Editing Laplace's method (section)

{{short description|Method for approximate evaluation of integrals}}
{{Distinguish|Laplace's approximation|Laplace smoothing}}
{{cleanup tone|date=March 2024}}
In [[mathematics]], '''Laplace's method''', named after [[Pierre-Simon Laplace]], is a technique used to approximate [[integral]]s of the form

:<math>\int_a^b e^{Mf(x)} \, dx,</math>

where <math>f</math> is a twice-[[Derivative|differentiable]] [[function (mathematics)|function]], <math>M</math> is a large [[number]], and the endpoints <math>a</math> and <math>b</math> could be infinite. This technique was originally presented in the book by {{harvtxt|Laplace|1774}}.

In [[Bayesian statistics]], [[Laplace's approximation]] can refer to either approximating the [[Normalizing constant|posterior normalizing constant]] with Laplace's method or approximating the posterior distribution with a [[Normal distribution|Gaussian]] centered at the [[Maximum a posteriori estimation|maximum a posteriori estimate]].<ref>{{cite journal| first1=Luke | last1=Tierney | first2=Joseph B. | last2=Kadane |title= Accurate Approximations for Posterior Moments and Marginal Densities |journal = J. Amer. Statist. Assoc. | year=1986 | volume=81 | number=393 |pages=82–86 | doi=10.1080/01621459.1986.10478240 }}</ref><ref>{{cite book |first1=M. Antónia |last1=Amaral Turkman |first2=Carlos Daniel |last2=Paulino |first3=Peter |last3=Müller |title=Computational Bayesian Statistics: An Introduction |location= |publisher=Cambridge University Press |year=2019 |isbn=978-1-108-70374-1 |chapter=Methods Based on Analytic Approximations |pages=150–171 }}</ref> Laplace approximations are used in the [[integrated nested Laplace approximations]] method for fast approximations of [[Bayesian inference]].