  Reversible jump Markov chain Monte Carlo (RJMCMC) proposals that achieve
reasonable acceptance rates and mixing are notoriously difficult to design in
most applications. Inspired by recent advances in deep neural network-based
normalizing flows and density estimation, we demonstrate an approach to enhance
the efficiency of RJMCMC sampling by performing transdimensional jumps
involving reference distributions. In contrast to other RJMCMC proposals, the
proposed method is the first to apply a non-linear transport-based approach to
construct efficient proposals between models with complicated dependency
structures. It is shown that, in the setting where exact transports are used,
our RJMCMC proposals have the desirable property that the acceptance
probability depends only on the model probabilities. Numerical experiments
demonstrate the efficacy of the approach.
