Summary: Consider a discrete bivariate random variable \((X,Y)\) with possible values \(1,2,\dots,I\) for \(X\) and \(1,2,\dots,J\) for \(Y\). Suppose that putative families of conditional distributions, for \(X\) given values of \(Y\) and of \(Y\) given values of \(X\), are available. After reviewing conditions for compatibility 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.