To Bridge, to Warp or to Wrap?

Important choices for efficient and accurate evaluation of marginal likelihoods by means of Monte Carlo simulation methods are studied for the case of highly non-elliptical posterior distributions. We focus on the situation where one makes use of importance sampling or the independence chain Metropolis-Hastings algorithm for posterior analysis. A comparative analysis is presented of possible advantages and limitations of different simulation techniques; of possible choices of candidate distributions and choices of target or warped target distributions; and finally of numerical standard errors. The importance of a robust and flexible estimation strategy is demonstrated where the complete posterior distribution is explored. In this respect, the adaptive mixture of Student-t distributions of Hoogerheide et al.(2007) works particularly well. Given an appropriately yet quickly tuned candidate, straightforward importance sampling provides the most efficient estimator of the marginal likelihood in the cases investigated in this paper, which include a non-linear regression model of Ritter and Tanner (1992) and a conditional normal distribution of Gelman and Meng (1991). A poor choice of candidate density may lead to a huge loss of efficiency where the numerical standard error may be highly unreliable.