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We present several Markov chain Monte Carlo simulation methods that have been widely used in recent years in econometrics and statistics. Among these is the Gibbs sampler, which has been of particular interest to econometricians. Although the paper summarizes some of the relevant theoretical literature, its emphasis is on the presentation and explanation of applications to important models that are studied in econometrics. We include a discussion of some implementation issues, the use of the methods in connection with the EM algorithm, and how the methods can be helpful in model specification questions. Many of the applications of these methods are of particular interest to Bayesians, but we also point out ways in which frequentist statisticians may find the techniques useful.

Monte Carlo maximum likelihood for normalized families of distributions (Geyer and Thompson, 1992) can be used for an extremely broad class of models. Given any family f h ` : ` 2 \Theta g of nonnegative integrable functions, maximum likelihood estimates in the family obtained by normalizing the the functions to integrate to one can be approximated by Monte Carlo, the only regularity conditions being a compactification of the parameter space such that the the evaluation maps ` 7! h ` (x) remain continuous. Then with probability one the Monte Carlo approximant to the log likelihood hypoconverges to the exact log likelihood, its maximizer converges to the exact maximum likelihood estimate, approximations to profile likelihoods hypoconverge to the exact profile, and level sets of the approximate likelihood (support regions) converge to the exact sets (in Painlev'e-Kuratowski set convergence). The same results hold when there are missing data (Thompson and Guo, 1991, Gelfand and Carlin, 19...

Markov chain Monte Carlo (the Metropolis-Hastings algorithm and the Gibbs sampler) is a general multivariate simulation method that permits sampling from any stochastic process whose density is known up to a constant of proportionality. It has recently received much attention as a method of carrying out Bayesian, likelihood, and frequentist inference in analytically intractable problems. Although many applications of Markov chain Monte Carlo do not need estimation of normalizing constants, three do: calculation of Bayes factors, calculation of likelihoods in the presence of missing data, and importance sampling from mixtures. Here reverse logistic regression is proposed as a solution to the problem of estimating normalizing constants, and convergence and asymptotic normality of the estimates are proved under very weak regularity conditions. Markov chain Monte Carlo is most useful when combined with importance reweighting so that a Monte Carlo sample from one distribution can be used fo...

We consider fixed scan Gibbs and block Gibbs samplers for a Bayesian hierarchical random effects model with proper conjugate priors. A drift condition given in Meyn and Tweedie (1993, Chapter 15) is used to show that these Markov chains are geometrically ergodic. Showing that a Gibbs sampler is geometrically ergodic is the first step towards establishing central limit theorems, which can be used to approximate the error associated with Monte Carlo estimates of posterior quantities of interest. Thus, our results will be of practical interest to researchers using these Gibbs samplers for Bayesian data analysis. Key words and phrases: Bayesian model, Central limit theorem, Drift condition, Markov chain, Monte Carlo, Rate of convergence, Variance Components AMS 1991 subject classifications: Primary 60J27, secondary 62F15 1 Introduction Gelfand and Smith (1990, Section 3.4) introduced the Gibbs sampler for the hierarchical one-way random effects model with proper conjugate priors. Rosen...

The Markov chain simulation method has been successfully used in many problems, including some that arise in Bayesian statistics. We give a self-contained proof of the convergence of this method in general state spaces under conditions that are easy to verify. Key words and phrases: Calculation of posterior distributions, ergodic theorem, successive substitution sampling.

Hierarchical models (HMs) provide a flexible framework for modeling data. The ongoing development of techniques like the EM algorithm and Markov chain Monte Carlo has enabled statisticians to make use of increasingly more complicated HMs over the last few decades. In this article, we consider Bayesian and frequentist versions of a general, two-stage HM, and describe several examples from the literature that illustrate its versatility. Some key aspects of the computational techniques that are currently used in conjunction with this HM are then examined in the context of McCullagh and Nelder's (1989) salamander data. Several areas that are ripe for new research are identified.

In a Bayesian analysis one fixes a prior on the unknown parameter, observes the data, and obtains the posterior distribution of the parameter given the data. For a number of problems the posterior cannot be obtained in closed form and one uses instead the Markov chain simulation method, which in effect produces a sequence of random variables distributed approximately from the posterior distribution. These random variables can be used to estimate the posterior or features of it like the posterior expectation and variance. Unfortunately, the Markov chain simulation method requires non-negligible computer time and this precludes consideration of a large number of priors and an interactive analysis. We present a computing environment within which one can interactively change the prior and immediately see the corresponding changes in the posterior. The environment is based on the objectoriented programming language LISP-STAT and an importance sampling procedure which enables one...

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Abstract—The performance of the IEEE 802.11 protocol based on the distributed coordination function (DCF) has been shown to be dependent on the number of competing terminals and the backoff parameters. Better performance can be expected if the parameters are adapted to the number of active users. In this paper we develop both off-line and online Bayesian signal processing algorithms to estimate the number of competing terminals. The estimation is based on the observed use of the channel and the number of competing terminals is modeled as a Markov chain with unknown transition matrix. The off-line estimator makes use of the Gibbs sampler whereas the first online estimator is based on the sequential Monte Carlo (SMC) technique. A deterministic variant of the SMC estimator is then developed, which is simpler to implement and offers superior performance. Finally a novel approximate maximum a posteriori (MAP) algorithm for hidden Markov models (HMM) with unknown transition matrix is proposed. Realistic IEEE 802.11 simulations using the ns-2 network simulator are provided to demonstrate the excellent performance of the proposed estimators. Index Terms—Gibbs sampler, hidden Markov model (HMM), IEEE 802.11 wireless networks, sequential Monte Carlo, unknown transition matrix. I.