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**COMPUTING BAYES FACTORS BY COMBINING SIMULATION AND
ASYMPTOTIC APPROXIMATIONS**

**Thomas J. DiCiccio, Robert E. Kass, Adrian Raftery and Larry Wasserman**

### Abstract:

*The Bayes factor is a ratio of two posterior
normalizing constants, which may be
difficult to compute. We compare several methods of estimating these
when it is possible to simulate observations from the posterior
distributions, via Markov chain Monte Carlo or other techniques.
The methods we study are all easily applied without
consideration of special features of the problem, assuming
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 corrections, 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 Meng and Wong (1993).
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 achieves an order of magnitude
improvement in accuracy.
**
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