  Finite mixture models are a useful statistical model class for clustering and
density approximation. In the Bayesian framework finite mixture models require
the specification of suitable priors in addition to the data model. These
priors allow to avoid spurious results and provide a principled way to define
cluster shapes and a preference for specific cluster solutions. A generic model
estimation scheme for finite mixtures with a fixed number of components is
available using Markov chain Monte Carlo (MCMC) sampling with data
augmentation. The posterior allows to assess uncertainty in a comprehensive
way, but component-specific posterior inference requires resolving the label
switching issue.
  In this paper we focus on the application of Bayesian finite mixture models
for clustering. We start with discussing suitable specification, estimation and
inference of the model if the number of components is assumed to be known. We
then continue to explain suitable strategies for fitting Bayesian finite
mixture models when the number of components is not known. In addition, all
steps required to perform Bayesian finite mixture modeling are illustrated on a
data example where a finite mixture model of multivariate Gaussian
distributions is fitted. Suitable prior specification, estimation using MCMC
and posterior inference are discussed for this example assuming the number of
components to be known as well as unknown.
