Optimal Proposal Design for Random Walk Type Metropolis Algorithms with Gaussian Random Field Priors

Details Optimal Proposal Design for Random Walk Type Metropolis Algorithms with Gaussian Random Field Priors Optimal Proposal Design for Random Walk Type Metropolis Algorithms with Gaussian Random Field Priors

2011-08-06

arxiv.org/abs/1108.1494v2

Mathematics

Statistics Theory

Major revision of the original version especially the introduction, slight change in the title, added a section on computation

Scientific paper

We study random walk based algorithms for posterior simulation in a large of class of Bayesian nonparametric problems with Gaussian random field or Gaussian process priors. Our emphasis is both on developing practical guidelines for the design and implementation of efficient algorithms for these naturally high dimensional problems, and on the development of rigorous underpinning theory. We illustrate via an example that, in designing algorithms for nonparametric problems, it is important to take advantage of the infinite dimensional structure inherent in both the prior and likelihood. Algorithms which ignore this structure can be very inefficient, even with large computing power. The novelty of our work is twofold. (i) We give one of the first results rigorously quantifying the efficiency of random walk based algorithms in infinite dimensional Bayesian nonparametric problems; in particular this work describes a modification of the standard random walk Metropolis algorithm which results in an order of magnitude efficiency gain, when measured in terms of the dimension of the approximation space. (ii) We develop a theory of simulated annealing for posterior simulation in an infinite dimensional framework and show how a noisy gradient descent algorithm can emerge, without explicitly computing the gradient, from certain carefully specified random walks when combined with a Metropolis-Hastings accept-reject mechanism. This theory extends results known in finite dimensions for finding local maxima using a gradient flow and hence is of independent interest.

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