Informed reversible jump algorithms

Incorporating information about the target distribution in proposal mechanisms generally increases the efficiency of Markov chain Monte Carlo algorithms, when compared with those based on naive random walks. Hamiltonian Monte Carlo represents a successful example of fixed-dimensional algorithms incorporating gradient information. In trans-dimensional algorithms, Green (2003) recommended to generate the parameter proposals during model switches from normal distributions with informative means and covariance matrices. These proposal distributions can be viewed as asymptotic approximations to the parameter distributions, where the limit is with regard to the sample size. Models are typically proposed using uninformed uniform distributions. In this paper, we build on the approach of Zanella (2020) for discrete spaces to incorporate information about neighbouring models. We rely on approximations to posterior model probabilities that are asymptotically exact. We prove that, in some scenarios, the samplers combining this approach with that of Green (2003) behave like those that use the exact model probabilities and sample from the parameter distributions, in the large sample regime. We show that the implementation of the proposed samplers is straightforward in some cases. The methodology is applied to a real-data example. The code is available online.