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43
Markov chains for exploring posterior distributions
 Annals of Statistics
, 1994
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Cited by 1049 (6 self)
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Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at
Regeneration in Markov Chain Samplers
, 1994
"... Markov chain sampling has received considerable attention in the recent literature, in particular in the context of Bayesian computation and maximum likelihood estimation. This paper discusses the use of Markov chain splitting, originally developed as a tool for the theoretical analysis of general s ..."
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Cited by 113 (5 self)
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Markov chain sampling has received considerable attention in the recent literature, in particular in the context of Bayesian computation and maximum likelihood estimation. This paper discusses the use of Markov chain splitting, originally developed as a tool for the theoretical analysis of general state space Markov chains, to introduce regeneration times into Markov chain samplers. This allows the use of regenerative methods for analyzing the output of these samplers, and can also provide a useful diagnostic of the performance of the samplers. The general approach is applied to several different samplers and is illustrated in a number of examples. 1 Introduction In Markov chain Monte Carlo, a distribution ß is examined by obtaining sample paths from a Markov chain constructed to have equilibrium distribution ß. This approach was introduced by Metropolis et al. (1953) and has recently received considerable attention as a method for examining posterior distributions in Bayesian infer...
Exact sampling from a continuous state space, Scandinavian
 Journal of Statistics
, 1998
"... ABSTRACT. Propp & Wilson (1996) described a protocol, called coupling from the past, for exact sampling from a target distribution using a coupled Markov chain Monte Carlo algorithm. In this paper we extend coupling from the past to various MCMC samplers on a continuous state space; rather than ..."
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Cited by 99 (7 self)
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ABSTRACT. Propp & Wilson (1996) described a protocol, called coupling from the past, for exact sampling from a target distribution using a coupled Markov chain Monte Carlo algorithm. In this paper we extend coupling from the past to various MCMC samplers on a continuous state space; rather than following the monotone sampling device of Propp & Wilson, our approach uses methods related to gammacoupling and rejection sampling to simulate the chain, and direct accounting of sample paths.
Two Estimators of the Mean of a Counting Process with Panel Count Data
, 1998
"... We study two estimators of the mean function of a counting process based on "panel count data". The setting for "panel count data" is one in which n independent subjects, each with a counting process with common mean function, are observed at several possibly di erent times durin ..."
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Cited by 25 (12 self)
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We study two estimators of the mean function of a counting process based on "panel count data". The setting for "panel count data" is one in which n independent subjects, each with a counting process with common mean function, are observed at several possibly di erent times during a study. Following a model proposed by Schick and Yu (1997), we allow the number of observation times, and the observation times themselves, to be random variables. Our goal is to estimate the mean function of the counting process. We show that the estimator of the mean function proposed by Sun and Kalbfleisch (1995) can be viewed as a pseudomaximum likelihood estimator when a nonhomogeneous Poisson process model is assumed for the counting process. We establish consistency of both the nonparametric pseudo maximum likelihood estimator of Sun and Kalbfleisch (1995) and the full maximum likelihood estimator, even if the underlying counting process is not a Poisson process. We also derive the asymptotic distribution of both estimators at a xed time t, and compare the resulting theoretical relative e ciency with nite sample relative efficiency by way of a limited montecarlo study.
An Iterative Monte Carlo Method for Nonconjugate Bayesian Analysis
 Statistics and Computing
, 1991
"... The Gibbs sampler has been proposed as a general method for Bayesian calculation in Gelfand and Smith (1990). However, the predominance of experience to date resides in applications assuming conjugacy where implementation is reasonably straightforward. This paper describes a tailored approximate rej ..."
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Cited by 20 (0 self)
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The Gibbs sampler has been proposed as a general method for Bayesian calculation in Gelfand and Smith (1990). However, the predominance of experience to date resides in applications assuming conjugacy where implementation is reasonably straightforward. This paper describes a tailored approximate rejection method approach for implementation of the Gibbs sampler when nonconjugate structure is present. Several challenging applications are presented for illustration.
Two likelihoodbased semiparametric estimation methods for panel count data with covariates
, 2005
"... We consider estimation in a particular semiparametric regression model for the mean of a counting process with “panel count ” data. The basic model assumption is that the conditional mean function of the counting process is of the form E{N(t)Z} = exp(β T 0 Z)Λ0(t) where Z is a vector of covariates ..."
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Cited by 20 (7 self)
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We consider estimation in a particular semiparametric regression model for the mean of a counting process with “panel count ” data. The basic model assumption is that the conditional mean function of the counting process is of the form E{N(t)Z} = exp(β T 0 Z)Λ0(t) where Z is a vector of covariates and Λ0 is the baseline mean function. The “panel count ” observation scheme involves observation of the counting process N for an individual at a random number K of random time points; both the number and the locations of these time points may differ across individuals. We study semiparametric maximum pseudolikelihood and maximum likelihood estimators of the unknown parameters (β0,Λ0) derived on the basis of a nonhomogeneous Poisson process assumption. The pseudolikelihood estimator is fairly easy to compute, while the maximum likelihood estimator poses more challenges from the computational perspective. We study asymptotic properties of both estimators assuming that the proportional mean model holds, but dropping the Poisson process assumption used to derive the estimators. In particular we establish asymptotic normality for the estimators of the regression parameter β0 under appropriate hypotheses. The results show that our estimation procedures are robust in the sense that the estimators converge to the truth regardless of the underlying counting process.
Bayesian and Frequentist Approaches to Parametric Predictive Inference
 BAYESIAN STATISTICS, J. M. BERNARDO , J. O. BERGER , A. P. DAWID , A. F. M. SMITH (EDS.)
, 1998
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Consistency of Markov chain quasiMonte Carlo on continuous state
, 2009
"... The random numbers driving Markov chain Monte Carlo (MCMC) simulation are usually modeled as independent U(0, 1) random variables. Tribble [31] reports substantial improvements when those random numbers are replaced by carefully balanced inputs from completely uniformly distributed sequences. The pr ..."
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Cited by 15 (4 self)
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The random numbers driving Markov chain Monte Carlo (MCMC) simulation are usually modeled as independent U(0, 1) random variables. Tribble [31] reports substantial improvements when those random numbers are replaced by carefully balanced inputs from completely uniformly distributed sequences. The previous theoretical justification for using anything other than IID U(0, 1) points shows consistency for estimated means, but only applies for discrete stationary distributions. We extend those results to some MCMC algorithms for continuous stationary distributions. The main motivation is the search for quasiMonte Carlo versions of MCMC. As a side benefit, the results also establish consistency for the usual method of using pseudorandom numbers in place of random ones. 1
General Strategies for Assessing Convergence of MCMC Algorithms Using Coupled Sample Paths
, 1995
"... this paper, we propose an extension of the Gibbs coupling algorithm that eliminates each of these difficulties. The proposed method for studying the convergence properties of general MCMC algorithms may be summarized as follows. For a specified MCMC algorithm, c chains are started from initial value ..."
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Cited by 5 (1 self)
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this paper, we propose an extension of the Gibbs coupling algorithm that eliminates each of these difficulties. The proposed method for studying the convergence properties of general MCMC algorithms may be summarized as follows. For a specified MCMC algorithm, c chains are started from initial values drawn at random from an overdispersed estimate of the stationary distribution. Each of the chains is updated according to its MCMC conditional distributions, except that updates are made jointly in a way that allows each pair of the c chains to couple at a random time. The iteration at which the c chains couple is recorded, and the process is repeated m times. Based on the m coupling times, the quantiles of the distribution of the coupling iteration are estimated and used to obtain a bound on the total variation distance of the MCMC iterates from the stationary distribution of the chain. Joint updates of the c chains are made either by a generalization of maximal coupling to multiple chains, or an approximation to this c chain Markovmaximal coupling obtained through simple mixture sampling. Like the coupling procedure proposed by Johnson (1996), the algorithm described above has strong connections to more theoretical studies of coupling in Markov chains and MCMC algorithms. Particularly relevant works in this direction include Doeblin (1933), Griffeath (1975), Pitman (1976), Goldstein (1979), Lindvall (1992), Meyn and Tweedie (1993), and Rosenthal (1995). Related work in the more general area of MCMC convergence diagnostics are reviewed in Besag et al 1995, and include Frigessi et al (1992), Gelman and Rubin (1992), Geyer (1992), Roberts (1992), Besag and Green (1993), Lund and Tweedie (1993), Smith and Roberts (1993), Garren and Smith (1994), Gelman et al (1994), and Robe...