Zigzag Path Connects Two Monte Carlo Samplers: Hamiltonian Counterpart to a Piecewise Deterministic Markov Process
本文建立了之字形采样器与基于拉普拉斯动量的哈密顿蒙特卡洛变体之间的重要联系，证明在动量刷新频率趋于无穷时，哈密顿之字形采样器强收敛于马尔可夫之字形过程，并通过对高维截断多元高斯分布的实验验证了其抑制扩散行为的优势。
Zigzag and other piecewise deterministic Markov process samplers have attracted significant interest for their non-reversibility and other appealing properties for Bayesian posterior computation. Hamiltonian Monte Carlo is another state-of-the-art sampler, exploiting fictitious momentum to guide Markov chains through complex target distributions. We establish an important connection between the zigzag sampler and a variant of Hamiltonian Monte Carlo based on Laplace-distributed momentum. The position and velocity component of the corresponding Hamiltonian dynamics travels along a zigzag path paralleling the Markovian zigzag process; however, the dynamics is non-Markovian in this position-velocity space as the momentum component encodes non-immediate pasts. This information is partially lost during a momentum refreshment step, in which we preserve its direction but re-sample magnitude. In the limit of increasingly frequent momentum refreshments, we prove that Hamiltonian zigzag converges strongly to its Markovian counterpart. This theoretical insight suggests that, when retaining full momentum information, Hamiltonian zigzag can better explore target distributions with highly correlated parameters by suppressing the diffusive behavior of Markovian zigzag. We corroborate this intuition by comparing performance of the two zigzag cousins on high-dimensional truncated multivariate Gaussians, including a 11,235-dimensional target arising from a Bayesian phylogenetic multivariate probit modeling of HIV virus data.