  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. However, due to the flexibility of these
models, the consequences of prior choices can be opaque. And so prior
specification can be relatively difficult. At the same time, prior choice 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.
