  This article establishes general conditions for posterior consistency of
Bayesian finite mixture models with a prior on the number of components. That
is, we provide sufficient conditions under which the posterior concentrates on
neighborhoods of the true parameter values when the data are generated from a
finite mixture over the assumed family of component distributions.
Specifically, we establish almost sure consistency for the number of
components, the mixture weights, and the component parameters, up to a
permutation of the component labels. The approach taken here is based on Doob's
theorem, which has the advantage of holding under extraordinarily general
conditions, and the disadvantage of only guaranteeing consistency at a set of
parameter values that has probability one under the prior. However, we show
that in fact, for commonly used choices of prior, this yields consistency at
Lebesgue-almost all parameter values -- which is satisfactory for most
practical purposes. We aim to formulate the results in a way that maximizes
clarity, generality, and ease of use.
