A NEW AUTOMATIC SAMPLER FOR TRANSDIMENSIONAL MARKOV CHAIN MONTE CARLO

Modern applied Bayesian statistics relies heavily on simulation methods for the analysis of posterior distributions.
Algorithms to simulate from a Markov chain having a prescribed limiting distribution are nowadays widely available and, for a large class of standard models, easy to implement. The case of
posterior distributions defined on a space of variable dimension, such as those arising in problems of variable selection for regression models, is considerably harder. In fact, although several algorithms to sample from a Markov chain with such a limiting distribution have been proposed in the literature - most notably the so-called Reversible Jump -, they are all difficult to implement on a routine basis and very ill-suited for an automatic use. In the case of reversible jump, the main drawback is the need to devise jump proposals in a clever way, so that the sampler moves reasonably fast between different regions of the support of the posterior. The calculation of the Jacobians entering the acceptance probabilities may also be unappealing for a certain class of practitioners. We propose a new approach to transdimensional Markov chain simulation, based on a simple geometric idea used to map spaces of different dimension onto a space having a fixed dimension. This makes it possible to use standard fixed-dimension samplers also in the transdimensional case. It should be noted that the map mentioned
above can be evaluated in an automatic way based only on the density of the target distribution in regions of different dimension. To make the approach even more automatic, we advocate the use of a multidimensional version of Adaptive Rejection Metropolis Sampling to simulate from the transformed target density. In this way the user has only to specify the set of posterior
densities on the different parts of the variable dimensional support. General purpose functions for implementing the proposed algorithms
have been written for the computer language R.