  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.
