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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 | import OAI.Probability.MatroidProphet.Main
namespace OAI
/-!
# Exact conditional fibers of a finite Bernoulli product law
A fixed outside mask is a genuine conditioning event. Restricting the full
product-law expectation to that event factors into its atom probability times
the expectation over the unrevealed coordinates. The unnormalized identity
also covers zero-probability fibers and endpoint Bernoulli rates.
-/
namespace MatroidProphet
open Finset
variable {α : Type*} [DecidableEq α]
/-- The weighted expectation of a single supported mask atom. -/
lemma bitsExpectation_mask_atom (q : α → ℝ) (V A : Finset α)
(hA : A ⊆ V) (c : ℝ) :
bitsExpectation q V (fun S => if S = A then c else 0) =
bitsWeight q V A * c := by
simp [bitsExpectation, mul_ite, hA]
/-- Restrict a full product-law expectation to one outside-coordinate fiber.
There are no positivity assumptions: null fibers are handled exactly. -/
theorem bitsExpectation_fixed_outside_event (q : α → ℝ)
(V G A : Finset α) (hG : G ⊆ V) (hA : A ⊆ V \ G)
(f : Finset α → ℝ) :
bitsExpectation q V (fun T => if T \ G = A then f T else 0) =
bitsWeight q (V \ G) A *
bitsExpectation q G (fun B => f (A ∪ B)) := by
rw [bitsExpectation_split q V G hG]
have hf : bitsExpectation q (V \ G)
(fun S => bitsExpectation q G
(fun B => if (S ∪ B) \ G = A then f (S ∪ B) else 0)) =
bitsExpectation q (V \ G) (fun S =>
if S = A then bitsExpectation q G (fun B => f (A ∪ B)) else 0) := by
apply bitsExpectation_congr
intro S hS
have hout (B : Finset α) (hB : B ⊆ G) : (S ∪ B) \ G = S := by
ext e
simp only [mem_sdiff, mem_union]
constructor
· rintro ⟨heS | heB, heG⟩
· exact heS
· exact False.elim (heG (hB heB))
· intro heS
exact ⟨Or.inl heS, (mem_sdiff.mp (hS heS)).2⟩
by_cases hSA : S = A
· subst S
rw [ite_eq_left rfl]
apply bitsExpectation_congr
intro B hB
rw [hout B hB, ite_eq_left rfl]
· rw [ite_eq_right hSA]
calc
_ = bitsExpectation q G (fun _ => 0) := by
apply bitsExpectation_congr
intro B hB
rw [hout B hB, ite_eq_right hSA]
_ = 0 := bitsExpectation_const q G 0
rw [hf]
exact bitsExpectation_mask_atom q (V \ G) A hA _
/-- The probability of a specified outside mask is exactly its product atom. -/
theorem bitsExpectation_fixed_outside_probability (q : α → ℝ)
(V G A : Finset α) (hG : G ⊆ V) (hA : A ⊆ V \ G) :
bitsExpectation q V (fun T => if T \ G = A then 1 else 0) =
bitsWeight q (V \ G) A := by
simpa only [bitsExpectation_const, mul_one] using
bitsExpectation_fixed_outside_event q V G A hG hA (fun _ => 1)
/-- On a nonnull outside fiber, the normalized conditional expectation is
exactly the fresh-coordinate expectation used in the accounting contract. -/
theorem bitsExpectation_condition_on_outside (q : α → ℝ)
(V G A : Finset α) (hG : G ⊆ V) (hA : A ⊆ V \ G)
(hpos : bitsWeight q (V \ G) A ≠ 0) (f : Finset α → ℝ) :
bitsExpectation q V (fun T => if T \ G = A then f T else 0) /
bitsExpectation q V (fun T => if T \ G = A then 1 else 0) =
bitsExpectation q G (fun B => f (A ∪ B)) := by
rw [bitsExpectation_fixed_outside_event q V G A hG hA f,
bitsExpectation_fixed_outside_probability q V G A hG hA]
simp [hpos]
/-- Interior coordinate rates make every outside atom positive, including
empty/full masks and the empty outside ground set. -/
lemma bitsWeight_pos_of_interior (q : α → ℝ) (V A : Finset α)
(hq0 : ∀ e ∈ V, 0 < q e) (hq1 : ∀ e ∈ V, q e < 1) :
0 < bitsWeight q V A := by
unfold bitsWeight
apply Finset.prod_pos
intro e he
split_ifs
· exact hq0 e he
· exact sub_pos.mpr (hq1 e he)
/-- Constant interior-rate specialization: no nonnull-fiber premise remains.
This applies in particular to the source's `t = 2⁻¹⁴⁰`. -/
theorem bitsExpectation_condition_on_outside_const
(t : ℝ) (ht0 : 0 < t) (ht1 : t < 1)
(V G A : Finset α) (hG : G ⊆ V) (hA : A ⊆ V \ G)
(f : Finset α → ℝ) :
bitsExpectation (fun _ => t) V (fun T => if T \ G = A then f T else 0) /
bitsExpectation (fun _ => t) V (fun T => if T \ G = A then 1 else 0) =
bitsExpectation (fun _ => t) G (fun B => f (A ∪ B)) := by
exact bitsExpectation_condition_on_outside (fun _ => t) V G A hG hA
(ne_of_gt (bitsWeight_pos_of_interior _ _ _ (fun _ _ => ht0) (fun _ _ => ht1))) f
end MatroidProphet
end OAI
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