Measuring Cluster Stability for Bayesian Nonparametrics Using the Linear Bootstrap

TL;DR

This paper introduces a fast, automatic linear bootstrap method to assess the stability of Bayesian nonparametric clustering, especially for complex models where traditional bootstrap is computationally expensive.

Contribution

The authors propose a novel linear bootstrap approach for efficiently estimating cluster stability in Bayesian nonparametric models, leveraging auto-differentiation for automation.

Findings

Linear bootstrap provides a fast approximation to traditional bootstrap.

The method is demonstrated on gene expression time course data.

Auto-differentiation enables automatic computation of stability estimates.

Abstract

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.