Bayesian Analysis

Data-Dependent Posterior Propriety of a Bayesian Beta-Binomial-Logit Model

Hyungsuk Tak and Carl N. Morris

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A Beta-Binomial-Logit model is a Beta-Binomial model with covariate information incorporated via a logistic regression. Posterior propriety of a Bayesian Beta-Binomial-Logit model can be data-dependent for improper hyper-prior distributions. Various researchers in the literature have unknowingly used improper posterior distributions or have given incorrect statements about posterior propriety because checking posterior propriety can be challenging due to the complicated functional form of a Beta-Binomial-Logit model. We derive data-dependent necessary and sufficient conditions for posterior propriety within a class of hyper-prior distributions that encompass those used in previous studies. When a posterior is improper due to improper hyper-prior distributions, we suggest using proper hyper-prior distributions that can mimic the behaviors of improper choices.

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Bayesian Anal., Volume 12, Number 2 (2017), 533-555.

First available in Project Euclid: 20 July 2016

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objective Bayes hierarchical models random effects overdispersion logistic regression beta-binomial uniform shrinkage prior

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Tak, Hyungsuk; Morris, Carl N. Data-Dependent Posterior Propriety of a Bayesian Beta-Binomial-Logit Model. Bayesian Anal. 12 (2017), no. 2, 533--555. doi:10.1214/16-BA1012.

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  • Albert, A. and Anderson, J. A. (1984). “On the Existence of Maximum Likelihood Estimates in Logistic Regression Models.” Biometrika, 71(1): 1–10.
  • Albert, J. H. (1988). “Computational Methods Using a Bayesian Hierarchical Generalized Linear Model.” Journal of the American Statistical Association, 83(404): 1037–1044.
  • Athreya, K. B. and Roy, V. (2014). “Monte Carlo Methods for Improper Target Distributions.” Electronic Journal of Statistics, 8(2): 2664–2692.
  • Christiansen, C. L. and Morris, C. N. (1997). “Hierarchical Poisson Regression Modeling.” Journal of the American Statistical Association, 92(438): 618–632.
  • Daniels, M. J. (1999). “A Prior for the Variance in Hierarchical Models.” The Canadian Journal of Statistics, 27(3): 567–578.
  • Efron, B. and Morris, C. N. (1975). “Data Analysis Using Stein’s Estimator and its Generalizations.” Journal of the American Statistical Association, 70(350): 311–319.
  • Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., and Rubin, D. B. (2013). Bayesian Data Analysis. CRC Press, 3rd edition.
  • 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(436): 1461–1473.
  • Jacobsen, M. (1989). “Existence and Unicity of MLEs in Discrete Exponential Family Distributions.” Scandinavian Journal of Statistics, 335–349.
  • Kahn, M. J. and Raftery, A. E. (1996). “Discharge Rates of Medicare Stroke Patients to Skilled Nursing Facilities: Bayesian Logistic Regression With Unobserved Heterogeneity.” Journal of the American Statistical Association, 91(433): 29–41.
  • 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(407): 717–726.
  • Morris, C. N. and Lysy, M. (2012). “Shrinkage Estimation in Multilevel Normal Models.” Statistical Science, 27(1): 115–134.
  • Morris, C. N. and Tang, R. (2011). “Estimating Random Effects via Adjustment for Density Maximization.” Statistical Science, 26(2): 271–287.
  • Natarajan, R. and Kass, R. E. (2000). “Reference Bayesian Methods for Generalized Linear Mixed Models.” Journal of the American Statistical Association, 95(449): 227–237.
  • R Development Core Team (2016). R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing.
  • Skellam, J. G. (1948). “A Probability Distribution Derived from the Binomial Distribution by Regarding the Probability of Success as Variable Between the Sets of Trials.” Journal of the Royal Statistical Society – Series B, 10: 257–261.
  • Speckman, P. L., Lee, J., and Sun, D. (2009). “Existence of the MLE and Propriety of Posteriors for a General Multinomial Choice Model.” Statistica Sinica, 731–748.
  • Strawderman, W. E. (1971). “Proper Bayes Minimax Estimators of the Multivariate Normal Mean.” The Annals of Mathematical Statistics, 42(1): 385–388.
  • Tierney, L. (1994). “Markov Chains for Exploring Posterior Distributions.” The Annals of Statistics, 22(4): 1701–1728.
  • Williams, D. A. (1982). “Extra-Binomial Variation in Logistic Linear Models.” Journal of the Royal Statistical Society – Series C, 31(2): 144–148.