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).
- 1 The idea of Laplace's method
- 2 General theory of Laplace's method
- 3 Other formulations
- 4 Laplace's method extension: Steepest descent
- 5 Further generalizations
- 6 Complex integrals
- 7 Example 1: Stirling's approximation
- 8 Example 2: parameter estimation and probabilistic inference
- 9 See also
- 10 References
The idea of 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 Mƒ(x0) and Mƒ(x) will stay the same (since Mƒ(x0)/Mƒ(x) = ƒ(x0)/ƒ(x)), but it will grow exponentially in the function (see figure)
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 .
We can expand ƒ(x) around x0 by Taylor's theorem,
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
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
(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
A generalization of this method and extension to arbitrary precision is provided by Fog (2008).
Formal statement and proof:
Assume that is a twice differentiable function on with the unique point such that . Assume additionally that .
Let . Then by the continuity of there exists such that if then . By Taylor's Theorem, for any , .
Then we have the following lower bound:
where the last equality was obtained by a change of variables . Remember that so that is why we can take the square root of its negation.
If we divide both sides of the above inequality by and take the limit we get:
since this is true for arbitrary we get the lower bound:
Note that this proof works also when or (or both).
The proof of the upper bound is similar to the proof of the lower bound but there are a few inconveniences. Again we start by picking an but in order for the proof to work we need small enough so that . Then, as above, by continuity of and Taylor's Theorem we can find so that if , then . Lastly, by our assumptions (assuming are finite) there exists an such that if , then .
Then we can calculate the following upper bound:
If we divide both sides of the above inequality by and take the limit we get:
Since is arbitrary we get the upper bound:
And combining this with the lower bound gives the result.
Note that the above proof obviously fails when or (or both). To deal with these cases, we need some extra assumptions. A sufficient (not necessary) assumption is that for , the integral is finite, and that the number as above exists (note that this must be an assumption in the case when the interval is infinite). The proof proceeds otherwise as above, but the integrals
must be approximated by
instead of as above, so that when we divide by , we get for this term
whose limit as is . The rest of the proof (the analysis of the interesting term) proceeds as above.
The given condition in the infinite interval case is, as said above, sufficient but not necessary. However, the condition is fulfilled in many, if not in most, applications: the condition simply says that the integral we are studying must be well-defined (not infinite) and that the maximum of the function at must be a "true" maximum (the number must exist). There is no need to demand that the integral is finite for but it is enough to demand that the integral is finite for some .
Laplace's approximation is sometimes written as
where is positive.
Importantly, the accuracy of the approximation depends on the variable of integration, that is, on what stays in and what goes into .
In the multivariate case where is a -dimensional vector and is a scalar function of , Laplace's approximation is usually written as:
where is the Hessian matrix of evaluated at .
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
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).
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.
For complex integrals in the form:
with t >> 1, we make the substitution t = iu and the change of variable s = c + ix to get the Laplace bilateral transform:
Example 1: Stirling's approximation
Laplace's method can be used to derive Stirling's approximation
for a large integer N.
From the definition of the Gamma function, we have
Now we change variables, letting
Plug these values back in to obtain
This integral has the form necessary for Laplace's method with
which is twice-differentiable:
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
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.
- 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, CiteSeerX: 10.1.1.91.2064.
- 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, doi:10.2307/2946540.
- 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, doi:10.1080/03610910701790269.
- 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).