Compatible Conditional Distributions

Abstract Consider two families of candidate conditional densities (or probability mass functions), f (x | y) ;y ∈ S y and f (y | x): x ∈ S x. This article investigates necessary and sufficient conditions for the existence of a joint density (or joint probability mass function) f (x, y) with the given families as its associated conditional densities. This supplements previous work that has addressed the question of uniqueness of f (x, y) assuming its existence.