Open Access
August 2004 A mixture representation of π with applications in Markov chain Monte Carlo and perfect sampling
James P. Hobert, Christian P. Robert
Ann. Appl. Probab. 14(3): 1295-1305 (August 2004). DOI: 10.1214/105051604000000305


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.


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James P. Hobert. Christian P. Robert. "A mixture representation of π with applications in Markov chain Monte Carlo and perfect sampling." Ann. Appl. Probab. 14 (3) 1295 - 1305, August 2004.


Published: August 2004
First available in Project Euclid: 13 July 2004

zbMATH: 1046.60062
MathSciNet: MR2071424
Digital Object Identifier: 10.1214/105051604000000305

Primary: 62C15
Secondary: 60J05

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

Rights: Copyright © 2004 Institute of Mathematical Statistics

Vol.14 • No. 3 • August 2004
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