Assessing Sensitivity to the Stick-Breaking Prior in Bayesian Nonparametrics
Two central questions in many probabilistic clustering problems is how many distinct clusters are present in a particular dataset, and which observations cluster together. Bayesian nonparametrics (BNP) addresses this question by placing a generative process on cluster assignment, making the number of distinct clusters present amenable to Bayesian inference. However, like all Bayesian approaches, BNP requires the specification of a prior, and this prior may favor a greater or fewer number of distinct clusters. In practice, it is important to establish that the prior is not too informative, particularly when---as is often the case in BNP---the particular form of the prior is chosen for mathematical convenience rather than because of a considered subjective belief.
We derive local sensitivity measures for assessing the impact of the prior based on Taylor expansions of a truncated variational Bayes approximation. Using a stick-breaking representation of a Dirichlet process, we consider perturbations both to the scalar concentration parameter and to the functional form of the stick-breaking distribution by embedding the stick-breaking density in the $L_p$ vector spaces of integrable functions. We prove that, though the variational optimum is directionally differentiable for all $1 \le p \le \infty$, the derivative provides a uniformly good approximation in a neighborhood of the original prior only with multiplicative perturbations and $p=\infty$.
We apply our methods to several real-world datasets, estimating sensitivity to the BNP prior specification of key posterior quantities. We show the accuracy of our approximation both for particular parametric and nonparametric perturbations, and demonstrate the usefulness of the influence function to find maximally influential alternative priors. [Joint work with Tamara Broderick].