The efficiency of Hamiltonian Monte Carlo (HMC) can suffer when sampling a distribution with a wide range of length scales, because the small step sizes needed for stability in high-curvature regions are inefficient elsewhere. To address this we present a delayed rejection (DR) variant: if an initial HMC trajectory is rejected, we make one or more subsequent proposals each using a step size geometrically smaller than the last. To reduce the cost of DR approaches, we extend the standard delayed rejection to a probabilistic framework wherein we do not make multiple proposals at every rejection, but allow the probability of a retry to depend on the probability of accepting the previous proposal. We test the scheme in several sampling tasks, including statistical applications and multiscale model distributions such as Neal's funnel. Delayed rejection enables sampling multiscale distributions for which standard approaches such as HMC fail to explore the tails, and improves performance five-fold over optimally-tuned HMC as measured by effective sample size per gradient evaluation. Even for simpler distributions, delayed rejection provides increased robustness to step size misspecification.

Chirag Modi

Alex Barnett

Bob Carpenter

Bayesian Analysis

贝叶斯分析是国际贝叶斯分析学会的电子期刊。它旨在发表广泛的文章，在某些理论或应用背景下展示或讨论贝叶斯方法。该期刊欢迎提交新的计算和统计方法的投稿；对现有方法的批判性审查和讨论；历史观点；重要科学或政策应用领域的描述；实例探究;以及实验设计、数据收集、数据共享或数据挖掘的方法。提交的评估基于内容的重要性和沟通的有效性。讨论论文通常由主编选择，或由编辑建议，在定期提交的论文中。此外，《期刊》鼓励个人作者提交手稿以作为讨论文件考虑。

Bayesian Analysis is the electronic journal of the International Society for Bayesian Analysis. It aims to publish a wide range of articles that showcase or discuss Bayesian methods in various theoretical or applied contexts. The journal welcomes submissions of new computational and statistical methods; critical reviews and discussions of existing methods; historical perspectives; descriptions of significant scientific or policy application areas; case studies; as well as methods for experimental design, data collection, data sharing, or data mining. Submitted papers are evaluated based on the importance of their content and the effectiveness of their communication. Discussion papers are typically selected by the editor-in-chief or recommended by editors as part of regularly submitted papers. Additionally, Bayesian Analysis encourages individual authors to submit manuscripts for consideration as discussion papers.

《Bayesian Analysis》是国际贝叶斯分析学会的电子期刊，旨在发表广泛的文章，展示或讨论贝叶斯方法在各种理论或应用背景下的应用。该期刊欢迎提交新的计算和统计方法的研究；对现有方法的批判性审查和讨论；历史视角；重要科学或政策应用领域的描述；实例研究；以及实验设计、数据收集、数据共享或数据挖掘的方法。提交的论文将根据内容的重要性和沟通的有效性进行评估。讨论论文通常由主编选择，或由编辑建议，作为定期提交的论文的一部分。此外，《Bayesian Analysis》鼓励个人作者提交手稿，以供作为讨论文件考虑。

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Delayed rejection Hamiltonian Monte Carlo for sampling multiscale distributions

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STATISTICS & PROBABILITY	1