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 ${\displaystyle f(x)}$ 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 Laplace (1774).

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