Bayes factors

Journal of the American Statistical Association

Volumn 90, Issue 430, 1995, Pages 773-795

  Kass, Robert E ^a   Raftery, Adrian E ^b  

^a CARNEGIE MELLON UNIVERSITY   (United States)

^b UNIVERSITY OF WASHINGTON   (United States)

Author keywords

Bayesian hypothesis tests; BIC; Importance sampling; Laplace method; Markov chain Monte Carlo; Model selection; Monte Carlo integration; Posterior model probabilities; Posterior odds; Quadrature; Schwarz criterion; Sensitivity analysis; Strength of evidence

Indexed keywords

EID: 84950934893     PISSN: 01621459     EISSN: 1537274X     Source Type: Journal    
DOI: 10.1080/01621459.1995.10476572     Document Type: Article

References (181)

1. Aitchison, J., and Dunsmore, I. R. (1975), Statistical Prediction Analysis, Cambridge, U.K.: Cambridge University Press.
2. Aitkin, M. (1991), “Posterior Bayes Factors” (with discussion), Journal of the Royal Statistical Society, Ser. B, 53, 111–142.
3. Akaike, H. (1973), “Information Theory and an Extension of the Maximum Likelihood Principle,” in Second International Symposium on Information Theory, eds. B. N. Petrox and F. Caski, Budapest: Akademiai Kiado, p. 267.
4. —— (1977), “On Entropy Maximization Principle,” in Applications of Statistics, ed. P. R. Krishaiah, Amsterdam: North-Holland, pp. 27–41.
5. —— (1983), “Information Measures and Model Selection,” Bulletin of the International Statistical Institute, 50, 277–290.
6. Akman, V. E., and Raftery, A. E. (1986a), “Bayes Factors for Non-Homogeneous Poisson Processes With Vague Prior Information,” Journal of the Royal Statistical Society, Ser. B, 48, 322–329.
7. —— (1986b), “Asymptotic Analysis of a Change-Point Poisson Process,” The Annals of Statistics, 14, 1583–1590.
8. Albert, J. H. (1990), “A Bayesian Test for a Two-Way Contingency Table Using Independence Priors,” Canadian Journal of Statistics, 4, 347–363.
9. Atkinson, A. C. (1978), “Posterior Probabilities for Choosing a Regression Model,” Biometrika, 65, 39–48.
10. Banfield, J. D., and Raftery, A. E. (1993), “Model-Based Gaussian and Non-Gaussian Clustering,” Biometrics, 49, 803–821.
11. Bartlett, M. S. (1957), “Comment on “A Statistical Paradox” by D. V. Lindley, Biometrika, 44, 533–534.
12. Berger, J. O. (1985), Statistical Decision Theory and Bayesian Analysis, New York: Springer-Verlag.
13. Berger, J. O., and Berry, D. A. (1988), “The Relevance of Stopping Rules in Statistical Inference,” in Statistical Decision Theory and Related Topics IV, Vol. 1, eds. S. S. Gupta and J. O. Berger, New York: Springer-Verlag, pp. 29–47.
14. Berger, J. O., and Delampady, M. (1987), “Testing Precise Hypotheses,” Statistical Science, 3, 317–352.
15. Berger, J. O., and Mortera, J. (1991), “Interpreting the Stars in Precise Hypothesis Testing,” International Statistical Review, 59, 337–353.
16. Berger, J. O., and Perrichi, L. R. (1993), “The Intrinsic Bayes Factor for Model Selection and Prediction,” technical report, Purdue University, Dept. of Statistics.
17. Berger, J. O., and Sellke, T. (1987), “Testing a Point Null Hypothesis: The Irreconcilability of P Values and Evidence,” Journal of the American Statistical Association, 82, 112–122.
18. Berger, J. O., and Wolpert, R. L. (1984), The Likelihood Principle, IMS Monograph Vol. 6, Hayward, CA: Institute of Mathematical Statistics.
19. Berry, D. A., Evett, I. W., and Pinchin, R. (1992), “Statistical Inference in Crime Investigations Using Deoxyribonucleic Acid Profiling” (with discussion), Applied Statistics, 41, 499–531.
20. Box, G. E. P. (1980), “Sampling and Bayes’s Inference in Scientific Modeling” (with discussion), Journal of the Royal Statistical Society, Ser. A, 143, 383–430.
21. Box, G. E. P., and Tiao, G. C. (1962), “A Further Look at Robustness via Bayes’s Theorem,” Biometrika, 49, 419–432.
22. Broemeling, L. L. (1985), Bayesian Inference for Linear Models, New York: Marcel Dekker.
23. Carlin, B. P., and Chib, S. (1993), Bayesian Model Choice via Markov Chain Monte Carlo,” Research Report 93-006, University of Minnesota, Division of Biostatistics.
24. Carlin, B. P., Kass, R. E., Lerch, J., and Huguenard, B. (1992), “Predicting Working Memory Failure: A Subjective Bayesian Approach to Model Selection,” Journal of the American Statistical Association, 87, 319–327.
25. Carlin, B. P., and Poison, N. G. (1991), “Inference for Nonconjugate Bayesian Models Using the Gibbs Sampler,” Canadian Journal of Statistics, 19, 399–405.
26. Chen, M.-H. (1992), “Importance Weighted Marginal Bayesian Posterior Density Estimation,” technical report, Purdue University, Dept. of Statistics.
27. Chow, G. C. (1981), “A Comparison of the Information and Posterior Probability Criteria for Model Selection,” Journal of Econometrics, 16, 21–33.
28. Cooper, G., and Herskovits, E. (1992), “A Bayesian Method for the Induction of Probabilistic Networks From Data,” Machine Learning, 9, 309–347.
29. Cornfield, J. (1966a), “A Bayesian Test of Some Classical Hypotheses, With Applications to Sequential Clinical Trials,” Journal of the American Statistical Association, 61, 577–594.
30. —— (1966b), “Sequential trials, sequential analysis, and the likelihood principle,” The American Statistician, 20, 18–23.
31. Cowell, R. G., Dawid, A. P., and Spiegelhalter, D. J. (1993), “Sequential Model Criticism in Probabilistic Expert Systems,” IEEE Transactions on Pattern Analysis and Machine Intelligence, 15, 209–219.
32. Cox, D. R. (1961), “Tests of Separate Families of Hypotheses,” in Proceedings of the Fourth Berkeley Symposium, 1, 105–123.
33. —— (1962), “Further Results on Tests of Separate Families of Hypotheses,” Journal of the Royal Statistical Society, Ser. B, 24, 406–424.
34. Dawid, A. P. (1984), “Present Position and Potential Developments: Some Personal Views. Statistical Theory. The Prequential Approach” (with discussion), Journal of the Royal Statistical Society, Ser. A, 147, 178–292.
35. —— (1992), “Prequential Analysis, Stochastic Complexity and Bayesian Inference,” in Bayesian Statistics 4, eds. J. M. Bernardo et al., Oxford, U.K.: Oxford Science Publications, pp. 109–125.
36. Dayal, H. H., and Dickey, J. M. (1976), “Bayes Factors for Behrens–Fisher Problems,” Sankhya, 38, 315–328.
37. De Bruijn, N. G. (1970), Asymptotic Methods in Analysis, Amsterdam: North-Holland.
38. DeGroot, M. H. (1970), Optimal Statistical Decisions. New York: McGraw-Hill.
39. Delampady, M., and Berger, J. O. (1990), “Lower Bounds on Bayes Factors for Multinomial Distribution, With Application to Chi-Squared Tests of Fit,” The Annals of Statistics, 18, 1295–1316.
40. Dempster, A. P. (1971), “Model Searching and Estimation in the Logic of Inference,” in Foundations of Statistical Inference, eds. V. P. Godambe and D. A. Sprott, Toronto: Holt, Rinehart and Winston.
41. Dickey, J. M. (1971), “The Weighted Likelihood Ratio, Linear Hypotheses on Normal Location Parameters,” Annals of Mathematical Statistics, 42, 204–223.
42. —— (1975), “Bayesian Alternatives to the F-Test and Least Squares Estimate in the Normal Linear Model,” in Studies in Bayesian Econometrics and Statistics, eds. S. E. Fieberg and A. Zellner, Amsterdam: North-Holland, pp. 515–554.
43. —— (1976), “Approximate Posterior Distributions,” Journal of the American Statistical Association, 71, 680–689.
44. —— (1980), “Approximate Coherence for Regression Model Inference,” in Bayesian Analysis in Econometrics and Statistics, ed. A. Zellner, Amsterdam: North-Holland, pp. 334–354.
45. Dickey, J. M., and Kadane, J. B. (1980), “Bayesian Decision Theory and the Simplification of Models,” in Evaluation of Econometric Models, eds. J. Kmenta and J. Ramsey, New York: Academic Press, pp. 245–268.
46. Dickey, J. M., and Lientz, B. P. (1970), “The Weighted Likelihood Ratio, Sharp Hypotheses About Chances, the Order of a Markov Chain,” Annals of Mathematical Statistics, 41, 214–226.
47. Draper, D. (1995), “Assessment and Propagation of Model Uncertainty” (with discussion), Journal of the Royal Statistical Society, Ser. B, 57, 45–98.
48. Draper, N. R., and Guttman, I. (1987), “A Common Model Selection Criterion,” Probability and Bayesian Statistics, ed. R. Viertl, New York: Plenum, pp. 139–150.
49. Edwards, W., Lindman, H., and Savage, L. J. (1963), “Bayesian Statistical Inference for Psychological Research,” Psychological Review, 70, 193–242.
50. Evans, M., and Swartz, T. (1995), “Methods for Approximating Integrals in Statistics With Special Emphasis on Bayesian Integration problems,” Statistical Science, to appear.
51. Evett, I. W. (1991), “Implementing Bayesian Methods in Forensic Science,” paper presented at the Fourth Valencia International Meeting on Bayesian Statistics.
52. Fairley, D. (1989), “Comment,” Statistical Science, 4, 381–383.
53. Fienberg, S. E., and Mason, W. M. M. (1979), “Identification and Estimation of Age-Period-Cohort Effects in the Analysis of Discrete Archival Data,” Sociological Methodology 1979 (K. F. Schuessler, ed.), San Francisco: Jossey-Bass, pp. 1–67.
54. Findley, D. F. (1991), “Counterexamples to Parsimony and BIC,” Annals of the Institute of Statistical Mathematics, 43, 505–514.
55. Fornell, C., and Rust, R. T. (1989), “Incorporating Prior Theory in Co-variance Structure Analysis: A Bayesian Approach,” Biometrika, 54, 249–259.
56. Freedman, D. A. (1983), “A Note on Screening Regression Equations,” The American Statistician, 37, 152–155.
57. Garthwaite, P. H. (1992), “Preposterior Expected Loss as a Scoring Rule for Prior Distributions,” technical report, University of Aberdeen, Dept. of Statistics.
58. Geisser, S., and Eddy, W. F. (1979), “A Predictive Approach to Model Selection,” Journal of the American Statistical Association, 74, 153–160.
59. Gelfand, A. E., and Dey, D. K. (1994), “Bayesian Model Choice: Asymptotics and Exact Calculations,” Journal of the Royal Statistical Society, Ser. B, 56, 501–514.
60. Gelfand, A. E., Dey, D. K., and Chang, H. (1992), “Model Determination Using Predictive Distributions With Implementations via Sampling-Based Methods,” in Bayesian Statistics 4, eds. J. M. Bernardo et al., Oxford, U.K.: Oxford University Press, pp. 147–167.
61. Genz, A., and Kass, R. E. (1993), “Subregion Adaptive Integration of Functions Having a Dominant Peak,” technical report, Carnegie Mellon University, Dept. of Statistics.
62. Geweke, J. (1989), “Bayesian Inference in Econometric Models Using Monte Carlo Integration,” Econometrica, 57, 1317–1340.
63. George, E. I., and McCulloch, R. E. (1993), “Variable Selection via Gibbs Sampling,” Journal of the American Statistical Association, 88, 881–889.
64. Gilovich, T., Vallone, R., and Tversky, A. (1985), “The Hot Hand in Basketball: On the Misperception of Random Sequences,” Cognitive Psychology, 17, 295–314.
65. Good, I. J. (1950), Probability and the Weighing of Evidence, London: Griffin.
66. —— (1952), “Rational Decisions,” Journal of the Royal Statistical Society, Ser. B, 14, 107–114.
67. —— (1958), “Significance Tests in Parallel and in Series,” Journal of the American Statistical Association, 53, 799–813.
68. —— (1967), “A Bayesian Significance Test for Multinomial Distributions,” Journal of the Royal Statistical Society, Ser. B, 29, 399–431.
69. —— (1983), “Explicativity, Corroboration, and the Relative Odds of Hypotheses (#846),” in Good Thinking—The Foundations of Probability and Its Applications, Minneapolis: University of Minnesota Press.
70. —— (1985), “Weight of Evidence: A Brief Survey,” in Bayesian Statistics 2, eds. J. M. Bernardo, M. H. DeGroot, D. V. Lindley, and A. F. M. Smith, New York: Elsevier, pp. 249–269.
71. —— (1992), “The Bayes/non-Bayes Compromise: A Brief Review,” Journal of the American Statistical Association, 87, 597–606.
72. Good, I. J., and Crook, J. F. (1987), “The Robustness and Sensitivity of the Mixed Dirichlet Bayesian Test for ‘Independence’ in Contingency Tables,” The Annals of Statistics, 15, 670–693.
73. Greaney, V., and Kelleghan, T. (1984), Equality of Opportunity in Irish Schools, Dublin: Educational Company.
74. Groenewald, P. C. N., and de Waall, D. J. (1989), “Bayesian Tests for Some Precise Hypotheses on Multi-Normal Means,” Australian Journal of Statistics, 31, 393–408.
75. Grusky, D. B., and Hauser, R. M. (1984), “Comparative Social Mobility Revisited: Models of Convergence and Divergence in 16 Countries,” American Sociological Review, 49, 19–38.
76. Gunel, E., and Dickey, J. (1974), “Bayes Factors for Independence in Contingency Tables,” Biometrika, 61, 545–557.
77. Hammersley, J. M., and Handscomb, D. C. (1964), Monte Carlo Methods, London: Chapman and Hall.
78. Hartigan, J. A. (1992), “Locally Uniform Prior Distributions,” technical report, Yale University, Dept. of Statistics.
79. Haughton, D. M. A. (1988), “On the Choice of a Model to Fit Data From an Exponential Family,” The Annals of Statistics, 16, 342–355.
80. Hodges, J. S. (1987), “Uncertainty, Policy Analysis and Statistics,” Statistical Science, 2, 259–291.
81. Hsiao, K. (1994), “Bayesian Tests of Extra-Binomial Variability With Emphasis on the Boundary Case,” Ph.D. dissertation, Carnegie Mellon University, Dept. of Statistics.
82. Jeffreys, H. (1935), “Some Tests of Significance, Treated by the Theory of Probability,” Proceedings of the Cambridge Philosophy Society, 31, 203–222.
83. —— (1961), Theory of Probability (3rd ed.), Oxford, U.K.: Oxford University Press.
84. Kadane, J. B., and Dickey, J. M. (1980), “Bayesian Decision Theory and the Simplification of Models,” in Evaluation of Econometric Models, eds. J. Kmenta and J. Ramsey, New York: Academic Press, 245–268.
85. Kadane, J. B., Dickey, J., Winkler, R., Smith, W., and Peters, S. (1980), “Interactive Elicitation of Opinion for A Normal Linear Model,” Journal of the American Statistical Association, 75, 845–854.
86. Kass, R. E. (1993), “Bayes Factors in Practice,” Statistician, 42, 551–560.
87. Kass, R. E., and Greenhouse, J. B. (1989), Comment on “Investigating Therapies of Potentially Great Benefit: ECMO,” by Ware (1989), Statistical Science, 4, 310–317.
88. Kass, R. E., and Raftery, A. E. (1993), “Bayes Factors and Model Uncertainty,” Technical Report 571, Carnegie Mellon University, Dept. of Statistics.
89. Kass, R. E., and Steffey, D. L. (1989), “Approximate Bayesian Methods in Conditionally Independent Hierarchical Models (Parametric Empirical Bayes Models),” Journal of the American Statistical Association, 84, 717–726.
90. Kass, R. E., Tierney, L., and Kadane, J. B. (1990), “The Validity of Posterior Asymptotic Expansions Based on Laplace’s Method,” in Bayesian and Likelihood Methods in Statistics and Econometrics, eds. S. Geisser, J. S. Hodges, S. J. Press, and A. Zellner, New York: North-Holland.
91. Kass, R. E., and Vaidyanathan, S. (1992), “Approximate Bayes Factors and Orthogonal Parameters, With Application to Testing Equality of Two Binomial Proportions,” Journal of the Royal Statistical Society, Ser. B, 54, 129–144.
92. Kass, R. E., and Wasserman, L. (1992a), “Improving the Laplace Approximation Using Posterior Simulation,” Technical Report 566, Carnegie Mellon University, Dept. of Statistics.
93. —— (1995), “A Reference Bayesian Test for Nested Hypotheses and Its Relationship to the Schwarz Criterion,” Journal of the American Statistical Association, to appear.
94. Katz, R. W. (1981), “On Some Criteria for Estimating the Order of a Markov Chain,” Technometrics, 23, 243–249.
95. Le, N. D., Raftery, A. E., and Martin, R. D. (1990), “Robust Order Selection for Autoregressive Models Using Robust Bayes Factors,” Technical Report 194, University of Washington, Dept. of Statistics.
96. Leamer, E. E. (1978), Specification Searches: Ad Hoc Inference With Non-experimental Data, New York: John Wiley.
97. —— (1983), “Model Choice and Specification Analysis,” in Handbook of Econometrics, Vol. 1, eds. Z. Griliches and M. D. Intriligator, Amsterdam: North-Holland.
98. Leonard, T. (1982), Comment on “A Simple Predictive Density Function,” Journal of the American Statistical Association, 77, 657–658.
99. Lempers, F. B. (1971), Posterior Probabilities of Alternative Linear Models, Rotterdam: University Press.
100. Lewis, S. M., and Raftery, A. E. (1994), “The Laplace-Metropolis Estimator for Bayes Factors via Posterior Simulation,” Technical Report 279, University of Washington, Dept. of Statistics.
101. Lindley, D. V. (1957), “A Statistical Paradox,” Biometrika, 44, 187–192.
102. —— (1961), “The Use of Prior Probability Distributions in Statistical Inference and Decisions,” in Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Prob., ed. J. Neyman, Berkeley, CA: University of California Press, pp. 453–468.
103. —— (1988), “Statistical Inference Concerning Hardy-Weinberg Equilibrium,” 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. 307–326.
104. Linhart, H., and Zucchini, W. (1986), Model Selection, New York: John Wiley.
105. Madigan, D., and Raftery, A. E. (1994), “Model Selection and Accounting for Model Uncertainty in Graphical Models Using Occam’s Window,” Journal of the American Statistical Association, 89, 1535–1546.
106. Madigan, D., Raftery, A. E., York, J. C., Bradshaw, J. M., and Almond, R. G. (1994), “Strategies for Graphical Model Selection,” in Selecting Models From Data: AI and Statistics IV, eds. P. Cheeseman and R. W. Oldford, New York: Springer-Verlag, pp. 91–100.
107. Madigan, D., and York, J. (1992), “Bayesian graphical models for discrete data,” Technical Report 259, University of Washington, Dept. of Statistics.
108. McCulloch, R. E., and Rossi, P. E. (1991), “A Bayesian Approach to Testing the Arbitrage Pricing Theory,” Journal of Econometrics, 49, 141–168.
109. —— (1993), “Bayes Factors for Nonlinear Hypotheses and Likelihood Distributions,” Biometrika, 79, 663–676.
110. Meng, X. L., and Wong, W. H. (1993), “Simulating Ratios of Normalizing Constants via a Simple Identity,” Technical Report 365, University of Chicago, Dept. of Statistics.
111. Miller, A. J. (1984), “Selection of Subsets of Regression Variables” (with discussion), Journal of the Royal Statistical Society, Ser. A, 147, 389–425.
112. —— (1990), Subset Selection in Regression, London: Chapman and Hall.
113. Mitchell, T. J., and Beauchamp, J. J. (1988), “Bayesian Variable Selection in Regression,” Journal of the American Statistical Association, 83, 1023–1032.
114. Mosteller, F., and Wallace, D. L. (1964), Inference and Disputed Authorship: The Federalist, Reading, MA: Addison-Wesley. Reprinted in 1985 by Springer-Verlag.
115. Moulton, B. R. (1991), “A Bayesian Approach to Regression Selection and Estimation, With Application to a Price Index for Radio Services,” Journal of Econometrics, 49, 169–193.
116. Neal, R. M. (1992), “Probabilistic Inference for Artificial Intelligence Using Monte Carlo Methods Based on Markov Chains,” technical report, University of Toronto, Dept. of Computer Science.
117. Newton, M. A., and Raftery, A. E. (1994), “Approximate Bayesian Inference by the Weighted Likelihood Bootstrap” (with discussion), Journal of the Royal Statistical Society, Ser. B, 56, 3–48.
118. O’Hagan, A. (1991), Contribution to the discussion of “Posterior Bayes Factors,” Journal of the Royal Statistical Society, Ser. B, 53, 137.
119. —— (1995), “Fractional Bayes Factors for Model Comparison” (with discussion), Journal of the Royal Statistical Society, Ser. B, 57
120. Parzen, E. (1974), “Some Recent Advances in Time Series Modeling,” IEEE Transactions on Automatic Control, 19, 723–730.
121. Pena, D., and Guttman, I. (1992), “Comparing Probabilistic Methods for Outlier Detection in Linear Models,” technical report, University of Toronto, Dept. of Statistics.
122. Pettit, L. I., and Young, K. D. S. (1990), “Measuring the Effect of Observations on Bayes Factors,” Biometrika, 77, 455–466.
123. Poirier, D. J. (1985), “Bayesian Hypothesis Testing in Linear Models With Continuously Induced Conjugate Priors Across Hypotheses,” in Bayesian Statistics 2, eds. J. M. Bernardo, M. H. DeGroot, D. V. Lindley, and A. F. M. Smith, New York: Elsevier, pp. 711–722.
124. Poison, N. G., and Roberts, G. O. (1994), “Bayes Factors for Discrete Observations From Diffusion Processes,” Biometrika, 81, 11–26.
125. Racine, A., Grieve, A. P., Fluhler, H., and Smith, A. F. M. (1986), “Bayesian Methods in Practice: Experiences in the Pharmaceutical Industry” (with discussion), Applied Statistics, 35, 93–150.
126. Raftery, A. E. (1986a), “A Note on Bayes Factors for Log-Linear Contingency Table Models With Vague Prior Information,” Journal of the Royal Statistical Society, Ser. B, 48, 249–250.
127. —— (1986b), “Choosing Models for Cross-Classifications,” American Sociological Review, 51, 145–146.
128. —— (1987), “Inference and Prediction for a General Order Statistic Model With Unknown Population Size,” Journal of the American Statistical Association, 82, 1163–1168.
129. —— (1988a), “Analysis of a Simple Debugging Model.” Applied Statistics, 37, 12–22.
130. —— (1988b), “Approximate Bayes Factors for Generalized Linear Models,” Technical Report 121, University of Washington, Dept. of Statistics.
131. —— (1989), “Are Ozone Exceedance Rates Decreasing?,” Statistical Science, 4, 378–381.
132. —— (1992), Discussion of “Model Determination Using Predictive Distributions With Implementations via Sampling-Based Methods,” in Bayesian Statistics 4, eds. J. M. Bernardo et al., Oxford, U.K.: Oxford University Press, 160–163.
133. —— (1993a), “Discussion of Three Papers on the Gibbs Sampler and Other Markov Chain Monte Carlo Methods,” Journal of the Royal Statistical Society, Ser. B, 53, 85.
134. —— (1993b), “Bayesian Model Selection in Structural Equation Models,” in Testing Structural Equation Models, eds. K. A. Bollen and J. S. Long, Beverly Hills, CA: Sage, pp. 163–180.
135. —— (1993c), “Approximate Bayes Factors and Accounting for Model Uncertainty in Generalized Linear Models,” Technical Report 255, University of Washington, Dept. of Statistics.
136. —— (1995a), “Hypothesis Testing and Model Selection with Posterior Simulation,” in Practical Markov Chain Monte Carlo, eds. W. R. Gilks, D. J. Spiegelhalter, and S. Richardson, London: Chapman and Hall.
137. —— (1995b), “Bayesian Model Selection in Social Research (with Discussion),” to appear in Sociological Methodology 1995 (P. V. Marden, ed.), Cambridge, MA: Blackwells.
138. Raftery, A. E., and Akman, V. E. (1986), “Bayesian Analysis of a Poisson Process With a Change-Point,” Biometrika, 73, 85–89.
139. Raftery, A. E., and Banfield, J. D. (1990), “Stopping the Gibbs Sampler, the Use of Morphology and Other Issues in Spatial Statistics,” Annals of the Institute of Statistical Mathematics, 43, 32–43.
140. Raftery, A. E., Givens, G. H., and Zeh, J. E. (1995), “Inference From a Deterministic Population Dynamics Model About Bowhead Whale, Balaena mysticetus, Replacement Yield” (with discussion), Journal of the American Statistical Association, 90, 402–416.
141. Raftery, A. E., and Hout, M. (1985), “Does Irish Education Approach the Meritocratic Ideal? A Logistic Analysis,” Economic and Social Review, 16, 115–140.
142. —— (1993), “Maximally Maintained Inequality: Expansion, Reform and Opportunity in Irish Education, 1921–1975,” Sociology of Education, 66, 41–62.
143. Raftery, A. E., Madigan, D. M., and Hoeting (1993), “Model Selection and Accounting for Model Uncertainty in Linear Regression Models,” Technical Report 262, University of Washington, Dept. of Statistics.
144. Raftery, A. E., Madigan, D., and Volinsky, C. T. (1995), “Accounting for Model Uncertainty in Survival Analysis Improves Predictive Performance (with discussion),” in Bayesian Statistics 5, eds. J. M. Bernardo et al., Oxford, U.K.: Oxford University Press.
145. Raftery, A. E., and Zeh, J. E. (1993), “Estimation of Bowhead Whale, Balaena mysticetus, Population Size” (with discussion), in Bayesian Statistics in Science and Technology: Case Studies, eds. C. Gatsonis, J. S. Hodges, R. E. Kass, and N. D. Singpurwalla, New York: Springer-Verlag, pp. 163–240.
146. Rissanen, J. (1987), “Stochastic Complexity,” Journal of the Royal Statistical Society, Ser. B, 49, 223–239.
147. Robert, C. P. (1993), “A Note on Jeffreys–Lindley Paradox,” Statistica Sinica, 3, 601–608.
148. Robert, C. P., and Caron, N. (1991), “Noninformative Bayesian Testing and Neutral Bayes Factors,” Technical Report 148, Laboratoire de Statistique Théorique et Appliquée, Université de Paris VI.
149. Rosenkranz, S. (1992), “The Bayes Factor for Model Evaluation in a Hierarchical Poisson Model for Area Counts,” Ph.D. dissertation, University of Washington, 1992, Dept. of Biostatistics.
150. Rossi, P. E. (1985), “Comparison of Alternative Functional Forms in Production,” Journal of Econometrics, 30, 345–361.
151. —— (1988), “Comparison of Dynamic Factor Demand Models: Dynamic Econometric Modeling,” in Proceedings of the Third International Symposium in Economic Theory & Econometrics, eds, W. A. Barnett, E. R. Berndt, and H. White, Cambridge, U.K.: Cambridge University Press, pp. 357–376.
152. Rubin, D. B. (1984), “Bayesianly Justifiable and Relevant Frequency Calculations for the Applied Statistician,” The Annals of Statistics, 12, 1151–1172.
153. Schotman, P., and van Dijk, H. (1991), “A Bayesian Analysis of the Unit Root in Real Exchange Rates,” Journal of Econometrics, 49, 195–238.
154. Schwarz, G. (1978), “Estimating the Dimension of a Model,” The Annals of Statistics, 6, 461–464.
155. Shibata, R. (1976), “Selection of the Order of an Autoregressive Model by Akaike’s Information Criterion,” Biometrika, 63, 117–126.
156. —— (1980), “Asymptotically Efficient Selection of the Order of the Model for Estimating Parameters of a Linear Process,” The Annals of Statistics, 8, 147–164.
157. —— (1981), “Optimal Selection of Regression Variables,” Biometrika, 68, 45–54.
158. Sklar, R., and Strauss, B. (1980), “Role of the uvrE Gene Product and of Inducible 0-Methylguanine Removal in the Induction of Mutations by N-methyl-N’-nitro-N-nitrosoguanidine in Escherichia coli,” Journal of Molecular Biology, 143, 345–363.
159. Slate, E. (1994), “Parameterizations for Natural Exponential Families With Quadratic Variance Functions,” Journal of the American Statistical Association, 89, 1471–1482.
160. Smith, A. F. M. (1991), Contribution to the discussion of “Posterior Bayes Factors,” Journal of the Royal Statistical Society, Ser. B, 53, 132–133.
161. Smith, A. F. M., and Roberts, G. O. (1993), “Bayesian Computation via the Gibbs Sampler and Related Markov Chain Monte Carlo Methods,” Journal of the Royal Statistical Society, Ser. B, 55, 3–23.
162. Smith, A. F. M., and Spiegelhalter, D. J. (1980), “Bayes Factors and Choice Criteria for Linear Models,” Journal of the Royal Statistical Society, Ser. B., 42, 213–220.
163. —— (1981), “Bayesian Approaches to Multivariate Structure,” in Interpreting Multivariate Data, ed. V. Barnett, Chichester, U.K.: John Wiley.
164. Smith, R. L. (1989), “Extreme Value Analysis of Environmental Time Series: An Application to Trend Detection in Ground-Level Ozone” (with discussion), Statistical Science, 4, 367–377.
165. Spiegelhalter, D. J., and Smith, A. F. M. (1982), “Bayes Factors for Linear and Log-Linear Models With Vague Prior Information,” Journal of the Royal Statistical Society, Ser. B, 44, 377–387.
166. Stewart, L. (1987), “Hierarchical Bayesian Analysis Using Monte Carlo Integration: Computing Posterior Distributions When There are Many Possible Models,” The Statistician, 36, 211–219.
167. Stone, M. (1979), “Comments on Model Selection Criteria of Akaike and Schwarz,” Journal of the Royal Statistical Society, Ser. B, 41, 276–278.
168. 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.
169. Thompson, E. A., and Wijsman, E. M. (1990), “The Gibbs Sampler on Extended Pedigrees: Monte Carlo Methods for the Genetic Analysis of Complex Traits,” Technical Report 193, University of Washington, Dept. of Statistics.
170. Tierney, L. (1989), “Bayesian Analysis in New-S and Lisp,” in Proceedings of the Section on Statistical Computing, American Statistical Association.
171. —— (1990), Lisp-Stat, New York: John Wiley.
172. Tierney, L., and Kadane, J. B. (1986), “Accurate Approximations for Posterior Moments and Marginal Densities,” Journal of the American Statistical Association, 81, 82–86.
173. 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.
174. Verdinelli, I., and Wasserman, L. (1993a), “Bayes Factors, Nuisance Parameters and Imprecise Tests,” Technical Report 570, Carnegie Mellon University, Dept. of Statistics.
175. —— (1993b), “A Note on Generalizing the Savage–Dickey Density Ratio,” Technical Report 573, Carnegie Mellon University, Dept. of Statistics.
176. Worsley, K. J. (1986), “Confidence Regions and Tests for a Change-Point in a Sequence of Exponential Family Random Variables,” Biometrika, 73, 91–104.
177. Zellner, A. (1971), An Introduction to Bayesian Inference in Econometrics, New York: John Wiley.
178. —— (1978), “Jeffreys–Bayes Posterior Odds Ratio and the Akaike Information Criterion for Discriminating Between Models,” Economics Letters, 1, 337–342.
179. —— (1984), “Posterior Odds Ratios for Regression Hypotheses: General Considerations and Some Specific Results,” in Basic Issues in Econometrics, Chicago: University of Chicago Press, pp. 275–305.
180. —— (1987), “Comment,” Statistical Science, 2, 339–341.
181. Zellner, A., and Siow (1980), “Posterior Odds for Selected Regression Hypotheses,” in Bayesian Statistics 1, eds. J. M. Bernardo et al., Valencia, Spain: Valencia University Press, pp. 585–603. Reply, pp. 638–643.