Partitioning some multivariate distributions

Abstract

A partition of a finite set is a collection of disjoint and exhaustive subsets of that set. This paper concerns the distributions occasioned by partitioning a multivariate distribution, in particular to find the resultant distribution within each partition subset, and the distribution between them. If the multivariate distribution has independent univariate components, then partitioning leads to independent univariate components within each partition subset. Consequently, our attention is drawn to distributions without independent univariate components. The multivariate normal distribution is often the first such distribution that one thinks of. When a partition consists of only two subsets, the normal distribution can be partitioned into a conditional of one subset conditioned on the other, and a marginal distribution on the second subset. And each of these can be recursively partitioned. The result, however, is a product of normal densities that apparently depends on the order in which the partitioning is done, although it cannot in fact so depend, as each such product re-expresses the original normal multivariate distribution. Among partitions, there are two extreme cases. The first is the coarsest possible partition, consisting of the whole space. Then trivially the distribution over this partition is the original multivariate distribution, whatever it was. The other extreme case is a finest possible partition, each subset of which has only a single component. Then the distribution on such a subset takes the value of the random variable with probability one. So the partitions at issue in this paper are the partitions in-between, neither the coarsest nor the finest.