OPTIMAL SCALING OF RANDOM WALK METROPOLIS ALGORITHMS WITH DISCONTINUOUS TARGET DENSITIES

• 成果类型: Article
• 署名作者: Neal, Peter; Roberts, Gareth; Yuen, Wai Kong
• 署名单位: University of Manchester; University of Warwick; Brock University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/11-AAP817
• 发表日期: 2012
• 页码: 1880-1927
• 关键词: hastings algorithms

摘要

We consider the optimal scaling problem for high-dimensional random walk Metropolis (RWM) algorithms where the target distribution has a discontinuous probability density function. Almost all previous analysis has focused upon continuous target densities. The main result is a weak convergence result as the dimensionality d of the target densities converges to infinity. In particular, when the proposal variance is scaled by d(-2), the sequence of stochastic processes formed by the first component of each Markov chain converges to an appropriate Langevin diffusion process. Therefore optimizing the efficiency of the RWM algorithm is equivalent to maximizing the speed of the limiting diffusion. This leads to an asymptotic optimal acceptance rate of e(-2) (=0.1353) under quite general conditions. The results have major practical implications for the implementation of RWM algorithms by highlighting the detrimental effect of choosing RWM algorithms over Metropolis-within-Gibbs algorithms.