We study posterior contraction behaviors for parameters of interest in the context of Bayesian mixture modeling, where the number of mixing components is unknown while the model itself may or may not be correctly specified. Posterior contraction rates are given under optimal transport distances for two popular types of prior specification: one requires explicitly a prior distribution on the number of mixture components, and a nonparametric Bayesian approach which places a prior on the space of mixing distributions. Paraphrasing George Box, all mixture models are misspecified, but some may be more interpretable than others — it will be shown that the modeling choice of kernel density functions plays perhaps the most impactful roles in determining the posterior contraction rates in the misspecified situations. Drawing on concrete parameter estimation rates I will highlight some aspects about the interesting tradeoffs between model expressiveness and interpretability that a statistical modeler must negotiate in the rich world of mixture modeling.
This work is joint with Aritra Guha and Nhat Ho.