= Quadratic posterior deviance moments
{title2=$p_D=\operatorname{tr}(J\Sigma)$}
Suppose the <Bayesian deviance> is locally $D(\theta)=D(\widehat\theta)+(\theta-\widehat\theta)^TJ(\theta-\widehat\theta)$ with positive-definite $J$, and the posterior is normal with mean $m$ and covariance $\Sigma$. Writing $b=m-\widehat\theta$, expectation and variance of the <quadratic form> give
$$
\mathbb E[D(\theta)\mid y]=D(\widehat\theta)+b^TJb+\operatorname{tr}(J\Sigma),
$$
$$
\operatorname{Var}[D(\theta)\mid y]=2\operatorname{tr}\{(J\Sigma)^2\}+4b^TJ\Sigma Jb.
$$
Consequently the usual <effective parameter count in DIC> is $p_D=\operatorname{tr}(J\Sigma)$, whereas half the deviance variance is $\operatorname{tr}\{(J\Sigma)^2\}+2b^TJ\Sigma Jb$. A locally flat prior gives $m=\widehat\theta$ and $\Sigma=J^{-1}$, so both counts equal the parameter dimension. Informative priors generally destroy their equality. The expectation formula requires only the first two moments; normality is used for the variance formula.
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