Distributions most nearly compatible with given families of conditional distributions

Consider a discrete bivariate random variable (X, Y) with possible values 1, 2, ...,I forX and 1, 2, ...J forY. Suppose that putative families of conditional distributions, forX given values ofY and ofY given values ofX, are available. After reviewing conditions for compatibiity of such conditional specifications of the distribution of (X, Y), attention is focussed on the incompatible case. The Kullback-Leibler information function is shown to provide a convenient measure of inconsistency. Using it, algorithms are provided for computing the joint distribution for (X, Y) that is least discrepant from the given inconsistent conditional specifications. Other discrepancy measures are briefly discussed.