Analysis of the maximal posterior partition in the Dirichlet Process Gaussian Mixture Model

TL;DR

This paper analyzes the properties of the MAP clustering in Gaussian Dirichlet process mixture models, revealing geometric and asymptotic characteristics that influence the estimated number of clusters.

Contribution

It provides theoretical insights into the structure and asymptotic behavior of the MAP partition in Gaussian DPMMs, including bounds on the number of clusters.

Findings

Almost disjoint convex hulls of clusters.

Bounded number of clusters intersecting a fixed region.

Asymptotic maximization of a simple function depending on covariance.

Abstract

Mixture models are a natural choice in many applications, but it can be difficult to place an a priori upper bound on the number of components. To circumvent this, investigators are turning increasingly to Dirichlet process mixture models (DPMMs). It is therefore important to develop an understanding of the strengths and weaknesses of this approach. This work considers the MAP (maximum a posteriori) clustering for the Gaussian DPMM (where the cluster means have Gaussian distribution and, for each cluster, the observations within the cluster have Gaussian distribution). Some desirable properties of the MAP partition are proved: `almost disjointness' of the convex hulls of clusters (they may have at most one point in common) and (with natural assumptions) the comparability of sizes of those clusters that intersect any fixed ball with the number of observations (as the latter goes to infinity). Consequently, the number of such clusters remains bounded. Furthermore, if the data arises from independent identically distributed sampling from a given distribution with bounded support then the asymptotic MAP partition of the observation space maximises a function which has a straightforward expression, which depends only on the within-group covariance parameter. As the operator norm of this covariance parameter decreases, the number of clusters in the MAP partition becomes arbitrarily large, which may lead to the overestimation of the number of mixture components.