= Solution
Conditional on the observed data, the preceding <multivariate normal distribution> approximation implies that
$$
Z=J^{1/2}(\theta-\widehat\theta)\ \dot\sim\ N_p(0,I).
$$
The deviance expansion consequently gives
$$
D(\theta)-D(\widehat\theta)\approx(\theta-\widehat\theta)^TJ(\theta-\widehat\theta)=Z^TZ.
$$
The sum of squares of $p$ independent standard <normal distribution> variables has <chi-squared distribution> with $p$ degrees of freedom. Hence, as a statement about the <posterior distribution> of the <Bayesian deviance>,
$$
\boxed{D(\theta)\mid y\ \dot\sim\ D(\widehat\theta)+\chi_p^2.}
$$
The data are fixed here and the parameter varies according to its posterior; this is not a claim about resampling the data.
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