PlanetMath: multidimensional Gaussian integral

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (197)
Orphanage
Unclass'd
Unproven (498)
Corrections (25)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

multidimensional Gaussian integral

(Theorem)

Let $\mathbf{x} = [x_1\ x_2\ \ldots\ x_n]^T$ and $d^n \mathbf{x} \equiv \prod_{i=1}^{n} d x_i$ .

Theorem 1 Let be a symmetric positive definite matrix and $f: R^n \to R$ , where $f(x) = \exp{(- \frac{1}{2} \mathbf{x}^T \mathbf{K}^{-1} \mathbf{x})}$ . Then

$\displaystyle \int e^{-\frac{1}{2} \mathbf{x}^T \mathbf{K}^{-1} \mathbf{x}} d^n \mathbf{x} = \left((2\pi)^n \vert\mathbf{K}\vert \right)^{\frac{1}{2}}$

(1)

where $\vert\mathbf{K}\vert = \det{\mathbf{K}}$ .

Proof. $\mathbf{K}^{-1}$ is real and symmetric (since $(\mathbf{K}^{-1})^{\mathrm{T}} = (\mathbf{K}^{\mathrm{T}})^{-1} = \mathbf{K}^{-1})$ . For convenience, let $\mathbf{A} = \mathbf{K}^{-1}$ . We can decompose $\mathbf{A}$ into $\mathbf{A} = \mathbf{T} \mathbf{\Lambda} \mathbf{T}^{-1}$ , where $\mathbf{T}$ is an orthonormal ( $\mathbf{T}^{\mathrm{T}} \mathbf{T} = \mathbf{I}$ ) matrix of the eigenvectors of $\mathbf{A}$ and $\mathbf{\Lambda}$ is a diagonal matrix of the eigenvalues of $\mathbf{A}$ . Then

$\displaystyle \int e^{-\frac{1}{2} \mathbf{x}^{\mathrm{T}} \mathbf{A} \mathbf{x... ...hrm{T}} \mathbf{T} \mathbf{\Lambda} \mathbf{T}^{-1} \mathbf{x}} d^n \mathbf{x}.$

(2)

Because $\mathbf{T}$ is orthonormal, we have $\mathbf{T}^{-1} = \mathbf{T}^{\mathrm{T}}$ . Now define a new vector variable $\mathbf{y} \equiv \mathbf{T}^{\mathrm{T}} \mathbf{x}$ , and substitute:

$\displaystyle \int e^{-\frac{1}{2} \mathbf{x}^{\mathrm{T}} \mathbf{T} \mathbf{\Lambda} \mathbf{T}^{-1} \mathbf{x}} d^n \mathbf{x}$	$\displaystyle = \int e^{-\frac{1}{2} \mathbf{x}^{\mathrm{T}} \mathbf{T} \mathbf{\Lambda} \mathbf{T}^{\mathrm{T}} \mathbf{x}} d^n \mathbf{x}$	(3)
	$\displaystyle = \int e^{-\frac{1}{2} \mathbf{y}^{\mathrm{T}} \mathbf{\Lambda}\mathbf{y}} \vert\mathbf{J}\vert d^n \mathbf{y}$	(4)

where $\vert\mathbf{J}\vert$ is the determinant of the Jacobian matrix $J_{mn} = \frac{\partial{x_m}}{\partial{y_n}}$ . In this case, $\mathbf{J} = \mathbf{T}$ and thus $\vert\mathbf{J}\vert = 1$ .

Since $\mathbf{\Lambda}$ is diagonal, the integral may be separated into the product of independent Gaussian distributions, each of which we can integrate separately using the well-known formula

$\displaystyle \int e^{-\frac{1}{2} a t^2} dt = \left(\frac{2 \pi}{a}\right)^{\frac{1}{2}}.$

(5)

Carrying out this program, we get

$\displaystyle \int e^{-\frac{1}{2} \mathbf{y}^{\mathrm{T}} \mathbf{\Lambda}\mathbf{y}} d^n \mathbf{y}$	$\displaystyle = \prod_{k=1}^{n} \int e^{-\frac{1}{2} \lambda_k y_k^2} d y_k$	(6)
	$\displaystyle = \prod_{k=1}^{n} \left(\frac{2 \pi}{\lambda_k}\right)^{\frac{1}{2}}$	(7)
	$\displaystyle = \left(\frac{(2 \pi)^n}{\prod_{k=1}^{n}\lambda_k}\right)^{\frac{1}{2}}$	(8)
	$\displaystyle = \left(\frac{(2 \pi)^n}{\vert\mathbf{\Lambda}\vert}\right)^{\frac{1}{2}}.$	(9)

Now, we have $\vert\mathbf{A}\vert = \vert\mathbf{T} \mathbf{\Lambda} \mathbf{T}^{-1}\vert =... ...ert\mathbf{\Lambda}\vert \vert\mathbf{T}^{-1}\vert = \vert\mathbf{\Lambda}\vert$ , so this becomes

$\displaystyle \int e^{-\frac{1}{2} \mathbf{x}^{\mathrm{T}} \mathbf{A} \mathbf{x... ...\mathbf{x} = \left(\frac{(2 \pi)^n}{\vert\mathbf{A}\vert}\right)^{\frac{1}{2}}.$

(10)

Substituting back in for $\mathbf{K}^{-1}$ , we get

$\displaystyle \int e^{-\frac{1}{2} \mathbf{x}^{\mathrm{T}} \mathbf{K}^{-1} \mat... ...ght)^{\frac{1}{2}} = \left((2 \pi)^n \vert\mathbf{K}\vert\right)^{\frac{1}{2}},$

(11)

as promised.

"multidimensional Gaussian integral" is owned by Mathprof. [ full author list (3) | owner history (3) ]

(view preamble)

Log in to rate this entry.
(view current ratings)

Cross-references: distributions, Gaussian, independent, product, separated, integral, diagonal, Jacobian matrix, determinant, variable, vector, diagonal matrix, eigenvectors, orthonormal, real, proof, matrix, symmetric
There is 1 reference to this entry.

This is version 19 of multidimensional Gaussian integral, born on 2002-02-13, modified 2006-10-14.
Object id is 1914, canonical name is MultidimensionalGaussianIntegral.
Accessed 11003 times total.

Classification:

AMS MSC:	60B11 (Probability theory and stochastic processes :: Probability theory on algebraic and topological structures :: Probability theory on linear topological spaces)
	62H10 (Statistics :: Multivariate analysis :: Distribution of statistics)
	62H99 (Statistics :: Multivariate analysis :: Miscellaneous)

Pending Errata and Addenda

None.[ View all 11 ]

Discussion

forum policy

No messages.

Interact