World Library  
Flag as Inappropriate
Email this Article

Laplace's method

Article Id: WHEBN0000642006
Reproduction Date:

Title: Laplace's method  
Author: World Heritage Encyclopedia
Language: English
Subject: Asymptotic analysis, Perturbation theory, LaplacesDemon, Laplace principle (large deviations theory), Saddle point
Collection:
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Laplace's method

See also: Additive smoothing (Laplace smoothing) a method of smoothing of a statistical estimator

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

\int_a^b\! e^{M f(x)} \, dx

where ƒ(x) is some twice-differentiable function, M is a large number, and the integral endpoints a and b could possibly be infinite. This technique was originally presented in Laplace (1774, pp. 366–367).

Contents

  • The idea of Laplace's method 1
  • General theory of Laplace's method 2
  • Other formulations 3
  • Laplace's method extension: Steepest descent 4
  • Further generalizations 5
  • Complex integrals 6
  • Example 1: Stirling's approximation 7
  • Example 2: parameter estimation and probabilistic inference 8
  • See also 9
  • Notes 10
  • References 11

The idea of Laplace's method

The function e(x), in blue, is shown on top for M = 0.5, and at the bottom for M = 3. Here, ƒ(x) = sin x/x, with a global maximum at x0 = 0. It is seen that as M grows larger, the approximation of this function by a Gaussian function (shown in red) is getting better. This observation underlies Laplace's method.

Assume that the function ƒ(x) has a unique global maximum at x0. Then, the value ƒ(x0) will be larger than other values ƒ(x). If we multiply this function by a large number M, the ratio between (x0) and (x) will stay the same (since (x0)/(x) = ƒ(x0)/ƒ(x)), but it will grow exponentially in the function (see figure)

e^{M f(x)}. \,

Thus, significant contributions to the integral of this function will come only from points x in a neighborhood of x0, which can then be estimated.

General theory of Laplace's method

To state and motivate the method, we need several assumptions. We will assume that x0 is not an endpoint of the interval of integration, that the values ƒ(x) cannot be very close to ƒ(x0) unless x is close to x0, and that the second derivative f''(x_0)<0.

We can expand ƒ(x) around x0 by Taylor's theorem,

\begin{align} f(x) & = f(x_0) + f'(x_0)(x-x_0) + \\ & + \frac{1}{2} f''(x_0)(x-x_0)^2 + R \end{align}

where R = O\left((x-x_0)^3\right).

Since ƒ has a global maximum at x0, and since x0 is not an endpoint, it is a stationary point, so the derivative of ƒ vanishes at x0. Therefore, the function ƒ(x) may be approximated to quadratic order

f(x) \approx f(x_0) - \frac{1}{2} |f''(x_0)| (x-x_0)^2

for x close to x0 (recall that the second derivative is negative at the global maximum ƒ(x0)). The assumptions made ensure the accuracy of the approximation

\int_a^b\! e^{M f(x)}\, dx\approx e^{M f(x_0)}\int_a^b e^{-M|f''(x_0)| (x-x_0)^2/2} \, dx

(see the picture on the right). This latter integral is a Gaussian integral if the limits of integration go from −∞ to +∞ (which can be assumed because the exponential decays very fast away from x0), and thus it can be calculated. We find

\int_a^b\! e^{M f(x)}\, dx\approx \sqrt{\frac{2\pi}{M|f''(x_0)|}}e^{M f(x_0)} \text { as } M\to\infty. \,

A generalization of this method and extension to arbitrary precision is provided by Fog (2008).

Formal statement and proof:

Assume that f(x) is a twice differentiable function on [a,b] with x_0 \in [a,b] the unique point such that f(x_0) = \max_ f(x) . Assume additionally that f''(x_0)<0 .

Then,

\lim_{n \to +\infty} \frac{\int_a^b e^{nf(x)} \, dx}{e^{nf(x_0)} \sqrt{\frac{2 \pi}{n (-f''(x_0))}}} =1

This method relies on 4 basic concepts such as

Based on these four concepts, we can derive the relative error of this Laplace's method.

Other formulations

Laplace's approximation is sometimes written as

\begin{align} & \int_a^b\! h(x) e^{M g(x)}\, dx \\ & \approx \sqrt{\frac{2\pi}{M|g''(x_0)|}} h(x_0) e^{M g(x_0)} \text { as } M\to\infty \, \end{align}

where h is positive.

Importantly, the accuracy of the approximation depends on the variable of integration, that is, on what stays in g(x) and what goes into h(x).[1]

In the multivariate case where \mathbf{x} is a d-dimensional vector and f(\mathbf{x}) is a scalar function of \mathbf{x}, Laplace's approximation is usually written as:

\begin{align} & \int e^{M f(\mathbf{x})}\, d\mathbf{x} \approx \\ & \approx \left(\frac{2\pi}{M}\right)^{d/2} |-H(f)(\mathbf{x}_0)|^{-1/2} e^{M f(\mathbf{x}_0)} \\ & \text { as } M\to\infty \, \end{align}

where H(f)(\mathbf{x}_0) is the Hessian matrix of f evaluated at \mathbf{x}_0 and where |\cdot| denotes matrix determinant. Analogously to the univariate case, the Hessian is required to be negative definite.[2]

By the way, although \mathbf{x} denotes a d-dimensional vector, the term d\mathbf{x} denotes an Infinitesimal volume here, i.e. d\mathbf{x} := dx_1dx_2\cdots dx_d.

Laplace's method extension: Steepest descent

In extensions of Laplace's method, complex analysis, and in particular Cauchy's integral formula, is used to find a contour of steepest descent for an (asymptotically with large M) equivalent integral, expressed as a line integral. In particular, if no point x0 where the derivative of ƒ vanishes exists on the real line, it may be necessary to deform the integration contour to an optimal one, where the above analysis will be possible. Again the main idea is to reduce, at least asymptotically, the calculation of the given integral to that of a simpler integral that can be explicitly evaluated. See the book of Erdelyi (1956) for a simple discussion (where the method is termed steepest descents).

The appropriate formulation for the complex z-plane is

\begin{align} & \int_a^b\! e^{M f(z)}\, dz \approx \\ & \approx \sqrt{\frac{2\pi}{-Mf''(z_0)}}e^{M f(z_0)} \\ & \text{ as } M\to\infty. \, \end{align}

for a path passing through the saddle point at z0. Note the explicit appearance of a minus sign to indicate the direction of the second derivative: one must not take the modulus. Also note that if the integrand is meromorphic, one may have to add residues corresponding to poles traversed while deforming the contour (see for example section 3 of Okounkov's paper Symmetric functions and random partitions).

Further generalizations

An extension of the steepest descent method is the so-called nonlinear stationary phase/steepest descent method. Here, instead of integrals, one needs to evaluate asymptotically solutions of Riemann–Hilbert factorization problems.

Given a contour C in the complex sphere, a function ƒ defined on that contour and a special point, say infinity, one seeks a function M holomorphic away from the contour C, with prescribed jump across C, and with a given normalization at infinity. If ƒ and hence M are matrices rather than scalars this is a problem that in general does not admit an explicit solution.

An asymptotic evaluation is then possible along the lines of the linear stationary phase/steepest descent method. The idea is to reduce asymptotically the solution of the given Riemann–Hilbert problem to that of a simpler, explicitly solvable, Riemann–Hilbert problem. Cauchy's theorem is used to justify deformations of the jump contour.

The nonlinear stationary phase was introduced by Deift and Zhou in 1993, based on earlier work of Its. A (properly speaking) nonlinear steepest descent method was introduced by Kamvissis, K. McLaughlin and P. Miller in 2003, based on previous work of Lax, Levermore, Deift, Venakides and Zhou.

The nonlinear stationary phase/steepest descent method has applications to the theory of soliton equations and integrable models, random matrices and combinatorics.

Complex integrals

For complex integrals in the form:

\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} g(s)e^{st} \,ds

with t >> 1, we make the substitution t = iu and the change of variable s = c + ix to get the bilateral Laplace transform:

\frac{1}{2 \pi}\int_{-\infty}^\infty g(c+ix)e^{-ux}e^{icu} \, dx.

We then split g(c+ix) in its real and complex part, after which we recover u = t / i. This is useful for inverse Laplace transforms, the Perron formula and complex integration.

Example 1: Stirling's approximation

Laplace's method can be used to derive Stirling's approximation

N!\approx \sqrt{2\pi N} N^N e^{-N}\,

for a large integer N.

From the definition of the Gamma function, we have

N! = \Gamma(N+1)=\int_0^\infty e^{-x} x^N \, dx.

Now we change variables, letting

x = N z \,

so that

dx = N \, dz.

Plug these values back in to obtain

\begin{align} N! & = \int_0^\infty e^{-N z} \left(N z \right)^N N \, dz \\ & = N^{N+1}\int_0^\infty e^{-N z} z^N \, dz \\ & = N^{N+1}\int_0^\infty e^{-N z} e^{N\ln z} \, dz \\ & = N^{N+1}\int_0^\infty e^{N(\ln z-z)} \, dz. \end{align}

This integral has the form necessary for Laplace's method with

f \left( z \right) = \ln{z}-z

which is twice-differentiable:

f'(z) = \frac{1}{z}-1,\,
f''(z) = -\frac{1}{z^2}.\,

The maximum of ƒ(z) lies at z0 = 1, and the second derivative of ƒ(z) has the value −1 at this point. Therefore, we obtain

N! \approx N^{N+1}\sqrt{\frac{2\pi}{N}} e^{-N}=\sqrt{2\pi N} N^N e^{-N}.\,

Example 2: parameter estimation and probabilistic inference

Azevedo-Filho & Shachter 1994 reviews Laplace's method results (univariate and multivariate) and presents a detailed example showing the method used in parameter estimation and probabilistic inference under a Bayesian perspective. Laplace's method is applied to a meta-analysis problem from the medical domain, involving experimental data, and compared to other techniques.

See also

Notes

  1. ^ Butler, Ronald W (2007). Saddlepoint approximations and applications. Cambridge University Press.  
  2. ^ MacKay, David J. C. (September 2003). Information Theory, Inference and Learning Algorithms. Cambridge: Cambridge University Press.  

References

  • Azevedo-Filho, A.; Shachter, R. (1994), "Laplace's Method Approximations for Probabilistic Inference in Belief Networks with Continuous Variables", in Mantaras, R.; Poole, D., Uncertainty in Artificial Intelligence, San Francisco, CA: Morgan Kauffman, .  
  • Deift, P.; Zhou, X. (1993), "A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation", Ann. of Math. 137 (2): 295–368,  .
  • Erdelyi, A. (1956), Asymptotic Expansions, Dover .
  • Fog, A. (2008), "Calculation Methods for Wallenius' Noncentral Hypergeometric Distribution", Communications in Statistics, Simulation and Computation 37 (2): 258–273,  .
  • Kamvissis, S.; McLaughlin, K. T.-R.; Miller, P. (2003), "Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrödinger Equation", Annals of Mathematics Studies (Princeton University Press) 154 .
  • Laplace, P. S. (1774). Memoir on the probability of causes of events. Mémoires de Mathématique et de Physique, Tome Sixième. (English translation by S. M. Stigler 1986. Statist. Sci., 1(19):364–378).

This article incorporates material from saddle point approximation on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.