File size: 4,167 Bytes
f205175 | 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 | import OAI.Probability.CriticalZ3.Conditioning
namespace OAI
open MeasureTheory ProbabilityTheory Classical Filter
open scoped ENNReal BigOperators Matrix
namespace InfiniteConditioning
open MeasureTheory ProbabilityTheory Classical
variable {E : Type} [Countable E]
noncomputable def conditionalProbability (p : E → Set.Icc (0 : ℝ) 1) (J : Finset E)
(C : Set (E → Bool)) (ξ : E → Bool) : ℝ :=
(ProductCylinder.law (fixedParameter p J ξ)).real C
omit [Countable E] in
theorem conditionalProbability_determined (p : E → Set.Icc (0 : ℝ) 1) (J : Finset E)
(C : Set (E → Bool)) :
ProductCylinder.DeterminedBy J (conditionalProbability p J C) := by
intro ω ω' h
unfold conditionalProbability
rw [fixedParameter_congr p J h]
/-- Reveal-wise tower/Markov bound, valid also for histories of zero mass. -/
theorem conditional_failure_mass (p : E → Set.Icc (0 : ℝ) 1) (J : Finset E)
(H C : Set (E → Bool))
(hH : ProductCylinder.DeterminedBy (F := Prop) J (fun ω => ω ∈ H))
(hC : MeasurableSet C) {δ : ℝ} (_hδ : 0 ≤ δ) :
δ * (ProductCylinder.law p).real
{ω | ω ∈ H ∧ conditionalProbability p J C ω ≤ 1-δ} ≤
(ProductCylinder.law p).real (H ∩ Cᶜ) := by
let ξ (a : J → Bool) := ProductCylinder.extend J a
let A (a : J → Bool) : Set (E → Bool) :=
if ξ a ∈ H ∧ conditionalProbability p J C (ξ a) ≤ 1-δ then Set.univ else ∅
let B (a : J → Bool) : Set (E → Bool) := if ξ a ∈ H then Cᶜ else ∅
have hA (a) : MeasurableSet (A a) := by dsimp only [A]; split_ifs <;> measurability
have hB (a) : MeasurableSet (B a) := by
dsimp only [B]; split_ifs
· exact hC.compl
· exact MeasurableSet.empty
have heA : {ω | ω ∈ H ∧ conditionalProbability p J C ω ≤ 1-δ} =
{ω | ω ∈ A (J.restrict ω)} := by
ext ω
have hh := ProductCylinder.determinedBy_eq_extend hH ω
have hc := ProductCylinder.determinedBy_eq_extend (conditionalProbability_determined p J C) ω
change conditionalProbability p J C ω = conditionalProbability p J C (ξ (J.restrict ω)) at hc
change (ω ∈ H) = (ξ (J.restrict ω) ∈ H) at hh
simp only [Set.mem_ofPred_eq,A,hh,hc]
split_ifs <;> simp_all
have heB : H ∩ Cᶜ = {ω | ω ∈ B (J.restrict ω)} := by
ext ω
have hh := ProductCylinder.determinedBy_eq_extend hH ω
change (ω ∈ H) = (ξ (J.restrict ω) ∈ H) at hh
simp only [Set.mem_inter_iff,Set.mem_ofPred_eq,B,hh]
split_ifs <;> simp_all
rw [heA,heB,measure_select p J A hA,measure_select p J B hB,Finset.mul_sum]
apply Finset.sum_le_sum
intro a _
have hcomp : (ProductCylinder.law (fixedParameter p J (ξ a))).real Cᶜ =
1-conditionalProbability p J C (ξ a) := by
simpa only [conditionalProbability,probReal_univ] using
(measureReal_compl (μ := ProductCylinder.law (fixedParameter p J (ξ a))) hC)
have hlocal : δ*(ProductCylinder.law (fixedParameter p J (ξ a))).real (A a) ≤
(ProductCylinder.law (fixedParameter p J (ξ a))).real (B a) := by
by_cases hha : ξ a ∈ H
· simp only [A,B,hha,true_and,ite_true]
split_ifs with hc
· rw [probReal_univ,mul_one,hcomp]; linarith
· rw [measureReal_empty,mul_zero]; exact measureReal_nonneg
· simp only [A,B,hha,false_and,ite_false,measureReal_empty,mul_zero,le_refl]
let hw := (ProductCylinder.law p).real (pattern J (ξ a))
calc
_ = hw*(δ*(ProductCylinder.law (fixedParameter p J (ξ a))).real (A a)) := by ring
_ ≤ _ := mul_le_mul_of_nonneg_left hlocal measureReal_nonneg
theorem conditional_failure_bound (p : E → Set.Icc (0 : ℝ) 1) (J : Finset E)
(H C : Set (E → Bool))
(hH : ProductCylinder.DeterminedBy (F := Prop) J (fun ω => ω ∈ H))
(hC : MeasurableSet C) {δ ε : ℝ} (hδ : 0 < δ)
(hfail : (ProductCylinder.law p).real (H ∩ Cᶜ) ≤ ε) :
(ProductCylinder.law p).real {ω | ω ∈ H ∧ conditionalProbability p J C ω ≤ 1-δ} ≤ ε/δ := by
apply (le_div_iff₀ hδ).mpr
simpa only [mul_comm] using (conditional_failure_mass p J H C hH hC hδ.le).trans hfail
end InfiniteConditioning
end OAI
|