Matrix Normal Distribution
These functions provide the density and random number generation for the matrix normal distribution.
This is data or parameters in the form of a matrix with n rows and k columns.
This is mean matrix with n rows and k columns.
This is a n x n positive-definite scale matrix.
This is a k x k positive-definite scale matrix.
Application: Continuous Multivariate Matrix
Density: p(theta) = exp(-0.5tr[V^(-1)(X-M)'U^(-1)(X-M)])/[(2*pi)^(nk/2)|V|^(n/2)|U|(k/2)]
Notation 1: theta ~ MN[n x k](M, U, V)
Notation 2: p(theta) = MN[n x k](theta | M, U, V)
Parameter 1: location n x k matrix M
Parameter 2: positive-definite n x n scale matrix U
Parameter 3: positive-definite k x k scale matrix V
Mean: E(theta) = M
The matrix normal distribution is also called the matrix Gaussian, matrix-variate normal, or matrix-variate Gaussian distribution. It is a generalization of the multivariate normal distribution to matrix-valued random variables.
An example of the use of a matrix normal distribution is multivariate regression, in which there is a j x k matrix of regression effects of j predictors for k dependent variables. For univariate regression, having only one dependent variable, the j regression effects may be multivariate normally distributed. For multivariate regression, this multivariate normal distribution may be extended to a matrix normal distribution to account for relationships of the regression effects across k dependent variables. In this example, the matrix normal distribution is the conjugate prior distribution for these regression effects.
The matrix normal distribution has two covariance matrices, one for the rows and one for the columns. When U is diagonal, the rows are independent. When V is diagonal, the columns are independent.
dmatrixnorm gives the density and
rmatrixnorm generates random deviates.
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