Our goal is to discuss general conditions under which candidate families of conditional distributions for X given Y and Y given X are compatible in the sense that there exists a valid joint distribution for the bivariate random variable (X, Y) with the given families as its corresponding conditional distributions. The concepts are most easily visualized when densities exist and, in particular, are most transparent when X, Y are discrete and each has only a finite set of possible values. We begin our discussion in such a setting.
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