Slice sampling. (With discussions and rejoinder).

*(English)*Zbl 1051.65007The 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.

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.

Reviewer: Jaromir Antoch (Praha)

##### MSC:

65C60 | Computational problems in statistics (MSC2010) |

65C05 | Monte Carlo methods |

62D05 | Sampling theory, sample surveys |

65C40 | Numerical analysis or methods applied to Markov chains |

##### Keywords:

slice sampling; Markov chain Monte Carlo method; adaptive methods; Gibbs sampling; Metropolis algorithm; overrelaxation; dynamical methods; reflective slice sampling##### Software:

BUGS
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##### References:

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[35] | and Schmeiser (1998). For single-variable slice sampling, the variation of slice sampling proposed by Neal operates analogously to Gibbs sampling in the sense that to obtain the next point x1, y is generated from the conditional distribution [y|x0] given the current point x0 and then x1 is drawn from [x|y]. Both [y|x0] and [x|y] are uniform distributions. Since the closed form of the support of [x|y] is not available, sampling directly from [x|y] is not possible. A clever development is Neal’s sophisticated (but relatively expensive) sampling procedure to generate x1 from the ”slice” S = x : y < f (x). In Chen and Schmeiser (1998), we proposed random-direction interior point (RDIP), a general sampler designed to be ”black box” in the sense that the user need not tune the sampler to the problem. RDIP samples from the uniform distribution defined over the region U below the curve of the surface defined by f (x). Both slice sampling and RDIP require evaluations of f (x). Slice sampling, however, can be more expensive than RDIP because slice sampling requires evaluating f (x) more than once per iteration. The intention of RDIP’s design is to use as much free information as possible. For the high-dimensional case, the hy perrectangle idea in slice sampling could be inefficient. For example, suppose f (x) is the bivariate normal density with a high correlation. Then, the hy perrectangle idea essentially mimics the Gibbs sampler, which suffers slow convergence; see Chen and Schmeiser (1993) for a detailed discussion. Aligning the hy perrectangle (or ellipses) to the shape of f (x), along the lines of Kaufman and Smith (1998), seems like a good idea. As Neal mentions, the computational efficiency of our ”black-box” sampler RDIP depends on the normalization constant. Our goal was to be automatic and reasonably efficient, rather than to tune the sampler to the problem. If, however, |

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[48] | REDMOND, WASHINGTON 98052 E-MAIL: t-odowns@microsoft.com Rosenthal (1999). Generalizing just a little from the setting described in Section 4 of the paper, suppose that our target density can be written as the methodology of Roberts and Tweedie (2000). As an illustration of these results, it can be shown that, for the case where f0 is constant (i.e., in the single-variable slice sampler), and f1 is a real-valued log-concave function, 525 iterations suffice for convergence from all starting points x with f1(x) 0.01 supy f1(y). Similar results can be deduced for multidimensional log-concave distributions but the bounds worsen as dimension increases reflecting a genuine curse of dimensionality in this problem (despite the fact that this is inherently a twodimensional Gibbs sampler). To counteract this issue, Roberts and Rosenthal (2002) introduces the polar slice sampler in d dimensions, where f0(x) is chosen can be constructed in the spirit of Kendall and Møller (2000). Reversibility of the slice sampler allows easy simulation of these processes backwards in time to identify the starting point of the maximal and minimal chains. The beauty of the perfect slice sampling construction relies on the possibility of coupling the maximal and minimal chains even on a continuous state space. This is achieved because, thanks to monotonicity, the minimal horizontal slice (i.e., the one defined by the minimal chain) is alway s a superset of the maximal horizontal slice. If, when sampling over the minimal horizontal slice, a point is selected that belongs to the intersection of the minimal and the maximal horizontal slices, instantaneous coupling happens. Examples of applications given in Mira, Møller and Roberts (2001) include the Ising model on a two-dimensional grid at the critical temperature and various other automodels. In Casella, Mengersen, Robert and Titterington (2002) a further application of the perfect slice sampler construction to mixture of distributions is studied. |

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[67] | which appear in Damien, Wakefield and Walker (1999). I took Neal’s example in Section 8 and used a many-variable slice sampler on it. In fact I took a latent variable for each data point, the idea criticized by Neal. The overall joint density is given by DAMIEN, P., WAKEFIELD, J. C. and WALKER, S. G. (1999). Gibbs sampling for Bayesian nonconjugate and hierarchical models by using auxiliary variables. J. R. Stat. Soc. Ser. B Stat. Methodol. 61 331-344. JSTOR: · Zbl 0913.62028 |

[68] | ROBERTS, G. O. and ROSENTHAL, J. S. (1999). Convergence of slice sampler Markov chains. J. R. Statist. Soc. Ser. B 61 643-660. JSTOR: · Zbl 0929.62098 |

[69] | BATH, BA2 7AY UNITED KINGDOM E-MAIL: S.G.Walker@bath.ac.uk CARACCIOLO, S., PELISSETTO, A. and SOKAL, A. D. (1994). A general limitation on Monte Carlo algorithms of Metropolis ty pe. Phy s. Rev. Lett. 72 179-182. · Zbl 0973.65500 |

[70] | CHEN, M.-H. and SCHMEISER, B. W. (1998). Toward black-box sampling: A random-direction interior-point Markov chain approach. J. Comput. Graph. Statistics 7 1-22. JSTOR: |

[71] | DAMIEN, P., WAKEFIELD, J. C. and WALKER, S. G. (1999). Gibbs sampling for Bayesian nonconjugate and hierarchical models by using auxiliary variables. J. R. Stat. Soc. Ser. B Stat. Methodol. 61 331-344. JSTOR: · Zbl 0913.62028 |

[72] | EDWARDS, R. G. and SOKAL, A. D. (1988). Generalization of the Fortuin-Kasteley n-Swendsen- Wang representation and Monte Carlo algorithm. Phy s. Rev. D 38 2009-2012. |

[73] | GREEN, P. J. and MIRA, A. (2001). Delay ed rejection in reversible jump Metropolis-Hastings. Biometrika 88 1035-1053. JSTOR: · Zbl 1099.60508 |

[74] | MIRA, A. (1998). Ordering, splicing and splitting Monte Carlo Markov chains. Ph.D. dissertation, School of Statistics, Univ. Minnesota. |

[75] | NEAL, R. M. (1997). Markov chain Monte Carlo methods based on ”slicing” the density function. Technical Report 9722, Dept. Statistics, Univ. Toronto. |

[76] | ROBERTS, G. O. and ROSENTHAL, J. S. (2002). The polar slice sampler. Stoch. Models 18 257-280. · Zbl 1006.65004 |

[77] | TIERNEY, L. and MIRA, A. (1999). Some adaptive Monte Carlo methods for Bayesian inference. Statistics in Medicine 18 2507-2515. |

[78] | TORONTO, ONTARIO CANADA M5S 3G3 E-MAIL: radford@stat.utoronto.ca |

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