Importance sampling schemes for evidence approximation in mixture models

TL;DR

This paper introduces two importance sampling schemes, including a Rao-Blackwellised approach, to improve evidence approximation in mixture models, addressing label switching issues and reducing computational costs.

Contribution

It proposes novel importance sampling methods for evidence estimation in mixture models, enhancing efficiency and accuracy while mitigating label switching challenges.

Findings

Dual importance sampling is a valid and more efficient estimator.

Approximate importance functions can reduce computational workload.

The proposed methods improve evidence estimation accuracy.

Abstract

The marginal likelihood is a central tool for drawing Bayesian inference about the number of components in mixture models. It is often approximated since the exact form is unavailable. A bias in the approximation may be due to an 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 label switching, as all possible label permutations must be simulated by a chain in order to converge and hence overcome the bias. In an importance sampling approach, imposing label switching to the importance function results in an exponential increase of the computational cost with the number of components. In this paper, two importance sampling schemes are proposed through choices for the importance function; a MLE proposal and a Rao-Blackwellised importance function. The second scheme is called dual importance sampling. We demonstrate that this dual importance sampling is a valid estimator of the evidence and moreover show that the statistical efficiency of estimates increases. To reduce the induced high demand in computation, the original importance function is approximated but a suitable approximation can produce an estimate with the same precision and with reduced computational workload.