Delayed rejection in reversible jump Metropolis-Hastings

In a Metropolis-Hastings algorithm, rejection of proposed moves is an intrinsic part of ensuring that the chain converges to the intended target distribution. However, persistent rejection, perhaps in particular parts of the state space, may indicate that locally the proposal distribution is badly calibrated to the target. As an alternative to careful off-line tuning of state-dependent proposals, the basic algorithm can be modified so that, on rejection, a second attempt to move is made. A different proposal can be generated from a new distribution that is allowed to depend on the previously rejected proposal. We generalise this idea of delaying the rejection and adapting the proposal distribution, due to Tierney & Mira (1999), to generate a more flexible class of methods that applies in particular to a variable-dimension setting. The approach is illustrated by two pedagogical examples and a more realistic application to a changepoints analysis for point processes.