Laplace's method

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

∫ a b e M f ( x ) d x , {\displaystyle \int _{a}^{b}e^{Mf(x)}\,dx,}

where f {\displaystyle f} is a twice-differentiable function, M {\displaystyle M} is a large number, and the endpoints a {\displaystyle a} and b {\displaystyle b} may 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. Laplace approximations are used in the integrated nested Laplace approximations method for fast approximations of Bayesian inference.

Concept

Let the function f ( x ) {\displaystyle f(x)} have a unique global maximum at x 0 {\displaystyle x_{0}} . M > 0 {\displaystyle M>0} is a constant here. The following two functions are considered:

g ( x ) = M f ( x ) , h ( x ) = e M f ( x ) . {\displaystyle {\begin{aligned}g(x)&=Mf(x),\\h(x)&=e^{Mf(x)}.\end{aligned}}}

Then, x 0 {\displaystyle x_{0}} is the global maximum of g {\displaystyle g} and h {\displaystyle h} as well. Hence:

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