A note on compatibility of conditional autoregressive models

Suppose that, to assess the joint distribution of a random vector (X1,…,Xn), one selects the kernels Q1,…,Qn with Qi to be regarded as a possible conditional distribution for Xi given (Xj:j≠i); Q1,…,Qn are compatible if there exists a joint distribution for (X1,…,Xn) with conditionals Q1,…,Qn. Similarly, Q1,…,Qn 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 Q1,…,Qn are characterized under some assumptions on their functional form. The characterization applies, in particular, if each Qi belongs to a one parameter exponential family. Special attention is paid to Gaussian conditional autoregressive models.