Multiple-try Metropolis (MTM) is a popular Markov chain Monte Carlo method with the appealing feature of being amenable to parallel computing. At each iteration, it samples several candidates for the next state of the Markov chain and randomly selects one of them based on a weight function. The canonical weight function is proportional to the target density. We show both theoretically and empirically that this weight function induces pathological behaviours in high dimensions, especially during the convergence phase. We propose to instead use weight functions akin to the locally-balanced proposal distributions of Zanella (2020), thus yielding MTM algorithms that do not exhibit those pathological behaviours. To theoretically analyse these algorithms, we study the high-dimensional performance of ideal schemes that can be thought of as MTM algorithms which sample an infinite number of candidates at each iteration, as well as the discrepancy between such schemes and the MTM algorithms which sample a finite number of candidates. Our analysis unveils a strong distinction between the convergence and stationary phases: in the former, local balancing is crucial and effective to achieve fast convergence, while in the latter, the canonical and novel weight functions yield similar performance. Numerical experiments include an application in precision medicine involving a computationally-expensive forward model, which makes the use of parallel computing within MTM iterations beneficial.

翻译：多尝试Metropolis（MTM）是一种流行的马尔可夫链蒙特卡洛方法，其显著特点是易于并行计算。在每次迭代中，该方法为马尔可夫链的下一状态采样多个候选值，并基于权重函数随机选择其中一个。规范权重函数与目标密度成正比。我们从理论和实验两方面证明，该权重函数在高维空间中会引发病态行为，尤其是在收敛阶段。我们建议改用类似Zanella（2020）提出的局部平衡提议分布的权重函数，从而得到不表现这些病态行为的MTM算法。为从理论上分析这些算法，我们研究了理想方案的高维性能，这些方案可被视为每次迭代采样无限个候选值的MTM算法，同时分析了这类方案与采样有限个候选值的MTM算法之间的差异。我们的分析揭示了收敛阶段与平稳阶段之间的显著区别：在前者中，局部平衡对实现快速收敛至关重要且有效，而在后者中，规范权重函数与新型权重函数性能相近。数值实验包括一项精准医学应用，其中涉及计算代价高昂的正向模型，这使得MTM迭代中使用并行计算更具优势。