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f0f05de | 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 | import OAI.Combinatorics.BalancedRyser.Probability.ConditionalOrderVariance
import OAI.Combinatorics.BalancedRyser.Profiles.ProfileDetectionConcentration
namespace OAI
open MeasureTheory ProbabilityTheory
open scoped BigOperators
namespace Balanced.ConditionalOrderDetection
open ConditionalOrderVariance UncoveredSampling
variable {F : Type*} [Field F] [Fintype F] [DecidableEq F]
noncomputable def planeObservable (U V : Finset (Line F))
(I : Finset U) (J : Finset V) (p : OrderPair U V) : ℝ :=
planeFraction (populationLines U (I.map p.1.toEmbedding))
(populationLines V (J.map p.2.toEmbedding))
noncomputable def targetObservable (U V : Finset (Line F))
(I : Finset U) (J : Finset V) (x : U) (p : OrderPair U V) : ℝ :=
targetFraction x.val (populationLines U (I.map p.1.toEmbedding))
(populationLines V (J.map p.2.toEmbedding))
private theorem finite_memLp {Ω : Type*} [Fintype Ω]
[MeasurableSpace Ω] [MeasurableSingletonClass Ω]
(μ : Measure Ω) [IsFiniteMeasure μ] (f : Ω → ℝ) : MemLp f 2 μ := by
apply MemLp.of_bound (measurable_of_countable f).aestronglyMeasurable
(∑ ω, |f ω|)
apply Filter.Eventually.of_forall
intro ω
simp only [Real.norm_eq_abs]
exact Finset.single_le_sum (fun z _ => abs_nonneg (f z)) (Finset.mem_univ ω)
section Conditional
variable (U V : Finset (Line F))
variable [MeasurableSpace (OrderPair U V)] [MeasurableSingletonClass (OrderPair U V)]
variable (I : Finset U) (J : Finset V) (i x : U)
theorem unmarked_probability_le_of_mean_gap
(hUV : Disjoint U V) (hi : i ∉ I) {ε : ℝ} (hε : 0 < ε)
(hgap : 2 * ε <
|(∫ p, targetObservable U V I J x p ∂(conditionedOrderLaw U V i x).toMeasure) -
∫ p, planeObservable U V I J p ∂(conditionedOrderLaw U V i x).toMeasure|) :
(conditionedOrderLaw U V i x).toMeasure.real
{p | |targetObservable U V I J x p - planeObservable U V I J p| ≤ ε} ≤
32 * ((I.card : ℝ) + J.card) / ((Fintype.card F : ℝ)^2 * ε^2) := by
have h := ProfileDetectionConcentration.unmarked_probability_le_of_mean_gap
(finite_memLp _ (targetObservable U V I J x))
(finite_memLp _ (planeObservable U V I J)) hε hgap
have htarget := target_conditioned_order_measure_variance_le U V hUV I J i x hi
have hplane := plane_conditioned_order_measure_variance_le U V I J i x hi
calc
_ ≤ 4 * (variance (targetObservable U V I J x)
(conditionedOrderLaw U V i x).toMeasure +
variance (planeObservable U V I J)
(conditionedOrderLaw U V i x).toMeasure) / ε^2 := h
_ ≤ 4 * (4 * ((I.card : ℝ) + J.card) / (Fintype.card F : ℝ)^2 +
4 * ((I.card : ℝ) + J.card) / (Fintype.card F : ℝ)^2) / ε^2 := by
gcongr
· exact htarget
· exact hplane
_ = _ := by ring
theorem unmarked_probability_le_of_profile_gap
(hUV : Disjoint U V) (hi : i ∉ I) {a b ε : ℝ}
(hε : 0 < ε) (hgap : 3 * ε < |a - b|)
(hTa : |(∫ p, targetObservable U V I J x p
∂(conditionedOrderLaw U V i x).toMeasure) - a| ≤ ε / 2)
(hPb : |(∫ p, planeObservable U V I J p
∂(conditionedOrderLaw U V i x).toMeasure) - b| ≤ ε / 2) :
(conditionedOrderLaw U V i x).toMeasure.real
{p | |targetObservable U V I J x p - planeObservable U V I J p| ≤ ε} ≤
32 * ((I.card : ℝ) + J.card) / ((Fintype.card F : ℝ)^2 * ε^2) := by
apply unmarked_probability_le_of_mean_gap U V I J i x hUV hi hε
have h₁ := abs_sub_le a
(∫ p, targetObservable U V I J x p ∂(conditionedOrderLaw U V i x).toMeasure) b
have h₂ := abs_sub_le
(∫ p, targetObservable U V I J x p ∂(conditionedOrderLaw U V i x).toMeasure)
(∫ p, planeObservable U V I J p ∂(conditionedOrderLaw U V i x).toMeasure) b
rw [abs_sub_comm a
(∫ p, targetObservable U V I J x p ∂(conditionedOrderLaw U V i x).toMeasure)] at h₁
linarith
theorem marked_probability_ge_of_profile_gap
(hUV : Disjoint U V) (hi : i ∉ I) {a b ε : ℝ}
(hε : 0 < ε) (hgap : 3 * ε < |a - b|)
(hTa : |(∫ p, targetObservable U V I J x p
∂(conditionedOrderLaw U V i x).toMeasure) - a| ≤ ε / 2)
(hPb : |(∫ p, planeObservable U V I J p
∂(conditionedOrderLaw U V i x).toMeasure) - b| ≤ ε / 2) :
1 - 32 * ((I.card : ℝ) + J.card) / ((Fintype.card F : ℝ)^2 * ε^2) ≤
(conditionedOrderLaw U V i x).toMeasure.real
{p | ε < |targetObservable U V I J x p - planeObservable U V I J p|} := by
rw [ProfileDetectionConcentration.marked_probability_eq
(finite_memLp _ (targetObservable U V I J x))
(finite_memLp _ (planeObservable U V I J))]
exact sub_le_sub_left
(unmarked_probability_le_of_profile_gap U V I J i x hUV hi hε hgap hTa hPb) 1
theorem unmarked_probability_le_scaled
(hUV : Disjoint U V) (hi : i ∉ I) {a b ε C : ℝ}
(hε : 0 < ε) (hgap : 3 * ε < |a - b|)
(hTa : |(∫ p, targetObservable U V I J x p
∂(conditionedOrderLaw U V i x).toMeasure) - a| ≤ ε / 2)
(hPb : |(∫ p, planeObservable U V I J p
∂(conditionedOrderLaw U V i x).toMeasure) - b| ≤ ε / 2)
(hsize : (I.card : ℝ) + J.card ≤ C * Fintype.card F) :
(conditionedOrderLaw U V i x).toMeasure.real
{p | |targetObservable U V I J x p - planeObservable U V I J p| ≤ ε} ≤
32 * C / ((Fintype.card F : ℝ) * ε^2) := by
have hq : (Fintype.card F : ℝ) ≠ 0 := by positivity
calc
_ ≤ 32 * ((I.card : ℝ) + J.card) / ((Fintype.card F : ℝ)^2 * ε^2) :=
unmarked_probability_le_of_profile_gap U V I J i x hUV hi hε hgap hTa hPb
_ ≤ 32 * (C * Fintype.card F) / ((Fintype.card F : ℝ)^2 * ε^2) := by
gcongr
_ = _ := by field_simp
end Conditional
end Balanced.ConditionalOrderDetection
end OAI
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