The Deviance Information Criterion

A widely used statistic for comparing models in a Bayesian framework is the Deviance Information Criterion. It can be calculated automatically in WinBUGS, just by clicking the right button. The results of the DIC calculations for our two models are summarized in Table 3.

Table 3: DIC calculations for the ABO example.

Model	Dbar	Dhat	pD	DIC
$f > 0$	24.900	22.319	2.581	24.480
$f = 0$	27.827	25.786	2.041	29.869

Dbar and Dhat are measures of how well the model fits the data. Dbar is the posterior mean log likelihood, i.e., the average of the log likelihood values calculated from the parameters in each sample from the posterior. Dhat is the log likelihood at the posterior mean, i.e., the log likelihood calcuated when all of the parameters are set to their posterior mean. pD is a measure of model complexity, roughly speaking the number of parameters in the model. DIC is a composite measure of how well the model does. It's a compromise between fit and complexity, and smaller DICs are preferred. A difference of more than 7-10 units is regarded as strong evidence in favor of the model with the smaller DIC.

In this case the difference in DIC values is about 5.5, so we have some evidence for $f > 0$ model for these data, even though they are from a human population. But the evidence is not very strong. This is consistent with the weak evidence for a departure from Hardy-Weinberg that was revealed in the $\chi^2$ analysis.