| import Mathlib |
| import OAI.Computability.VertexCover.Repetition.SelectedMarginal |
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| namespace OAI |
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| namespace UniqueGames.Foundations.Repetition |
| open scoped BigOperators |
| open Games Information |
| noncomputable section |
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| theorem isProbability_comp_equiv {A B : Type*} [Fintype A] [Fintype B] |
| (e : A ≃ B) (p : B → ℝ) (hp : IsProbability p) : |
| IsProbability (fun a => p (e a)) := by |
| constructor |
| · intro a |
| exact hp.1 _ |
| · rw [e.sum_comp] |
| exact hp.2 |
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| theorem partialRevealMarginal_isProbability |
| {I T X Y : Type*} [Fintype I] [DecidableEq I] [Fintype T] |
| [Fintype X] [Fintype Y] [DecidableEq X] [DecidableEq Y] |
| (μ : FiniteDistribution (X × Y)) (j : I) |
| (likelihood : T → (I → X × Y) → ℝ) |
| (hp : IsProbability (fullRevealMarginal μ j likelihood)) : |
| IsProbability (partialRevealMarginal μ j likelihood) := by |
| apply (maskedJoint_isProbability_iff _).mp |
| have h := isProbability_comp_equiv (revealSplitEquiv (T := T) j).symm _ hp |
| have he : (fun z => fullRevealMarginal μ j likelihood |
| ((revealSplitEquiv (T := T) j).symm z)) = |
| maskedJoint (partialRevealMarginal μ j likelihood) := by |
| funext z |
| exact fullRevealMarginal_merge μ j likelihood z.1.1.1 z.1.1.2 z.1.2 z.2 |
| rw [he] at h |
| exact h |
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| variable {Q₁ Q₂ A₁ A₂ : Type*} |
| [Fintype Q₁] [Fintype Q₂] [Fintype A₁] [Fintype A₂] |
| [DecidableEq Q₁] [DecidableEq Q₂] {n : Nat} |
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| abbrev SelectedCommonData (selected : Finset (Fin n)) |
| (j : {i : Fin n // i ∉ selected}) := |
| ((selected → Q₁ × Q₂) × SelectedLabels (A₁ := A₁) (A₂ := A₂) selected) × |
| ({i : {i : Fin n // i ∉ selected} // i ≠ j} → Q₁ ⊕ Q₂) |
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| theorem selected_fullReveal_isProbability (G : Game Q₁ Q₂ A₁ A₂) |
| (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂)) |
| (selected : Finset (Fin n)) (positive : 0 < G.selectedSuccess strategy selected) |
| (j : {i : Fin n // i ∉ selected}) : |
| IsProbability (fullRevealMarginal G.questions j |
| (selectedOutsideLikelihood G strategy selected)) := by |
| have h := isProbability_comp_equiv |
| (selectedObservationEquiv (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected).symm |
| _ (selectedRawMarginal_isProbability G strategy selected positive j) |
| have he : (fun z => selectedRawMarginal G strategy selected j |
| ((selectedObservationEquiv (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected).symm z)) = |
| fullRevealMarginal G.questions j (selectedOutsideLikelihood G strategy selected) := by |
| funext z |
| exact selectedRawMarginal_fullReveal G strategy selected j z |
| rw [he] at h |
| exact h |
|
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| def selectedCommonLaw (G : Game Q₁ Q₂ A₁ A₂) |
| (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂)) |
| (selected : Finset (Fin n)) (positive : 0 < G.selectedSuccess strategy selected) |
| (j : {i : Fin n // i ∉ selected}) : |
| FiniteDistribution (SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) |
| (A₁ := A₁) (A₂ := A₂) selected j × (Q₁ × Q₂)) := |
| toGameLaw (partialRevealMarginal G.questions j (selectedOutsideLikelihood G strategy selected)) |
| (partialRevealMarginal_isProbability G.questions j _ |
| (selected_fullReveal_isProbability G strategy selected positive j)) |
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| end |
| end UniqueGames.Foundations.Repetition |
| end |
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| end |
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| end |
| end |
| end |
| end |
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| end OAI |
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