Cluster and Feature Modeling from Combinatorial Stochastic Processes

One of the focal points of the modern literature on Bayesian nonparametrics has been the problem of clustering, or partitioning, where each data point is modeled as being associated with one and only one of some collection of groups called clusters or partition blocks. Underlying these Bayesian nonparametric models are a set of interrelated stochastic processes, most notably the Dirichlet process and the Chinese restaurant process. In this paper we provide a formal development of an analogous problem, called feature modeling, for associating data points with arbitrary nonnegative integer numbers of groups, now called features or topics. We review the existing combinatorial stochastic process representations for the clustering problem and develop analogous representations for the feature modeling problem. These representations include the beta process and the Indian buffet process as well as new representations that provide insight into the connections between these processes. We thereby bring the same level of completeness to the treatment of Bayesian nonparametric feature modeling that has previously been achieved for Bayesian nonparametric clustering.Comment: Published in at http://dx.doi.org/10.1214/13-STS434 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

A Tutorial on Bayesian Nonparametric Models

A key problem in statistical modeling is model selection, how to choose a model at an appropriate level of complexity. This problem appears in many settings, most prominently in choosing the number ofclusters in mixture models or the number of factors in factor analysis. In this tutorial we describe Bayesian nonparametric methods, a class of methods that side-steps this issue by allowing the data to determine the complexity of the model. This tutorial is a high-level introduction to Bayesian nonparametric methods and contains several examples of their application.Comment: 28 pages, 8 figure