A generalized Bayes framework for probabilistic clustering

TL;DR

This paper introduces a generalized Bayes framework for clustering that unifies loss-based and model-based methods, enabling uncertainty quantification without strict likelihood assumptions.

Contribution

It proposes a Gibbs posterior approach for clustering, allowing uncertainty estimation and interpretation of existing algorithms within this probabilistic framework.

Findings

Unified framework for clustering and uncertainty quantification.

Efficient algorithms for point estimation and sampling.

Interpretation of existing methods like k-means as generalized Bayes estimators.

Abstract

Loss-based clustering methods, such as k-means and its variants, are standard tools for finding groups in data. However, the lack of quantification of uncertainty in the estimated clusters is a disadvantage. Model-based clustering based on mixture models provides an alternative, but such methods face computational problems and large sensitivity to the choice of kernel. This article proposes a generalized Bayes framework that bridges between these two paradigms through the use of Gibbs posteriors. In conducting Bayesian updating, the log likelihood is replaced by a loss function for clustering, leading to a rich family of clustering methods. The Gibbs posterior represents a coherent updating of Bayesian beliefs without needing to specify a likelihood for the data, and can be used for characterizing uncertainty in clustering. We consider losses based on Bregman divergence and pairwise similarities, and develop efficient deterministic algorithms for point estimation along with sampling algorithms for uncertainty quantification. Several existing clustering algorithms, including k-means, can be interpreted as generalized Bayes estimators under our framework, and hence we provide a method of uncertainty quantification for these approaches.