= Laplace approximation
{c}
{title2=$\int e^{nh(\theta)}a(\theta)d\theta\sim e^{nh(\widehat\theta)}a(\widehat\theta)(2\pi/n)^{p/2}|-h''(\widehat\theta)|^{-1/2}$}
At a unique interior maximum with positive-definite negative <Hessian matrix>, a quadratic expansion and rescaling by $\sqrt n$ give the displayed Gaussian integral, provided the tails are negligible. In a regular <Bayesian inference> it approximates the integrated <likelihood> by the <likelihood> at its dominant point times the prior density and inverse square root of local curvature. Boundary modes, singular Hessians, and multiple equally dominant modes need modified formulas.
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