Electronic Journal of Statistics

Estimating the spectral gap of a trace-class Markov operator

Qian Qin, James P. Hobert, and Kshitij Khare

Full-text: Open access

Abstract

The utility of a Markov chain Monte Carlo algorithm is, in large part, determined by the size of the spectral gap of the corresponding Markov operator. However, calculating (and even approximating) the spectral gaps of practical Monte Carlo Markov chains in statistics has proven to be an extremely difficult and often insurmountable task, especially when these chains move on continuous state spaces. In this paper, a method for accurate estimation of the spectral gap is developed for general state space Markov chains whose operators are non-negative and trace-class. The method is based on the fact that the second largest eigenvalue (and hence the spectral gap) of such operators can be bounded above and below by simple functions of the power sums of the eigenvalues. These power sums often have nice integral representations. A classical Monte Carlo method is proposed to estimate these integrals, and a simple sufficient condition for finite variance is provided. This leads to asymptotically valid confidence intervals for the second largest eigenvalue (and the spectral gap) of the Markov operator. In contrast with previously existing techniques, our method is not based on a near-stationary version of the Markov chain, which, paradoxically, cannot be obtained in a principled manner without bounds on the spectral gap. On the other hand, it can be quite expensive from a computational standpoint. The efficiency of the method is studied both theoretically and empirically.

Article information

Source
Electron. J. Statist., Volume 13, Number 1 (2019), 1790-1822.

Dates
Received: February 2018
First available in Project Euclid: 6 June 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1559786481

Digital Object Identifier
doi:10.1214/19-EJS1563

Mathematical Reviews number (MathSciNet)
MR3959873

Zentralblatt MATH identifier
07080062

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

Keywords
Data augmentation algorithm eigenvalues Hilbert-Schmidt operator Markov chain Monte Carlo

Rights
Creative Commons Attribution 4.0 International License.

Citation

Qin, Qian; Hobert, James P.; Khare, Kshitij. Estimating the spectral gap of a trace-class Markov operator. Electron. J. Statist. 13 (2019), no. 1, 1790--1822. doi:10.1214/19-EJS1563. https://projecteuclid.org/euclid.ejs/1559786481


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References

  • Ahues, M., Largillier, A. and Limaye, B. (2001)., Spectral Computations for Bounded Operators. CRC Press.
  • Albert, J. H. and Chib, S. (1993). Bayesian analysis of binary and polychotomous response data., Journal of the American statistical Association 88 669–679.
  • Billingsley, P. (1995)., Probability and Measure, 3 ed. John Wiley & Sons.
  • Brislawn, C. (1988). Kernels of trace class operators., Proceedings of the American Mathematical Society 104 1181–1190.
  • Chakraborty, S. and Khare, K. (2017). Convergence properties of Gibbs samplers for Bayesian probit regression with proper priors., Electronic Journal of Statistics 11 177–210.
  • Chakraborty, S. and Khare, K. (2019+). Consistent estimation of the spectrum of trace class data augmentation algorithms., Bernoulli, to appear.
  • Chan, K. S. and Geyer, C. J. (1994). Discussion: Markov chains for exploring posterior distributions., Annals of Statistics 22 1747–1758.
  • Choi, H. M. and Román, J. C. (2017). Analysis of Polya-Gamma Gibbs sampler for Bayesian logistic analysis of variance., Electronic Journal of Statistics 11 326–337.
  • Conway, J. B. (1990)., A Course in Functional Analysis, second ed. Springer-Verlag.
  • Conway, J. B. (2000)., A Course in Operator Theory. American Mathematical Soc.
  • Diaconis, P., Khare, K. and Saloff-Coste, L. (2008). Gibbs sampling, exponential families and orthogonal polynomials (with discussion)., Statistical Science 23 151–200.
  • Diaconis, P. and Stroock, D. (1991). Geometric bounds for eigenvalues of Markov chains., Annals of Applied Probability 1 36–61.
  • Erdös, J. (1974). On the trace of a trace class operator., Bulletin of the London Mathematical Society 6 47–50.
  • Fernandez, C. and Steel, M. F. (1999). Multivariate Student-t regression models: Pitfalls and inference., Biometrika 86 153–167.
  • Garren, S. T. and Smith, R. L. (2000). Estimating the second largest eigenvalue of a Markov transition matrix., Bernoulli 6 215–242.
  • Gohberg, I., Goldberg, S. and Krupnik, N. (2012)., Traces and Determinants of Linear Operators 116. Birkhäuser.
  • Helmberg, G. (2014)., Introduction to Spectral Theory in Hilbert Space. Elsevier.
  • Hobert, J. P. (2011). The data augmentation algorithm: Theory and methodology. In, Handbook of Markov Chain Monte Carlo (S. Brooks, A. Gelman, G. Jones and X. L. Meng, eds.) Chapman & Hall/CRC Press.
  • Hobert, J. P. and Marchev, D. (2008). A theoretical comparison of the data augmentation, marginal augmentation and PX-DA algorithms., Annals of Statistics 36 532–554.
  • Khare, K. and Hobert, J. P. (2011). A spectral analytic comparison of trace-class data augmentation algorithms and their sandwich variants., Annals of Statistics 39 2585–2606.
  • Koltchinskii, V. and Giné, E. (2000). Random matrix approximation of spectra of integral operators., Bernoulli 6 113–167.
  • Kontoyiannis, I. and Meyn, S. P. (2012). Geometric ergodicity and the spectral gap of non-reversible Markov chains., Probability Theory and Related Fields 154 327–339.
  • Lawler, G. F. and Sokal, A. D. (1988). Bounds on the $L^2$ spectrum for Markov chains and Markov processes: A generalization of Cheeger’s inequality., Transactions of the American Mathematical Society 309 557–580.
  • Liu, C. (1996). Bayesian robust multivariate linear regression with incomplete data., Journal of the American Statistical Association 91 1219–1227.
  • Liu, J. S., Wong, W. H. and Kong, A. (1994). Covariance Structure of the Gibbs Sampler with Applications to the Comparisons of Estimators and Augmentation Schemes., Biometrika 81 27–40.
  • Liu, J. S. and Wu, Y. N. (1999). Parameter expansion for data augmentation., Journal of the American Statistical Association 94 1264–1274.
  • Mira, A. and Geyer, C. J. (1999). Ordering Monte Carlo Markov chains., Technical Report 632, School of Statistics, University of Minnesota.
  • Pal, S., Khare, K. and Hobert, J. P. (2017). Trace class Markov chains for Bayesian inference with generalized double Pareto shrinkage priors., Scandinavian Journal of Statistics 44 307–323.
  • Qin, Q. and Hobert, J. P. (2018). Trace-class Monte Carlo Markov chains for Bayesian multivariate linear regression with non-Gaussian errors., Journal of Multivariate Analysis 166 335–345.
  • Roberts, G. O. and Rosenthal, J. S. (1997). Geometric ergodicity and hybrid Markov chains., Electronic Communications in Probability 2 13–25.
  • Roy, V. and Hobert, J. P. (2007). Convergence rates and asymptotic standard errors for Markov chain Monte Carlo algorithms for Bayesian probit regression., Journal of the Royal Statistical Society: Series B (Statistical Methodology) 69 607–623.
  • Sinclair, A. and Jerrum, M. (1989). Approximate counting, uniform generation and rapidly mixing Markov chains., Information and Computation 82 93–133.
  • Tanner, M. A. and Wong, W. H. (1987). The calculation of posterior distributions by data augmentation (with discussion)., Journal of the American statistical Association 82 528–540.
  • van Dyk, D. A. and Meng, X.-L. (2001). The art of data augmentation (with discussion)., Journal of Computational and Graphical Statistics 10 1–50.
  • Zhang, L., Khare, K. and Xing, Z. (2019). Trace class Markov chains for the Normal-Gamma Bayesian shrinkage model., Electronic Journal of Statistics 13 166–207.