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e63eb7f | 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 | import OAI.Combinatorics.EditApproximation.Probability.AdaptiveSeedLaw
import OAI.Combinatorics.EditApproximation.Geometry.Accuracy
namespace OAI
/-! The final fresh draw selects the coordinate of the actual incoming seed.
This keeps the dependence of the finite draw type on that table explicit. -/
namespace EditApproximation
open MeasureTheory ProbabilityTheory
section Final
variable {Seed : Type*} [Fintype Seed]
[MeasurableSpace Seed] [MeasurableSingletonClass Seed]
{Draw : Seed → Type*} [∀ seed, Finite (Draw seed)]
[∀ seed, MeasurableSpace (Draw seed)] [∀ seed, MeasurableSingletonClass (Draw seed)]
(drawLaw : ∀ seed, Measure (Draw seed)) [∀ seed, IsProbabilityMeasure (drawLaw seed)]
(incoming : Measure Seed) [IsProbabilityMeasure incoming]
def freshSeedValue (value : ∀ seed, Draw seed → ℝ)
(pair : Seed × (∀ seed, Draw seed)) : ℝ := value pair.1 (pair.2 pair.1)
theorem freshSeed_bad_probability (good : Seed → Prop) :
(incoming.prod (Measure.pi drawLaw)).real
{pair : Seed × (∀ seed, Draw seed) | ¬ good pair.1} = incoming.real {seed | ¬ good seed} := by
classical
let : Fintype (Seed × (∀ seed, Draw seed)) := Fintype.ofFinite _
rw [← finite_integral_test (incoming.prod (Measure.pi drawLaw)) (fun pair => ¬ good pair.1)]
rw [integral_prod _ (.of_finite)]
change (∫ seed, (∫ _ : ∀ seed, Draw seed, if ¬ good seed then (1 : ℝ) else 0
∂Measure.pi drawLaw) ∂incoming) = _
simp only [integral_const, probReal_univ, one_smul]
exact finite_integral_test incoming (fun seed => ¬ good seed)
theorem freshSeed_good_excess (good : Seed → Prop) (value : ∀ seed, Draw seed → ℝ)
(D : ℕ) (ell : ℝ) (hell : 0 < ell)
(hmoment : ∀ seed, good seed →
(∫ draw, max (value seed draw - D) 0 ∂drawLaw seed) ≤ (D : ℝ) / ell) :
(∫ pair, goodExcess {pair : Seed × (∀ seed, Draw seed) | good pair.1}
(freshSeedValue value) D pair ∂incoming.prod (Measure.pi drawLaw)) ≤ (D : ℝ) / ell := by
classical
rw [integral_prod _ (.of_finite)]
have hinner (seed : Seed) :
(∫ draws : ∀ seed, Draw seed,
goodExcess {pair : Seed × (∀ seed, Draw seed) | good pair.1}
(freshSeedValue value) D (seed, draws) ∂Measure.pi drawLaw) ≤ (D : ℝ) / ell := by
by_cases hg : good seed
· simp only [goodExcess, Set.indicator, Set.mem_ofPred_eq, hg, ite_true,
freshSeedValue]
rw [integral_comp_eval (μ := drawLaw) (i := seed)
(f := fun draw => max (value seed draw - D) 0)
(measurable_of_finite _).aestronglyMeasurable]
exact hmoment seed hg
· simp only [goodExcess, Set.indicator, Set.mem_ofPred_eq, hg, ite_false,
integral_zero]
exact div_nonneg (Nat.cast_nonneg _) hell.le
have h := integral_mono (μ := incoming) (.of_finite) (integrable_const _) hinner
simpa only [integral_const, probReal_univ, one_smul] using h
theorem freshSeed_answer_accuracy (good : Seed → Prop) (value : ∀ seed, Draw seed → ℝ)
{N D : ℕ} {ell epsilon : ℝ} (hDN : D ≤ N) (hD : 0 < D)
(hell : 0 < ell) (hepsilon : 0 < epsilon) (haccuracy : 10 ≤ ell * epsilon)
(hseed : incoming.real {seed | ¬ good seed} ≤ 1 / 10)
(hlower : ∀ seed, good seed → ∀ draw, (D : ℝ) ≤ value seed draw)
(hmoment : ∀ seed, good seed →
(∫ draw, max (value seed draw - D) 0 ∂drawLaw seed) ≤ (D : ℝ) / ell) :
(4 / 5 : ℝ) ≤ (incoming.prod (Measure.pi drawLaw)).real
{pair | D ≤ integerAnswer N (freshSeedValue value pair) ∧
(integerAnswer N (freshSeedValue value pair) : ℝ) ≤ (1 + epsilon) * D} := by
classical
apply final_answer_accuracy (incoming.prod (Measure.pi drawLaw)) (measurable_of_finite _)
hDN hD hell hepsilon haccuracy
· change (incoming.prod (Measure.pi drawLaw)).real {pair | ¬ good pair.1} ≤ 1 / 10
rw [freshSeed_bad_probability]
exact hseed
· intro pair hg
exact hlower pair.1 hg (pair.2 pair.1)
· exact .of_finite
· exact freshSeed_good_excess drawLaw incoming good value D ell hell hmoment
end Final
end EditApproximation
end OAI
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