Suppose that, to assess the joint distribution of a random vector (X-1,...,X-n), one selects the kernels Q(1),...,Q(n) with Q(1) to be regarded as a possible conditional distribution for X-i, given (X, : j not equal i);,Q(n) are compatible if there exists a joint distribution for (X-1,...,X-n) with conditionals Q(1),...,Q(n). Similarly, are improperly compatible if they can be obtained, according to the usual rule, with an improper distribution in place of a probability distribution. In this paper, compatibility and improper compatibility of Q(1),...,Q(n) are characterized under some assumptions on their functional form. The characterization applies, in particular, if each Q(i) belongs to a one parameter exponential family. Special attention is paid to Gaussian conditional autoregressive models. (C) 2017 Elsevier B.V. All rights reserved.