RANGES OF POSTERIOR MEASURES FOR SOME CLASSES OF PRIORS WITH SPECIFIED MOMENTS

In Bayesian analysis, it is often difficult to completely specify the prior distribution. One way of expressing uncertainty about the prior is by specifying only the first few moments of the parameters or some functions of parameters. Thus the prior is restricted to a certain class but is otherwise arbitrary. We examine the ranges of certain posterior Bayesian measures when the priors belong to that class. The method is used in a contingency table example, where prior moments of interesting functions of cell probabilities are specified. Exploiting the convex structure of the class, we obtain posterior bounds by a low dimensional minimization or maximization and we propose possible solutions to computational problems. We examine the general problem theoretically and we find conditions which ensure that the posterior measures have finite bounds. We also consider the case when the priors are restricted to a smaller class, by requiring them to be bell shaped.