  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.
