 Solution

Using the same Bayesian deviance convention, the deviance information criterion and Akaike information criterion areBoth combine a lack-of-fit term with a complexity penalty; smaller values are preferred among comparable models. Under the locally flat prior distribution and the regular multivariate normal distribution approximation, $D(\overline{\theta })\approx D(\hat{\theta })$ and $p_D\approx p$, soAll fitted parameters counted in the likelihood, including unknown dispersion parameters, belong in $p$. The equivalence need not hold for informative priors, nonregular models, or different choices of the likelihood's latent-variable representation.