Abstract: This paper provides new versions of necessary and sufficient conditions for compatibility of finite discrete conditional distributions, and of the uniqueness for those compatible conditional distributions. We note that the ratio matrix (the matrix in Arnold and Press (1989)), after interchanging its rows and/or columns, can be rearranged to be an irreducible block diagonal matrix. We find that checking compatibility is equivalent to inspecting whether every block on the diagonal has a rank one positive extension, and that the necessary and sufficient conditions of the uniqueness, if the given conditional densities are compatible, is that the ratio matrix itself is irreducible. We show that each joint density, if it exists, corresponds to a rank one positive extension of the ratio matrix, and we characterize the set of all possible joint densities. Finally, we provide algorithms for checking compatibility, for checking uniqueness, and for constructing densities.
Key words and phrases: Compatibility, conditional density, irreducible block diagonal matrix, joint density, marginal density, rank one positive extension matrix, ratio matrix, uniqueness.