On the construction of a joint distribution given two discrete conditionals

Consider a two-dimensional discrete random variable (X, Y ) with possible values 1, 2, . . . , I for X and 1, 2, . . . , J for Y . For specifying the distribution of (X, Y ), suppose both conditional distributions of X given Y and Y given X are specified. In this paper, we address the problem of determining whether a given set of constraints involving marginal and conditional probabilities and expectations of functions are compatible or most nearly compatible. To this end, we incorporate all those information with the Kullback-Leibler (K-L) divergence and power divergence statistics to obtain the most nearly compatible probability distribution when the two conditionals are not compatible, under the discrete set up. Finally, a comparative study is carried out between the K-L divergence and power divergence statistics for some illustrative examples.