*(English)*Zbl 1051.65007

The author describes a class of slice sampling methods that can be applied to a wide variety of distributions. Section 2 summarizes general-purpose Markov chain sampling methods as the Gibbs sampling, the adaptive rejection sampling, the adaptive rejection Metropolis sampling etc.

Section 3 presents the basic ideas of a slice sampling and thoroughly discusses different predecessors more or less connected to it. The principal message of the paper is concentrated in chapters 4–7. At first, simple variable slice samplings methods are described. Then the author concentrates on multivariate slice sampling methods and reflective slice sampling. An example forms the final section.

I liked the paper and I must say that despite it is a paper for Annals of Statistics, the author really concentrates on the ideas and not on the formal proofs as is typical for this journal. I am sure that everybody who want to get an idea of what is slice sampling will be satisfied.

The paper is complemented by an interesting discussion prepared by Ming-Hui Chen, B. W. Schmeiser, O. B. Downs, A. Mira, G. O. Roberts, J. Skilling, D. J. C. MacKey and G. S. Walker.

##### MSC:

65C60 | Computational problems in statistics |

65C05 | Monte Carlo methods |

62D05 | Statistical sampling theory, sample surveys |

65C40 | Computational Markov chains (numerical analysis) |