Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-08T16:25:31.354Z Has data issue: false hasContentIssue false
  • English
  • Français

Comparison of hit-and-run, slice sampler and random walk Metropolis

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

Published online by Cambridge University Press:  16 January 2019

Daniel Rudolf  and
Mario Ullrich
Daniel Rudolf*
Affiliation:
Universität Göttingen
Mario Ullrich*
Affiliation:
Universität Linz
*
* Postal address: Felix-Bernstein-Institute for Mathematical Statistics in the Biosciences, Universität Göttingen, Goldschmidstraße 7, 37077 Göttingen, Germany. Email address: [email protected]
** Postal address: Universität Linz, Altenberger Straße 69, 4040 Linz, Austria. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Different Markov chains can be used for approximate sampling of a distribution given by an unnormalized density function with respect to the Lebesgue measure. The hit-and-run, (hybrid) slice sampler, and random walk Metropolis algorithm are popular tools to simulate such Markov chains. We develop a general approach to compare the efficiency of these sampling procedures by the use of a partial ordering of their Markov operators, the covariance ordering. In particular, we show that the hit-and-run and the simple slice sampler are more efficient than a hybrid slice sampler based on hit-and-run, which, itself, is more efficient than a (lazy) random walk Metropolis algorithm.

Keywords

Hit-and-runslice samplerrandom walk Metropoliscovariance ordering

MSC classification

Primary: 60J22: Computational methods in Markov chains
Secondary: 65C05: Monte Carlo methods 60J05: Discrete-time Markov processes on general state spaces
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 4 , December 2018 , pp. 1186 - 1202
Copyright
Copyright © Applied Probability Trust 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Andersen, C. and Diaconis, P. (2007). Hit and run as a unifying device. J. Soc. Fr. Stat. 148, 5–28.Google Scholar
[2]Andrieu, C. and Vihola, M. (2016). Establishing some order amongst exact approximations of MCMCs. Ann. Appl. Prob. 26, 2661–2696.Google Scholar
[3]Baxendale, P. (2005).Renewal theory and computable convergence rates for geometrically ergodic Markov chains. Ann. Appl. Prob. 15, 700–738.Google Scholar
[4]Bélisle, C., Romeijn, H. and Smith, R. (1993).Hit-and-run algorithms for generating multivariate distributions. Math. Operat. Res. 18, 255–266.Google Scholar
[5]Brooks, S., Gelman, A., Jones, G. and Meng, X. (eds) (2011).Handbook of Markov chain Monte Carlo. Chapman and Hall, Boca Raton.Google Scholar
[6]Casella, G. and Robert, C. (2004). Monte Carlo Statistical Methods, 2nd edn. Springer, New York.Google Scholar
[7]Chen, M. (2005). Eigenvalues, inequalities, and ergodic theory. In Probability and its Applications,Springer, LondonGoogle Scholar
[8]Diaconis, P. and Freedman, D. (1997). On the hit and run process. Tech. Rep. University of California497, 1–16.Google Scholar
[9]Fill, J. and Khan, J. (2013 ). Comparison inequalities and fastest mixing Markov chains. Ann. App. Prob. 23, 1778–1816.Google Scholar
[10]Higdon, D. (1998). Auxiliary variable methods for Markov chain Monte Carlo with applications. J. Amer. Statist. Assoc. 93, 585–595.Google Scholar
[11]Jarner, S. and Hansen, E. (2000). Geometric ergodicity of Metropolis algorithms. Stoch. Process. Appl. 85, 341–361.Google Scholar
[12]Kaufman, D. and Smith, R. (1998). Direction choice for accelerated convergence in hit-and-run sampling. Operat. Res. 46, 84–95.Google Scholar
[13]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, 1–11.Google Scholar
[14]Kreyszig, E. (1989). Introductory Functional Analysis with Applications. John Wiley, New York.Google Scholar
[15]Latuszyński, K. and Rudolf, D. (2014). Convergence of hybrid slice sampling via spectral gap. Preprint. Available at https://arxiv.org/abs/1409.2709.Google Scholar
[16]Lovász, L. (1999). Hit-and-run mixes fast. Math. Program. 86, 443–461.Google Scholar
[17]Lovász, L. and Vempala, S. (2006). Fast algorithms for logconcave functions: sampling, rounding, integration and optimization.. In Proc. 47th Annual IEEE Symposium on Foundations of Computer Science,Berkeley, CA, pp. 57–68.Google Scholar
[18]Lovász, L. and Vempala, S. (2006). Hit-and-run from a corner. SIAM J. Comput. 35, 985–1005.Google Scholar
[19]Lovász, L. and Vempala, S. (2007). The geometry of logconcave functions and sampling algorithms. Random Structures Algorithms 30, 307–358.Google Scholar
[20]Mathé, P. and Novak, E. (2007). Simple Monte Carlo and the Metropolis algorithm. J. Complexity 23, 673–696.Google Scholar
[21]Mengersen, K. and Tweedie, R. (1996). Rates of convergence of the Hastings and Metropolis algorithms. Ann. Statist. 24, 101–121.Google Scholar
[22]Mira, A. (2001). Ordering and improving the performance of Monte Carlo Markov chains. Statist. Sci. 16, 340–350.Google Scholar
[23]Mira, A. and Leisen, F. (2009). Covariance ordering for discrete and continuous time Markov chains. Statistica Sinica 19, 651–666.Google Scholar
[24]Mira, A. and Tierney, L. (2002). Efficiency and convergence properties of slice samplers. Scand. J. Statist. 29, 1–12.Google Scholar
[25]Mira, A., Møller, J. and Roberts, G. (2001). Perfect slice samplers. J. R. Statist. Soc. B 63, 593–606.Google Scholar
[26]Montenegro, R. and Tetali, P. (2006). Mathematical aspects of mixing times in Markov chains. Found. Trends Theor. Comput. Sci. 1, 121 pp.Google Scholar
[27]Neal, P., Roberts, G. and Yuen, W. (2012). Optimal scaling of random walk Metropolis algorithms with discontinuous target densities. Ann. Appl. Prob. 22, 1880–1927.Google Scholar
[28]Neal, R. (2003). Slice sampling. Ann. Statist. 31, 705–767.Google Scholar
[29]Peskun, P. (1973). Optimum Monte-Carlo sampling using Markov chains. Biometrika 60, 607–612.Google Scholar
[30]Roberts, G. and Rosenthal, J. (1999). Convergence of slice sampler Markov chains. J. R. Statist. Soc. B 61, 643–660.Google Scholar
[31]Roberts, G. and Rosenthal, J. (2001). Optimal scaling for various Metropolis–Hastings algorithms. Statist. Sci. 16, 351–367.Google Scholar
[32]Roberts, G. and Rosenthal, J. (2002). The polar slice sampler. Stoch. Models 18, 257–280.Google Scholar
[33]Roberts, G. and Tweedie, R. (1996). Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms. Biometrika 83, 95–110.Google Scholar
[34]Roberts, G., Gelman, A. and Gilks, W. (1997). Weak convergence and optimal scaling of random walk Metropolis algorithms. Ann. Appl. Prob. 7, 110–120.Google Scholar
[35]Rudolf, D. (2012). Explicit error bounds for Markov chain Monte Carlo. Dissertationes Math. 485, 93 pp.Google Scholar
[36]Rudolf, D. (2013). Hit-and-run for numerical integration. In Monte Carlo and Quasi-Monte Carlo Methods 2012 (Springer Proc. Math. Statist. 65), eds J. Dick, F. Y. Kuo, G. Peters, and I. H. Sloan. Springer, Berlin, pp. 597–612.Google Scholar
[37]Rudolf, D. and Ullrich, M. (2013). Positivity of hit-and-run and related algorithms. Electron. Commun. Prob. 18, 1–8.Google Scholar
[38]Smith, R. (1984). Efficient Monte Carlo procedures for generating points uniformly distributed over bounded regions. Operat. Res. 32, 1296–1308.Google Scholar
[39]Tierney, L. (1998). A note on Metropolis–Hastings kernels for general state spaces. Ann. Appl. Prob. 8, 1–9.Google Scholar
[40]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
[41]Ullrich, M. (2014). Rapid mixing of Swendsen–Wang dynamics in two dimensions. Dissertationes Math. 502, 64 pp.Google Scholar