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This article studies the convergence properties of trans-dimensional MCMCalgorithms when the total number of models is finite. It is shown that, forreversible and some non-reversible trans-dimensional Markov chains, under mildconditions, geometric convergence is guaranteed if the Markov chains associatedwith the within-model moves are geometrically ergodic. This result is proved inan $L^2$ framework using the technique of Markov chain decomposition. While thetechnique was previously developed for reversible chains, this work extends itto the point that it can be applied to some commonly used non-reversiblechains. Under geometric convergence, a central limit theorem holds for ergodicaverages, even in the absence of Harris ergodicity. This allows for theconstruction of simultaneous confidence intervals for features of the targetdistribution. This procedure is rigorously examined in a trans-dimensionalsetting, and special attention is given to the case where the asymptoticcovariance matrix in the central limit theorem is singular. The theory andmethodology herein are applied to reversible jump algorithms for two Bayesianmodels: an autoregression with Laplace errors and unknown model order, and aprobit regression with variable selection.

Geometric ergodicity of trans-dimensional Markov chain Monte Carlo algorithms