Markov chain Monte Carlo algorithms with sequential proposals
This work addresses efficiency and mixing issues in MCMC sampling for statistical inference, offering incremental improvements over existing methods like HMC and bouncy particle samplers.
The authors tackled the problem of improving Markov chain Monte Carlo (MCMC) sampling efficiency by introducing a sequential-proposal framework, which led to novel methods that automatically tune trajectories to avoid inefficiencies and showed favorable numerical efficiency compared to existing methods like NUTS, with better mixing in multimodal scenarios.
We explore a general framework in Markov chain Monte Carlo (MCMC) sampling where sequential proposals are tried as a candidate for the next state of the Markov chain. This sequential-proposal framework can be applied to various existing MCMC methods, including Metropolis-Hastings algorithms using random proposals and methods that use deterministic proposals such as Hamiltonian Monte Carlo (HMC) or the bouncy particle sampler. Sequential-proposal MCMC methods construct the same Markov chains as those constructed by the delayed rejection method under certain circumstances. In the context of HMC, the sequential-proposal approach has been proposed as extra chance generalized hybrid Monte Carlo (XCGHMC). We develop two novel methods in which the trajectories leading to proposals in HMC are automatically tuned to avoid doubling back, as in the No-U-Turn sampler (NUTS). The numerical efficiency of these new methods compare favorably to the NUTS. We additionally show that the sequential-proposal bouncy particle sampler enables the constructed Markov chain to pass through regions of low target density and thus facilitates better mixing of the chain when the target density is multimodal.