  Bayesian nonparametric mixture models are widely used to cluster
observations. However, one major drawback of the approach is that the estimated
partition often presents unbalanced clusters' frequencies with only a few
dominating clusters and a large number of sparsely-populated ones. This feature
translates into results that are often uninterpretable unless we accept to
ignore a relevant number of observations and clusters. Interpreting the
posterior distribution as penalized likelihood, we show how the unbalance can
be explained as a direct consequence of the cost functions involved in
estimating the partition. In light of our findings, we propose a novel Bayesian
estimator of the clustering configuration. The proposed estimator is equivalent
to a post-processing procedure that reduces the number of sparsely-populated
clusters and enhances interpretability. The procedure takes the form of
entropy-regularization of the Bayesian estimate. While being computationally
convenient with respect to alternative strategies, it is also theoretically
justified as a correction to the Bayesian loss function used for point
estimation and, as such, can be applied to any posterior distribution of
clusters, regardless of the specific model used.
