Dealing with label switching in mixture models under genuine multimodality

The fitting of finite mixture models is an ill-defined estimation problem as completely
different parameterizations can induce similar mixture distributions. This
leads to multiple modes in the likelihood which is a problem for frequentist maximum
likelihood estimation, and complicates statistical inference of Markov chain
Monte Carlo draws in Bayesian estimation. For the analysis of the posterior density
of these draws a suitable separation into different modes is desirable. In addition, a
unique labelling of the component specific estimates is necessary to solve the label
switching problem. This paper presents and compares two approaches to achieve
these goals: relabelling under multimodality and constrained clustering. The algorithmic
details are discussed and their application is demonstrated on artificial and
real-world data.