  This paper proposes solutions to three issues pertaining to the estimation of
finite mixture models with an unknown number of components: the
non-identifiability induced by overfitting the number of components, the mixing
limitations of standard Markov Chain Monte Carlo (MCMC) sampling techniques,
and the related label switching problem. An overfitting approach is used to
estimate the number of components in a finite mixture model via a Zmix
algorithm. Zmix provides a bridge between multidimensional samplers and test
based estimation methods, whereby priors are chosen to encourage extra groups
to have weights approaching zero. MCMC sampling is made possible by the
implementation of prior parallel tempering, an extension of parallel tempering.
Zmix can accurately estimate the number of components, posterior parameter
estimates and allocation probabilities given a sufficiently large sample size.
The results will reflect uncertainty in the final model and will report the
range of possible candidate models and their respective estimated probabilities
from a single run. Label switching is resolved with a computationally
light-weight method, Zswitch, developed for overfitted mixtures by exploiting
the intuitiveness of allocation-based relabelling algorithms and the precision
of label-invariant loss functions. Four simulation studies are included to
illustrate Zmix and Zswitch, as well as three case studies from the literature.
All methods are available as part of the R package Zmix, which can currently be
applied to univariate Gaussian mixture models
