COMPUTING BAYES FACTORS BY COMBINING SIMULATION AND ASYMPTOTIC APPROXIMATIONS

The Bayes factor is a ratio of two posterior normalizing constants. wh ich may be difficult to compute. We compare several methods of estimat ing Bayes factors when it is possible to simulate observations from th e posterior distributions, via Markov chain Monte Carlo or other techn iques. The methods that we study are all easily applied without consid eration of special features of the problem, provided that each posteri or distribution is well behaved in the sense of having a single domina nt mode. We consider a simulated version of Laplace's method, a simula ted version of Bartlett correction, importance sampling, and a recipro cal importance sampling technique. We also introduce local volume corr ections for each of these. In addition, we apply the bridge sampling m ethod of Meng and Wong. We find that a simulated version of Laplace's method, with local volume correction, furnishes an accurate approximat ion that is especially useful when likelihood function evaluations are costly. A simple bridge sampling technique in conjunction with Laplac e's method often achieves an order of magnitude improvement in accurac y.