  A new class of Markov chain Monte Carlo (MCMC) algorithms, based on
simulating piecewise deterministic Markov processes (PDMPs), have recently
shown great promise: they are non-reversible, can mix better than standard MCMC
algorithms, and can use subsampling ideas to speed up computation in big data
scenarios. However, current PDMP samplers can only sample from posterior
densities that are differentiable almost everywhere, which precludes their use
for model choice. Motivated by variable selection problems, we show how to
develop reversible jump PDMP samplers that can jointly explore the discrete
space of models and the continuous space of parameters. Our framework is
general: it takes any existing PDMP sampler, and adds two types of
trans-dimensional moves that allow for the addition or removal of a variable
from the model. We show how the rates of these trans-dimensional moves can be
calculated so that the sampler has the correct invariant distribution.
Simulations show that the new samplers can mix better than standard MCMC
algorithms. Our empirical results show they are also more efficient than
gradient-based samplers that avoid model choice through use of continuous
spike-and-slab priors which replace a point mass at zero for each parameter
with a density concentrated around zero.
