Laplace's method - Knowlpedia

Laplace's method approximates integrals of the form

I(n)=\int_a^b e^{n f(x)}\,g(x)\,dx

for large $n$ , when the main contribution comes from a neighborhood of the point where $f$ is maximal.

Let $a<b$ be finite. Assume that $f\in C^2([a,b])$ has a unique global maximizer $x_0\in(a,b)$ , that $f''(x_0)<0$ , and that $g$ is continuous on $[a,b]$ with $g(x_0)\ne 0$ . Then, as $n\to\infty$ ,

\int_a^b e^{n f(x)}\,g(x)\,dx \sim e^{n f(x_0)}\,g(x_0)\,\sqrt{\frac{2\pi}{n\,|f''(x_0)|}}.

Why it works

Near $x_0$ , the second-order Taylor approximation is

f(x) = f(x_0) + \tfrac12 f''(x_0)(x-x_0)^2 + \text{higher-order terms}.

For large $n$ , the factor $e^{n f(x)}$ is sharply peaked at $x_0$ . Replacing the exponent locally by its quadratic part yields the displayed Gaussian factor. More refined expansions require stronger smoothness and retain higher-order Taylor terms.

Multidimensional version (informal)

For $x\in\mathbb{R}^d$ ,

I(n)=\int_{\Omega} e^{n f(x)} g(x)\,dx,

if suitable global hypotheses hold and $f$ has a unique interior maximizer $x_0$ with negative-definite Hessian $H=\nabla^2f(x_0)$ , then the corresponding leading term is

I(n)\sim e^{n f(x_0)}\,g(x_0)\,(2\pi/n)^{d/2}\,(\det(-H))^{-1/2}.

Common variants and caveats

If the maximum occurs at an endpoint ( $x_0=a$ or $x_0=b$ ), the leading order usually changes, often to an $n^{-1}$ scale rather than $n^{-1/2}$ .
If there are multiple maximizers, the leading term is typically the sum of contributions from each (when they are well-separated and nondegenerate).
If $f''(x_0)=0$ , the maximum is degenerate and the power of $n$ depends on the first nonzero higher derivative.

Where it shows up

Normal approximations and local limit behavior.
Large- $n$ asymptotics of combinatorial sums via integral representations.
As the real-variable building block for the complex steepest-descent/saddle-point techniques (see saddle-point-method).