This article studies the convergence properties of trans-dimensional MCMC algorithms when the total number of models is finite. It is shown that, for reversible and some non-reversible trans-dimensional Markov chains, under mild conditions, geometric convergence is guaranteed if the Markov chains associated with the within-model moves are geometrically ergodic. This result is proved in an $L^2$ framework using the technique of Markov chain decomposition. While the technique was previously developed for reversible chains, this work extends it to the point that it can be applied to some commonly used non-reversible chains. The theory herein is applied to reversible jump algorithms for three Bayesian models: a probit regression with variable selection, a Gaussian mixture model with unknown number of components, and an autoregression with Laplace errors and unknown model order.

翻译：本文研究了当模型总数有限时跨维MCMC算法的收敛性质。研究表明，对于可逆及某些不可逆的跨维马尔可夫链，在温和条件下，若与模型内转移相关的马尔可夫链具有几何遍历性，则可保证几何收敛性。该结果在$L^2$框架下通过马尔可夫链分解技术得以证明。虽然该技术先前仅针对可逆链开发，但本研究将其扩展至可应用于某些常用不可逆链的程度。本文理论应用于三个贝叶斯模型的可逆跳转算法：变量选择的概率回归、分量数未知的高斯混合模型，以及具有拉普拉斯误差和未知模型阶数的自回归。