= Solution
The alternative effective count is
$$
\boxed{p_V=\frac12\operatorname{Var}\{D(\theta)\mid y\}.}
$$
Under the locally flat-prior approximation, the constant $D(\widehat\theta)$ 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 $b=m-\widehat\theta$ gives
$$
p_V=\operatorname{tr}\{(J\Sigma)^2\}+2b^TJ\Sigma Jb,
\qquad p_D=\operatorname{tr}(J\Sigma).
$$
They 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.
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