  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.
