Zigzag Path Connects Two Monte Carlo Samplers: Hamiltonian Counterpart to a Piecewise Deterministic Markov Process
连接两类蒙特卡洛采样器的折线路径：分段确定性马尔可夫过程的哈密顿对应AI 解读 ↓
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连接两类蒙特卡洛采样器的折线路径：分段确定性马尔可夫过程的哈密顿对应
- 研究问题
- 采用拉普拉斯动量的哈密顿采样与马尔可夫折线采样之间有什么联系，保留动量历史如何影响相关参数空间的探索？
- 方法与思路
- 作者构造哈密顿折线动力学，在保持动量方向时刷新其大小，并研究刷新愈加频繁时向马尔可夫折线过程的收敛。
- 材料陈述的贡献
- 摘要称证明了两类折线过程的强收敛联系，并用高维截断高斯目标比较性能，支持保留动量改善相关参数探索的直觉。
- 阅读关注点
- 应核对强收敛的条件与对象、动量刷新机制，以及高维实验中的调参、计算成本和采样效率比较是否充分。
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- 仅据版本未注明的索引摘要，不能把给定目标分布上的比较推广为普遍优势；摘要未说明收敛条件或具体效率结果。
原文摘要
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.