Matrix normal distribution

Matrix normal distribution
Matrix normal distribution
parameters:  \mathbf{M}\, - mean
\mathbf{\Omega} \, - row covariance
\mathbf{\Sigma} \, - column covariance.
Parameters are matrices (all of them).
support: \mathbf{W}\! is a matrix
pdf: (see Notebox below)
mean: \mathbf{M}
Notebox Probability density function:
\frac{\exp\left(    -\frac{1}{2}    \mbox{tr}\left[      {\boldsymbol  \Omega}^{-1}      (\mathbf{X} - \mathbf{M})^{T}      {\boldsymbol  \Sigma}^{-1}      (\mathbf{X} - \mathbf{M})    \right]  \right)}{(2\pi)^{np/2} |{\boldsymbol \Omega}|^{n/2}  |{\boldsymbol  \Sigma}|^{p/2}}

The matrix normal distribution is a probability distribution that is a generalization of the normal distribution to matrix-valued random variables.

Contents

Definition

The probability density function for the random matrix X (n × p) that follows the matrix normal distribution has the form:

p(\mathbf{X}|\mathbf{M}, {\boldsymbol \Omega}, {\boldsymbol \Sigma})= \frac{\exp\left(    -\frac{1}{2}    \mbox{tr}\left[      {\boldsymbol  \Omega}^{-1}      (\mathbf{X} - \mathbf{M})^{T}      {\boldsymbol  \Sigma}^{-1}      (\mathbf{X} - \mathbf{M})    \right]  \right)}{(2\pi)^{np/2} |{\boldsymbol \Omega}|^{n/2}  |{\boldsymbol  \Sigma}|^{p/2}}

where M is n × p, Ω is p × p and Σ is n × n. There are several ways to define the two covariance matrices. One possibility is


    {\boldsymbol  \Sigma} = E[  (\mathbf{X} - \mathbf{M})(\mathbf{X} - \mathbf{M})^{T}]\;,\;

    {\boldsymbol  \Omega} = E[  (\mathbf{X} - \mathbf{M})^{T} (\mathbf{X} - \mathbf{M})]/c

where c is a constant which depends on Σ and ensures appropriate power normalization.

The matrix normal is related to the multivariate normal distribution in the following way:

\mathbf{X} \sim MN_{n\times p}(\mathbf{M}, {\boldsymbol \Omega}, {\boldsymbol \Sigma}),

if and only if


    \mathrm{vec}\;\mathbf{X} \sim N_{np}(\mathrm{vec}\;\mathbf{M}, 
    {\boldsymbol \Omega}\otimes{\boldsymbol \Sigma}),

where \otimes denotes the Kronecker product and \mathrm{vec}\;\mathbf{M} denotes the vectorization of \mathbf{M}.

Example

Matrix Normal random variables arise from a sample identically distributed multivariate Normal random variables with possible dependence between the vectors. For example, if \mathbf{Y} is an (n × p) matrix whose rows are independent with distribution N_{p}(\mathbf{0}, {\boldsymbol \Sigma}) then \mathbf{Y} \sim MN_{n\times p}(\mathbf{0}, \mathbf{I}_n, {\boldsymbol \Sigma}), where \mathbf{I}_n is the (n × n) identity matrix. On the other hand, the columns of \mathbf{X=}{\boldsymbol \Gamma}\mathbf{Y} are dependent but identically distributed multivariate Normal random variables. Furthermore, \mathbf{X+M}\sim MN_{n\times p}(\mathbf{M}, {\boldsymbol \Omega, \Sigma}), where {\boldsymbol \Omega = \Gamma\Gamma}^T.

Relation to other distributions

Dawid (1981) provides a discussion of the relation of the matrix-valued normal distribution to other distributions, including the Wishart distribution, Inverse Wishart distribution and a "matrix-t distribution", but uses different notation from that employed here.

See also

References


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