Title: Flexible Mixture Modeling on Constrained Spaces,
Abstract: This paper addresses challenges in flexibly modeling multimodal data that lie
on constrained spaces. Applications include climate or crime measurements in a
geographical area, or flow-cytometry experiments, where unsuitable recordings
are discarded. A simple approach to modeling such data is through the use of
mixture models, with each component following an appropriate truncated
distribution. Problems arise when the truncation involves complicated
constraints, leading to difficulties in specifying the component distributions,
and in evaluating their 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 components and truncated mixtures of components. In both
situations, we describe exact Markov chain Monte Carlo sampling algorithms, as
well as approximations that bound the number of rejected samples, achieving
computational efficiency and lower variance at the cost of asymptotic bias.
Overall, our methodology only requires practitioners to provide an indicator
function for the set of interest. We present results on simulated data and
apply our algorithm to two problems, one involving flow-cytometry data, and the
other, crime recorded in the city of Chicago.