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A mixture representation of \(\pi\) with applications in Markov chain Monte Carlo and perfect sampling. (English) Zbl 1046.60062
Summary: Let \(X=\{X_n:n=0,1,2,\dots\}\) be an irreducible, positive recurrent Markov chain with invariant probability measure \(\pi\). We show that if \(X\) satisfies a one-step minorization condition, then \(\pi\) 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 K. B. Athreya and P. Ney [Trans. Am. Math. Soc. 245, 493–501 (1978; Zbl 0397.60053)] and E. Nummelin [Z. Wahrscheinlichkeitstheorie Verw. Geb. 43, 309–318 (1978; Zbl 0364.60104)]. When the small set in the minorization condition is the entire state space, our mixture representation of \(\pi\) reduces to a simple formula, first derived by L. A. Breyer and G. O. Roberts [Methodol. Comput. Appl. Probab. 3, 161–177 (2001; Zbl 0999.60069)] 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 D. J. Murdoch and P. J. Green’s multigamma coupler [Scand. J. Stat. 25, 483–502 (1998; Zbl 0921.62020)] and D. B. Wilson’s read-once CFTP algorithm [Random Struct. Algorithms 16, 85–113 (2000; Zbl 0952.60072)]. 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 \(\pi\) 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 \(X\).

MSC:
60J05 Discrete-time Markov processes on general state spaces
62E15 Exact distribution theory in statistics
65C05 Monte Carlo methods
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