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Convergence of hybrid slice sampling via spectral gap

Part of: Markov processes Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  25 July 2024

Krzysztof Łatuszyński  and
Daniel Rudolf Open the ORCID record for Daniel Rudolf [Opens in a new window]
Krzysztof Łatuszyński*
Affiliation:
University Warwick
Daniel Rudolf*
Affiliation:
Universität Passau
*
*Postal address: Department of Statistics, CV47AL Coventry, United Kingdom. Email address: K.G.Latuszynski@warwick.ac.uk
**Postal address: Universität Passau, Innstraße 33, 94032 Passau, Germany. Email address: daniel.rudolf@uni-passau.de
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Abstract

It is known that the simple slice sampler has robust convergence properties; however, the class of problems where it can be implemented is limited. In contrast, we consider hybrid slice samplers which are easily implementable and where another Markov chain approximately samples the uniform distribution on each slice. Under appropriate assumptions on the Markov chain on the slice, we give a lower bound and an upper bound of the spectral gap of the hybrid slice sampler in terms of the spectral gap of the simple slice sampler. An immediate consequence of this is that the spectral gap and geometric ergodicity of the hybrid slice sampler can be concluded from the spectral gap and geometric ergodicity of the simple version, which is very well understood. These results indicate that robustness properties of the simple slice sampler are inherited by (appropriately designed) easily implementable hybrid versions. We apply the developed theory and analyze a number of specific algorithms, such as the stepping-out shrinkage slice sampling, hit-and-run slice sampling on a class of multivariate targets, and an easily implementable combination of both procedures on multidimensional bimodal densities.

Keywords

Slice samplerspectral gapgeometric ergodicity

MSC classification

Primary: 60J22: Computational methods in Markov chains
Secondary: 65C05: Monte Carlo methods 60J05: Discrete-time Markov processes on general state spaces
Type
Original Article
Information
Advances in Applied Probability , First View , pp. 1 - 27
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Baxendale, P. (2005). Renewal theory and computable convergence rates for geometrically ergodic Markov chains. Ann. Appl. Prob. 15, 700–738.CrossRefGoogle Scholar
Bednorz, W. and Łatuszyński, K. (2007). A few remarks on ‘Fixed-width output analysis for Markov chain Monte Carlo’ by Jones et al. J. Amer. Statist. Assoc. 102, 1485–1486.CrossRefGoogle Scholar
Bélisle, C., Romeijn, H. and Smith, R. (1993). Hit-and-run algorithms for generating multivariate distributions. Math. Operat. Res. 18, 255–266.CrossRefGoogle Scholar
Besag, J. and Green, P. (1993). Spatial statistics and Bayesian computation. J. R. Statist. Soc. B [Statist. Methodology] 55, 25–37.CrossRefGoogle Scholar
Diaconis, P. and Freedman, D. (1997). On the hit and run process. Tech. rep. 497, Department of Statistics, University of California, Berkeley, 16 pp.Google Scholar
Edwards, R. and Sokal, A. (1988). Generalization of the Fortuin–Kasteleyn–Swendsen–Wang representation and Monte Carlo algorithm. Phys. Rev. D 38, article no. 2009.CrossRefGoogle ScholarPubMed
Flegal, J. and Jones, G. (2010). Batch means and spectral variance estimators in Markov chain Monte Carlo. Ann. Statist. 38, 1034–1070.CrossRefGoogle Scholar
Geyer, C. (1992). Practical Markov chain Monte Carlo. Statist. Sci. 7, 473–483.Google Scholar
Higdon, D. (1998). Auxiliary variable methods for Markov chain Monte Carlo with applications. J. Amer. Statist. Assoc. 93, 585–595.CrossRefGoogle Scholar
Hobert, J., Jones, G., Presnell, B. and Rosenthal, J. (2002). On the applicability of regenerative simulation in Markov chain Monte Carlo. Biometrika 89, 731–743.CrossRefGoogle Scholar
Jones, G., Haran, M., Caffo, B. and Neath, R. (2006). Fixed-width output analysis for Markov chain Monte Carlo. J. Amer. Statist. Assoc. 101, 1537–1547.CrossRefGoogle Scholar
Kaufman, D. and Smith, R. (1998). Direction choice for accelerated convergence in hit-and-run sampling. Operat. Res. 46, 84–95.CrossRefGoogle Scholar
Kiatsupaibul, S., Smith, R. and Zabinsky, Z. (2011). An analysis of a variation of hit-and-run for uniform sampling from general regions. ACM Trans. Model. Comput. Simul. 21, 16:1–16:11.CrossRefGoogle Scholar
Kipnis, C. and S, V. (1986). Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Commun. Math. Phys. 104, 1–19.CrossRefGoogle Scholar
Kontoyiannis, I. and Meyn, S. (2012). Geometric ergodicity and the spectral gap of non-reversible Markov chains. Prob. Theory Relat. Fields 154, 327–339.CrossRefGoogle Scholar
Lovász, L. (1999). Hit-and-run mixes fast. Math. Program. 86, 443–461.Google Scholar
Lovász, L. and Vempala, S. (2006). Hit-and-run from a corner. SIAM J. Comput. 35, 985–1005.CrossRefGoogle Scholar
Meyn, S. and Tweedie, R. (2009). Markov Chains and Stochastic Stability, 2nd edn. Cambridge University Press.CrossRefGoogle Scholar
Mira, A. (2001). Ordering and improving the performance of Monte Carlo Markov chains. Statist. Sci. 16, 340–350.CrossRefGoogle Scholar
Mira, A., Møller, J. and Roberts, G. (2001). Perfect slice samplers. J. R. Statist. Soc. B [Statist. Methodology] 63, 593–606.CrossRefGoogle Scholar
Mira, A. and Tierney, L. (2002). Efficiency and convergence properties of slice samplers. Scand. J. Statist. 29, 1–12.CrossRefGoogle Scholar
Murray, I., Adams, R. and MacKay, D. (2010). Elliptical slice sampling. In Proceedings of the 13th International Conference on Artificial Intelligence and Statistics, Journal of Machine Learning Research, pp. 541–548.Google Scholar
Natarovskii, V., Rudolf, D. and Sprungk, B. (2021). Geometric convergence of elliptical slice sampling. In Proceedings of the 38th International Conference on Machine Learning, Proceedings of Machine Learning Research 139, pp. 7969–7978.Google Scholar
Natarovskii, V., Rudolf, D. and Sprungk, B. (2021). Quantitative spectral gap estimate and Wasserstein contraction of simple slice sampling. Ann. Appl. Prob. 31, 806–825.CrossRefGoogle Scholar
Neal, R. (1993). Probabilistic inference using Markov chain Monte Carlo methods. Department of Computer Science, University of Toronto.Google Scholar
Neal, R. (2003). Slice sampling. Ann. Statist. 31, 705–767.CrossRefGoogle Scholar
Novak, E. and Rudolf, D. (2014). Computation of expectations by Markov chain Monte Carlo methods. In Extraction of Quantifiable Information from Complex Systems, Springer, Cham, pp. 397–411.CrossRefGoogle Scholar
Roberts, G. and Rosenthal, J. (1997). Geometric ergodicity and hybrid Markov chains. Electron. Commun. Prob. 2, 13–25.CrossRefGoogle Scholar
Roberts, G. and Rosenthal, J. (1999). Convergence of slice sampler Markov chains. J. R. Statist. Soc. B [Statist. Methodology] 61, 643–660.CrossRefGoogle Scholar
Roberts, G. and Rosenthal, J. (2002). The polar slice sampler. Stoch. Models 18, 257–280.CrossRefGoogle Scholar
Roberts, G. and Rosenthal, J. (2008). Variance bounding Markov chains. Ann. Appl. Prob. 18, 1201–1214.CrossRefGoogle Scholar
Rudin, W. (1991). Functional Analysis, 2nd edn. McGraw-Hill, New York.Google Scholar
Rudolf, D. (2012). Explicit error bounds for Markov chain Monte Carlo. Dissertationes Math. 485, 93 pp.CrossRefGoogle Scholar
Rudolf, D. (2013). Hit-and-run for numerical integration. In Monte Carlo and Quasi-Monte Carlo Methods 2012, eds J. Dick, F. Kuo, G. Peters, and I. Sloan, Springer, Berlin, pp. 565–579.CrossRefGoogle Scholar
Rudolf, D. and Ullrich, M. (2013). Positivity of hit-and-run and related algorithms. Electron. Commun. Prob. 18, 1–8.CrossRefGoogle Scholar
Rudolf, D. and Ullrich, M. (2018). Comparison of hit-and-run, slice sampler and random walk Metropolis. J. Appl. Prob. 55, 1186–1202.CrossRefGoogle Scholar
Schär, P., Habeck, M. and Rudolf, D. (2023). Gibbsian polar slice sampling. In Proceedings of the 40th International Conference on Machine Learning, Proceedings of Machine Learning Research 202, pp. 30204–30223.Google Scholar
Smith, R. (1984). Efficient Monte Carlo procedures for generating points uniformly distributed over bounded regions. Operat. Res. 32, 1296–1308.CrossRefGoogle Scholar
Ullrich, M. (2012). Rapid mixing of Swendsen–Wang and single-bond dynamics in two dimensions. Preprint. Available at https://arxiv.org/abs/1202.6321.Google Scholar
Ullrich, M. (2014). Rapid mixing of Swendsen–Wang dynamics in two dimensions. Dissertationes Math. 502, 64 pp.CrossRefGoogle Scholar
Wang, J. and Swendsen, R. (1987). Nonuniversal critical dynamics in Monte Carlo simulations. Phys. Rev. Lett. 58, article no. 86.Google Scholar
Werner, D. (2005). Funktionalanalysis. Springer, Berlin.Google Scholar