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9eeb9c7 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 | import Mathlib.Probability.Independence.Conditional
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Real
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Tactic
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal Classical
namespace Invariant233.ConditionalAvoidance
variable {Ω ι : Type*} {m : MeasurableSpace Ω} [mΩ : MeasurableSpace Ω]
variable {μ : Measure Ω} [IsFiniteMeasure μ]
omit mΩ [IsFiniteMeasure μ] in
theorem measurableSet_filter_selection_eq (I S : Finset ι) (P : ι → Ω → Prop)
(hP : ∀ i ∈ I, MeasurableSet[m] {ω | P i ω}) :
MeasurableSet[m] {ω | I.filter (fun i => P i ω) = S} := by
classical
by_cases hSI : S ⊆ I
· have heq : {ω | I.filter (fun i => P i ω) = S} =
⋂ i ∈ I, {ω | P i ω ↔ i ∈ S} := by
ext ω
simp only [Set.mem_ofPred_eq, Set.mem_iInter]
constructor
· intro h i hi
have := Finset.ext_iff.mp h i
simpa only [Finset.mem_filter, hi, true_and] using this
· intro h
ext i
by_cases hi : i ∈ I
· simpa only [Finset.mem_filter, hi, true_and] using h i hi
· simp only [Finset.mem_filter, hi, false_and, false_iff]
exact fun his => hi (hSI his)
rw [heq]
apply MeasurableSet.biInter I.countable_toSet
intro i hi
by_cases his : i ∈ S
· simpa only [his, iff_true] using hP i hi
· simpa only [his, iff_false, ← Set.compl_ofPred] using (hP i hi).compl
· have heq : {ω | I.filter (fun i => P i ω) = S} = (∅ : Set Ω) := by
apply Set.eq_empty_iff_forall_notMem.mpr
intro ω hω
exact hSI (hω ▸ Finset.filter_subset _ _)
rw [heq]
exact MeasurableSet.empty
theorem condProb_compl (hm : m ≤ mΩ) {A : Set Ω} (hA : MeasurableSet A) :
μ⟦Aᶜ | m⟧ =ᵐ[μ] fun ω => 1 - (μ⟦A | m⟧) ω := by
have heq : Aᶜ.indicator (fun _ : Ω => (1 : ℝ)) =
(fun _ => (1 : ℝ)) - A.indicator (fun _ => (1 : ℝ)) := by
ext ω
by_cases hω : ω ∈ A <;> simp [hω]
change μ[Aᶜ.indicator (fun _ => (1 : ℝ)) | m] =ᵐ[μ] _
rw [heq]
have hsub := condExp_sub (μ := μ) (integrable_const (1 : ℝ))
((integrable_const (1 : ℝ)).indicator hA) m
filter_upwards [hsub] with ω hω
simpa only [Pi.sub_apply, condExp_const hm] using hω
omit [IsFiniteMeasure μ] in
theorem condProb_nonneg (A : Set Ω) :
∀ᵐ ω ∂μ, 0 ≤ (μ⟦A | m⟧) ω := by
apply condExp_nonneg
exact Eventually.of_forall fun ω => by
by_cases hω : ω ∈ A <;> simp [hω]
theorem condProb_le_on_of_ae_inter_subset (hm : m ≤ mΩ)
{H K B : Set Ω} (hH : MeasurableSet H) (hK : MeasurableSet[m] K)
(hB : MeasurableSet B) (hsub : (H ∩ K : Set Ω) ≤ᵐ[μ] B) :
∀ᵐ ω ∂μ, ω ∈ K → (μ⟦H | m⟧) ω ≤ (μ⟦B | m⟧) ω := by
have hpull := condExp_indicator (μ := μ)
((integrable_const (1 : ℝ)).indicator hH) hK
have hmono := condExp_mono (μ := μ) (m := m)
(((integrable_const (1 : ℝ)).indicator hH).indicator (hm _ hK))
((integrable_const (1 : ℝ)).indicator hB)
(hsub.mono fun ω hsubω => show
K.indicator (H.indicator (fun _ => (1 : ℝ))) ω ≤
B.indicator (fun _ => (1 : ℝ)) ω from by
by_cases hωK : ω ∈ K
· by_cases hωH : ω ∈ H
· have hωB : ω ∈ B := hsubω ⟨hωH, hωK⟩
simp [hωK, hωH, hωB]
· simp [hωK, hωH, Set.indicator_nonneg]
· simp [hωK, Set.indicator_nonneg])
filter_upwards [hpull, hmono] with ω hp hmω hωK
simpa only [hp, Set.indicator_of_mem hωK] using hmω
theorem condProb_le_on_of_inter_subset (hm : m ≤ mΩ)
{H K B : Set Ω} (hH : MeasurableSet H) (hK : MeasurableSet[m] K)
(hB : MeasurableSet B) (hsub : H ∩ K ⊆ B) :
∀ᵐ ω ∂μ, ω ∈ K → (μ⟦H | m⟧) ω ≤ (μ⟦B | m⟧) ω :=
condProb_le_on_of_ae_inter_subset hm hH hK hB (Eventually.of_forall hsub)
theorem prod_one_sub_le_exp (S : Finset ι) (q : ι → ℝ) {p : ℝ}
(hq : ∀ i ∈ S, 0 ≤ 1 - q i) (hp : ∀ i ∈ S, p ≤ q i) :
(∏ i ∈ S, (1 - q i)) ≤ Real.exp (-p * S.card) := by
calc
(∏ i ∈ S, (1 - q i)) ≤ ∏ _i ∈ S, Real.exp (-p) := by
apply Finset.prod_le_prod₀ hq
intro i hi
calc
1 - q i ≤ 1 - p := sub_le_sub_left (hp i hi) 1
_ ≤ Real.exp (-p) := by linarith [Real.add_one_le_exp (-p)]
_ = Real.exp (-p) ^ S.card := by simp
_ = Real.exp (-p * S.card) := by rw [← Real.exp_nat_mul]; congr 1; ring
section ConditionalIndependence
variable [StandardBorelSpace Ω]
theorem condProb_avoid_eq_prod (hm : m ≤ mΩ) (A : ι → Set Ω)
(hA : ∀ i, MeasurableSet (A i)) (hind : iCondIndepSet m hm A μ)
(S : Finset ι) :
μ⟦⋂ i ∈ S, (A i)ᶜ | m⟧ =ᵐ[μ]
fun ω => ∏ i ∈ S, (1 - (μ⟦A i | m⟧) ω) := by
have hmA : ∀ i, MeasurableSpace.generateFrom {A i} ≤ mΩ := by
intro i
apply MeasurableSpace.generateFrom_le
intro s hs
rcases Set.mem_singleton_iff.mp hs with rfl
exact hA i
have hi := (iCondIndepSet_iff_iCondIndep m hm A μ).mp hind
have hprod := (iCondIndep_iff m hm _ hmA μ).mp hi S
(f := fun i => (A i)ᶜ) (fun i _ =>
(MeasurableSpace.measurableSet_generateFrom (Set.mem_singleton (A i))).compl)
have hcomp : ∀ᵐ ω ∂μ, ∀ i ∈ S,
(μ⟦(A i)ᶜ | m⟧) ω = 1 - (μ⟦A i | m⟧) ω := by
exact (ae_ball_iff S.countable_toSet).mpr (fun i _ => condProb_compl hm (hA i))
filter_upwards [hprod, hcomp] with ω hω hc
rw [hω, Finset.prod_apply]
exact Finset.prod_congr rfl hc
theorem condProb_avoid_random_selection_le_exp_ae
(hm : m ≤ mΩ) (I : Finset ι) (A : ι → Set Ω)
(hA : ∀ i, MeasurableSet (A i)) (hind : iCondIndepSet m hm A μ)
(J : Ω → Finset ι)
(hJ : ∀ S : Finset ι, MeasurableSet[m] {ω | J ω = S})
(hJI : ∀ ω, J ω ⊆ I) {H : Set Ω} (hH : MeasurableSet H)
{p r : ℝ} (hp : 0 ≤ p)
(hposterior : ∀ i ∈ I, ∀ᵐ ω ∂μ, i ∈ J ω → p ≤ (μ⟦A i | m⟧) ω)
(hcard : ∀ᵐ ω ∂μ, ω ∈ H → r ≤ ((J ω).card : ℝ))
(havoid : ∀ᵐ ω ∂μ, ω ∈ H → ∀ i ∈ J ω, ω ∉ A i) :
∀ᵐ ω ∂μ, (μ⟦H | m⟧) ω ≤ Real.exp (-p * r) := by
classical
have hlocal : ∀ S ∈ I.powerset, ∀ᵐ ω ∂μ,
J ω = S → (μ⟦H | m⟧) ω ≤ Real.exp (-p * r) := by
intro S hS
have hSI : S ⊆ I := Finset.mem_powerset.mp hS
by_cases hSr : r ≤ (S.card : ℝ)
· have hsub : (H ∩ {ω | J ω = S} : Set Ω) ≤ᵐ[μ] (⋂ i ∈ S, (A i)ᶜ : Set Ω) := by
filter_upwards [havoid] with ω ha hω
change ω ∈ (⋂ i ∈ S, (A i)ᶜ : Set Ω)
simp only [Set.mem_iInter, Set.mem_compl_iff]
intro i hi
exact ha hω.1 i (hω.2.symm ▸ hi)
have hloc := condProb_le_on_of_ae_inter_subset (μ := μ) hm hH (hJ S)
(MeasurableSet.biInter S.countable_toSet (fun i _ => (hA i).compl)) hsub
have hprod := condProb_avoid_eq_prod (μ := μ) hm A hA hind S
have hpost : ∀ᵐ ω ∂μ, ∀ i ∈ S,
i ∈ J ω → p ≤ (μ⟦A i | m⟧) ω :=
(ae_ball_iff S.countable_toSet).mpr (fun i hi => hposterior i (hSI hi))
have hnonneg : ∀ᵐ ω ∂μ, ∀ i ∈ S, 0 ≤ 1 - (μ⟦A i | m⟧) ω := by
apply (ae_ball_iff S.countable_toSet).mpr
intro i _
filter_upwards [condProb_nonneg (μ := μ) (m := m) (A i)ᶜ,
condProb_compl hm (hA i)] with ω hn hc
simpa only [hc] using hn
filter_upwards [hloc, hprod, hpost, hnonneg] with ω hl he hpo hnn hωS
calc
(μ⟦H | m⟧) ω ≤ (μ⟦⋂ i ∈ S, (A i)ᶜ | m⟧) ω := hl hωS
_ = ∏ i ∈ S, (1 - (μ⟦A i | m⟧) ω) := he
_ ≤ Real.exp (-p * S.card) := prod_one_sub_le_exp S _ hnn
(fun i hi => hpo i hi (hωS.symm ▸ hi))
_ ≤ Real.exp (-p * r) := Real.exp_le_exp.mpr (by nlinarith)
· have hsub : (H ∩ {ω | J ω = S} : Set Ω) ≤ᵐ[μ] (∅ : Set Ω) := by
filter_upwards [hcard] with ω hc hω
have heq : J ω = S := hω.2
exact (hSr (by simpa only [heq] using hc hω.1)).elim
have hloc := condProb_le_on_of_ae_inter_subset (μ := μ) hm hH (hJ S)
MeasurableSet.empty hsub
filter_upwards [hloc] with ω hl hωS
have hz : (μ⟦H | m⟧) ω ≤ 0 := by simpa using hl hωS
exact hz.trans (Real.exp_pos _).le
have hall := (ae_ball_iff I.powerset.countable_toSet).mpr hlocal
filter_upwards [hall] with ω hω
exact hω (J ω) (Finset.mem_powerset.mpr (hJI ω)) rfl
theorem condProb_avoid_random_selection_le_exp
(hm : m ≤ mΩ) (I : Finset ι) (A : ι → Set Ω)
(hA : ∀ i, MeasurableSet (A i)) (hind : iCondIndepSet m hm A μ)
(J : Ω → Finset ι)
(hJ : ∀ S : Finset ι, MeasurableSet[m] {ω | J ω = S})
(hJI : ∀ ω, J ω ⊆ I) {H : Set Ω} (hH : MeasurableSet H)
{p r : ℝ} (hp : 0 ≤ p)
(hposterior : ∀ i ∈ I, ∀ᵐ ω ∂μ, i ∈ J ω → p ≤ (μ⟦A i | m⟧) ω)
(hcard : ∀ ω ∈ H, r ≤ ((J ω).card : ℝ))
(havoid : ∀ ω ∈ H, ∀ i ∈ J ω, ω ∉ A i) :
∀ᵐ ω ∂μ, (μ⟦H | m⟧) ω ≤ Real.exp (-p * r) :=
condProb_avoid_random_selection_le_exp_ae hm I A hA hind J hJ hJI hH hp
hposterior (Eventually.of_forall hcard) (Eventually.of_forall havoid)
theorem condProb_half_selection_le_exp
(hm : m ≤ mΩ) (I : Finset ι) (A : ι → Set Ω)
(hA : ∀ i, MeasurableSet (A i)) (hind : iCondIndepSet m hm A μ)
(J : Ω → Finset ι)
(hJ : ∀ S : Finset ι, MeasurableSet[m] {ω | J ω = S})
(hJI : ∀ ω, J ω ⊆ I) {H : Set Ω} (hH : MeasurableSet H)
{ρ M : ℝ} (hρ : 0 ≤ ρ) (hI : M / 20 ≤ (I.card : ℝ))
(hposterior : ∀ i ∈ I, ∀ᵐ ω ∂μ, i ∈ J ω → ρ / 10 ≤ (μ⟦A i | m⟧) ω)
(hcard : ∀ ω ∈ H, (I.card : ℝ) / 2 ≤ ((J ω).card : ℝ))
(havoid : ∀ ω ∈ H, ∀ i ∈ J ω, ω ∉ A i) :
∀ᵐ ω ∂μ, (μ⟦H | m⟧) ω ≤ Real.exp (-M * ρ / 400) := by
have h := condProb_avoid_random_selection_le_exp hm I A hA hind J hJ hJI hH
(div_nonneg hρ (by norm_num : (0 : ℝ) ≤ 10)) hposterior hcard havoid
filter_upwards [h] with ω hω
apply hω.trans
apply Real.exp_le_exp.mpr
nlinarith
theorem condProb_avoid_intensity_le_exp_ae
(hm : m ≤ mΩ) (I : Finset ι) (A : ι → Set Ω)
(hA : ∀ i, MeasurableSet (A i)) (hind : iCondIndepSet m hm A μ)
(J : Ω → Finset ι)
(hJ : ∀ S : Finset ι, MeasurableSet[m] {ω | J ω = S})
(hJI : ∀ ω, J ω ⊆ I) {H : Set Ω} (hH : MeasurableSet H)
{ρ M L : ℝ} (hρ : 0 ≤ ρ) (hL : L ≤ M * ρ)
(hI : M / 20 ≤ (I.card : ℝ))
(hposterior : ∀ i ∈ I, ∀ᵐ ω ∂μ, i ∈ J ω → ρ / 10 ≤ (μ⟦A i | m⟧) ω)
(hcard : ∀ᵐ ω ∂μ, ω ∈ H → (I.card : ℝ) / 2 ≤ ((J ω).card : ℝ))
(havoid : ∀ᵐ ω ∂μ, ω ∈ H → ∀ i ∈ J ω, ω ∉ A i) :
∀ᵐ ω ∂μ, (μ⟦H | m⟧) ω ≤ Real.exp (-L / 400) := by
have h := condProb_avoid_random_selection_le_exp_ae hm I A hA hind J hJ hJI hH
(div_nonneg hρ (by norm_num : (0 : ℝ) ≤ 10)) hposterior hcard havoid
filter_upwards [h] with ω hω
apply hω.trans
apply Real.exp_le_exp.mpr
have hi := mul_le_mul_of_nonneg_right hI hρ
nlinarith
theorem iCondIndepSet_preimages {β : ι → Type*}
[mβ : ∀ i, MeasurableSpace (β i)]
(hm : m ≤ mΩ) (f : ∀ i, Ω → β i) (hf : ∀ i, Measurable (f i))
(hind : iCondIndepFun m hm f μ) (C : ∀ i, Set (β i))
(hC : ∀ i, MeasurableSet (C i)) :
iCondIndepSet m hm (fun i => f i ⁻¹' C i) μ := by
rw [iCondIndepSet_iff m hm _ (fun i => (hf i) (hC i)) μ]
intro S
exact (iCondIndepFun_iff_condExp_inter_preimage_eq_mul mβ f hf).mp hind S
(fun i _ => hC i)
end ConditionalIndependence
end Invariant233.ConditionalAvoidance
end
end OAI
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