Posterior mean

The posterior mean is the expected value of a parameter under its posterior distribution; under squared-error loss it is the Bayes estimator.

Write the log posterior kernel as $H_n(\theta )=\log p(\theta )+{\sum }_i\log f(Y_i\mid \theta )$, with a unique concentrating interior mode $\tilde{\theta }$ and $J_n=-H_n^{\prime \prime }(\tilde{\theta })>0$. Apply Laplace's method to the numerator and denominator of the expected value under the posterior distribution. Their common leading Gaussian factor cancels. Under smoothness and localization, the first correction is $g^{\prime \prime }(\tilde{\theta })/(2J_n)+g^{\prime }(\tilde{\theta })H_n^{\prime \prime \prime }(\tilde{\theta })/(2J_n^2)$, of order $n^{-1}$ when the posterior curvature and log-kernel derivatives have their usual order $n$.