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.

翻译：Z字形及其他分段确定性马尔可夫过程采样器因其非可逆性以及在贝叶斯后验计算中的优良特性而备受关注。哈密顿蒙特卡洛是另一种前沿采样器，其通过引入虚拟动量引导马尔可夫链穿越复杂的目标分布。本文在Z字形采样器与一种基于拉普拉斯分布动量的哈密顿蒙特卡洛变体之间建立了重要联系。对应哈密顿动力学的位置-速度分量沿着与马尔可夫Z字形过程平行的Z字形路径运动；然而，在该位置-速度空间中动力学是非马尔可夫的，因为动量分量编码了非即时的历史信息。这部分信息在动量更新步骤中会部分丢失——在此步骤中我们保留其方向但重新采样其大小。我们证明，在动量更新频率趋于无穷的极限下，哈密顿Z字形会强收敛于其马尔可夫对应版本。这一理论洞见表明，当保留完整的动量信息时，哈密顿Z字形能通过抑制马尔可夫Z字形的扩散行为，更有效地探索具有高度相关参数的目标分布。我们通过比较两种Z字形算法在高维截断多元高斯分布（包括一个基于HIV病毒数据的贝叶斯系统发育多元概率单位模型所产生的11,235维目标分布）上的性能，验证了该直觉。