Laplace's method

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In mathematics, Laplace's method, named after Pierre-Simon Laplace, is a technique used to approximate integrals of the form

${\displaystyle \int _{a}^{b}e^{Mf(x)}\,dx,}$

where ${\displaystyle f}$ is a twice-differentiable function, ${\displaystyle M}$ is a large number, and the endpoints ${\displaystyle a}$ and ${\displaystyle b}$ could be infinite. This technique was originally presented in the book by Laplace (1774).

In Bayesian statistics, Laplace's approximation can refer to either approximating the posterior normalizing constant with Laplace's method or approximating the posterior distribution with a Gaussian centered at the maximum a posteriori estimate.^[1][2] Laplace approximations are used in the integrated nested Laplace approximations method for fast approximations of Bayesian inference.