  We study Bayesian estimation of finite mixture models in a general setup
where the number of components is unknown and allowed to grow with the sample
size. An assumption on growing number of components is a natural one as the
degree of heterogeneity present in the sample can grow and new components can
arise as sample size increases, allowing full flexibility in modeling the
complexity of data. This however will lead to a high-dimensional model which
poses great challenges for estimation. We novelly employ the idea of a sample
size dependent prior in a Bayesian model and establish a number of important
theoretical results. We first show that under mild conditions on the prior, the
posterior distribution concentrates around the true mixing distribution at a
near optimal rate with respect to the Wasserstein distance. Under a separation
condition on the true mixing distribution, we further show that a better and
adaptive convergence rate can be achieved, and the number of components can be
consistently estimated. Furthermore, we derive optimal convergence rates for
the higher-order mixture models where the number of components diverges
arbitrarily fast. In addition, we suggest a simple recipe for using Dirichlet
process (DP) mixture prior for estimating the finite mixture models and provide
theoretical guarantees. In particular, we provide a novel solution for adopting
the number of clusters in a DP mixture model as an estimate of the number of
components in a finite mixture model. Simulation study and real data
applications are carried out demonstrating the utilities of our method.
