Journal of Statistical Theory and Practice

, Volume 4, Issue 1, pp 85–110 | Cite as

On Objective Priors for Testing Hypotheses About Some Poisson Models

  • Dongming JiangEmail author
  • S. Sivaganesan


In the absence of prior information, use of non-informative proper priors is often crucial for testing hypotheses, when using the Bayesian approach. In this context, the use of objective priors such as intrinsic priors and Zellner’s g-priors have gained much interest. In this paper, we consider the use of these priors for testing hypotheses about means and regression coefficients when observations come from Poisson distributions. We first derive an intrinsic prior for testing the equality of several Poisson means. We then focus on g-priors, giving a new motivation, based on shrinkage and minimal training sample arguments, for a mixture g-prior recommended by Liang, Paulo, Molina, Clyde and Berger (2008) for normal linear models. Using the same motivation, we propose a mixture g-prior for Poisson regression model. While the proposed g prior is similar to the one used by Wang and George (2007), it is also different in certain aspects. Specifically, we show that the Bayes factor derived from the proposed prior is consistent. We also provide examples using simulated and real data.

AMS Subject Classification

62F15 62F03 62F05 


Objective Bayes Intrinsic priors g-priors Poisson means Poisson regression 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Agresti, A., 1996. An Introduction to Categorical Data Analysis. Wiley, New York.zbMATHGoogle Scholar
  2. Berger, J.O., 2006. The case for objective Bayesian analysis. Bayesian Analysis, 1 (3), 385–402.MathSciNetCrossRefGoogle Scholar
  3. Berger, J.O., Pericchi, L.R., 1996a. The intrinsic Bayes factor for model selection and prediction. Journal of the American Statistical Association, 91, 109–122.MathSciNetCrossRefGoogle Scholar
  4. Berger, J.O., Pericchi, L.R., 1996b. On the justification of default and intrinsic Bayes factors. In Modeling and Prediction, Lee, J.C., Lee, C.F., Wei, K.C. (editors), Springer-Verlag, New York, 276–293.zbMATHGoogle Scholar
  5. Berger, J.O., Pericchi, L.R., 1998. Accurate and stable Bayesian model selection: the median intrinsic Bayes factor. Sankhyā B, 60, 1–18.MathSciNetzbMATHGoogle Scholar
  6. Berger, J.O., Pericchi, L.R., 2001. Objective Bayesian methods for model selection: Introduction and comparison. In Model Selection, Lahiri, P. (editor), Institute of Mathematical Statistics Lecture Notes — Monograph Series, Beachwood, Ohio, 38, 135–207.MathSciNetCrossRefGoogle Scholar
  7. Brockmann, H.J., 1996. Satellite Male Groups in Horseshoe Crabs, Limulus polyphemus. Ethology, 102, 1–21.CrossRefGoogle Scholar
  8. Casella, G., Moreno, E., 2006. Objective Bayesian variable selection. Journal of the American Statistical Association, 101 (473), 157–167.MathSciNetCrossRefGoogle Scholar
  9. Daniels, M.J., 1999. A prior for the variance in hierarchical models. Canadian Journal of Statistics, 27, 569–580.MathSciNetCrossRefGoogle Scholar
  10. Fernández, C., Ley, E., Steel, M.F., 2001. Benchmark priors for Bayesian model averaging. J. Econometrics, 100, 381–427.MathSciNetCrossRefGoogle Scholar
  11. Kass, R.E., Tierney, L., Kadane, J.B., 1990. The validity of posterior expansions based on Laplace’s method. In Essays in Honor of George Bernard, Geisser, S., Hodges, J.S., Press, S.J., Zellner, A. (editors), North Holland, Amsterdam, 473–488.Google Scholar
  12. Kass, R.E., Wasserman, L., 1995. A reference Bayesian test for nested hypotheses and its relationship to the Schwarz criterion. Journal of the American Statistical Association, 90, 928–934.MathSciNetCrossRefGoogle Scholar
  13. Liang, F., Paulo, R., Molina, G., Clyde, M.A., Berger, J.O., 2008. Mixtures of g-priors for Bayesian Variable Selection. Journal of the American Statistical Association, 481, 410–423.MathSciNetCrossRefGoogle Scholar
  14. Neter, J., Kutner, M.H., Wasserman, W., Nachtsheim, C.J., 1996. Applied Linear Statistical Models, McGraw-Hill, New York.Google Scholar
  15. Wang, X., George, E.I., 2007. Adaptive Bayesian criteria in variable selection for generalized linear models. Statistica Sinica, 17, 667–690.MathSciNetzbMATHGoogle Scholar
  16. Zellner, A., Siow, A., 1980. Posterior odds ratios for selected regression hypotheses. In Bayesian Statistics 1, Bernardo, J.M., DeGroot, M.H., Lindley, D.V., Smith, A.F.M. (editors), Valencia University Press, 585–603.Google Scholar
  17. Zellner, A., 1986. On assessing prior distributions and Bayesian regression analysis with g-prior distributions. In Bayesian Inference and Decision Techniques: Essays in Honor of Bruno de Finetti, Goel, P.K., Zellner, A. (editors), North-Holland, Amsterdam, 233–243.zbMATHGoogle Scholar

Copyright information

© Grace Scientific Publishing 2010

Authors and Affiliations

  1. 1.University of CincinnatiCincinnatiUSA

Personalised recommendations