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Marginally Specified Priors for Nonparametric Bayesian Estimation

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.