Difficulties in drawing inferences with finite-mixture models: A simple example with a simple solution

Likelihood functions from finite mixture models have many unusual features. Maximum likelihood (ML) estimates may behave poorly over repeated samples, and the abnormal shape of the likelihood often makes it difficult to assess the uncertainty in parameter estimates. Bayesian inference via Markov chain Monte Carlo (MCMC) can be a useful alternative to ML, but the component labels may switch during the MCMC run, making the output difficult to interpret. Two basic methods for handling the label-switching problem have been proposed: imposing constraints on the parameter space and cluster-based relabeling of the simulated parameters. We have found that label switching may also be reduced by supplying small amounts of prior information that are asymmetric with respect to the mixture components. Simply assigning one observation to each component a priori may effectively eliminate the problem. Using a very simple example-a univariate sample from a mixture of two exponentials-we evaluate the performance of likelihood and MCMC-based estimates and intervals over repeated sampling. Our simulations show that MCMC performs much better than ML if the label-switching problem is adequately addressed, and that asymmetric prior information performs as well as or better than the other proposed methods.