Bayesian Mixture Labeling by Highest Posterior Density

A fundamental problem for Bayesian mixture model analysis is label switching, which occurs as a result of the nonidentifiability of the mixture components under symmetric priors. We propose two labeling methods to solve this problem. The first method, denoted by PM(ALG), is based on the posterior modes and an ascending algorithm generically denoted ALG. We use each Markov chain Monte Carlo sample as the starting point in an ascending algorithm, and label the sample based on the mode of the posterior to which it converges. Our natural assumption here is that the samples converged to the same mode should have the same labels. The PM(ALG) labeling method has some computational advantages over other popular labeling methods. Additionally, it automatically matches the "ideal" labels in the highest posterior density credible regions. The second method does labeling by maximizing the normal likelihood of the labeled Gibbs samples. Using a Monte Carlo simulation study and a real dataset, we demonstrate the success of our new methods in dealing with the label switching problem.