Compatibility conditions for a set of conditional Gaussian distributions

Suppose that a n-dimensional (random) vector is partitioned into m sets of components as X=(X1,...,Xm), where the dimension of Xi is ni, i.e., dim(Xi)=ni, for i=1,...,m and [summation operator]i=1mni=n, with m Gaussian conditionalswhere X-i=(X1,...,Xi-1,Xi+1,...,Xm), dim(A'i,j)=(ni,nj), dim([Psi]i,i)=(ni,ni), and [Psi]i,i>0, for i=1,...,m. We present a set of simple conditions on Ai,j and [Psi]i,i for i,j=1,...,m and j[not equal to]i under which the conditionals are compatible. The conditions are obtained based on the idea that essentially considers the inverse of the problem in terms of the compatibility of the likelihood functions of the "parameters" [alpha]i,Ai,j, and [Psi]i,is. An illustrative example is also provided.