| import Mathlib |
| import OAI.Computability.MinUncut.Games.JointHintSampling |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open scoped BigOperators |
| namespace MinUncut.Outer |
| open MinUncut.Inner OuterSmoothness MeasureTheory |
| attribute [local instance] Classical.propDecidable BinaryFourier.dualFintype |
| variable {Name I S : Type*} [Fintype I] [Fintype S] |
|
|
| abbrev EdgeStatistic (Name I : Type*) [Fintype I] := |
| (U : I → Equation Name) → (hidden : I → Bool) → (pos : I → Fin 3) → ℝ |
|
|
| def edgeMean (equations : S → Equation Name) (k : ℕ) (F : EdgeStatistic Name I) : ℝ := |
| 𝔼 H : FixedSets I k, 𝔼 a : I → S, 𝔼 pos : I → Fin 3, |
| F (fun i => equations (a i)) (hiddenSet H.val) pos |
|
|
| lemma edgeMean_mono (equations : S → Equation Name) (k : ℕ) {F G : EdgeStatistic Name I} |
| (h : ∀ U hidden pos, F U hidden pos ≤ G U hidden pos) : |
| edgeMean equations k F ≤ edgeMean equations k G := by |
| exact Finset.expect_le_expect (fun H _ => Finset.expect_le_expect (fun a _ => |
| Finset.expect_le_expect (fun pos _ => h _ _ _))) |
|
|
| lemma edgeMean_add (equations : S → Equation Name) (k : ℕ) (F G : EdgeStatistic Name I) : |
| edgeMean equations k (fun U h pos => F U h pos+G U h pos) = |
| edgeMean equations k F+edgeMean equations k G := by |
| simp only [edgeMean,Finset.expect_add_distrib] |
|
|
| lemma edgeMean_mul (equations : S → Equation Name) (k : ℕ) (F : EdgeStatistic Name I) (t : ℝ) : |
| edgeMean equations k (fun U h pos => t*F U h pos) = t*edgeMean equations k F := by |
| simp only [edgeMean,← Finset.mul_expect] |
|
|
| lemma edgeMean_expect {X : Type*} [Fintype X] (equations : S → Equation Name) (k : ℕ) |
| (F : X → EdgeStatistic Name I) : |
| (𝔼 x, edgeMean equations k (F x)) = edgeMean equations k (fun U h pos => 𝔼 x,F x U h pos) := by |
| unfold edgeMean |
| rw [Finset.expect_comm] |
| congr 1 |
| funext H |
| rw [Finset.expect_comm] |
| congr 1 |
| funext a |
| rw [Finset.expect_comm] |
|
|
| lemma edgeMean_const [Nonempty S] (equations : S → Equation Name) {k : ℕ} |
| (hk : k≤Fintype.card I) (t : ℝ) : edgeMean (I := I) equations k (fun _ _ _ => t)=t := by |
| let : Nonempty (FixedSets I k) := fixedSets_nonempty hk |
| simp only [edgeMean,Fintype.expect_const] |
|
|
| end MinUncut.Outer |
|
|
| end |
| end |
|
|
| end OAI |
|
|
