= Solution
Using the same <Bayesian deviance> convention, the <deviance information criterion> and <Akaike information criterion> are
$$
\boxed{\operatorname{DIC}=\overline D+p_D=D(\overline\theta)+2p_D,\qquad
\operatorname{AIC}=D(\widehat\theta)+2p.}
$$
Both 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, $D(\overline\theta)\approx D(\widehat\theta)$ and $p_D\approx p$, so
$$
\boxed{\operatorname{DIC}\approx D(\widehat\theta)+2p=\operatorname{AIC}.}
$$
All 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.
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