LAPLACE APPROXIMATIONS FOR NATURAL EXPONENTIAL-FAMILIES WITH CUTS

Standard and fully exponential form Laplace approximations to marginal densities are described and conditions under which these give exact a nswers are investigated. A general result is obtained and is subsequen tly applied in the case of natural exponential families with cuts, in order to derive the marginal posterior density of the mean parameter c orresponding to the cut, the canonical parameter corresponding to the complement of the cut and transformations of these, Important cases of families for which a cut exists and the approximations are exact are presented as examples.