On Shorter Tempering Paths in Non-Reversible Parallel tempering - Danielsson,Erik (Master's Theses in Mathematical Sciences, 2026)
Parallel tempering (PT) is a classical Markov chain Monte Carlo algorithm for sampling from challenging distributions. It employs a sequence of distributions interpolating between a simple reference distribution and a complex target distribution to aid sampling at the target. Non-reversible parallel tempering (NRPT) is a variant of PT that has been shown to outperform standard, reversible variants of PT in past work. In NRPT, the performance of the algorithm increases with the number of parallel chains but eventually saturates at a level determined by the global communication barrier, a PT-specific divergence between the tempering endpoints that is inherent to the standard, linear tempering path. In this thesis we consider two approaches for decreasing the global communication barrier and improving the performance of NRPT. In the first approach, we study Bayesian problems and check whether the intermediate distributions, the tempered posteriors, retain statistical properties of the target posterior distribution. We show via two counterexamples that replacing the posterior with the tempered posterior is not justified in general, and conclude that the use of tempered posteriors for inference requires additional restrictions on the Bayesian model. In the second approach, we develop new methods for optimizing tempering paths. Previous work on path optimization in NRPT used a path variant that can produce improper intermediate distributions. We construct two families of proper paths and analyze them numerically. Our results show that path optimization can be a viable approach to reduce the global communication barrier, as both path families outperform the linear NRPT algorithm on some models. For one path family, however, the numerical experiments show that path optimization can worsen violations of the model conditions needed to establish the communication barrier. We therefore conclude that not only properness, but also the interaction with the model conditions should be considered when constructing tempering paths.
Keywords: Markov chain Monte Carlo; Parallel tempering; Bayesian inference; Tempered posterior; Stochastic optimization