 Solution

For retained Markov chain Monte Carlo draws ${\theta }^{(1)},\dots ,{\theta }^{(S)}$, calculate both the parameter average $\hat{m}=S^{-1}{\sum }_s{\theta }^{(s)}$ and the average deviance ${\hat{D}}_{\mathrm{av}}=S^{-1}{\sum }_{s}D({\theta }^{(s)})$. Two estimates areThe first uses the approximate equality of the posterior mean and maximum-likelihood estimator. The second subtracts the mean $p$ of the chi-squared distribution deviance excess. These are different operations: evaluation after averaging versus averaging after evaluation. Both rely on a well-mixed chain and the regular locally flat-prior approximation. The smallest sampled deviance is an upper bound on the true minimum, but merely taking that minimum can be inefficient in high dimension because few draws approach the mode closely.