File size: 12,705 Bytes
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 | import Mathlib
import OAI.Probability.GaussianInformation.Kernel.NoiseLaw
namespace OAI
open MeasureTheory ProbabilityTheory Metric Set
open scoped ENNReal
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal
namespace KernelPotential.DriftTools
lemma compensated_log_bound {α : Type*} [MeasurableSpace α]
(μ : Measure α) [IsProbabilityMeasure μ] (f q c : α → ℝ)
(hf : ∀ᵐ x ∂μ, 0 < f x) (hq : ∀ᵐ x ∂μ, 0 < q x)
(hc : ∀ᵐ x ∂μ, 0 ≤ c x)
(hu : AEStronglyMeasurable (fun x => q x*f x/(1+c x)) μ)
(hmass : (∫⁻ x, ENNReal.ofReal (q x*f x/(1+c x)) ∂μ) ≤ 1)
(hflog : Integrable (fun x => Real.log (f x)) μ)
(hqlog : Integrable (fun x => Real.log (q x)) μ)
(hclog : Integrable (fun x => Real.log (1+c x)) μ) :
(∫ x,Real.log (f x) ∂μ) ≤
(∫ x,Real.log (1+c x) ∂μ) - ∫ x,Real.log (q x) ∂μ := by
have hu0 : ∀ᵐ x ∂μ, 0 ≤ q x*f x/(1+c x) := by
filter_upwards [hf,hq,hc] with x hfx hqx hcx
positivity
have hui : Integrable (fun x => q x*f x/(1+c x)) μ :=
⟨hu,(hasFiniteIntegral_iff_ofReal hu0).mpr (hmass.trans_lt (by norm_num))⟩
have huint : (∫ x,q x*f x/(1+c x) ∂μ) ≤ 1 := by
rw [← ofReal_integral_eq_lintegral_ofReal hui hu0] at hmass
exact (ENNReal.ofReal_le_one).mp hmass
have hpoint : ∀ᵐ x ∂μ, Real.log (f x) ≤
q x*f x/(1+c x)-1+Real.log (1+c x)-Real.log (q x) := by
filter_upwards [hf,hq,hc] with x hfx hqx hcx
have hcp : 0 < 1+c x := by linarith
have h := Real.log_le_sub_one_of_pos (div_pos (mul_pos hqx hfx) hcp)
rw [Real.log_div (mul_ne_zero hqx.ne' hfx.ne') hcp.ne',Real.log_mul hqx.ne' hfx.ne'] at h
linarith
have hbound := integral_mono_ae hflog (((hui.sub (integrable_const 1)).add hclog).sub hqlog) hpoint
simp only [Pi.sub_apply,Pi.add_apply] at hbound
have he : (∫ x,q x*f x/(1+c x)-1+Real.log (1+c x)-Real.log (q x) ∂μ) =
(∫ x,q x*f x/(1+c x) ∂μ)-1+(∫ x,Real.log (1+c x) ∂μ)-(∫ x,Real.log (q x) ∂μ) := by
rw [integral_sub (f := fun x => q x*f x/(1+c x)-1+Real.log (1+c x))
(g := fun x => Real.log (q x)) ((hui.sub (integrable_const 1)).add hclog) hqlog,
integral_add (f := fun x => q x*f x/(1+c x)-1) (g := fun x => Real.log (1+c x))
(hui.sub (integrable_const 1)) hclog,
integral_sub (f := fun x => q x*f x/(1+c x)) (g := fun _ => (1:ℝ)) hui (integrable_const 1)]
simp
rw [he] at hbound
linarith
lemma compensation_lintegral {α β ι : Type*} [MeasurableSpace α] [MeasurableSpace β]
[Fintype ι] (ν : Measure α) (Λ : Measure β)
(K : β → α → ℝ≥0∞) (hxK : ∀ z, Measurable (K z))
(κ : β → ι → ℝ≥0∞) (hκ : ∀ z, ∑ w,κ z w=1)
(x : ι → α → ℝ≥0∞) (hx : ∀ w, Measurable (x w))
(q : β → ℝ≥0∞) (hq : ∫⁻ z,q z ∂Λ=1) :
(∫⁻ z,∑ w, ∫⁻ s, K z s * κ z w * q z * x w s /
(1+∫⁻ u,K z u*x w u ∂ν) ∂ν ∂Λ) ≤ 1 := by
calc
_ ≤ ∫⁻ z,q z ∂Λ := by
apply lintegral_mono
intro z
calc
_ = ∑ w,κ z w*q z*((∫⁻ s,K z s*x w s ∂ν)/(1+∫⁻ u,K z u*x w u ∂ν)) := by
apply Finset.sum_congr rfl
intro w _
let C := ∫⁻ u,K z u*x w u ∂ν
calc
_ = ∫⁻ s, (κ z w*q z/(1+C))*(K z s*x w s) ∂ν := by
apply lintegral_congr; intro s; dsimp [C]; simp only [div_eq_mul_inv]; ring
_ = (κ z w*q z/(1+C))*C :=
lintegral_const_mul _ ((hxK z).mul (hx w))
_ = _ := by dsimp [C]; simp only [div_eq_mul_inv]; ring
_ ≤ ∑ w,κ z w*q z*1 := by
apply Finset.sum_le_sum
intro w _
gcongr
exact ENNReal.div_le_of_le_mul (by simp)
_ = q z := by simp only [mul_one,← Finset.sum_mul,hκ z,one_mul]
_ = 1 := hq
lemma log_young {u v a : ℝ} (hu : 0 ≤ u) (hv : 0 < v) (ha : 0 < a) :
u*Real.log v ≤ u*Real.log u-u*Real.log a-u+a*v := by
by_cases hu0 : u=0
· simp only [hu0,zero_mul,sub_zero,zero_add]
positivity
have hup : 0 < u := lt_of_le_of_ne hu (Ne.symm hu0)
have h := Real.log_le_sub_one_of_pos (div_pos (mul_pos ha hv) hup)
rw [Real.log_div (mul_ne_zero ha.ne' hv.ne') hu0,Real.log_mul ha.ne' hv.ne'] at h
have hh := mul_le_mul_of_nonneg_left h hu
have hc : u*(a*v/u-1)=a*v-u := by field_simp
rw [hc] at hh
nlinarith
lemma weighted_log_moment {α : Type*} [MeasurableSpace α]
(μ : Measure α) (u v : α → ℝ) (hu : ∀ᵐ x ∂μ, 0 ≤ u x)
(hv : ∀ᵐ x ∂μ, 0 < v x) (hui : Integrable u μ) (hvi : Integrable v μ)
(huli : Integrable (fun x => u x*Real.log (u x)) μ)
(huvi : Integrable (fun x => u x*Real.log (v x)) μ)
{r G : ℝ} (hr : 0 < r) (hG : 0 < G)
(hur : ∫ x,u x ∂μ=r) (hvG : (∫ x,v x ∂μ) ≤ G) :
(∫ x,u x*Real.log (v x) ∂μ) ≤
(∫ x,u x*Real.log (u x) ∂μ)-r*Real.log (r/G) := by
have hpoint : ∀ᵐ x ∂μ,u x*Real.log (v x) ≤
u x*Real.log (u x)-u x*Real.log (r/G)-u x+(r/G)*v x := by
filter_upwards [hu,hv] with x hx hy
exact log_young hx hy (div_pos hr hG)
have hright : Integrable (fun x => u x*Real.log (u x)-u x*Real.log (r/G)-u x+(r/G)*v x) μ :=
((huli.sub (hui.mul_const _)).sub hui).add (hvi.const_mul _)
have h := integral_mono_ae huvi hright hpoint
rw [integral_add (f := fun x => u x*Real.log (u x)-u x*Real.log (r/G)-u x)
(g := fun x => (r/G)*v x) ((huli.sub (hui.mul_const _)).sub hui) (hvi.const_mul _),
integral_sub (f := fun x => u x*Real.log (u x)-u x*Real.log (r/G)) (g := u)
(huli.sub (hui.mul_const _)) hui,
integral_sub (f := fun x => u x*Real.log (u x)) (g := fun x => u x*Real.log (r/G)) huli (hui.mul_const _),
integral_mul_const,integral_const_mul,hur] at h
have hb := mul_le_mul_of_nonneg_left hvG (div_nonneg hr.le hG.le)
rw [div_mul_cancel₀ _ hG.ne'] at hb
linarith
lemma integral_log_le_log_of_integral_le {α : Type*} [MeasurableSpace α]
(μ : Measure α) [IsProbabilityMeasure μ] (v : α → ℝ)
(hv : ∀ᵐ x ∂μ,0 < v x) (hvi : Integrable v μ)
(hvlog : Integrable (fun x => Real.log (v x)) μ)
{a : ℝ} (ha : 0 < a) (hva : (∫ x,v x ∂μ) ≤ a) :
(∫ x,Real.log (v x) ∂μ) ≤ Real.log a := by
have hpoint : ∀ᵐ x ∂μ,Real.log (v x) ≤ v x/a-1+Real.log a := by
filter_upwards [hv] with x hx
have h := Real.log_le_sub_one_of_pos (div_pos hx ha)
rw [Real.log_div hx.ne' ha.ne'] at h
linarith
have h := integral_mono_ae hvlog (((hvi.div_const a).sub (integrable_const 1)).add (integrable_const (Real.log a))) hpoint
have he : (∫ x,v x/a-1+Real.log a ∂μ) = (∫ x,v x ∂μ)/a-1+Real.log a := by
rw [integral_add (f := fun x => v x/a-1) (g := fun _ => Real.log a)
((hvi.div_const a).sub (integrable_const 1)) (integrable_const _),
integral_sub (f := fun x => v x/a) (g := fun _ => (1:ℝ)) (hvi.div_const a) (integrable_const 1),
integral_div]
simp
change (∫ x,Real.log (v x) ∂μ) ≤ (∫ x,v x/a-1+Real.log a ∂μ) at h
rw [he] at h
have hb : (∫ x,v x ∂μ)/a ≤ 1 := (div_le_one ha).mpr hva
linarith
lemma pow_one_add_le {x : ℝ} (hx : 0 ≤ x) (k : ℕ) :
(1+x)^k ≤ 2^k*(1+x^k) := by
have h : 1+x ≤ 2*max 1 x := by have := le_max_left (1:ℝ) x; have := le_max_right (1:ℝ) x; linarith
calc
_ ≤ (2*max 1 x)^k := pow_le_pow_left₀ (by positivity) h k
_ = 2^k*(max 1 x)^k := mul_pow _ _ _
_ ≤ 2^k*(1+x^k) := by
gcongr
rcases le_total x 1 with hh|hh
· rw [max_eq_left hh,one_pow]; have := pow_nonneg hx k; linarith
· rw [max_eq_right hh]; linarith
lemma integrable_of_bounded {α : Type*} [MeasurableSpace α] (μ : Measure α)
[IsFiniteMeasure μ] (f : α → ℝ) (hf : AEStronglyMeasurable f μ)
{C : ℝ} (hC : ∀ᵐ x ∂μ, ‖f x‖ ≤ C) : Integrable f μ :=
(integrable_const C).mono' hf hC
lemma integral_weighted_log_moment {α : Type*} [MeasurableSpace α]
(μ : Measure α) [IsProbabilityMeasure μ] (u g : α → ℝ)
(hu : Measurable u) (hg : Measurable g)
(hu0 : ∀ x,0 ≤ u x) (hu1 : ∀ x,u x ≤ 1)
(hg0 : ∀ x,0 ≤ g x) {M G : ℝ} (hgM : ∀ x,g x ≤ M) (hG : 0 ≤ G)
(k : ℕ) (hk : 0 < k) (hmoment : (∫ x,(g x)^k ∂μ) ≤ G^k) :
(∫ x,u x*Real.log (1+g x) ∂μ) ≤
-(∫ x,u x ∂μ)*Real.log (∫ x,u x ∂μ)/(k:ℝ) +
(∫ x,u x ∂μ)*Real.log (4*(1+G)) := by
let r : ℝ := ∫ x,u x ∂μ
have hui : Integrable u μ := integrable_of_bounded μ u hu.aestronglyMeasurable (C := 1)
(Filter.Eventually.of_forall (fun x => by rw [Real.norm_eq_abs,abs_of_nonneg (hu0 x)]; exact hu1 x))
have hr : 0 ≤ r := integral_nonneg hu0
have hgMi : Integrable (fun x => (g x)^k) μ :=
integrable_of_bounded μ _ (hg.pow_const k).aestronglyMeasurable (C := M^k)
(Filter.Eventually.of_forall (fun x => by rw [Real.norm_eq_abs,abs_of_nonneg (pow_nonneg (hg0 x) k)]; exact pow_le_pow_left₀ (hg0 x) (hgM x) k))
have hv0 (x : α) : 0 < (1+g x)^k := pow_pos (by linarith [hg0 x]) _
have hvi : Integrable (fun x => (1+g x)^k) μ :=
integrable_of_bounded μ _ ((measurable_const.add hg).pow_const k).aestronglyMeasurable (C := (1+M)^k)
(Filter.Eventually.of_forall (fun x => by rw [Real.norm_eq_abs,abs_of_pos (hv0 x)]; exact pow_le_pow_left₀ (by linarith [hg0 x]) (by linarith [hgM x]) k))
have hlogi : Integrable (fun x => Real.log (1+g x)) μ :=
integrable_of_bounded μ _ (measurable_const.add hg).log.aestronglyMeasurable (C := Real.log (1+M))
(Filter.Eventually.of_forall (fun x => by
rw [Real.norm_eq_abs,abs_of_nonneg (Real.log_nonneg (by linarith [hg0 x]))]
exact Real.log_le_log (by linarith [hg0 x]) (by linarith [hgM x])))
have huli : Integrable (fun x => u x*Real.log (u x)) μ := by
obtain ⟨C,hC⟩ := ((isCompact_Icc (a := (0:ℝ)) (b := 1)).image Real.continuous_mul_log).isBounded.exists_norm_le
exact integrable_of_bounded μ _ (Real.continuous_mul_log.measurable.comp hu).aestronglyMeasurable
(Filter.Eventually.of_forall (fun x => hC _ (mem_image_of_mem _ ⟨hu0 x,hu1 x⟩)))
have huli0 : (∫ x,u x*Real.log (u x) ∂μ) ≤ 0 := integral_nonpos (fun x =>
mul_nonpos_of_nonneg_of_nonpos (hu0 x) (Real.log_nonpos (hu0 x) (hu1 x)))
have huwlog : Integrable (fun x => u x*Real.log (1+g x)) μ :=
hlogi.bdd_mul hu.aestronglyMeasurable (Filter.Eventually.of_forall (fun x => by
rw [Real.norm_eq_abs,abs_of_nonneg (hu0 x)]; exact hu1 x))
have huvi : Integrable (fun x => u x*Real.log ((1+g x)^k)) μ := by
simp_rw [Real.log_pow]
convert huwlog.const_mul (k:ℝ) using 1
ext x
ring
by_cases hr0 : r=0
· have huae : u =ᵐ[μ] 0 := (integral_eq_zero_iff_of_nonneg_ae (Filter.Eventually.of_forall hu0) hui).mp hr0
have hz : (∫ x,u x*Real.log (1+g x) ∂μ)=0 := by
apply integral_eq_zero_of_ae
filter_upwards [huae] with x hx
simp [hx]
change _ ≤ -r*Real.log r/(k:ℝ)+r*Real.log (4*(1+G))
rw [hz,hr0]; simp
have hrp : 0 < r := lt_of_le_of_ne hr (Ne.symm hr0)
have hvG : (∫ x,(1+g x)^k ∂μ) ≤ (4*(1+G))^k := by
calc
_ ≤ ∫ x,2^k*(1+(g x)^k) ∂μ := integral_mono hvi
(((integrable_const 1).add hgMi).const_mul _) (fun x => pow_one_add_le (hg0 x) k)
_ = 2^k*(1+∫ x,(g x)^k ∂μ) := by rw [integral_const_mul,integral_add (integrable_const 1) hgMi]; simp
_ ≤ 2^k*(1+G^k) := by gcongr
_ ≤ 2^k*(2*(1+G)^k) := by gcongr; have h1 : (1:ℝ) ≤ (1+G)^k := one_le_pow₀ (by linarith); have h2 : G^k ≤ (1+G)^k := pow_le_pow_left₀ hG (by linarith) k; linarith
_ ≤ 2^k*(2^k*(1+G)^k) := by gcongr; exact le_self_pow₀ (by norm_num) (by omega)
_ = _ := by rw [mul_pow,show (4:ℝ)=2*2 by norm_num,mul_pow]; ring
have h := weighted_log_moment μ u (fun x => (1+g x)^k)
(Filter.Eventually.of_forall hu0) (Filter.Eventually.of_forall hv0) hui hvi huli huvi hrp
(by positivity : 0 < (4*(1+G))^k) rfl hvG
have he : (∫ x,u x*Real.log ((1+g x)^k) ∂μ) = (k:ℝ)*(∫ x,u x*Real.log (1+g x) ∂μ) := by
rw [← integral_const_mul]
apply integral_congr_ae
filter_upwards [] with x
rw [Real.log_pow]; ring
rw [he, Real.log_div hrp.ne' (by positivity), Real.log_pow] at h
change _ ≤ -r*Real.log r/(k:ℝ)+r*Real.log (4*(1+G))
have hkR : 0 < (k:ℝ) := by exact_mod_cast hk
apply (mul_le_mul_iff_right₀ hkR).mp
have heq : (k:ℝ)*(-r*Real.log r/(k:ℝ)+r*Real.log (4*(1+G))) =
-r*Real.log r+r*((k:ℝ)*Real.log (4*(1+G))) := by field_simp
rw [heq]
nlinarith
end KernelPotential.DriftTools
end OAI
|