Limit theorems for the zig-zag process
Limit theorems for the zig-zag process
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DOI：
10.1017/apr.2017.22
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发表时间：
2016-07
影响因子：
1.2
通讯作者：
J. Bierkens;A. Duncan
中科院分区：
文献类型：
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作者：
J. Bierkens;A. Duncan
摘要马尔可夫链蒙特卡罗（MCMC）方法为复杂概率分布的抽样统计提供了一个重要的工具。虽然MCMC的标准方法涉及构建离散时间可逆马尔可夫链，其过渡内核是通过Metropolis-Hastings算法获得的，但最近人们对基于分段确定性马尔可夫过程（PDMP）的替代方案感兴趣。其中一种方法是基于Bierkens和Roberts（2016）中介绍的zig-zag过程，该过程被证明为大数据体系中的采样提供了高度可扩展的采样方案;参见Bierkens et al.（2016）。在本文中，我们研究的锯齿形采样器的性能，侧重于一维的情况下。特别是，我们确定的条件下，一个中心极限定理举行，并证明了渐近方差。此外，我们研究了切换率对zig-zag过程扩散率的影响，确定了切换率趋于∞时的扩散极限。根据我们的研究结果，我们比较现有的Monte Carlo方法，分析和模拟的锯齿形采样器的性能。
Abstract Markov chain Monte Carlo (MCMC) methods provide an essential tool in statistics for sampling from complex probability distributions. While the standard approach to MCMC involves constructing discrete-time reversible Markov chains whose transition kernel is obtained via the Metropolis–Hastings algorithm, there has been recent interest in alternative schemes based on piecewise deterministic Markov processes (PDMPs). One such approach is based on the zig-zag process, introduced in Bierkens and Roberts (2016), which proved to provide a highly scalable sampling scheme for sampling in the big data regime; see Bierkens et al. (2016). In this paper we study the performance of the zig-zag sampler, focusing on the one-dimensional case. In particular, we identify conditions under which a central limit theorem holds and characterise the asymptotic variance. Moreover, we study the influence of the switching rate on the diffusivity of the zig-zag process by identifying a diffusion limit as the switching rate tends to ∞. Based on our results we compare the performance of the zig-zag sampler to existing Monte Carlo methods, both analytically and through simulations.