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General state space Markov chains and MCMC algorithm
 PROBABILITY SURVEYS
, 2004
"... This paper surveys various results about Markov chains on general (noncountable) state spaces. It begins with an introduction to Markov chain Monte Carlo (MCMC) algorithms, which provide the motivation and context for the theory which follows. Then, sufficient conditions for geometric and uniform e ..."
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Cited by 190 (38 self)
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This paper surveys various results about Markov chains on general (noncountable) state spaces. It begins with an introduction to Markov chain Monte Carlo (MCMC) algorithms, which provide the motivation and context for the theory which follows. Then, sufficient conditions for geometric and uniform ergodicity are presented, along with quantitative bounds on the rate of convergence to stationarity. Many of these results are proved using direct coupling constructions based on minorisation and drift conditions. Necessary and sufficient conditions for Central Limit Theorems (CLTs) are also presented, in some cases proved via the Poisson Equation or direct regeneration constructions. Finally, optimal scaling and weak convergence results for MetropolisHastings algorithms are discussed. None of the results presented is new, though many of the proofs are. We also describe some Open Problems.
Layered multishift coupling for use in perfect sampling algorithms (with a primer to CFTP
 Fields Institute Communications Series, American Mathematical Society
, 2000
"... Abstract. In this article we describe a new coupling technique which is useful in a variety of perfect sampling algorithms. A multishift coupler generates a random function f() so that for each x ∈ R, f(x) − x is governed by the same fixed probability distribution, such as a normal distribution. We ..."
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Cited by 25 (1 self)
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Abstract. In this article we describe a new coupling technique which is useful in a variety of perfect sampling algorithms. A multishift coupler generates a random function f() so that for each x ∈ R, f(x) − x is governed by the same fixed probability distribution, such as a normal distribution. We develop the class of layered multishift couplers, which are simple and have several useful properties. For the standard normal distribution, for instance, the layered multishift coupler generates an f() which (surprisingly) maps an interval of length ℓ to fewer than 2+ℓ/2.35 points — useful in applications which perform computations on each such image point. The layered multishift coupler improves and simplifies algorithms for generating perfectly random samples from several distributions, including the autogamma distribution, posterior distributions for Bayesian inference, and the steady state distribution for certain storage systems. We also use the layered multishift coupler to develop a Markovchain based perfect sampling algorithm for the autonormal distribution. At the request of the organizers, we begin by giving a primer on CFTP (coupling from the past); CFTP and Fill’s algorithm are the two predominant techniques for generating perfectly random samples using coupled Markov chains.
Perfect sampling of the master equation for gene regulatory networks
 Biophysical journal
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Advances and Applications in Perfect Sampling by
"... The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline. Schneider, Ulrike (Ph.D., Applied Mathematics) Advances and Applications in Perfect Sampling ..."
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The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline. Schneider, Ulrike (Ph.D., Applied Mathematics) Advances and Applications in Perfect Sampling Thesis directed by Prof. Jem Corcoran Perfect sampling algorithms are Markov Chain Monte Carlo (MCMC) methods without statistical error. The latter are used when one needs to get samples from certain (nonstandard) distributions. This can be accomplished by creating a Markov chain that has the desired distribution as its stationary distribution, and by running sample paths ”for a long time”, i.e. until the chain is believed to be in equilibrium. The question ”how long is long enough? ” is generally hard to answer and the assessment of convergence is a major concern when applying MCMC schemes. This issue completely vanishes with the use of perfect sampling algorithms which – if applicable – enable exact simulation from the stationary distribution of a Markov chain.