| import Mathlib.Algebra.BigOperators.Group.Finset.Basic | |
| import Mathlib.Basic.Real.Basic | |
| namespace OAI | |
| /-! | |
| # Finite conditional seed kernels | |
| The observation map can include the branch and the sacrificed mask. Its domain | |
| is the *complete* source seed, so conditioning does not discard unused coins. | |
| These definitions are independent of every proposed selection-rule proof. | |
| -/ | |
| namespace MatroidProphet.Secretary | |
| open scoped BigOperators | |
| variable {α β : Type*} [Fintype α] [DecidableEq α] [DecidableEq β] | |
| /-- The source mass of a fiber of the observation map. -/ | |
| noncomputable def fiberMass (p : α → ℝ) (f : α → β) (b : β) : ℝ := | |
| ∑ a, if f a = b then p a else 0 | |
| /-- The conditional source mass, totalized by a fixed point mass on empty fibers. | |
| The fallback only concerns observations of source probability zero. -/ | |
| noncomputable def fiberKernel (p : α → ℝ) (f : α → β) (fallback : α) | |
| (b : β) (a : α) : ℝ := | |
| if fiberMass p f b = 0 then | |
| if a = fallback then 1 else 0 | |
| else if f a = b then p a / fiberMass p f b else 0 | |
| end MatroidProphet.Secretary | |
| end OAI | |