  New sampling algorithms based on simulating continuous-time stochastic
processes called piece-wise deterministic Markov processes (PDMPs) have shown
considerable promise. However, these methods can struggle to sample from
multi-modal or heavy-tailed distributions. We show how tempering ideas can
improve the mixing of PDMPs in such cases. We introduce an extended
distribution defined over the state of the posterior distribution and an
inverse temperature, which interpolates between a tractable distribution when
the inverse temperature is 0 and the posterior when the inverse temperature is
1. The marginal distribution of the inverse temperature is a mixture of a
continuous distribution on [0,1) and a point mass at 1: which means that we
obtain samples when the inverse temperature is 1, and these are draws from the
posterior, but sampling algorithms will also explore distributions at lower
temperatures which will improve mixing. We show how PDMPs, and particularly the
Zig-Zag sampler, can be implemented to sample from such an extended
distribution. The resulting algorithm is easy to implement and we show
empirically that it can outperform existing PDMP-based samplers on challenging
multimodal posteriors.
