The Polya Tree Sampler: Toward Efficient and Automatic Independent Metropolis–Hastings Proposals

Abstract

We present a simple, efficient, and computationally cheap sampling method for exploring an unnormalized multivariate density on ℝd, such as a posterior density, called the Polya tree sampler. The algorithm constructs an independent proposal based on an approximation of the target density. The approximation is built from a set of (initial) support points—data that act as parameters for the approximation—and the predictive density of a finite multivariate Polya tree. In an initial “warming-up” phase, the support points are iteratively relocated to regions of higher support under the target distribution to minimize the distance between the target distribution and the Polya tree predictive distribution. In the “sampling” phase, samples from the final approximating mixture of finite Polya trees are used as candidates which are accepted with a standard Metropolis–Hastings acceptance probability. Several illustrations are presented, including comparisons of the proposed approach to Metropolis-within-Gibbs and delayed rejection adaptive Metropolis algorithm. This article has supplementary material online.