Open AccessHierarchical Models and Tuning of Random Walk Metropolis Algorithms
Author(s) -
Mylène Bédard
Publication year - 2019
Publication title -
journal of probability and statistics
Language(s) - English
Resource type - Journals
eISSN - 1687-9538
pISSN - 1687-952X
DOI - 10.1155/2019/8740426
Subject(s) - random walk , mathematics , algorithm , convergence (economics) , generator (circuit theory) , state space , gaussian , metropolis–hastings algorithm , position (finance) , markov chain , bayesian probability , block (permutation group theory) , markov chain monte carlo , mathematical optimization , power (physics) , statistics , physics , geometry , finance , quantum mechanics , economics , economic growth
We obtain weak convergence and optimal scaling results for the random walk Metropolis algorithm with a Gaussian proposal distribution. The sampler is applied to hierarchical target distributions, which form the building block of many Bayesian analyses. The global asymptotically optimal proposal variance derived may be computed as a function of the specific target distribution considered. We also introduce the concept of locally optimal tunings, i.e., tunings that depend on the current position of the Markov chain. The theorems are proved by studying the generator of the first and second components of the algorithm and verifying their convergence to the generator of a modified RWM algorithm and a diffusion process, respectively. The rate at which the algorithm explores its state space is optimized by studying the speed measure of the limiting diffusion process. We illustrate the theory with two examples. Applications of these results on simulated and real data are also presented.