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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 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 | import OAI.Probability.MatroidProphet.SafeParity
import OAI.Probability.MatroidProphet.Residual.Statistics
namespace OAI
namespace MatroidProphet
open Finset Pivots
variable {α : Type*} [Fintype α] [DecidableEq α]
attribute [local instance] Classical.propDecidable
omit [DecidableEq α] in
lemma nominalBirth_eq_iff_departure (M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (D C : ℕ → Set α) (h : ℕ) (d : α) (hd : d ∉ M.closure ∅) (b : ℤ) :
nominalBirth M hE κ D C h d = b ↔
d ∈ nominalPath M hE κ D C h b ∧ d ∉ nominalPath M hE κ D C h (b-1) := by
rw [mem_nominal_iff_birth_le M hE κ D C h d hd b,
mem_nominal_iff_birth_le M hE κ D C h d hd (b-1)]
omega
omit [DecidableEq α] in
lemma isSafeSample_iff_window (M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (h : ℕ) (U : Set α) (I C T : Finset α)
(hIU : (I : Set α) ⊆ U) (d : α) (hdI : d ∈ I) (hd : d ∉ M.closure ∅) (parity : Bool) :
IsSafeSample M hE κ D G h d C T parity ↔
∃ b : ParityWindow (activation h) (boolParity parity),
d ∈ safeLayerSet M hE κ D (fun i => (C : Set α) ∩ (G i : Set α))
(fun i => (T : Set α) ∩ (G i : Set α)) h U (I : Set α) (boolParity parity) b := by
constructor
· intro hs
obtain ⟨hlo, hhi, hp, hex⟩ := hs
let birth := nominalBirth M hE κ D (fun i => (C : Set α) ∩ (G i : Set α)) h d
have hbwin : birth ∈ Set.Icc (activation h-2) 1 := by
change activation h+2 ≤ birth at hlo
change birth ≤ 0 at hhi
constructor <;> omega
have hbpar : birth % 2 = ((boolParity parity).val : ℤ) := by
rw [boolParity_val]
exact hp
refine ⟨⟨⟨birth, hbwin⟩, hbpar⟩, ?_⟩
have hdep := (nominalBirth_eq_iff_departure M hE κ D _ h d hd birth).mp rfl
exact ⟨⟨hdI, hIU hdI, hlo, hhi, hdep.1, hdep.2⟩, hex⟩
· rintro ⟨b, hb⟩
rcases hb with ⟨⟨_, _, hlo, hhi, hmem, hnot⟩, hex⟩
have hbirth := (nominalBirth_eq_iff_departure M hE κ D _ h d hd b.val.val).mpr ⟨hmem, hnot⟩
unfold IsSafeSample
rw [hbirth]
have hp := b.property.trans (boolParity_val parity)
exact ⟨hlo, hhi, hp, hex⟩
theorem safeSampleCount_eq_layerStatistic
{α : Type u_1} [Fintype α] [DecidableEq α] (M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (h : ℕ) (U : Set α) (I C T : Finset α)
(hIU : (I : Set α) ⊆ U) (hIL : ∀ d ∈ I, d ∉ M.closure ∅) (parity : Bool) :
safeSampleCount M hE κ D G h I C T parity =
(safeLayerStatistic M hE κ D (fun i => (C : Set α) ∩ (G i : Set α))
(fun i => (T : Set α) ∩ (G i : Set α)) h U (I : Set α) (boolParity parity) : ℝ) := by
let W := ParityWindow (activation h) (boolParity parity)
let L : W → Set α := safeLayerSet M hE κ D (fun i => (C : Set α) ∩ (G i : Set α))
(fun i => (T : Set α) ∩ (G i : Set α)) h U (I : Set α) (boolParity parity)
have hcount (b : W) : (L b).ncard = (I.filter (fun d => d ∈ L b)).card := by
have hs : L b = ((I.filter (fun d => d ∈ L b) : Finset α) : Set α) := by
ext d
constructor
· intro hd
exact mem_filter.mpr ⟨hd.1.1, hd⟩
· intro hd
exact (mem_filter.mp hd).2
exact (congrArg Set.ncard hs).trans (Set.ncard_coe_finset _)
have hlabel (d : α) (hd : d ∈ I) :
(∑ b : W, if d ∈ L b then (1:ℝ) else 0) = if IsSafeSample M hE κ D G h d C T parity then 1 else 0 := by
rw [sum_unique_indicators]
· have he : (∃ b ∈ (univ : Finset W), d ∈ L b) ↔ IsSafeSample M hE κ D G h d C T parity := by
simp only [mem_univ, true_and]
exact (isSafeSample_iff_window M hE κ D G h U I C T hIU d hd (hIL d hd) parity).symm
simp only [he]
· intro b hb c hc hdb hdc
by_contra hbc
have hdisj := nominalLayerSet_disjoint M hE κ D (fun i => (C : Set α) ∩ (G i : Set α))
h U (boolParity parity) hbc
exact Set.disjoint_left.mp hdisj hdb.1.2 hdc.1.2
calc
safeSampleCount M hE κ D G h I C T parity =
∑ d ∈ I, if IsSafeSample M hE κ D G h d C T parity then (1:ℝ) else 0 := by
rw [safeSampleCount_eq_card]
simp [safeSampleSet]
_ = ∑ d ∈ I, ∑ b : W, if d ∈ L b then (1:ℝ) else 0 :=
sum_congr rfl (fun d hd => (hlabel d hd).symm)
_ = ∑ b : W, ∑ d ∈ I, if d ∈ L b then (1:ℝ) else 0 := sum_comm
_ = ∑ b : W, ((L b).ncard : ℝ) := by
apply sum_congr rfl
intro b hb
rw [hcount]
simp
_ = _ := by simp only [safeLayerStatistic, Nat.cast_sum]; rfl
theorem safeLayerStatistic_lower (M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (hκ : 0 < κ) (D : ℕ → Set α) (G : ℕ → Finset α)
(hG : Pairwise (fun i j => Disjoint (G i) (G j))) (q : α → ℝ)
(hq0 : ∀ e, 0 ≤ q e) (hq1 : ∀ e, q e ≤ 1)
(h : ℕ) (U I : Set α) (hIU : I ⊆ U) (hI : M.Indep I) :
(((2:ℝ)^12)⁻¹) / 4 * I.ncard - (D h).ncard / (κ:ℝ) ≤
bitsExpectation q univ (fun C => bitsExpectation q univ (fun T =>
fairParityExpectation (fun parity => (safeLayerStatistic M hE κ D
(fun i => (C : Set α) ∩ (G i : Set α)) (fun i => (T : Set α) ∩ (G i : Set α))
h U I (boolParity parity) : ℝ)))) := by
let J := I.toFinset
have hJI : (J : Set α) = I := by simp [J]
have hJcard : J.card = I.ncard := by rw [← Set.ncard_coe_finset, hJI]
have hJ : M.Indep (J : Set α) := hJI.symm ▸ hI
have hJU : (J : Set α) ⊆ U := hJI.symm ▸ hIU
have hl := safeSampleCount_lower M hE κ hκ D G hG q hq0 hq1 h J hJ
rw [hJcard] at hl
have heq : bitsExpectation q univ (fun C => bitsExpectation q univ (fun T =>
fairParityExpectation (safeSampleCount M hE κ D G h J C T))) =
bitsExpectation q univ (fun C => bitsExpectation q univ (fun T =>
fairParityExpectation (fun parity => (safeLayerStatistic M hE κ D
(fun i => (C : Set α) ∩ (G i : Set α)) (fun i => (T : Set α) ∩ (G i : Set α))
h U I (boolParity parity) : ℝ)))) := by
apply bitsExpectation_congr
intro C hC
apply bitsExpectation_congr
intro T hT
congr 1
funext parity
rw [safeSampleCount_eq_layerStatistic M hE κ D G h U J C T hJU
(fun d hd => indep_not_loop M J hJ hd) parity, hJI]
exact hl.trans_eq heq
end MatroidProphet
end OAI
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