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 expressed in terms of sums over partitions of the set of integers {1,...,n}.