Comment on article by Celeux et al.

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".