Prior-based Bayesian information criterion

ABSTRACT

We present a new approach to model selection and Bayes factor determination, based on Laplace expansions (as in BIC), which we call Prior-based Bayes Information Criterion (PBIC). In this approach, the Laplace expansion is only done with the likelihood function, and then a suitable prior distribution is chosen to allow exact computation of the (approximate) marginal likelihood arising from the Laplace approximation and the prior. The result is a closed-form expression similar to BIC, but now involves a term arising from the prior distribution (which BIC ignores) and also incorporates the idea that different parameters can have different effective sample sizes (whereas BIC only allows one overall sample size n). We also consider a modification of PBIC which is more favourable to complex models.

KEYWORDS:

• Bayes factors
• model selection
• Cauchy priors
• consistency
• effective sample size
• Fisher information
• Laplace expansions
• robust priors

Acknowledgements

This research was begun under the auspices of the 2004–2005 SAMSI program on Latent Variables in the Social Sciences.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

M. J. Bayarri's research was supported by the Spanish Ministry of Education and Science [grant number MTM2010-19528]; James Berger's research was supported by USA National Science Foundation [grant numbers DMS-1007773 and DMS-1407775]; Woncheol Jang's research was supported by the National Research Foundation of Korea (NRF) grants funded by the Korea government (MSIP), No. 2014R1A4A1007895 and No. 2017R1A2B2012816; Luis Pericchi's research was supported by grant CA096297/CA096300 from the USA National Cancer Institute of the National Institutes of Health.