Cahiers du CEREMADE

Unité Mixte de Recherche du C.N.R.S. N°7534
Abstract : et f be an irreducible, aperiodic, Harris recurrent Markov chain with invariant probability measure p. We show that if a minorization condition can be established for f, then p can be represented as an infinite mixture. Making exact draws from this mixture (and hence from p) requires the ability to simulate two different random variables, T* and Nt. When f is uniformly ergodic, simulating these random variables is straightforward and the resulting algorithm turns out to be equivalent to possessivecite Murdoch and Green's 1998 multigamma coupler. Thus, we are able to achieve perfect sampling in the uniformly ergodic case without any appeal to coupling or backward simulation. When f is not uniformly ergodic, simulating T* is more problematic. Fortunately, T* is univariate and has support N. We construct an estimate of the mass function of T* and an asymptotic bound on the error of this estimate. These results are used to address rigorously the issue of burn-in. Specifically, we form an approximation of p whose error can be controlled and which can be used as an initial distribution for f. We illustrate this with a realistic example. Finally, we provide a simple Markov chain Monte Carlo (MCMC) algorithm whose stationary distribution is that of T*. Finding a perfect sampler based on our MCMC algorithm for the univariate, discrete T* is tantamount to the ability to sample exactly from p in the non-uniformly ergodic case.
HOBERT J.P., ROBERT Christian P.
Université de PARIS - DAUPHINE
Place du Maréchal de Lattre De Tassigny - 75775 PARIS CEDEX 16 - FRANCE
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