Laplace's method

Asymptotic evaluation of integrals dominated by a single interior maximizer of the exponent.

Laplace’s method

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. It is the real-variable analogue of the complex saddle-point-method .

One-dimensional interior-maximum theorem

Assume:

$f$ is twice continuously differentiable on $(a,b)$ and has a unique maximizer $x_0\in(a,b)$ ,
$f''(x_0)<0$ ,
$g$ is continuous at $x_0$ and $g(x_0)\ne 0$ .

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)|}}.

More refined expansions are obtained by keeping higher-order terms of the local Taylor expansion.

Sketch of why it works

Near the maximizer $x_0$ , expand $f$ using a second-order Taylor approximation:

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 $x_0$ , and the integral is well-approximated by replacing the exponent with its quadratic part and extending the local integral to all of $\mathbb{R}$ , yielding a Gaussian integral.

Multidimensional version (informal)

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

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

if $f$ has a unique interior maximizer $x_0$ with negative definite Hessian $H = \nabla^2 f(x_0)$ , then typically

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 $n^{-1}$ -type 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 $f''(x_0)=0$ (degenerate maximum), the scale becomes $n^{-\alpha}$ for some $\alpha$ determined by the first nonzero derivative at $x_0$ .

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 ).