Roberts, Gareth O.; Rosenthal, Jeffrey S. The polar slice sampler. (English) Zbl 1006.65004 Stoch. Models 18, No. 2, 257-280 (2002). The slice sampler is a specialized type of Markov chain Monte Carlo (MCMC) auxiliary method. This interesting paper focuses closely on the relationship between the dimension of the target density and certain level set conditions. In particular, the authors analyze simple slice sampler factorization on log-concave densities. The investigation leads to the introduction of the polar slice sampler, an algorithm that effectively preserves the robust convergence properties of the slice sampler for the log-concave densities and for arbitrary dimensional distributions.Furthermore, a simple comparison of computing times is presented that shows that the polar slice sampler performs well in comparison with the rejection sampler and the simple slice sampler. A major limitation of all slice sampler algorithms is the fact that their implementation is often difficult. However, the authors demonstrate that the polar slice sampler has very good convergence properties, provided the X-updating step can be feasibly implemented.Section 2 introduces the slice sampler and describes some of its basic properties. In Section 3, the performance of the uniform simple slice sampler in high dimensions is investigated, and it is shown empirically that even for spherically symmetric, log-concave densities, the algorithm’s performance deteriorates markedly as dimension increases. The polar slice sampler is motivated and constructed in Section 4, and studied theoretically in Section 5. Section 6 gives empirical study of the performance of the polar slice sampler for a particular example and gives comparisons with the uniform simple slice sampler and a corresponding rejection sampling algorithm. Section 7 offers some concluding remarks. Reviewer: Jaromir Antoch (Praha) Cited in 1 ReviewCited in 9 Documents MSC: 65C40 Numerical analysis or methods applied to Markov chains Keywords:MCMC; slice sampler; polar slice sampler; uniform simple slice sampler; log-concave densities; Monte Carlo simulations; Markov chain Monte Carlo method; algorithm; convergence; rejection sampler; performance PDFBibTeX XMLCite \textit{G. O. Roberts} and \textit{J. S. Rosenthal}, Stoch. Models 18, No. 2, 257--280 (2002; Zbl 1006.65004) Full Text: DOI References: [1] Edwards R.G., Phys. Rev. Lett. 38 pp 2009– (1988) [2] Besag J.E., J. R. Stat. Soc. Ser. B 55 pp 25– (1993) [3] DOI: 10.2307/2670110 · Zbl 0953.62103 · doi:10.2307/2670110 [4] DOI: 10.1111/1467-9868.00179 · Zbl 0913.62028 · doi:10.1111/1467-9868.00179 [5] DOI: 10.1111/1467-9868.00198 · Zbl 0929.62098 · doi:10.1111/1467-9868.00198 [6] DOI: 10.1093/biomet/83.1.95 · Zbl 0888.60064 · doi:10.1093/biomet/83.1.95 [7] DOI: 10.1016/S0304-4149(98)00085-4 · Zbl 0961.60066 · doi:10.1016/S0304-4149(98)00085-4 [8] Roberts G.O., JAP 37 pp 359– (2000) [9] DOI: 10.2307/2291067 · Zbl 0824.60077 · doi:10.2307/2291067 [10] DOI: 10.1287/moor.21.1.182 · Zbl 0847.60053 · doi:10.1287/moor.21.1.182 [11] DOI: 10.1111/1467-9868.00301 · Zbl 0993.65015 · doi:10.1111/1467-9868.00301 [12] DOI: 10.1214/aoap/1028903381 · Zbl 0935.60054 · doi:10.1214/aoap/1028903381 [13] Hardy G.H., Inequalities (1934) · Zbl 0010.10703 [14] Mira A., Scand. J. Stat. [15] DOI: 10.1111/1467-9469.00250 · Zbl 0985.60067 · doi:10.1111/1467-9469.00250 [16] DOI: 10.1023/A:1008982016666 · doi:10.1023/A:1008982016666 [17] DOI: 10.1214/aoap/1034625254 · Zbl 0876.60015 · doi:10.1214/aoap/1034625254 [18] DOI: 10.1214/ss/1015346320 · Zbl 1127.65305 · doi:10.1214/ss/1015346320 [19] DOI: 10.2307/2291683 · Zbl 0869.62066 · doi:10.2307/2291683 [20] DOI: 10.1214/ss/1177009937 · Zbl 0955.60526 · doi:10.1214/ss/1177009937 [21] DOI: 10.1023/A:1008820505350 · doi:10.1023/A:1008820505350 [22] DOI: 10.2307/2347565 · Zbl 0825.62407 · doi:10.2307/2347565 [23] DOI: 10.1103/PhysRevLett.58.86 · doi:10.1103/PhysRevLett.58.86 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.