Reversible jump, birth-and-death and more general continuous time Markov chain Monte Carlo samplers. (English) Zbl 1063.62133
Summary: Reversible jump methods are the most commonly used Markov chain Monte Carlo tool for exploring variable dimension statistical models. Recently, however, an alternative approach based on birth-and-death processes has been proposed by M. Stephens [Ann. Stat. 28, No. 1, 40–74 (2000; Zbl 1106.62316)] for mixtures of distributions. We show that the birth-and-death setting can be generalized to include other types of continuous time jumps like split-and-combine moves in the spirit of S. Richardson and P. J. Green [J. R. Stat. Soc., Ser. B 59, 731–792 (1997; Zbl 0891.62020)].
We illustrate these extensions both for mixtures of distributions and for hidden Markov models. We demonstrate the strong similarity of reversible jump and continuous time methodologies by showing that, on appropriate rescaling of time, the reversible jump chain converges to a limiting continuous time birth-and-death process. A numerical comparison in the setting of mixtures of distributions highlights this similarity.
We illustrate these extensions both for mixtures of distributions and for hidden Markov models. We demonstrate the strong similarity of reversible jump and continuous time methodologies by showing that, on appropriate rescaling of time, the reversible jump chain converges to a limiting continuous time birth-and-death process. A numerical comparison in the setting of mixtures of distributions highlights this similarity.
MSC:
| 62M99 | Inference from stochastic processes |
| 65C40 | Numerical analysis or methods applied to Markov chains |
| 60J80 | Branching processes (Galton-Watson, birth-and-death, etc.) |
References:
| [1] | Baum L. E., Ann. Math. Statist. 41 pp 164– (1970) |
| [2] | Breiman L., Probability (1992) · doi:10.1137/1.9781611971286 |
| [3] | Cappe O., J. Am. Statist. Ass. 95 pp 1282– (2000) |
| [4] | Casella G., Biometrika 83 pp 81– (1996) |
| [5] | Celeux G., J. Am. Statist. Ass. 95 pp 957– (2000) |
| [6] | P. Clifford, and G. Nicholls (1994 ) Comparison of birth-and-death and Metropolis-Hastings Markov Chain Monte Carlo for the Strauss process . Department of Statistics, University of Oxford, Oxford. |
| [7] | DOI: 10.1111/1467-9892.00068 · Zbl 0919.62100 · doi:10.1111/1467-9892.00068 |
| [8] | Geweke J., Econometrica 57 pp 1317– (1989) |
| [9] | Geyer C. J., Scand. J. Statist. 21 pp 359– (1994) |
| [10] | Green P. J., Biometrika 82 pp 711– (1995) |
| [11] | Grenander U., J. R. Statist. Soc. 56 pp 549– (1994) |
| [12] | Hurn M., J. Comput. Graph. Statist. 12 pp 1– (2003) |
| [13] | Karr A. F., Z. Wahrsch. Ver. Geb. 33 pp 41– (1975) |
| [14] | Phillips D. B., Markov Chain Monte Carlo in Practice pp 215– (1996) · doi:10.1007/978-1-4899-4485-6_13 |
| [15] | Preston C. J., Bull. Inst. Int. Statist. 46 pp 371– (1976) |
| [16] | DOI: 10.1111/1467-9868.00095 · doi:10.1111/1467-9868.00095 |
| [17] | Ripley B. D., J. R. Statist. Soc. 39 pp 172– (1977) |
| [18] | Robert C. P., J. Statist. Comput. Simuln 64 pp 327– (1999) |
| [19] | DOI: 10.1111/1467-9868.00219 · Zbl 0941.62090 · doi:10.1111/1467-9868.00219 |
| [20] | Roeder K., J. Am. Statist. Ass. 85 pp 617– (1990) |
| [21] | Stephens M., Ann. Statist. 28 pp 40– (2000) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
