BAYESIAN NONPARAMETRIC METHODS FOR DATA FROM A UNIMODAL DENSITY

A strongly unimodal density with mode-theta is one that is non-decreasing on (- infinity, theta) and non-increasing on (theta, infinity). Brunner and Lo (1989) have described Bayesian procedures for sampling from a unimodal density, assuming only that it is symmetric about an unknown mode-theta. Here, the case where the unimodal density need not be symmetric is considered. The unimodal density is first written as a mixture with mixing distribution G. Placing a Dirichlet process prior on the unknown mixing distribution G and an arbitrary prior on the unknown mode-theta, the posterior distribution of the pair (theta, G) is obtained; the marginal posterior distribution of theta and the posterior expectation of G are "pressed in terms of sums over partitions of the set of integers {1,...,n}.