  Clustering procedures typically estimate which data points are clustered
together, a quantity of primary importance in many analyses. Often used as a
preliminary step for dimensionality reduction or to facilitate interpretation,
finding robust and stable clusters is often crucial for appropriate for
downstream analysis. In the present work, we consider Bayesian nonparametric
(BNP) models, a particularly popular set of Bayesian models for clustering due
to their flexibility. Because of its complexity, the Bayesian posterior often
cannot be computed exactly, and approximations must be employed. Mean-field
variational Bayes forms a posterior approximation by solving an optimization
problem and is widely used due to its speed. An exact BNP posterior might vary
dramatically when presented with different data. As such, stability and
robustness of the clustering should be assessed.
  A popular mean to assess stability is to apply the bootstrap by resampling
the data, and rerun the clustering for each simulated data set. The time cost
is thus often very expensive, especially for the sort of exploratory analysis
where clustering is typically used. We propose to use a fast and automatic
approximation to the full bootstrap called the "linear bootstrap", which can be
seen by local data perturbation. In this work, we demonstrate how to apply this
idea to a data analysis pipeline, consisting of an MFVB approximation to a BNP
clustering posterior of time course gene expression data. We show that using
auto-differentiation tools, the necessary calculations can be done
automatically, and that the linear bootstrap is a fast but approximate
alternative to the bootstrap.
