 Laplace approximation

For a smooth phase and a continuous amplitude nonzero 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.