Hamiltonian Monte Carlo (HMC) is widely used for sampling from high-dimensional target distributions with probability density known up to proportionality. While HMC possesses favorable dimension scaling properties, it encounters challenges when applied to strongly multimodal distributions. Traditional tempering methods, commonly used to address multimodality, can be difficult to tune, particularly in high dimensions. In this study, we propose a method that combines a tempering strategy with Hamiltonian Monte Carlo, enabling efficient sampling from high-dimensional, strongly multimodal distributions. Our approach involves proposing candidate states for the constructed Markov chain by simulating Hamiltonian dynamics with time-varying mass, thereby searching for isolated modes at unknown locations. Moreover, we develop an automatic tuning strategy for our method, resulting in an automatically-tuned, tempered Hamiltonian Monte Carlo (ATHMC). Unlike simulated tempering or parallel tempering methods, ATHMC provides a distinctive advantage in scenarios where the target distribution changes at each iteration, such as in the Gibbs sampler. We numerically show that our method scales better with increasing dimensions than an adaptive parallel tempering method and demonstrate its efficacy for a variety of target distributions, including mixtures of log-polynomial densities and Bayesian posterior distributions for a sensor network self-localization problem.

翻译：哈密顿蒙特卡洛（HMC）广泛用于从概率密度已知（至多差一个比例常数）的高维目标分布中采样。尽管HMC具有良好的维度缩放特性，但在应用于强多峰分布时会遇到挑战。传统用于处理多峰性的回火方法通常难以调参，尤其是在高维情况下。本研究提出一种将回火策略与哈密顿蒙特卡洛相结合的方法，能够高效地从高维强多峰分布中采样。我们的方法通过模拟具有时变质量的哈密顿动力学来为构建的马尔可夫链提出候选状态，从而在未知位置搜索孤立的峰。此外，我们为该方开发了一种自动调参策略，形成了自动调参的回火哈密顿蒙特卡洛（ATHMC）。与模拟回火或并行回火方法不同，ATHMC在目标分布在每次迭代中发生变化（例如在吉布斯采样器中）的场景下具有独特优势。我们通过数值实验表明，与自适应并行回火方法相比，我们的方法在维度增加时具有更好的可扩展性，并证明了其对多种目标分布的有效性，包括对数多项式密度混合分布以及传感器网络自定位问题的贝叶斯后验分布。