Compatible Conditional Distributions

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.