A risk-based approach to optimal clustering under random labeled point processes

Typically, optimization in clustering is relative to a heuristic metric, rather than relative to a definition of error with respect to a probabilistic model to make clustering rigorously predictive. To address this, we develop a general risk- based formulation for clustering that parallels classical Bayes decision theory for classification, transforming clustering from a subjective activity to an objective operation. We develop a general analytic procedure to find an optimal clustering operator, called a Bayes clusterer, which corresponds to the Bayes classifier in classification theory. In particular, we address Gaussian models, and discuss fundamental limits of performance in clustering.