NuZZ：一般模型的数值Zig-Zag采样
NuZZ: numerical Zig-Zag sampling for general models
摘要：
马尔可夫链蒙特卡罗（MCMC）算法是计算统计学中的一个关键算法，随着数据集的增大和模型的复杂化，许多流行的MCMC算法由于计算量太大而变得不实用。最近的进展已经取得了这个问题，通过开发MCMC算法的基础上分段确定性马尔可夫过程（PDMP），不可逆的过程，可以被设计为收敛的速度是独立的数据的大小。虽然可以理解的是，在这些结果之后出现了大量的理论研究，但到目前为止，PDMP仅用于某些梯度可以被限制的模型，这在许多统计环境中是不可能的。针对Zig-Zag过程，提出了适用于一般统计模型的数值Zig-Zag（NuZZ）算法，该算法不需要对对数后验的梯度进行限制.这使我们能够进行数值实验：（i）Zig-Zag动力学如何在具有共同挑战性特征的一些测试问题上表现出来;以及（ii）目标和采样分布之间的误差如何作为不同MCMC算法（包括NuZZ）的计算工作量的函数而演变。此外，由于具体的NuZZ算法，我们能够给出一个明确的约束之间的瓦瑟斯坦距离的精确后，其数值扰动对应的用户指定的数值公差的NuZZ。
英文摘要：
Markov chain Monte Carlo (MCMC) is a key algorithm in computational statistics, and as datasets grow larger and models grow more complex, many popular MCMC algorithms become too computationally expensive to be practical. Recent progress has been made on this problem through development of MCMC algorithms based on Piecewise Deterministic Markov Processes (PDMPs), irreversible processes that can be engineered to converge at a rate which is independent of the size of data. While there has understandably been a surge of theoretical studies following these results, PDMPs have so far only been implemented for models where certain gradients can be bounded, which is not possible in many statistical contexts. Focusing on the Zig-Zag process, we present the Numerical Zig-Zag (NuZZ) algorithm, which is applicable to general statistical models without the need for bounds on the gradient of the log posterior. This allows us to perform numerical experiments on: (i) how the Zig-Zag dynamics behaves on some test problems with common challenging features; and (ii) how the error between the target and sampled distributions evolves as a function of computational effort for different MCMC algorithms including NuZZ. Moreover, due to the specifics of the NuZZ algorithms, we are able to give an explicit bound on the Wasserstein distance between the exact posterior and its numerically perturbed counterpart in terms of the user-specified numerical tolerances of NuZZ.
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