= Posterior expectation by Laplace approximation
{title2=$\mathbb E[g(\theta)\mid Y]\approx g(\widetilde\theta)$}
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 $\widetilde\theta$ and $J_n=-H_n''(\widetilde\theta)>0$. 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 $g''(\widetilde\theta)/(2J_n)+g'(\widetilde\theta)H_n'''(\widetilde\theta)/(2J_n^2)$, of order $n^{-1}$ when the posterior curvature and log-kernel derivatives have their usual order $n$.
Back to article page
