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. Yet, the practical relevance of this value is uncertain due to the restrictive conditions underlying its derivation. We synthesise previous theoretical advances in extending the 0.234 acceptance rate to more general settings, and demonstrate the applicability and generalizability of the 0.234 theory for practitioners with a comprehensive empirical simulation study on a variety of 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. Experiments on parallel tempering 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，由于推导该值所依赖的严格条件，其实际相关性尚不确定。我们综合了先前将0.234接受率推广至更一般设置的理论进展，并通过在不同案例上进行的全面实证模拟研究——考察接受率如何影响期望平方跳跃距离（ESJD）——展示了0.234理论对实践者的适用性与普适性。我们的实验表明，即使在较低维度下，对于各种建议分布和多峰目标分布（无论其是否具有独立同分布乘积密度），RWM的0.234接受率最优性表现出惊人的稳健性。并行回火实验也表明，理想化的0.234逆温度间隔在低维和非独立同分布乘积目标密度下可能近似最优，且以交换接受率为0.234给出的间隔构建逆温度阶梯是一种可行策略。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接受率启发式方法的适用性均会下降，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。