
We consider the following estimation problem. Let X1,..., XN be a sequence of N independent random vectors with common but unknown probability distribution F. The { Xi } are not observed. Let Y1,..., YN be another sequence of N independent random vectors which are observed. Assume the conditional probability P(Yi  Xi) is known for i = 1,..., N. The problem is to estimate the probability distribution F given the { Yi }. The main theoretical result is that the maximum likelihood estimate of F can be found in the space of discrete distributions with no more that N support points. An elegant proof of this will be given using results of convexity theory in a topological vector space. The practical numerical problem of actually determining the discrete maximum likelihood estimator of F is still not solved in a satisfactory manner. Examples from the area of applied pharmacokinetics will be given.

