Nonreversible MCMC from conditional invertible transforms: a complete recipe with convergence guarantees
This work provides crucial theoretical guarantees for researchers and practitioners developing and applying nonreversible MCMC methods, ensuring their validity and convergence.
This paper addresses the lack of systematic treatment for ensuring the validity of nonreversible Markov Chain Monte Carlo (MCMC) kernels, especially those using complex invertible transforms. It develops general tools and provides simple, practically verifiable conditions to guarantee the desired invariance property and convergence for these algorithms.
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.