The Annals of Applied Probability

A mixture representation of π with applications in Markov chain Monte Carlo and perfect sampling

James P. Hobert and Christian P. Robert

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Let X={Xn:n=0,1,2,} be an irreducible, positive recurrent Markov chain with invariant probability measure π. We show that if X satisfies a one-step minorization condition, then π can be represented as an infinite mixture. The distributions in the mixture are associated with the hitting times on an accessible atom introduced via the splitting construction of Athreya and Ney [Trans. Amer. Math. Soc. 245 (1978) 493–501] and Nummelin [Z. Wahrsch. Verw. Gebiete 43 (1978) 309–318]. When the small set in the minorization condition is the entire state space, our mixture representation of π reduces to a simple formula, first derived by Breyer and Roberts [Methodol. Comput. Appl. Probab. 3 (2001) 161–177] from which samples can be easily drawn. Despite the fact that the derivation of this formula involves no coupling or backward simulation arguments, the formula can be used to reconstruct perfect sampling algorithms based on coupling from the past (CFTP) such as Murdoch and Green’s [Scand. J. Statist. 25 (1998) 483–502] Multigamma Coupler and Wilson’s [Random Structures Algorithms 16 (2000) 85–113] Read-Once CFTP algorithm. In the general case where the state space is not necessarily 1-small, under the assumption that X satisfies a geometric drift condition, our mixture representation can be used to construct an arbitrarily accurate approximation to π from which it is straightforward to sample. One potential application of this approximation is as a starting distribution for a Markov chain Monte Carlo algorithm based on X.

Article information

Ann. Appl. Probab., Volume 14, Number 3 (2004), 1295-1305.

First available in Project Euclid: 13 July 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62C15: Admissibility
Secondary: 60J05: Discrete-time Markov processes on general state spaces

Burn-in drift condition geometric ergodicity Kac’s theorem minorization condition Multigamma Coupler Read-Once CFTP regeneration split chain


Hobert, James P.; Robert, Christian P. A mixture representation of π with applications in Markov chain Monte Carlo and perfect sampling. Ann. Appl. Probab. 14 (2004), no. 3, 1295--1305. doi:10.1214/105051604000000305.

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  • Athreya, K. B. and Ney, P. (1978). A new approach to the limit theory of recurrent Markov chains. Trans. Amer. Math. Soc. 245 493–501.
  • Breyer, L. A. and Roberts, G. O. (2001). Catalytic perfect simulation. Methodol. Comput. Appl. Probab. 3 161–177.
  • Douc, R., Moulines, E. and Rosenthal, J. S. (2004). Quantitative bounds on convergence of time-inhomogeneous Markov chains. Ann. Appl. Probab. 14(4).
  • Fill, J. A. (1998). An interruptible algorithm for perfect sampling via Markov chains. Ann. Appl. Probab. 8 131–162.
  • Hobert, J. P., Jones, G. L., Presnell, B. and Rosenthal, J. S. (2002). On the applicability of regenerative simulation in Markov chain Monte Carlo. Biometrika 89 731–743.
  • Jones, G. L. and Hobert, J. P. (2001). Honest exploration of intractable probability distributions via Markov chain Monte Carlo. Statist. Sci. 16 312–334.
  • Jones, G. L. and Hobert, J. P. (2004). Sufficient burn-in for Gibbs samplers for a hierarchical random effects model. Ann. Statist. 32 784–817.
  • Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.
  • Meyn, S. P. and Tweedie, R. L. (1994). Computable bounds for geometric convergence rates of Markov chains. Ann. Appl. Probab. 4 981–1011.
  • Murdoch, D. J. and Green, P. J. (1998). Exact sampling from a continuous state space. Scand. J. Statist. 25 483–502.
  • Nummelin, E. (1978). A splitting technique for Harris recurrent Markov chains. Z. Wahrsch. Verw. Gebiete 43 309–318.
  • Nummelin, E. (1984). General Irreducible Markov Chains and Non-Negative Operators. Cambridge Univ. Press.
  • Propp, J. G. and Wilson, D. B. (1996). Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures Algorithms 9 223–252.
  • Roberts, G. O. and Tweedie, R. L. (1999). Bounds on regeneration times and convergence rates for Markov chains. Stochastic Process. Appl. 80 211–229. [Corrigendum (2001) 91 337–338.]
  • Rosenthal, J. S. (1995). Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Statist. Assoc. 90 558–566.
  • Wilson, D. B. (2000). How to couple from the past using a read-once source of randomness. Random Structures Algorithms 16 85–113.