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
