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 cla ss but is otherwise arbitrary. We examine the ranges of certain poster ior Bayesian measures when the priors belong to that class. The method is used in a contingency table example, where prior moments of intere sting functions of cell probabilities are specified. Exploiting the co nvex structure of the class, we obtain posterior bounds by a low dimen sional minimization or maximization and we propose possible solutions to computational problems. We examine the general problem theoreticall y and we find conditions which ensure that the posterior measures have finite bounds. We also consider the case when the priors are restrict ed to a smaller class, by requiring them to be bell shaped.