## Bayesian Analysis

### Comment on article by Celeux et al.

Martyn Plummer

#### Abstract

The Deviance Information Criterion (DIC) for model choice was introduced by Spiegelhalter et al. (2002) as a Bayesian analogue of the Akaike Information Criterion. The aim of DIC is not to identify the "true" probability model, but to find a parsimonious description of the data $Y$ in terms of parameters $\theta$. The parameters are of lower dimension than the data, either because $\theta$ is restricted to a low-dimensional subspace, or because it has a highly structured prior. A penalty function $p_D$ measures the "effective number of parameters" of the model, and this is added to a measure of fit -- the expected deviance -- to give the DIC. Given a set of models, the one with the smallest DIC has the best balance between goodness of fit and model complexity.

DIC has received a mixed reception, as shown by the discussion of Spiegelhalter et al. (2002). On the one hand, it gives a pragmatic solution to the problem of model choice, and is now routinely available in the software WinBUGS (Spiegelhalter et al. 2004). On the other hand, a number of technical and conceptual difficulties with the criterion remain. Celeux et al. (2006) investigate these difficulties in the context of missing data models, and in particular mixture models. They have produced 8 variations on the theme of DIC. Some of these variations address the problem of finding a good "plug-in" estimate of $\theta$, which is necessary for the calculation of the penalty $p_D$. Others are innovations that provide a way of calculating DIC in missing data models, which might otherwise be intractable. I have attempted to classify these criteria according to their level of "focus".

#### Article information

Source
Bayesian Anal. Volume 1, Number 4 (2006), 681-686.

Dates
First available in Project Euclid: 22 June 2012

https://projecteuclid.org/euclid.ba/1340370936

Digital Object Identifier
doi:10.1214/06-BA122C

Mathematical Reviews number (MathSciNet)
MR2282200

Zentralblatt MATH identifier
1331.62340

#### Citation

Plummer, Martyn. Comment on article by Celeux et al. Bayesian Anal. 1 (2006), no. 4, 681--686. doi:10.1214/06-BA122C. https://projecteuclid.org/euclid.ba/1340370936

#### References

• Celeux, G., Forbes, F., Robert, C. P., and Titterington, D. M. (2006). "Deviance information criteria for missing data models." Bayesian Analysis (in press).
• Plummer, M. (2002). "Discussion of Speigelhalter et al." Journal of the Royal Statistical Society - Series B, 64: 620.
• Richardson, S. and Green, P. (1997). "On Bayesian analysis of mixtures with an unknown number of components (with discussion)." Journal of the Royal Statistical Society - Series B, 59: 731–758.
• Spiegelhalter, D. J., Best, N. G., Carlin, B. R., and van der Linde, A. (2002). "Bayesian measures of complexity and fit." Journal of the Royal Statistical Society - Series B, 64: 583–616.
• Spiegelhalter, D. J., Thomas, A., Best, N., and Lunn, D. (2004). WinBUGS User Manual, Version 2.0.