Easy Computation of Bayes Factors and Normalizing Constants for Mixture Models via Mixture Importance Sampling
Mary Jane Emond, Adrian E. Raftery, Russell John Steele · 2001
We propose a method for approximating integrated likelihoods, or posterior normalizing constants, in finite mixture models, for which analytic approximations such as the Laplace method are invalid. Integrated likelihoods are key components of Bayes factors and of the posterior model probabilities used in Bayesian model averaging. The method starts by formulating the model in terms of the unobserved group memberships, Z, and making these, rather than the model parameters, the variables of integration. The integral is then evaluated using importance sampling over the Z. The tricky part is choosing the importance sampling function, and we study the use of mixtures as importance sampling functions. We propose two forms of this: defensive mixture importance sampling (DMIS), and Z-distance importance sampling. We choose the parameters of the mixture adaptively, and we show how this can be done so as to approximately minimize the variance of the approximation to the integral.