul_nonneg hc hG) k hk hgmoment
end KernelPotential.DriftTools
end OAI
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