Brazilian Journal of Probability and Statistics

On geometric ergodicity of additive and multiplicative transformation-based Markov Chain Monte Carlo in high dimensions

Kushal Kr. Dey and Sourabh Bhattacharya

Full-text: Open access

Abstract

Recently Dutta and Bhattacharya (Statistical Methodology 16 (2014) 100–116) introduced a novel Markov Chain Monte Carlo methodology that can simultaneously update all the components of high-dimensional parameters using simple deterministic transformations of a one-dimensional random variable drawn from any arbitrary distribution defined on a relevant support. The methodology, which the authors refer to as transformation-based Markov Chain Monte Carlo (TMCMC), greatly enhances computational speed and acceptance rate in high-dimensional problems. Two significant transformations associated with TMCMC are additive and multiplicative transformations. Combinations of additive and multiplicative transformations are also of much interest. In this work, we investigate geometric ergodicity associated with additive and multiplicative TMCMC, along with their combinations, assuming that the target distribution is multi-dimensional and belongs to the super-exponential family; we also illustrate their efficiency in practice with simulation studies.

Article information

Source
Braz. J. Probab. Stat., Volume 30, Number 4 (2016), 570-613.

Dates
Received: July 2014
Accepted: July 2015
First available in Project Euclid: 13 December 2016

Permanent link to this document
https://projecteuclid.org/euclid.bjps/1481619618

Digital Object Identifier
doi:10.1214/15-BJPS295

Mathematical Reviews number (MathSciNet)
MR3582391

Zentralblatt MATH identifier
1359.60095

Keywords
Acceptance rate geometric ergodicity high dimension mixture proposal distribution transformation-based Markov Chain Monte Carlo

Citation

Dey, Kushal Kr.; Bhattacharya, Sourabh. On geometric ergodicity of additive and multiplicative transformation-based Markov Chain Monte Carlo in high dimensions. Braz. J. Probab. Stat. 30 (2016), no. 4, 570--613. doi:10.1214/15-BJPS295. https://projecteuclid.org/euclid.bjps/1481619618


Export citation

References

  • Dey, K. K. and Bhattacharya, S. (2016). A brief tutorial on Transformation based Markov Chain Monte Carlo and optimal scaling of the additive transformation. Brazilian Journal of Probability and Statistics. To appear. Available at arXiv:1307.1446.
  • Dutta, S. (2012). Multiplicative random walk Metropolis–Hastings on the real line. Sankhya B 74, 315–342.
  • Dutta, S. and Bhattacharya, S. (2014). Markov Chain Monte Carlo based on deterministic transformations. Statistical Methodology 16, 100–116. Also available at arXiv:1106.5850. Supplement available at arXiv:1306.6684.
  • Jarner, S. F. and Hansen, E. (2000). Geometric ergodicity of Metropolis algorithms. Stochastic Processes and their Applications 85, 341–361.
  • Jarner, S. F. and Roberts, G. O. (2002). Polynomial convergence rates of Markov chains. Annals of Applied Probability 12, 224–247.
  • Jarner, S. F. and Roberts, G. O. (2007). Convergence of heavy-tailed Monte Carlo Markov chain algorithms. Scandinavian Journal of Statistics 34, 781–815.
  • Johnson, L. T. and Geyer, C. J. (2012). Variable transformation to obtain geometric ergodicity in the random-walk Metropolis algorithm. The Annals of Statistics 40, 3050–3076.
  • Jones, G. L. and Hobert, J. P. (2001). Honest exploration of intractable probability distributions via Markov chain Monte Carlo. Statistical Science 16, 312–334.
  • Mengersen, K. L. and Tweedie, R. L. (1996). Rates of convergence of the Hastings and Metropolis algorithms. The Annals of Statistics 24, 101–121.
  • Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. London: Springer.
  • Roberts, G. O. and Tweedie, R. L. (1996). Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms. Biometrika 83, 95–110.
  • 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, 110–120.