Concave-Convex PDMP-based sampling

Abstract

Recently non-reversible samplers based on simulating piecewise deterministic Markov processes (PDMPs) have shown potential for efficient sampling in Bayesian inference problems. However, there remains a lack of guidance on how to best implement these algorithms. If implemented poorly, the computational costs of simulating event times can out-weigh the statistical efficiency of the non-reversible dynamics. Drawing on the adaptive rejection literature, we propose the concave-convex adaptive thinning approach for simulating a piecewise deterministic Markov process (CC-PDMP). This approach provides a general guide for constructing bounds that may be used to facilitate PDMP-based sampling. A key advantage of this method is its additive structure -adding concave-convex decompositions yields a concave-convex decomposition. This facilitates swapping priors, simple implementation and computationally efficient thinning. In particular, our approach is well suited to local PDMP simulation where known conditional independence of the target can be exploited for potentially huge computational gains. We provide an R package for implementing the CC-PDMP approach and illustrate how our method outperforms existing approaches to simulating events in the PDMP literature.

Sticky PDMP samplers for sparse and local inference problems

“…This scheme is referred as adaptive thinning in Bouchard-Côtè et al (2018). More sophisticated and potentially efficient thinning schemes have been proposed, see Sutton and Fearnhead (2021). The simulation of unfreezing times is easier: once the i-th component hits zero then it sticks at zero for a time that is exponentially distributed with parameter κ i |v i |.…”

“…If the upper bound is not tight to the Poisson rate λ, the procedure induce extra computational costs that can deteriorate the performance of the sampler. Several numerical schemes have been recently proposed trying to address this issue, see for example Pagani et al, 2020, Corbella, Spencer, and Roberts (2022),Bertazzi and Bierkens (2022) andSutton and Fearnhead (2021).…”

Sticky PDMP samplers for sparse and local inference problems

“…This scheme is referred as adaptive thinning in Bouchard-Côté, Vollmer, and Doucet (2018). More sophisticated and potentially efficient thinning schemes have been proposed, see Sutton and Fearnhead (2021). The simulation of unfreezing times is easier: once the i-th component hits zero then it sticks at zero for a time that is exponentially distributed with parameter κ i |v i |.…”

Continuously-Tempered PDMP Samplers

“…These PDMP samplers have a number of advantages, including non-reversible dynamics (which is known to improve mixing relative to reversible processes [9,1]), and the ability to reduce computation-per-iteration by either leveraging sparsity structure in the model [5,18] or using only sub-samples of the data to approximate the log-likelihood at each iteration (whilst still guaranteeing sampling from the target [2]). However, like other MCMC algorithms, particularly those that use gradient information, these PDMP samplers can struggle to mix for multi-modal target distributions, or for heavy-tailed targets [20].…”