用于离散结构与不可逆系统发育MCMC的Zig-zag采样

Sticky PDMP samplers for sparse and local inference problems

We construct a new class of efficient Monte Carlo methods based on continuous-time piecewise deterministic Markov processes (PDMPs) suitable for inference in high dimensional sparse models, i.e. models for which there is prior knowledge that many coordinates are likely to be exactly $0$. This is achieved with the fairly simple idea of endowing existing PDMP samplers with 'sticky' coordinate axes, coordinate planes etc. Upon hitting those subspaces, an event is triggered during which the process sticks to the subspace, this way spending some time in a sub-model. This results in non-reversible j…

PDMP Monte Carlo methods for piecewise-smooth densities

There has been substantial interest in developing Markov chain Monte Carlo algorithms based on piecewise-deterministic Markov processes. However existing algorithms can only be used if the target distribution of interest is differentiable everywhere. The key to adapting these algorithms so that they can sample from to densities with discontinuities is defining appropriate dynamics for the process when it hits a discontinuity. We present a simple condition for the transition of the process at a discontinuity which can be used to extend any existing sampler for smooth densities, and give specifi…

A Note on the Polynomial Ergodicity of the One-Dimensional Zig-Zag process

We prove polynomial ergodicity for the one-dimensional Zig-Zag process on heavy tailed targets and identify the exact order of polynomial convergence of the process when targeting Student distributions.

Zigzag路径连接两个蒙特卡洛采样器：分段确定性马尔可夫过程的哈密顿对应物

Zigzag path connects two Monte Carlo samplers: Hamiltonian counterpart to a piecewise deterministic Markov process

本文建立了zigzag采样器与基于Laplace动量的Hamiltonian Monte Carlo变体之间的理论联系，证明在频繁动量刷新下前者强收敛于后者，并表明保留完整动量的Hamiltonian zigzag能更好探索高相关参数分布。

Speed Up Zig-Zag

Zig-Zag is Piecewise Deterministic Markov Process, efficiently used for simulation in an MCMC setting. As we show in this article, it fails to be exponentially ergodic on heavy tailed target distributions. We introduce an extension of the Zig-Zag process by allowing the process to move with a non-constant speed function $s$, depending on the current state of the process. We call this process Speed Up Zig-Zag (SUZZ). We provide conditions that guarantee stability properties for the SUZZ process, including non-explosivity, exponential ergodicity in heavy tailed targets and central limit theorem.…

The Zig-Zag Process and Super-Efficient Sampling for Bayesian Analysis of Big Data

Standard MCMC methods can scale poorly to big data settings due to the need to evaluate the likelihood at each iteration. There have been a number of approximate MCMC algorithms that use sub-sampling ideas to reduce this computational burden, but with the drawback that these algorithms no longer target the true posterior distribution. We introduce a new family of Monte Carlo methods based upon a multi-dimensional version of the Zig-Zag process of (Bierkens, Roberts, 2017), a continuous time piecewise deterministic Markov process. While traditional MCMC methods are reversible by construction (a…

用于贝叶斯推断的Zig-Zag算法的大样本缩放分析

Large sample scaling analysis of the Zig-Zag algorithm for Bayesian inference

该研究针对子采样下Zig-Zag算法收敛慢、混合差的问题，开展大样本缩放分析，证明瞬态轨迹可由ODE近似、给出平稳阶段弱收敛结果，证实合适控制变量下采样成本为O(1)，较传统方法提速O(n)。

Automatic Zig-Zag sampling in practice

Novel Monte Carlo methods to generate samples from a target distribution, such as a posterior from a Bayesian analysis, have rapidly expanded in the past decade. Algorithms based on Piecewise Deterministic Markov Processes (PDMPs), non-reversible continuous-time processes, are developing into their own research branch, thanks their important properties (e.g., correct invariant distribution, ergodicity, and super-efficiency). Nevertheless, practice has not caught up with the theory in this field, and the use of PDMPs to solve applied problems is not widespread. This might be due, firstly, to se…