  Prior specification for nonparametric Bayesian inference involves the
difficult task of quantifying prior knowledge about a parameter of high, often
infinite, dimension. Realistically, a statistician is unlikely to have informed
opinions about all aspects of such a parameter, but may have real information
about functionals of the parameter, such the population mean or variance. This
article proposes a new framework for nonparametric Bayes inference in which the
prior distribution for a possibly infinite-dimensional parameter is decomposed
into two parts: an informative prior on a finite set of functionals, and a
nonparametric conditional prior for the parameter given the functionals. Such
priors can be easily constructed from standard nonparametric prior
distributions in common use, and inherit the large support of the standard
priors upon which they are based. Additionally, posterior approximations under
these informative priors can generally be made via minor adjustments to
existing Markov chain approximation algorithms for standard nonparametric prior
distributions. We illustrate the use of such priors in the context of
multivariate density estimation using Dirichlet process mixture models, and in
the modeling of high-dimensional sparse contingency tables.
