Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 36 / 3 / c / Solution

Conditional on the observed data, the preceding multivariate normal distribution approximation implies thatThe deviance expansion consequently givesThe 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,The data are fixed here and the parameter varies according to its posterior; this is not a claim about resampling the data.