Population-based reversible jump Markov chain Monte Carlo. (English) Zbl 1158.62019

Jasra, Ajay; Stephens, David A.; Holmes, Christopher C.

Biometrika 94, No. 4, 787-807 (2007).

Summary: We present an extension of population-based Markov chain Monte Carlo to the transdimensional case. A major challenge is that of simulating from high- and transdimensional target measures. In such cases, Markov chain Monte Carlo methods may not adequately traverse the support of the target; the simulation results will be unreliable. We develop population methods to deal with such problems, and give a result proving the uniform ergodicity of these population algorithms, under mild assumptions. This result is used to demonstrate the superiority, in terms of the convergence rate, of a population transition kernel over a reversible jump sampler for a Bayesian variable selection problem.
We also give an example of a population algorithm for a Bayesian multivariate mixture model with an unknown number of components. This is applied to gene expression data of 1000 data points in six dimensions and it is demonstrated that our algorithm outperforms some competing Markov chain samplers. In this example, we show how to combine the methods of parallel chains, tempering [C. J. Geyer and E. A. Thompson, J. Am. Stat. Assoc. 90, No. 431, 909–920 (1995; Zbl 0850.62834)], snooker algorithms [W. R. Gilks et al., Statistician 43, 179–189 (1994)], constrained sampling and delayed rejection [P. J. Green and A. Mira, Biometrika 88, No. 4, 1035–1053 (2001; Zbl 1099.60508)].

MSC:

62F15	Bayesian inference
62P10	Applications of statistics to biology and medical sciences; meta analysis
65C40	Numerical analysis or methods applied to Markov chains
65C05	Monte Carlo methods

Keywords:

Bayesian variable selection; gene expression data; mixture model; reversible jump Markov chain Monte Carlo; uniform ergodicity