Understanding predictive information criteria for Bayesian models
- 6.8k Downloads
We review the Akaike, deviance, and Watanabe-Akaike information criteria from a Bayesian perspective, where the goal is to estimate expected out-of-sample-prediction error using a bias-corrected adjustment of within-sample error. We focus on the choices involved in setting up these measures, and we compare them in three simple examples, one theoretical and two applied. The contribution of this paper is to put all these information criteria into a Bayesian predictive context and to better understand, through small examples, how these methods can apply in practice.
KeywordsAIC DIC WAIC Cross-validation Prediction Bayes
We thank two reviewers for helpful comments and the National Science Foundation, Institute of Education Sciences, and Academy of Finland (grant 218248) for partial support of this research.
- Akaike, H.: Information theory and an extension of the maximum likelihood principle. In: Petrov, B.N., Csaki, F. (eds.) Proceedings of the Second International Symposium on Information Theory, pp. 267–281. Akademiai Kiado, Budapest (1973). Reprinted in: Kotz, S. (ed.) Breakthroughs in Statistics, pp. 610–624. Springer, New York (1992) Google Scholar
- Burnham, K.P., Anderson, D.R.: Model Selection and Multimodel Inference: a Practical Information Theoretic Approach. Springer, New York (2002) Google Scholar
- Dempster, A.P.: The direct use of likelihood for significance testing. In: Proceedings of Conference on Foundational Questions in Statistical Inference, Department of Theoretical Statistics: University of Aarhus, pp. 335–352 (1974) Google Scholar
- Gelman, A., Carlin, J.B., Stern, H.S., Rubin, D.B.: Bayesian Data Analysis, 2nd edn. CRC Press, London (2003) Google Scholar
- Rubin, D.B.: Estimation in parallel randomized experiments. J. Educ. Stat. 6, 377–401 (1981) Google Scholar
- Spiegelhalter, D.J., Best, N.G., Carlin, B.P., van der Linde, A.: Bayesian measures of model complexity and fit (with discussion). J. R. Stat. Soc. B (2002) Google Scholar
- Spiegelhalter, D., Thomas, A., Best, N., Gilks, W., Lunn, D.: BUGS: Bayesian inference using Gibbs sampling. MRC Biostatistics Unit, Cambridge, England (1994, 2003). http://www.mrc-bsu.cam.ac.uk/bugs/
- Stone, M.: An asymptotic equivalence of choice of model cross-validation and Akaike’s criterion. J. R. Stat. Soc. B 36, 44–47 (1977) Google Scholar