Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at

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.

. I describe a simple procedure for investigating the convergence properties of Markov Chain Monte Carlo sampling schemes. The procedure employs multiple runs from a sampler, using the same random deviates for each run. When the sample paths from all sequences converge, it is argued that approximate equilibrium conditions hold. The procedure also provides a simple diagnostic for detecting modes in multimodal posteriors. Several examples of the procedure are provided. In Ising models, the relation between the correlation parameter and the convergence rate of rudimentary Gibbs samplers is investigated. In another example, the effects of multiple modes on the convergence of coupled paths are explored using mixtures of bivariate normal distributions. The technique is also used to evaluate the convergence properties of a Gibbs sampling scheme applied to a model for rat growth rates (Gelfand et al 1990). Acknowledgements I would like to thank Steve MacEachern, Julian Besag, Donald Rubin, A...

this paper are motivated and aimed at analyzing some common types of survival data from different medical studies. We will center our attention to the following topics.

Introduction In order to use Markov chain Monte Carlo, MCMC, it is necessary to determine how long the simulation needs to be run. It is also a good idea to discard a number of initial "burnin " simulations, since from an arbitrary starting point it would be unlikely that the initial simulations came from the stationary distribution intended for the Markov chain. Also, consecutive simulations from Markov chains are dependent, sometimes highly so. Since saving all simulations can require a large amount of storage, researchers using MCMC sometimes prefer saving only every third, fifth, tenth, etc. simulation, especially if the chain is highly dependent. This is sometimes referred to as thinning the chain. While neither burn-in nor thinning are mandatory practices, they both reduce the amount of data saved from a MCMC run. In this chapter, we outline a way of determining in advance the number of iterations needed for a given level of precision in a MCMC algorithm.

Blind linear system identication consists in estimating the parameters of a linear timeinvariant

In Bayesian inference, a Bayes factor is defined as the ratio of posterior odds versus prior odds where posterior odds is simply a ratio of the normalizing constants of two posterior densities. In many practical problems, the two posteriors have different dimensions. For such cases, the current Monte Carlo methods such as the bridge sampling method (Meng and Wong 1996), the path sampling method (Gelman and Meng 1994), and the ratio importance sampling method (Chen and Shao 1994) cannot directly be applied. In this article, we extend importance sampling, bridge sampling, and ratio importance sampling to problems of different dimensions. Then we find global optimal importance sampling, bridge sampling, and ratio importance sampling in the sense of minimizing asymptotic relative mean-square errors of estimators. Implementation algorithms, which can asymptotically achieve the optimal simulation errors, are developed and two illustrative examples are also provided.

This paper deals with the analysis of multivariate survival data from a Bayesian perspective using Markov Chain Monte Carlo methods. Metropolis along with Gibbs algorithm (Metropolis et al., 1953; Muller, 1991) is used to calculate some of the marginal posteriors. Multivariate survival model is proposed since, survival times within the same `group' are correlated as a consequence of a frailty random block effect (Vaupel et al., 1979). The conditional proportional hazards model of Clayton and Cuzick (1985) is used with a martingale structured prior process (Arjas and Gasbarra, 1994) for the discretized baseline hazard. Besides the calculation of the marginal posteriors of the parameters of interest, this paper presents some Bayesian EDA diagnostic techniques to detect model adequacy. The methodology is exemplified with the kidney infection data where the times to infections within the same patients are expected to be correlated. Key Words: Autocorrelated prior process, credible regions,...

The problem of marginal density estimation for a multivariate density function f(x) can be generally stated as a problem of density function estimation for a random vector λ(x) of dimension lower than that of x. In this paper, we propose a technique, the so-called continuous Contour Monte Carlo (CCMC) algorithm, for solving this problem. CCMC can be viewed as a continuous version of the contour Monte Carlo (CMC) algorithm recently proposed in the literature. CCMC abandons the use of sample space partitioning and incorporates the techniques of kernel density estimation into its simulations. CCMC is more general than other marginal density estimation algorithms. First, it works for any density functions, even for those having a rugged or unbalanced energy landscape. Second, it works for any transformation λ(x) regardless of the availability of the analytical form of the inverse transformation. In this paper, CCMC is applied to estimate the unknown normalizing constant function for a spatial autologistic model, and the estimate is then used in a Bayesian analysis for the spatial autologistic model in place of the true normalizing constant function. Numerical results on the US cancer mortality data indicate that the Bayesian method can produce much more accurate estimates than the MPLE and MCMLE methods for the parameters of the spatial autologistic model.

(Statistics) BAYESIAN NONPARAMETRIC MIXTURE MODELING by Guoliang Cao Institute of Statistics and Decision Sciences Duke University Date: Approved: Mike West, Advisor Donald Burdick Michael Lavine Peter Muller Dennis A. Turner An abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Institute of Statistics and Decision Sciences in the Graduate School of Duke University Abstract This dissertation explores a Bayesian nonparametric approach to mixture modeling and the use of the Gibbs sampling scheme to approximate posterior estimates. The predictive distribution is modeled as a mixture of normal distributions by using a Dirichlet process prior for the unknown means and variances. The definition and some properties of mixtures of Dirichlet processes are reviewed. Analytically evaluating the predictive distribution is very tedious and difficult in this case. An approximation based on Monte Carlo integration is pro...