Conditionally Specified Multivariate Discrete Distributions

Abstract

To determine a joint distribution from a collection of conditional distributions is called conditionally specified models. Arnold et al. (2001) provides up-to-date accounts for many theoretical issues related to conditionally specified distributions. The first issue is to verify that a solution exists, which is also referred to as compatibility among conditional densities. When compatibility is confirmed, the next step is to assert the uniqueness of the joint distribution. Afterward, how to compute the joint from the conditionals is also of great interest. This paper is devoted exclusively to conditionally specified discrete distributions, because all the above issues can be easily answered with an encompassing decomposition. Hence, I will discuss the decomposition first. Instead of probabilities itself, my approach is based on the interaction parameters of the decomposition. Therefore, my approach is entirely different from that of Arnold et al. (2001). We prove that (a) two conditional densities are compatible if their overlapping interaction parameters are the same; (b) the joint is uniquely determined only when the set of interaction parameters is complete; and (c) the joint can be computed without iterations or simulations. Examples will be shown to illustrate the computations.