Some new formulae for posterior expectations and Bartlett corrections

Some new accurate approximations for posterior expectations and Bartlett corrections are derived. These approximations are modifications of formulae based on signed root log-likelihood ratios obtained in Sweeting (1996) and are designed to address two problems that arise in the practical application of these formulae in the multiparameter case. The first problem is a computational one associated with inversion of signed root log-likelihood ratios. The second concerns the form of the posterior expectation formula, which is not in a particularly convenient form for the computation of predictive densities. The theory is illustrated by two examples.