Sampling-based approaches to calculating marginal densities

Journal of the American Statistical Association

Volumn 85, Issue 410, 1990, Pages 398-409

  Gelfand, Alan E ^a   Smith, Adrian F M ^b  

^a UNIVERSITY OF CONNECTICUT   (United States)

^b IMPERIAL COLLEGE LONDON   (United Kingdom)

Author keywords

Conditional probability structure; Data augmentation; Gibbs sampler; Hierarchical models; Importance sampling; Missing data; Monte Carlo sampling; Posterior distributions; Stochastic substitution; Variance components

Indexed keywords

EID: 84950453304     PISSN: 01621459     EISSN: 1537274X     Source Type: Journal    
DOI: 10.1080/01621459.1990.10476213     Document Type: Article

References (25)

1. Besag, J. (1974), “Spatial Interaction and the Statistical Analysis of Lattice Systems” (with discussion), Journal of the Royal Statistical Society, Ser. B, 36, 192–326.
2. Box, G. E. P., and Tiao, G. C. (1973), Bayesian Inference in Statistical Analysis, Reading, MA: Addison-Wesley.
3. Dempster, A., Laird, N., and Rubin, D. B. (1977), “Maximum Likelihood From Incomplete Data Via the EM Algorithm” (with discussion), Journal of the Royal Statistical Society, Ser. B, 39, 1–38.
4. Devroye, L. (1986), Non-uniform Random Variate Generation, New York: Springer-Verlag.
5. 1 View, New York: John Wiley.
6. Gaver, D., and O’Muircheartaigh, I. (1987), “Robust Empirical Bayes Analysis of Event Rates,” Technometrics, 29, 1–15.
7. Geman, S., and Geman, D. (1984), “Stochastic Relaxation, Gibbs Distributions and the Bayesian Restoration of Images,” IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 721–741.
8. Geweke, J. (1988), “Antithetic Acceleration of Monte Carlo Integration in Bayesian Inference,” Journal of Econometrics, 38, 73–90.
9. Glick, N. (1974), “Consistency Conditions for Probability Estimators and Integrals of Density Estimators,” Utilitas Mathematica, 6, 61–74.
10. Hartley, H. (1958), “Maximum Likelihood Estimation From Incomplete Data,” Biometrics, 14, 174–194.
11. Hastings, W. K. (1970), “Monte Carlo Sampling Methods Using Markov Chains and Their Applications,” Biometrika, 87, 97–109.
12. Morris, C. (1983a), “Natural Exponential Families With Quadratic Variance Functions: Statistical Theory,” The Annals of Statistics, 11, 515–529.
13. —— (1983b), “Parametric Empirical Bayes Inference: Theory and Applications,” Journal of the American Statistical Association, 78, 47–59.
14. —— (1987), “Determining the Accuracy of Bayesian Empirical Bayes Estimates in Familiar Exponential Families,” in Statistical Decision Theory and Related Topics (Vol. 4), eds. S. S. Gupta and J. O. Berger, New York: Springer-Verlag, pp. 251–264.
15. Naylor, J. C., and Smith, A. F. M. (1982), “Applications of a Method for the Efficient Computation of Posterior Distributions,” Applied Statistics, 31, 214–225.
16. —— (1988), “Econometric Illustrations of Novel Numerical Integration Strategies for Bayesian Inferences,” Journal of Econometrics, 38, 103–126.
17. Rall, L. (1969), Computational Solution of Non-linear Operator Equations, New York: John Wiley.
18. Ripley, B. (1987), Stochastic Simulation, New York: John Wiley.
19. Rubin, D. B. (1987), Comment on “The Calculation of Posterior Distributions by Data Augmentation,” by M. A. Tanner and W. H. Wong, Journal of the American Statistical Association, 82, 543–546.
20. —— (1988), “Using the SIR Algorithm to Simulate Posterior Distributions,” in Bayesian Statistics 3, eds. J. M. Bernardo, M. H. DeGroot, D. V. Lindley, and A. F. M. Smith, Oxford, U.K.: Oxford University Press, pp. 395–402.
21. Shaw, J. E. H. (1988), “A Quasirandom Approach to Integration in Bayesian Statistics,” The Annals of Statistics, 16, 895–914.
22. Smith, A. F. M., Skene, A. M., Shaw, J. E. H., and Naylor, J. C. (1987), “Progress With Numerical and Graphical Methods for Bayesian Statistics,” The Statistician, 36, 75–82.
23. Smith, A. F. M., Skene, A. M., Shaw, J. E. H., Naylor, J. C., and Dransfield, M. (1985), “The Implementation of the Bayesian Paradigm,” Communications in Statistics—Theory and Methods, 14, 1079–1102.
24. Tanner, M., and Wong, W. (1987), “The Calculation of Posterior Distributions by Data Augmentation” (with discussion), Journal of the American Statistical Association, 82, 528–550.
25. Tierney, L., and Kadane, J. (1986), “Accurate Approximations for Posterior Moments and Marginal Densities,” Journal of the American Statistical Association, 81, 82–86.