A bivariate distribution can sometimes be characterized completely by properties of its conditional distributions. The present article surveys available research in this area. Questions of compatibility of conditional specifications are addressed as are characterizations of distributions based on their having conditional distributions that are members of prescribed parametric families of distributions. The topics of compatibility and near compatibility of conditional distributions are discussed. Estimation strategies for conditionally specified distributions are summarized. Additionally, certain conditionally specified densities are shown to provide convenient flexible conjugate prior families in certain multi- parameter Bayesian settings.
"Conditionally Specified Distributions: An Introduction (with comments and a rejoinder by the authors)." Statist. Sci. 16 (3) 249 - 274, August 2001. https://doi.org/10.1214/ss/1009213728