The Bayes factor is a ratio of two posterior normalizing constants, which may be difficult to compute. We compare several methods of estimating Bayes factors when it is possible to simulate observations from the posterior distributions, via Markov chain Monte Carlo or other techniques. The methods that we study are all easily applied without consideration of special features of the problem, provided that each posterior distribution is well behaved in the sense of having a single dominant mode.
We consider a simulated version of Laplace’s method, a simulated version of Bartlett correction, importance sampling, and a reciprocal importance sampling technique. We also introduce local volume corrections for each of these. In addition, we apply the bridge sampling method of X-L. Meng and W.H. Wong [Stat. Sin. 6, 831–860 (1996; Zbl 0857.62017)]. We find that a simulated version of Laplace’s method, with local volume correction, furnishes an accurate approximation that is especially useful when likelihood function evaluations are costly. A simple bridge sampling technique in conjunction with Laplace’s method often achieves an order of magnitude improvement in accuracy.