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The marginal likelihood is a central tool for drawing Bayesian inferenceabout the number of components in mixture models. It is often approximatedsince the exact form is unavailable. A bias in the approximation may be due toan incomplete exploration by a simulated Markov chain (e.g., a Gibbs sequence)of the collection of posterior modes, a phenomenon also known as lack of labelswitching, as all possible label permutations must be simulated by a chain inorder to converge and hence overcome the bias. In an importance samplingapproach, imposing label switching to the importance function results in anexponential increase of the computational cost with the number of components.In this paper, two importance sampling schemes are proposed through choices forthe importance function; a MLE proposal and a Rao-Blackwellised importancefunction. The second scheme is called dual importance sampling. We demonstratethat this dual importance sampling is a valid estimator of the evidence andmoreover show that the statistical efficiency of estimates increases. To reducethe induced high demand in computation, the original importance function isapproximated but a suitable approximation can produce an estimate with the sameprecision and with reduced computational workload.

Importance sampling schemes for evidence approximation in mixture models