Bayesian Analysis

Integral Priors and Constrained Imaginary Training Samples for Nested and Non-nested Bayesian Model Comparison

Juan Antonio Cano and Diego Salmerón

Full-text: Open access

Abstract

In Bayesian model selection when the prior information on the parameters of the models is vague default priors should be used. Unfortunately, these priors are usually improper yielding indeterminate Bayes factors that preclude the comparison of the models. To calibrate the initial default priors Cano et al. (2008) proposed integral priors as prior distributions for Bayesian model selection. These priors were defined as the solution of a system of two integral equations that under some general assumptions has a unique solution associated with a recurrent Markov chain. Later, in Cano et al. (2012b) integral priors were successfully applied in some situations where they are known and they are unique, being proper or not, and it was pointed out how to deal with other situations. Here, we present some new situations to illustrate how this new methodology works in the cases where we are not able to explicitly find the integral priors but we know they are proper and unique (one-sided testing for the exponential distribution) and in the cases where recurrence of the associated Markov chains is difficult to check. To deal with this latter scenario we impose a technical constraint on the imaginary training samples space that virtually implies the existence and the uniqueness of integral priors which are proper distributions. The improvement over other existing methodologies comes from the fact that this method is more automatic since we only need to simulate from the involved models and their posteriors to compute very well behaved Bayes factors.

Article information

Source
Bayesian Anal., Volume 8, Number 2 (2013), 361-380.

Dates
First available in Project Euclid: 24 May 2013

Permanent link to this document
https://projecteuclid.org/euclid.ba/1369407556

Digital Object Identifier
doi:10.1214/13-BA812

Mathematical Reviews number (MathSciNet)
MR3066945

Zentralblatt MATH identifier
1329.62118

Keywords
Bayesian model selection Bayes factor intrinsic priors integral priors

Citation

Cano, Juan Antonio; Salmerón, Diego. Integral Priors and Constrained Imaginary Training Samples for Nested and Non-nested Bayesian Model Comparison. Bayesian Anal. 8 (2013), no. 2, 361--380. doi:10.1214/13-BA812. https://projecteuclid.org/euclid.ba/1369407556


Export citation

References

  • Athreya, K. B., Doss, H., and Sethuraman, J. (1996). “On the convergence of the Markov chain simulation method.” The Annals of Statistics, 24: 69–100.
  • Berger, J. and Pericchi, L. R. (1996). “The intrinsic Bayes factor for model selection and prediction.” Journal of the American Statistical Association, 91: 109–122.
  • — (2004). “Training samples in objective Bayesian model selection.” The Annals of Statistics, 32: 841–869.
  • Cano, J. A., Carazo, C., and Salmerón, D. (2012a). “Bayesian model selection approach to the one way analysis of variance under homoscedasticity.” Computational Statistics, doi:10.1007/s00180-012-0339-8.
  • Cano, J. A., Iniesta, M., and Salmerón, D. (2012b). “Integral priors for Bayesian model selection: illustrative implementations and perspectives.” Technical report, Department of Statistics and Operational Research, University of Murcia.
  • Cano, J. A., Kessler, M., and Moreno, E. (2004). “On intrinsic priors for nonnested models.” Test, 13: 445–463.
  • Cano, J. A., Kessler, M., and Salmerón, D. (2007a). “Integral priors for the one way random effects model.” Bayesian Analysis, 2: 59–68.
  • — (2007b). “A synopsis of integral priors for the one way random effects model.” In Bayesian Statistics 8, 577–582. Oxford University Press.
  • Cano, J. A., Salmerón, D., and Robert, C. P. (2008). “Integral equation solutions as prior distributions for Bayesian model selection.” Test, 17: 493–504.
  • Moreno, E. (2005). “Objective Bayesian methods for one-sided testing.” Test, 14: 181–198.
  • Moreno, E., Bertolino, F., and Racugno, W. (1998). “An intrinsic limiting procedure for model selection and hypotheses testing.” Journal of the American Statistical Association, 93: 1451–1460.
  • Pérez, J. M. and Berger, J. (2002). “Expected posterior priors for model selection.” Biometrika, 89: 491–511.