File size: 3,011 Bytes
dd14c26 | 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 | import OAI.Probability.MatroidProphet.Main
namespace OAI
/-!
# Source accounting contracts
The finite-fiber expectation below expresses conditioning on all test coordinates
outside the focal group. In contrast to an expectation over the entire test mask,
it retains the conditional quantifier in `lem:all-orders` (accounting.tex).
-/
namespace MatroidProphet
open Set Finset
variable {α : Type*} [Fintype α] [DecidableEq α]
/-- Fix every test bit outside `G`, and average only the fresh bits inside `G`.
The contracted base may depend arbitrarily on the fixed outside mask. -/
theorem fixed_outside_thinning_rank
(M : Matroid α) (hE : M.E = Set.univ) (t : ℝ) (ht0 : 0 ≤ t) (ht1 : t ≤ 1)
(G A : Finset α) (hAG : Disjoint A G) (S : Set α) (hS : S ⊆ (G : Set α))
(P : Finset α → Set α)
(hP : ∀ B : Finset α, B ⊆ G → P (A ∪ B) = P A) :
t * (conditionalRank M S (P A) : ℝ) ≤
bitsExpectation (fun _ => t) G
(fun B => (conditionalRank M (S ∩ ((A ∪ B : Finset α) : Set α))
(P (A ∪ B)) : ℝ)) := by
classical
have hdis : S ∩ (A : Set α) = ∅ := by
apply Set.eq_empty_iff_forall_notMem.mpr
rintro e ⟨heS, heA⟩
exact Finset.disjoint_left.mp hAG heA (hS heS)
have hb := independent_thinning_rank M hE G S.toFinset
(fun e he => hS (by simpa only [Set.mem_toFinset] using he)) (P A) t ht0 ht1
calc
_ ≤ bitsExpectation (fun _ => t) G
(fun B => (conditionalRank M (S ∩ (B : Set α)) (P A) : ℝ)) := by
simpa only [Finset.coe_inter, Set.coe_toFinset] using hb
_ = _ := by
apply bitsExpectation_congr
intro B hB
rw [hP B hB]
rw [Finset.coe_union, Set.inter_union_distrib_left, hdis, Set.empty_union]
/-- The source's test-thinning inequality conditional on the complete outside
configuration, not on an arrival order and not merely after averaging it away. -/
theorem fixed_outside_finalRankStatistic
(M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (D C : ℕ → Set α) (G : ℕ → Finset α)
(hG : Pairwise (fun i j => Disjoint (G i) (G j)))
(h final : ℕ) (O : Set α) (ε : Fin 2) (t : ℝ) (ht0 : 0 ≤ t) (ht1 : t ≤ 1)
(A : Finset α) (hA : Disjoint A (G h)) :
t * (finalRankStatistic M hE κ D C
(fun j => (A : Set α) ∩ (G j : Set α)) h final (G h) O Set.univ ε : ℝ) ≤
bitsExpectation (fun _ : α => t) (G h) (fun B =>
(finalRankStatistic M hE κ D C
(fun j => ((A ∪ B : Finset α) : Set α) ∩ (G j : Set α)) h final
(G h) O ((A ∪ B : Finset α) : Set α) ε : ℝ)) := by
simp only [finalRankStatistic, Set.inter_univ, Nat.cast_sum,
bitsExpectation_sum, mul_sum]
apply Finset.sum_le_sum
intro b hb
apply fixed_outside_thinning_rank M hE t ht0 ht1 (G h) A hA
· exact fun e he => nominalLayerSet_subset M hE κ D C h (G h) ε b he.1
· intro B hB
rw [lowerCompetition_ignore_focal M hE κ D C G hG b.val.val h A B hB]
end MatroidProphet
end OAI
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