Brazilian Journal of Probability and Statistics

A note on Bayesian robustness for count data

Jairo A. Fúquene and Moises Delgado

Full-text: Open access

Abstract

The usual Bayesian approach for count data is Gamma/Poisson conjugate analysis. However, in this conjugate analysis the influence of the prior distribution could be dominant even when prior and likelihood are in conflict. Our proposal is an analysis based on the Cauchy prior for natural parameter in exponential families. In this work, we show that the Cauchy/Poisson posterior model is a robust model for count data in contrast with the usual conjugate Bayesian approach Gamma/Poisson model. We use the polynomial tails comparison theorem given in (Bayesian Anal. 4 (2009) 817–843) that gives easy-to-check conditions to ensure prior robustness. In short, this means that when the location of the prior and the bulk of the mass of the likelihood get further apart (a situation of conflict between prior and likelihood information), Bayes theorem will cause the posterior distribution to discount the prior information. Finally, we analyze artificial data sets to investigate the robustness of the Cauchy/Poisson model.

Article information

Source
Braz. J. Probab. Stat. Volume 26, Number 3 (2012), 279-287.

Dates
First available in Project Euclid: 5 April 2012

Permanent link to this document
https://projecteuclid.org/euclid.bjps/1333632165

Digital Object Identifier
doi:10.1214/10-BJPS134

Mathematical Reviews number (MathSciNet)
MR2911706

Zentralblatt MATH identifier
1239.62019

Keywords
Exponential family polynomial tails comparison theorem robust priors Cauchy/Poisson model

Citation

Fúquene, Jairo A.; Delgado, Moises. A note on Bayesian robustness for count data. Braz. J. Probab. Stat. 26 (2012), no. 3, 279--287. doi:10.1214/10-BJPS134. https://projecteuclid.org/euclid.bjps/1333632165


Export citation

References

  • Abramowitz, M. and Stegun, I. A. (1992). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover.
  • Berger, J. O. (1985). Statistical Decision Theory and Bayesian Analysis, 2nd ed. Springer-Verlag.
  • Dawid, A. P. (1973). Posterior expectations for large observations. Biometrika 60, 664–667.
  • Fúquene, J. (2009). Robust Bayesian priors in clinical trials: An R package for practitioners. Biometric Brazilian Journal 27, 627–643.
  • Fúquene, J. A., Cook, J. D. and Pericchi, L. R. (2009). A case for robust Bayesian priors with applications to clinical trials. Bayesian Analysis 4, 817–843.
  • Gelman, A., Jakulin, A., Pittau, M. G. and Su, Y.-S. (2008). A weakly informative default prior distribution for logistic and other regression models. The Annals of Applied Statistics 2, 1360–1383.
  • O’Hagan, A. (1979). On outlier rejection phenomena in Bayes inference. Journal of the Royal Statistical Society, Ser. B 41, 358–367.
  • Perez, M. E. and Pericchi, L. R. (2009). The case for a fully robust hierarchical Bayesian analysis of clinical trials. Technical Report, Wharton School of Business.
  • Pericchi, L. R., Sanso, B. and Smith, A. F. M. (1993). Posterior cumulant relationships in Bayesian inference involving the exponential family. Journal of the American Statistical Association 88, 1419–1426.
  • Pericchi, L. R. and Smith, A. F. M. (1992). Exact and approximate posterior moments for a normal location parameter. Journal of the Royal Statistical Society, Ser. B 54, 793–804.
  • R Development Core Team (2010). R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing. ISBN 3-900051-07-0. Available at http://www.R-project.org.
  • Gamerman, D. and Lopes, H. F. (2006). Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference, 2nd ed. London: Chapman & Hall.