Posterior computation with the Gibbs zig-zag sampler

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…

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。

The Coordinate Sampler: A Non-Reversible Gibbs-like MCMC Sampler

In this article, we derive a novel non-reversible, continuous-time Markov chain Monte Carlo (MCMC) sampler, called Coordinate Sampler, based on a piecewise deterministic Markov process (PDMP), which can be seen as a variant of the Zigzag sampler. In addition to proving a theoretical validation for this new sampling algorithm, we show that the Markov chain it induces exhibits geometrical ergodicity convergence, for distributions whose tails decay at least as fast as an exponential distribution and at most as fast as a Gaussian distribution. Several numerical examples highlight that our coordina…

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…

使用弹性哈密顿动力学的MCMC：一种统一的哈密顿蒙特卡洛和分段确定性马尔可夫过程采样框架

MCMC using $\textit{bouncy}$ Hamiltonian dynamics: A unifying framework for Hamiltonian Monte Carlo and piecewise deterministic Markov process samplers

本文提出基于弹性哈密顿动力学的无拒绝Metropolis提案框架，将哈密顿蒙特卡洛与分段确定性马尔可夫过程采样统一，提升贝叶斯推断的扩展性与创新性。

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.

Adaptive schemes for piecewise deterministic Monte Carlo algorithms

The Bouncy Particle sampler (BPS) and the Zig-Zag sampler (ZZS) are continuous time, non-reversible Monte Carlo methods based on piecewise deterministic Markov processes. Experiments show that the speed of convergence of these samplers can be affected by the shape of the target distribution, as for instance in the case of anisotropic targets. We propose an adaptive scheme that iteratively learns all or part of the covariance matrix of the target and takes advantage of the obtained information to modify the underlying process with the aim of increasing the speed of convergence. Moreover, we def…

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.…

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)。

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…

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…

Continuously-Tempered PDMP Samplers

New sampling algorithms based on simulating continuous-time stochastic processes called piece-wise deterministic Markov processes (PDMPs) have shown considerable promise. However, these methods can struggle to sample from multi-modal or heavy-tailed distributions. We show how tempering ideas can improve the mixing of PDMPs in such cases. We introduce an extended distribution defined over the state of the posterior distribution and an inverse temperature, which interpolates between a tractable distribution when the inverse temperature is 0 and the posterior when the inverse temperature is 1. Th…