Bayesian Analysis

Ergodic averages for monotone functions using upper and lower dominating processes

Kerrie Mengersen and Jesper M{\o}ller

Full-text: Open access

Abstract

We show how the mean of a monotone function (defined on a state space equipped with a partial ordering) can be estimated, using ergodic averages calculated from upper and lower dominating processes of a stationary irreducible Markov chain. In particular, we do not need to simulate the stationary Markov chain and we eliminate the problem of whether an appropriate burn-in is determined or not. Moreover, when a central limit theorem applies, we show how confidence intervals for the mean can be estimated by bounding the asymptotic variance of the ergodic average based on the equilibrium chain. Our methods are studied in detail for three models using Markov chain Monte Carlo methods and we also discuss various types of other models for which our methods apply.

Article information

Source
Bayesian Anal. Volume 2, Number 4 (2007), 761-781.

Dates
First available in Project Euclid: 22 June 2012

Permanent link to this document
https://projecteuclid.org/euclid.ba/1340370714

Digital Object Identifier
doi:10.1214/07-BA231

Mathematical Reviews number (MathSciNet)
MR2361974

Zentralblatt MATH identifier
1331.62157

Keywords
Asymptotic variance Bayesian models Burn-in Ergodic average Ising model Markov chain Monte Carlo Mixture model Monotonocity Perfect simulation Random walk Spatial models Upper and lower dominating processes

Citation

M{\o}ller, Jesper; Mengersen, Kerrie. Ergodic averages for monotone functions using upper and lower dominating processes. Bayesian Anal. 2 (2007), no. 4, 761--781. doi:10.1214/07-BA231. https://projecteuclid.org/euclid.ba/1340370714


Export citation

References

  • Baddeley, A. J. and van Lieshout, M. N. M. (1995). "Area-interaction point processes." Annals of the Institute of Statistical Mathematics, 46: 601–619.
  • Besag, J. (1974). "Spatial interaction and the statistical analysis of lattice systems (with discussion)." Journal of the Royal Statistical Society Series B, 36: 192–326.
  • Brooks, S., Fan, Y., and Rosenthal, J. (2002). "Perfect forward simulation via simulation tempering." Technical report, Department of Statistics, University of Cambridge.
  • Casella, G., K., M., Robert, C., and Titterington, D. (2002). "Perfect slice samplers for mixtures of distributions." Journal of the Royal Statistical Soceity Series B, 64(4): 777–790.
  • Chan, K. and Geyer, C. (1994). "Discussion of “Markov chains for exploring posterior distributions”." Annals of Statistics, 22: 1747–1758.
  • Foss, S. and Tweedie, R. (1998). "Perfect simulation and backward coupling." Stochastic Models, 14: 187–203.
  • Geman, S. and Geman, D. (1984). "Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images." IEEE Transactions on Pattern Analysis and Machine Intelligence, 6: 721–741.
  • Geyer, C. (1992). "Practical Monte Carlo Markov chain (with discussion)." Statistical Science, 7: 473–511.
  • –- (1996). "Estimation and optimization of functions." In Gilks, W., Richardson, S., and Spiegelhalter, D. (eds.), Markov Chain Monte Carlo in Practice, 241–258. London: Chapman and Hall.
  • Häggström, O. and Nelander, K. (1998). "Exact sampling from anti-monotone systems." Statistica Neerlandica, 52: 360–380.
  • Häggström, O., van Lieshout, M. N. M., and Møller, J. (1999). "Characterization results and Markov chain Monte Carlo algorithms including exact simulation for some spatial point processes." Bernoulli, 5: 641–659.
  • Kendall, W. (1998). "Perfect simulation for the area-interaction point process." In Heyde, C. and Accardi, L. (eds.), Proability Towards 2000, 218–234. New York: Springer-Verlag.
  • Kendall, W. and Møller, J. (2000). "Perfect simulation using dominating processes on ordered spaces, with application to locally stable point processes." Advances in Applied Probability, 32: 844–85.
  • Meyn, S. and Tweedie, R. (1993). Markov Chains and Stochastic Stability. Springer-Verlag.
  • Mira, A., Møller, J., and Roberts, G. (2001). "Perfect slice samplers." Journal of the Royal Statistical Society Series B, 63: 583–606.
  • Møller, J. (1999). "Perfect simulation of conditionally specified models." Journal of the Royal Statistical Society Series B, 61: 251–264.
  • Møller, J. and Nicholls, G. (1999). "Perfect simulation for sample-based inference." Technical Report R-99-2011, Department of Mathematical Sciences, Aalborg University.
  • Møller, J. and Waagepetersen, R. (2003). Statistical Inference and Simulation for Spatial Point Processes. Boca Raton: Chapman and Hall/CRC.
  • Priestly, M. (1981). Spectral Analysis and Time Series. London: Academic Press.
  • Propp, J. and Wilson, D. (1996). "Exact sampling with coupled Markov chains and applications to statistical mechanics." Random Structures and Algorithms, 9: 223–252.
  • Ripley, B. (1987). Stochastic Simulation. New York: John Wiley.
  • Robert, C. and Casella, G. (2004). Monte Carlo Statistical Methods. New York: Springer, 2 edition.
  • Roberts, G. and Rosenthal, J. (1998). "Markov chain Monte Carlo: Some practical implications of theoretical results (with discussion)." Canadian Journal of Statistics, 26: 5–32.
  • Swendson, R. and Wang, J. (1987). "Nonuniversal critical dynamics in Monte Carlo simulations." Physical Review Letters, 58: 86–88.
  • Widom, B. and Rowlinson, J. S. (1970). "A new model for the study of liquid-vapor phase transitions." Journal of Chemical Physics, 52: 1670–1684.
  • Wilson, D. (2000). "How to couple from the past using a read-once source of randomness." Random Structures and Algorithms, 16: 85–113.