COUNTEREXAMPLES FOR OPTIMAL SCALING OF METROPOLIS-HASTINGS CHAINS WITH ROUGH TARGET DENSITIES

• 成果类型: Article
• 署名作者: Vogrinc, Jure; Kendall, Wilfrid S.
• 署名单位: University of Warwick
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/20-AAP1612
• 发表日期: 2021
• 页码: 972-1019
• 关键词: transient phase; WEAK-CONVERGENCE; algorithms; EFFICIENCY; PROPOSALS; diffusion; contours

摘要

For sufficiently smooth targets of product form it is known that the variance of a single coordinate of the proposal in RWM (random walk Metropolis) and MALA (Metropolis adjusted Langevin algorithm) should optimally scale as n(-1) and as n(-1/3) with dimension n, and that the acceptance rates should be tuned to 0.234 and 0.574. We establish counterexamples to demonstrate that smoothness assumptions of the order of C-1(R) for RWM and C-3(R) for MALA are indeed required if these scaling rates are to hold. The counterexamples identify classes of marginal targets for which these guidelines are violated, obtained by perturbing a standard normal density (at the level of the potential for RWM and the second derivative of the potential for MALA) using roughness generated by a path of fractional Brownian motion with Hurst exponent H. For such targets there is strong evidence that RWM and MALA proposal variances should optimally be scaled as n(-1/H) and as n(-1/2+H) and will then obey anomalous acceptance rate guidelines. Useful heuristics resulting from this theory are discussed. The paper develops a framework capable of tackling optimal scaling results for quite general Metropolis-Hastings algorithms (possibly depending on a random environment).