Functional Compatibility, Markov Chains, and Gibbs Sampling with Improper Posteriors

References

• Arnold , B. C. and Press , S. 1989 . “Compatible Conditional Distributions” . Journal of the American Statistical Association , 84 : 152 – 156 .
   Web of Science ®Google Scholar
• Bartle , R. G. 1976 . The Elements of Real Analysis, , 2nd ed. , New York : Wiley .
   Google Scholar
• Berger , J. O. 1985 . Statistical Decision Theory and Bayesian Analysis , New York : Springer-Verlag .
   Google Scholar
• Besag , J. 1974 . “Spatial Interaction and the Statistical Analysis of Lattice Systems” . Journal of the Royal Statistical Society , 36 : 192 – 236 . (with discussion), Series B
   Google Scholar
• Billingsley , P. 1986 . Probability and Measure, , 2nd ed. , New York : Wiley .
   Google Scholar
• Casella , G. and George , E. I. 1992 . “Explaining the Gibbs Sampler” . The American Statistician , 46 : 167 – 174 .
   Web of Science ®Google Scholar
• Chaloner , K. 1987 . “A Bayesian Approach to the Estimation of Variance Components for the Unbalanced One-way Random Model” . Technometrics , 29 : 323 – 337 .
   Web of Science ®Google Scholar
• Chan , K. S. 1993 . “Asymptotic Behavior of the Gibbs Sampler” . Journal of the American Statistical Association , 88 : 320 – 326 .
   Web of Science ®Google Scholar
• Chan , K. S. and Geyer , C. J. 1994 . Comments on “Markov Chains for Exploring Posterior Distributions” . The Annals of Statistics , 22 : 1747 – 1757 .
   Web of Science ®Google Scholar
• Cowles , M. K. and Carlin , B. P. 1996 . “Markov Chain Monte Carlo Convergence Diagnostics: A Comparative Review” . Journal of the American Statistical Association , 91 : 883 – 904 .
   Web of Science ®Google Scholar
• Gelfand , A. E. , Hills , S. E. , Racine-Poon , A. and Smith , A. F. M. 1990 . “Illustration of Bayesian Inference in Normal Data Models Using Gibbs Sampling” . Journal of the American Statistical Association , 85 : 972 – 985 .
   Web of Science ®Google Scholar
• Gelfand , A. E. and Sahu , S. K. 1996 . “Identifiability, Propriety and Parameterization with Regard to Simulation-Based Fitting of Generalized Linear Mixed Models” , University of Connecticut, Dept. of Statistics . technical report
   Google Scholar
• Gelfand , A. E. and Smith , A. F. M. 1990 . “Sampling-Based Approaches to Calculating Marginal Densities” . Journal of the American Statistical Association , 85 : 398 – 409 .
   Web of Science ®Google Scholar
• Gelman , A. and Rubin , D. B. 1992 . “Inference from Iterative Simulation Using Multiple Sequences” . Statistical Science , 7 : 457 – 472 . (with discussion)
   Google Scholar
• Gelman , A. and Speed , T. P. 1993 . “Characterizing a Joint Probability Distribution by Conditionals” . Journal of the Royal Statistical Society , 55 : 185 – 188 . Series B
   Google Scholar
• Geweke , J. 1996 . Handbook of Computational Economics , Amsterdam : North-Holland . chap. 15
   Google Scholar
• Geyer , C. J. 1992 . “Practical Markov Chain Monte Carlo” . Statistical Science , 7 : 473 – 483 . (with discussion)
   Google Scholar
• Gourieroux , C. and Monfort , A. 1979 . “On the Characterization of a Joint Probability Distribution by Conditional Distributions” . Journal of Econometrics , 10 : 115 – 118 .
   Google Scholar
• Hill , B. M. 1965 . “Inference about Variance Components in the One-Way Model” . Journal of the American Statistical Association , 60 : 806 – 825 .
   Web of Science ®Google Scholar
• Hill , B. M. 1975 . “A Simple General Approach to Inference about the Tail of a Distribution” . The Annals of Statistics , 3 : 1163 – 1174 .
   Web of Science ®Google Scholar
• Hobert , J. P. 1994 . “Occurrences and Consequences of Nonpositive Markov Chains in Gibbs Sampling” , Biometrics Unit : Cornell University . unpublished PhD thesis
   Google Scholar
• Hobert , J. P. and Casella , G. 1996 . “The Effect of Improper Priors on Gibbs Sampling in Hierarchical Linear Mixed Models” . Journal of the American Statistical Association , 91 : 1461 – 1473 .
   Web of Science ®Google Scholar
• Ibrahim , J. G. and Laud , P. W. 1991 . “On Bayesian Analysis of Generalized Linear Models Using Jeffreys's Prior” . Journal of the American Statistical Association , 86 : 981 – 986 .
   Web of Science ®Google Scholar
• Kass , R. E. and Steffey , D. 1989 . “Approximate Bayesian Inference in Conditionally Independent Hierarchical Models (Parametric Empirical Bayes Models)” . Journal of the American Statistical Association , 84 : 717 – 726 .
   Web of Science ®Google Scholar
• Lehmann , E. L. 1983 . Theory of Point Estimation , Pacific Grove , CA : Wadsworth & Brook/Cole .
   Google Scholar
• Liu , J. S. , Wong , W. H. and Kong , A. 1994 . “Covariance Structure of the Gibbs Sampler with Applications to the Comparisons of Estimators and Augmentation Schemes” . Biometrika , 81 : 27 – 40 .
   Web of Science ®Google Scholar
• Meyn , S. P. and Tweedie , R. L. 1993 . Markov Chains and Stochastic Stability , London : Springer-Verlag .
   Google Scholar
• Natarajan , R. and McCulloch , C. E. 1995 . “A Note on the Existence of the Posterior Distribution for a Class of Mixed Models for Binomial Responses” . Biometrika , 82 : 639 – 643 .
   Web of Science ®Google Scholar
• Raftery , A. E. and Banfield , J. D. 1991 . Comment on “Bayesian Image Restoration, with Two Applications in Spatial Statistics” . Annals of the Institute of Statistical Mathematics , 43 : 32 – 43 .
   Web of Science ®Google Scholar
• Roberts , G. O. “Convergence Diagnostics of the Gibbs Sampler” . Bayesian Statistics 4. Proceedings of the Fourth Valencia International Meeting . Edited by: Bernardo , J. M. , Berger , J. O. , Dawid , A. P. and Smith , A. F. M. pp. 775 – 782 . Oxford : Oxford University Press .
   Google Scholar
• Robert , C. P. 1995 . “Convergence Control Methods for Markov chain Monte Carlo Algorithms” . Statistical Science , 10 : 231 – 253 .
   Web of Science ®Google Scholar
• Roberts , G. O. and Poison , N. G. 1994 . “On the Geometric Convergence of the Gibbs Sampler” . Journal of the Royal Statistical Society , 56 : 377 – 384 . Series B
   Google Scholar
• Roberts , G. O. , Sahu , S. K. and Gilks , W. R. 1995 . Comments on “Bayesian Computation and Stochastic Systems” . Statistical Science , 10 : 49 – 51 .
   Web of Science ®Google Scholar
• Tanner , M. A. 1996 . Tools for Statistical Inference: Methods for the Exploration of Posterior Distributions and Likelihood Functions, , 3rd ed. , New York : Springer .
   Google Scholar
• Tiao , G. C. and Tan , W. Y. 1965 . “Bayesian Analysis of Random-Effect Models in the Analysis of Variance. I: Posterior Distribution of Variance Components” . Biometrika , 52 : 37 – 53 .
   Web of Science ®Google Scholar
• Tierney , L. 1994 . “Markov Chains for Exploring Posterior Distributions” . The Annals of Statistics , 22 : 1701 – 1762 .
   Web of Science ®Google Scholar
• Tierney , L. , Kass , R. E. and Kadane , J. B. 1989 . “Fully Exponential Laplace Approximations to Expectations and Variances of Nonpositive Functions” . Journal of the American Statistical Association , 84 : 710 – 716 .
   Web of Science ®Google Scholar
• Wang , C. S. , Rutledge , J. J. and Gianda , D. 1993 . “Marginal Inferences about Variance Components in a Mixed Linear Model using Gibbs Sampling” . Genetique, Selection, Evolution , 25 : 41 – 62 .
   Web of Science ®Google Scholar