For random-walk Metropolis (RWM) and parallel tempering (PT) algorithms, an asymptotic acceptance rate of around 0.234 is known to be optimal in the high-dimensional limit. However, its practical relevance is uncertain due to restrictive derivation conditions. We synthesise previous theoretical advances in extending the 0.234 acceptance rate to more general settings, and demonstrate its applicability with a comprehensive empirical simulation study on examples examining how acceptance rates affect Expected Squared Jumping Distance (ESJD). Our experiments show the optimality of the 0.234 acceptance rate for RWM is surprisingly robust even in lower dimensions across various proposal and multimodal target distributions which may or may not have an i.i.d. product density. Parallel tempering experiments also show that the idealized 0.234 spacing of inverse temperatures may be approximately optimal for low dimensions and non i.i.d. product target densities, and that constructing an inverse temperature ladder with spacings given by a swap acceptance of 0.234 is a viable strategy. However, we observe the applicability of the 0.234 acceptance rate heuristic diminishes for both RWM and PT algorithms below a certain dimension which differs based on the target density, and that inhomogeneously scaled components in the target density further reduces its applicability in lower dimensions.

翻译：对于随机游走Metropolis（RWM）与并行回火（PT）算法，已知在高维极限下约0.234的渐近接受率是最优的。然而，由于其推导条件具有限制性，该结论的实际适用性尚不明确。我们综合了先前将0.234接受率推广至更一般设定下的理论进展，并通过一项全面的实证模拟研究，以考察接受率如何影响期望平方跳跃距离（ESJD）的实例，论证了其适用性。我们的实验表明，即使在较低维度下，对于各种建议分布以及可能具有或不具有独立同分布乘积密度的多峰目标分布，RWM的0.234接受率最优性表现出惊人的鲁棒性。并行回火实验也表明，对于低维和非独立同分布乘积目标密度，理想的0.234逆温度间隔可能是近似最优的，并且构建一个以交换接受率为0.234给出的间隔所定义的逆温度阶梯是一种可行的策略。然而，我们观察到，对于RWM和PT算法，当维度低于某个取决于目标密度的特定阈值时，0.234接受率启发式规则的适用性会减弱，并且目标密度中非均匀缩放的成分会进一步降低其在较低维度下的适用性。