Markov Chain Monte Carlo (MCMC) is a class of algorithms to sample complex and high-dimensional probability distributions. The Metropolis-Hastings (MH) algorithm, the workhorse of MCMC, provides a simple recipe to construct reversible Markov kernels. Reversibility is a tractable property that implies a less tractable but essential property here, invariance. Reversibility is however not necessarily desirable when considering performance. This has prompted recent interest in designing kernels breaking this property. At the same time, an active stream of research has focused on the design of novel versions of the MH kernel, some nonreversible, relying on the use of complex invertible deterministic transforms. While standard implementations of the MH kernel are well understood, the aforementioned developments have not received the same systematic treatment to ensure their validity. This paper fills the gap by developing general tools to ensure that a class of nonreversible Markov kernels, possibly relying on complex transforms, has the desired invariance property and leads to convergent algorithms. This leads to a set of simple and practically verifiable conditions.
翻译:Markov 链子蒙特卡洛( MCMC) 是一组用于抽样复杂和高维概率分布的算法。 MMC 的工作马MH 提供了一种简单的配方, 用于构建可逆的 Markov 内核。 Reviversity 是一种可移动的属性, 意味着这里的属性不那么容易移动, 但却是不可改变的。 但是, 在审议性能时, 反向性不一定是可取的。 这引起了最近对设计打破此属性的内核的兴趣。 与此同时, 活跃的研究流侧重于设计新版的 MH 内核, 一些不可逆的、 依赖复杂不可逆的确定性变形的配方。 虽然对 MH 内核的标准实施非常了解, 但上述发展没有受到同样的系统化处理以确保其有效性。 本文填补了这一空白, 开发了一般工具以确保一类不可逆的 Markov 内核, 可能依靠复杂的变形, 具有理想的可逆性属性, 并导致趋同式的算法。 这导致一套简单且可核实的系统化的方法。