Informed reversible jump algorithms

TL;DR

This paper introduces an informed reversible jump algorithm that uses asymptotically exact posterior model probability approximations to improve trans-dimensional MCMC sampling efficiency, demonstrated through theoretical analysis and a real-data example.

Contribution

It extends existing reversible jump algorithms by incorporating asymptotically exact model information, enhancing sampling efficiency in large-sample regimes.

Findings

Samplers behave like ideal ones with exact model probabilities in large samples.

Implementation of the proposed method is straightforward in some cases.

The methodology improves efficiency in model switching within MCMC.

Abstract

Incorporating information about the target distribution in proposal mechanisms generally produces efficient Markov chain Monte Carlo algorithms (or at least, algorithms that are more efficient than uninformed counterparts). For instance, it has proved successful to incorporate gradient information in fixed-dimensional algorithms, as seen with algorithms such as Hamiltonian Monte Carlo. In trans-dimensional algorithms, Green (2003) recommended to sample 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 ideal ones that use the exact model probabilities and sample from the correct 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.