  Selection among alternative theoretical models given an observed data set is
an important challenge in many areas of physics and astronomy. Reversible-jump
Markov chain Monte Carlo (RJMCMC) is an extremely powerful technique for
performing Bayesian model selection, but it suffers from a fundamental
difficulty: it requires jumps between model parameter spaces, but cannot
efficiently explore both parameter spaces at once. Thus, a naive jump between
parameter spaces is unlikely to be accepted in the MCMC algorithm and
convergence is correspondingly slow. Here we demonstrate an interpolation
technique that uses samples from single-model MCMCs to propose inter-model
jumps from an approximation to the single-model posterior of the target
parameter space. The interpolation technique, based on a kD-tree data
structure, is adaptive and efficient in modest dimensionality. We show that our
technique leads to improved convergence over naive jumps in an RJMCMC, and
compare it to other proposals in the literature to improve the convergence of
RJMCMCs. We also demonstrate the use of the same interpolation technique as a
way to construct efficient "global" proposal distributions for single-model
MCMCs without prior knowledge of the structure of the posterior distribution,
and discuss improvements that permit the method to be used in
higher-dimensional spaces efficiently.
