Bayes Prediction Density and Regression Estimation — A Semiparametric Approach

This paper is concerned with the Bayes estimation of an arbitrary multivariate density, f(x), x ∈Rk. Such an f(x) may be represented as a mixture of a given parametric family of densities h(x|θ) with support in Rk, where θ (in Rd) is chosen according to a mixing distribution G. We consider the semiparametric Bayes approach in which G, in turn, is chosen according to a Dirichlet process prior with given parameter a. We then specialize these results when f is expressed as a mixture of multivariate normal densities θ(x|μ, Λ) where μ is the mean vector and Λ is the precision matrix. The results are finally applied to estimating a regression parameter.