Independence structure of natural conjugate densities to exponential families and the Gibbs' sampler

In this paper the independence between a block of natural parameters and the complementary block of mean value parameters holding for densities which are natural conjugate to some regular exponential families is used to design in a convenient way a Gibbs' sampler with block updates. Even when the densities of interest are obtained by conditioning to zero a block of natural parameters in a density conjugate to a larger "saturated" model, the updates require only the computation of marginal distributions under the "unconditional" density. For exponential families which are closed under marginalization, including both the zero mean Gaussian family and the cross-classified Bernoulli family such an implementation of the Gibbs' sampler can be seen as an Iterative Proportional Fitting algorithm with random inputs.