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Let $\overline{\theta }=\mathbb{E}(\theta \mid y)$ and $\overline{D}=\mathbb{E}\{D(\theta )\mid y\}$. The usual effective parameter count in DIC isWith the locally flat prior distribution, $\overline{\theta }\approx \hat{\theta }$ and the mean of the chi-squared distribution excess is $p$. Thus $\overline{D}\approx D(\hat{\theta })+p$, while $D(\overline{\theta })\approx D(\hat{\theta })$, giving $p_D\approx p$.

More generally, the quadratic posterior deviance moments give the effective count. Within the same approximation, let the posterior covariance matrix be $\Sigma$. The expectation of a quadratic form giveswhereas evaluation at $\overline{\theta }$ omits the trace term. Therefore $p_D\approx \mathrm{tr}(J\Sigma )$. For a flat prior, $\Sigma \approx J^{-1}$ and the trace is $p$; informative prior shrinkage can reduce this effective count.