  This paper addresses challenges in flexibly modeling multimodal data that lie
on constrained spaces. Such data are commonly found in spatial applications,
such as climatology and criminology, where measurements are restricted to a
geographical area. Other settings include domains where unsuitable recordings
are discarded, such as flow-cytometry measurements. A simple approach to
modeling such data is through the use of mixture models, especially
nonparametric mixture models. Mixture models, while flexible and theoretically
well-understood, are unsuitable for settings involving complicated constraints,
leading to difficulties in specifying the component distributions and in
evaluating normalization constants. Bayesian inference over the parameters of
these models results in posterior distributions that are doubly-intractable. We
address this problem via an algorithm based on rejection sampling and data
augmentation. We view samples from a truncated distribution as outcomes of a
rejection sampling scheme, where proposals are made from a simple mixture model
and are rejected if they violate the constraints. Our scheme proceeds by
imputing the rejected samples given mixture parameters and then resampling
parameters given all samples. We study two modeling approaches: mixtures of
truncated Gaussians and truncated mixtures of Gaussians, along with their
associated Markov chain Monte Carlo sampling algorithms. We also discuss
variations of the models, as well as approximations to improve mixing, reduce
computational cost, and lower variance. We present results on simulated data
and apply our algorithms to two problems; one involving flow-cytometry data,
and the other, crime recorded in the city of Chicago.
