Bayesian Analysis

Designing Simple and Efficient Markov Chain Monte Carlo Proposal Kernels

Yuttapong Thawornwattana, Daniel Dalquen, and Ziheng Yang

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Abstract

We discuss a few principles to guide the design of efficient Metropolis–Hastings proposals for well-behaved target distributions without deeply divided modes. We illustrate them by developing and evaluating novel proposal kernels using a variety of target distributions. Here, efficiency is measured by the variance ratio relative to the independent sampler. The first principle is to introduce negative correlation in the MCMC sample or to reduce positive correlation: to propose something new, propose something different. This explains why single-moded proposals such as the Gaussian random-walk is poorer than the uniform random walk, which is in turn poorer than the bimodal proposals that avoid values very close to the current value. We evaluate three new bimodal proposals called Box, Airplane and StrawHat, and find that they have similar performance to the earlier Bactrian kernels, suggesting that the general shape of the proposal matters, but not the specific distributional form. We propose the “Mirror” kernel, which generates new values around the mirror image of the current value on the other side of the target distribution (effectively the “opposite” of the current value). This introduces negative correlations, leading in many cases to efficiency of >100%. The second principle, applicable to multidimensional targets, is that a sequence of well-designed one-dimensional proposals can be more efficient than a single d-dimensional proposal. Thirdly, we suggest that variable transformation be explored as a general strategy for designing efficient MCMC kernels. We apply these principles to a high-dimensional Gaussian target with strong correlations, a logistic regression problem and a molecular clock dating problem to illustrate their practical utility.

Article information

Source
Bayesian Anal., Volume 13, Number 4 (2018), 1037-1063.

Dates
First available in Project Euclid: 10 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.ba/1510282998

Digital Object Identifier
doi:10.1214/17-BA1084

Mathematical Reviews number (MathSciNet)
MR3855362

Zentralblatt MATH identifier
06989975

Keywords
asymptotic variance bimodal kernel Metropolis–Hastings algorithm Mirror kernel molecular clock dating variable transformation

Rights
Creative Commons Attribution 4.0 International License.

Citation

Thawornwattana, Yuttapong; Dalquen, Daniel; Yang, Ziheng. Designing Simple and Efficient Markov Chain Monte Carlo Proposal Kernels. Bayesian Anal. 13 (2018), no. 4, 1037--1063. doi:10.1214/17-BA1084. https://projecteuclid.org/euclid.ba/1510282998


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References

  • Adler, S. L. (1981). “Over-relaxation method for the Monte Carlo evaluation of the partition function for multiquadratic actions.” Physical Review. D. Particles and Fields, 23: 2901–2904.
  • Barone, P. and Frigessi, A. (1990). “Improving stochastic relaxation for Gaussian random fields.” Probability in the Engineering and Informational Sciences, 4(3): 369–389.
    Zentralblatt MATH: 1134.60347
    Digital Object Identifier: doi:10.1017/S0269964800001674
  • Bédard, M., Douc, R., and Moulines, E. (2012). “Scaling analysis of multiple-try MCMC methods.” Stochastic Processes and Their Applications, 122(3): 758–786.
    Zentralblatt MATH: 1239.60075
    Digital Object Identifier: doi:10.1016/j.spa.2011.11.004
  • Bédard, M., Douc, R., and Moulines, E. (2014). “Scaling analysis of delayed rejection MCMC methods.” Methodology and Computing in Applied Probability, 16(4): 811–838.
    Mathematical Reviews (MathSciNet): MR3270597
    Zentralblatt MATH: 1323.65001
    Digital Object Identifier: doi:10.1007/s11009-013-9326-y
  • Beskos, A., Pillai, N., Roberts, G., Sanz-Serna, J.-M., and Stuart, A. (2013). “Optimal tuning of the hybrid Monte Carlo algorithm.” Bernoulli, 19(5A): 1501–1534.
    Zentralblatt MATH: 1287.60090
    Digital Object Identifier: doi:10.3150/12-BEJ414
    Project Euclid: euclid.bj/1383661192
  • Beskos, A., Roberts, G., and Stuart, A. (2009). “Optimal scalings for local Metropolis-Hastings chains on nonproduct targets in high dimensions.” The Annals of Applied Probability, 19(3): 863–898.
    Mathematical Reviews (MathSciNet): MR2537193
    Zentralblatt MATH: 1172.60328
    Digital Object Identifier: doi:10.1214/08-AAP563
    Project Euclid: euclid.aoap/1245071013
  • Browne, W. J. (2006). “MCMC algorithms for constrained variance matrices.” Computational Statistics & Data Analysis, 50(7): 1655–1677.
    Mathematical Reviews (MathSciNet): MR2222038
    Zentralblatt MATH: 05381647
    Digital Object Identifier: doi:10.1016/j.csda.2005.02.008
  • Carpenter, B., Gelman, A., Hoffman, M., Lee, D., Goodrich, B., Betancourt, M., Brubaker, M., Guo, J., Li, P., and Riddell, A. (2017). “Stan: a probabilistic programming language.” Journal of Statistical Software, 76(1): 1–32.
  • Devroye, L. (1986). Non-uniform random variate generation. Springer-Verlag, New York.
    Zentralblatt MATH: 0593.65005
  • Frigessi, A., Gåsemyr, J., and Rue, H. (2000). “Antithetic coupling of two Gibbs sampler chains.” Annals of Statistics, 28(4): 1128–1149.
    Zentralblatt MATH: 1105.65303
    Digital Object Identifier: doi:10.1214/aos/1015956710
    Project Euclid: euclid.aos/1015956710
  • Gelfand, A. E., Smith, A. F. M., and Lee, T.-M. (1992). “Bayesian analysis of constrained parameter and truncated data problems using Gibbs sampling.” Journal of the American Statistical Association, 87(418): 523–532.
  • Gelman, A., Roberts, G. O., and Gilks, W. R. (1996). “Efficient Metropolis jumping rules.” In Bayesian statistics, 5, 599–607. Oxford Univ. Press, New York.
  • Geyer, C. J. (1992). “Practical Markov chain Monte Carlo.” Statistical Science, 7(4): 473–483.
  • Girolami, M. and Calderhead, B. (2011). “Riemann manifold Langevin and Hamiltonian Monte Carlo methods.” Journal of the Royal Statistical Society. Series B, Statistical Methodology, 73(2): 123–214. With discussion and a reply by the authors.
  • Hammersley, J. M. and Morton, K. W. (1956). “A new Monte Carlo technique: antithetic variates.” Mathematical Proceedings of the Cambridge Philosophical Society, 52: 449–475.
    Zentralblatt MATH: 0071.35404
    Digital Object Identifier: doi:10.1017/S0305004100031455
  • Hastings, W. K. (1970). “Monte Carlo sampling methods using Markov chains and their applications.” Biometrika, 57(1): 97–109.
    Zentralblatt MATH: 0219.65008
    Digital Object Identifier: doi:10.1093/biomet/57.1.97
  • Hoffman, M. D. and Gelman, A. (2014). “The no-U-turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo.” Journal of Machine Learning Research, 15: 1593–1623.
    Zentralblatt MATH: 1319.60150
  • Horai, S., Hayasaka, K., Kondo, R., Tsugane, K., and Takahata, N. (1995). “Recent African origin of modern humans revealed by complete sequences of hominoid mitochondrial DNAs.” Proceedings of the National Academy of Sciences of the United States of America, 92(2): 532–536.
  • Jukes, T. H. and Cantor, C. R. (1969). “Evolution of protein molecules.” In Munro, H. N. (ed.), Mammalian Protein Metabolism, volume 3, 21–132. Academic Press, New York.
  • Kemeny, J. G. and Snell, J. L. (1960). Finite Markov chains. D. Van Nostrand Co., Inc.
    Zentralblatt MATH: 0089.13704
  • Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., and Teller, E. (1953). “Equation of state calculations by fast computing machines.” The Journal of Chemical Physics, 21(6): 1087–1092.
  • Mira, A. (2001). “Efficiency of finite state space Monte Carlo Markov chains.” Statistics & Probability Letters, 54(4): 405–411.
    Zentralblatt MATH: 0989.60066
    Digital Object Identifier: doi:10.1016/S0167-7152(01)00115-8
  • Pasarica, C. and Gelman, A. (2010). “Adaptively scaling the Metropolis algorithm using expected squared jumped distance.” Statist. Sinica, 20(1): 343–364.
    Zentralblatt MATH: 1186.62038
  • Peskun, P. H. (1973). “Optimum Monte-Carlo sampling using Markov chains.” Biometrika, 60: 607–612.
    Zentralblatt MATH: 0271.62041
    Digital Object Identifier: doi:10.1093/biomet/60.3.607
  • Pillai, N. S., Stuart, A. M., and Thiéry, A. H. (2012). “Optimal scaling and diffusion limits for the Langevin algorithm in high dimensions.” The Annals of Applied Probability, 22(6): 2320–2356.
    Zentralblatt MATH: 1272.60053
    Digital Object Identifier: doi:10.1214/11-AAP828
    Project Euclid: euclid.aoap/1353695955
  • Roberts, G. O., Gelman, A., and Gilks, W. R. (1997). “Weak convergence and optimal scaling of random walk Metropolis algorithms.” The Annals of Applied Probability, 7(1): 110–120.
    Zentralblatt MATH: 0876.60015
    Digital Object Identifier: doi:10.1214/aoap/1034625254
    Project Euclid: euclid.aoap/1034625254
  • Roberts, G. O. and Rosenthal, J. S. (1998). “Optimal scaling of discrete approximations to Langevin diffusions.” Journal of the Royal Statistical Society. Series B, Statistical Methodology, 60(1): 255–268.
    Zentralblatt MATH: 0913.60060
    Digital Object Identifier: doi:10.1111/1467-9868.00123
  • Sherlock, C. and Roberts, G. (2009). “Optimal scaling of the random walk Metropolis on elliptically symmetric unimodal targets.” Bernoulli, 15(3): 774–798.
    Zentralblatt MATH: 1215.60047
    Digital Object Identifier: doi:10.3150/08-BEJ176
    Project Euclid: euclid.bj/1251463281
  • Thawornwattana, Y., Dalquen, D., and Yang, Z. (2017). “Supplementary Material of the “Designing Simple and Efficient Markov Chain Monte Carlo Proposal Kernels”.” Bayesian Analysis.
    Mathematical Reviews (MathSciNet): MR3855362
    Digital Object Identifier: doi:10.1214/17-BA1084
    Project Euclid: euclid.ba/1510282998
  • Tierney, L. (1994). “Markov chains for exploring posterior distributions.” Annals of Statistics, 22(4): 1701–1762.
    Mathematical Reviews (MathSciNet): MR1329166
    Zentralblatt MATH: 0829.62080
    Digital Object Identifier: doi:10.1214/aos/1176325750
    Project Euclid: euclid.aos/1176325750
  • Tierney, L. (1998). “A note on Metropolis-Hastings kernels for general state spaces.” The Annals of Applied Probability, 8(1): 1–9.
    Zentralblatt MATH: 0935.60053
    Digital Object Identifier: doi:10.1214/aoap/1027961031
    Project Euclid: euclid.aoap/1027961031
  • Wang, Z., Mohamed, S., and de Freitas, N. (2013). “Adaptive Hamiltonian and Riemann Manifold Monte Carlo.” In Proceedings of the 30th International Conference on Machine Learning (ICML), 1462–1470.
  • Yang, Z. and Rodríguez, C. E. (2013). “Searching for efficient Markov chain Monte Carlo proposal kernels.” Proceedings of the National Academy of Sciences of the United States of America, 110(48): 19307–19312.
    Zentralblatt MATH: 1292.60076
    Digital Object Identifier: doi:10.1073/pnas.1311790110

Supplemental materials

  • Supplementary Material of “Designing Simple and Efficient Markov Chain Monte Carlo Proposal Kernels”. (I) Efficiency curves for Box, Airplane and StrawHat kernels for a range of a values. (II) Two-dimensional Gaussian target example. (III) MCMC algorithms for the phylogenetic problem. (IV) Effect of μ∗ on efficiency.
    Digital Object Identifier: doi:10.1214/17-BA1084SUPP
    Supplemental PDF (483 KB)

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