 Solution

The alternative effective count isUnder the locally flat-prior approximation, the constant $D(\hat{\theta })$ contributes no variance, and the remaining chi-squared distribution term has variance $2p$. Hence $p_V\approx p$, the same calibration as the usual effective parameter count in DIC. It can be estimated by half the sample variance of deviance draws, without evaluating deviance at a parameter average.

This is a calibrated alternative, not a general identity with $p_D$. For example, with a quadratic deviance and a multivariate normal distribution posterior of mean $m$ and covariance $\Sigma$, putting $b=m-\hat{\theta }$ givesThey agree when $b=0$ and $\Sigma =J^{-1}$, but informative priors can change both the covariance and the displacement of the posterior mean. Their equality should therefore not be assumed outside the stated regime.