Electronic Communications in Probability

Fast mixing of metropolis-hastings with unimodal targets

James Johndrow and Aaron Smith

Full-text: Open access

Abstract

A well-known folklore result in the MCMC community is that the Metropolis-Hastings algorithm mixes quickly for any unimodal target, as long as the tails are not too heavy. Although we’ve heard this fact stated many times in conversation, we are not aware of any quantitative statement of this result in the literature, and we are not aware of any quick derivation from well-known results. The present paper patches this small gap in the literature, providing a generic bound based on the popular “drift-and-minorization” framework of [19]. Our main contribution is to study two sublevel sets of the Lyapunov function and use path arguments in order to obtain a sharper bound than what can typically be obtained from multistep minorization arguments.

Article information

Source
Electron. Commun. Probab., Volume 23 (2018), paper no. 71, 9 pp.

Dates
Received: 28 July 2018
Accepted: 23 September 2018
First available in Project Euclid: 16 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1539655259

Digital Object Identifier
doi:10.1214/18-ECP170

Mathematical Reviews number (MathSciNet)
MR3866044

Zentralblatt MATH identifier
06964414

Subjects
Primary: 60J05: Discrete-time Markov processes on general state spaces

Keywords
Markov chain Monte Carlo mixing geometric ergodicity

Rights
Creative Commons Attribution 4.0 International License.

Citation

Johndrow, James; Smith, Aaron. Fast mixing of metropolis-hastings with unimodal targets. Electron. Commun. Probab. 23 (2018), paper no. 71, 9 pp. doi:10.1214/18-ECP170. https://projecteuclid.org/euclid.ecp/1539655259


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References

  • [1] Persi Diaconis, Kshitij Khare, and Laurent Saloff-Coste, Gibbs sampling, exponential families and orthogonal polynomials, Statistical Science 23 (2008), no. 2, 151–178.
  • [2] Persi Diaconis and Laurent Saloff-Coste, Comparison theorems for reversible Markov chains, The Annals of Applied Probability 3 (1993), no. 3, 696–730.
  • [3] Persi Diaconis and Laurent Saloff-Coste, Logarithmic Sobolev inequalities for finite Markov chains, The Annals of Applied Probability 6 (1996), no. 3, 695–750.
  • [4] Persi Diaconis and Laurent Saloff-Coste, Nash inequalities for finite Markov chains, Journal of Theoretical Probability 9 (1996), no. 2, 459–510.
  • [5] Martin Hairer and Jonathan C. Mattingly, Yet another look at Harris’ ergodic theorem for Markov chains, Seminar on Stochastic Analysis, Random Fields and Applications VI, Springer, 2011, pp. 109–117.
  • [6] W. Keith Hastings, Monte Carlo sampling methods using Markov chains and their applications, Biometrika 57 (1970), no. 1, 97–109.
  • [7] Søren F. Jarner and Richard L. Tweedie, Necessary conditions for geometric and polynomial ergodicity of random-walk-type, Bernoulli 9 (2003), no. 4, 559–578.
  • [8] Søren F. Jarner and Wai Kong Yuen, Conductance bounds on the $l^{2}$ convergence rate of Metropolis algorithms on unbounded state spaces, Advances in Applied Probability 36 (2004), no. 1, 243–266.
  • [9] Søren F. Jarner and Ernst Hansen, Geometric ergodicity of Metropolis algorithms, Stochastic processes and their applications 85 (2000), no. 2, 341–361.
  • [10] James E. Johndrow, Aaron Smith, Natesh Pillai, and David B. Dunson, MCMC for imbalanced categorical data, Journal of the American Statistical Association (in press) (2018).
  • [11] Galin L. Jones and James P. Hobert, Honest exploration of intractable probability distributions via Markov chain Monte Carlo, Statistical Science 16 (2001), no. 4, 312–334.
  • [12] Rafail Khasminskii, Stochastic stability of differential equations, 2 ed., Springer, 2011.
  • [13] Gregory F. Lawler and Alan D. Sokal, Bounds on the ${L}_{2}$ spectrum for Markov chains and Markov processes: a generalization of Cheeger’s inequality, Transactions of the American Mathematical Society 309 (1988), no. 2, 557–580.
  • [14] David A. Levin, Yuval Peres, and Elizabeth L. Wilmer, Markov chains and mixing times, American Mathematical Soc., 2009.
  • [15] Kerrie L. Mengersen and Richard L. Tweedie, Rates of convergence of the Hastings and Metropolis algorithms, The Annals of Statistics 24 (1996), no. 1, 101–121.
  • [16] Nicholas Metropolis, Arianna W. Rosenbluth, Marshall N. Rosenbluth, Augusta H. Teller, and Edward Teller, Equation of state calculations by fast computing machines, The Journal of Chemical Physics 21 (1953), no. 6, 1087–1092.
  • [17] Sean P. Meyn and Richard L. Tweedie, Markov chains and stochastic stability, Springer, 1993.
  • [18] Gareth O. Roberts and Jeffrey S. Rosenthal, Geometric ergodicity and hybrid Markov chains, Electronic Communications in Probability 2 (1997), no. 2, 13–25.
  • [19] Jeffrey S. Rosenthal, Minorization conditions and convergence rates for Markov chain Monte Carlo, Journal of the American Statistical Association 90 (1995), no. 430, 558–566.
  • [20] Alistair Sinclair and Mark Jerrum, Approximate counting, uniform generation and rapidly mixing Markov chains, Information and Computation 82 (1989), no. 1, 93–133.
  • [21] Wai Kong Yuen, Applications of geometric bounds to the convergence rate of Markov chains on $\mathbb{R} ^{n}$, Stochastic Processes and their Applications 87 (2000), no. 1, 1–23.