The multivariate Gaussian distribution has the following form:

Among them,D mean vector, 是Covariance matrix,The ith row and the JTH column is the covariance of the ith variable and the JTH variable.Represents the determinant of the covariance matrix.

A two-dimensional Gaussian distribution is shown below (from Wikipedia), where each dimension is gaussian:

This paper is mainly about the origin of formula (1).

Prior knowledge: Jacobian matrix and Jacobian determinant

setPhi is a function whose input is a vectorAnd the output is a vector:

So the Jacobian is an M by N matrix:

Because the matrix describes the motion transformation in vector space, and the Jacobian is regarded asIs the pointConversion to pointOr from an N-dimensional Euclidean space to an m-dimensional Euclidean space.

If m is equal to n, you can define the JacobianThe determinant of theta, which is thetaJacobian determinant.

In the calculus substitution, that is, given the ratio of n dimensional volume from x to y,

Geometric meaning of two-dimensional Jacobian matrix

In the two-dimensional case (which is visually illustrated), the Jacobian represents the ratio of the area element in the XY plane to the area element in the UV plane.

set

The Jacobian is:

As the figure shows: dA represents the area of the parallelogram spanized by dx and dy. If du and dV are sufficiently close to 0, then dA:

Double integral substitution:

And the same goes for the n dimension.

Multivariate Gaussian distribution

Firstly, the univariate standard normal distribution is considered, and the probability density function is:

Then consider the n-dimensional independent standard Gaussian distribution, which is the joint distribution of n independent one-dimensional standard normal distribution random variables:

And just to make it easier, let’s write it in vector form, let’s sayFormula (3) is written as follows:

In general, let’s say 由 The linear transformation of is:

Among them A istheNonsingular matrix.It’s an n-dimensional vector

Can be the 用 Said:

Note that,Equation (6) Jacobian determinant of linear transformationis, so:

set,, according to the definition of joint probability distribution density, can be:

So the vectorThe joint probability probability density function of

So I get equation (1).

It can be seen that the multivariable Gaussian distribution is the extension of the univariate Gaussian distribution to multidimensional.

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