  Bayesian models based on the Dirichlet process and other stick-breaking
priors have been proposed as core ingredients for clustering, topic modeling,
and other unsupervised learning tasks. Prior specification is, however,
relatively difficult for such models, given that their flexibility implies that
the consequences of prior choices are often relatively opaque. Moreover, these
choices can have a substantial effect on posterior inferences. Thus,
considerations of robustness need to go hand in hand with nonparametric
modeling. In the current paper, we tackle this challenge by exploiting the fact
that variational Bayesian methods, in addition to having computational
advantages in fitting complex nonparametric models, also yield sensitivities
with respect to parametric and nonparametric aspects of Bayesian models. In
particular, we demonstrate how to assess the sensitivity of conclusions to the
choice of concentration parameter and stick-breaking distribution for
inferences under Dirichlet process mixtures and related mixture models. We
provide both theoretical and empirical support for our variational approach to
Bayesian sensitivity analysis.
