dist.Multivariate.Normal.Precision: Multivariate Normal Distribution: Precision Parameterization

In LaplacesDemon: Complete Environment for Bayesian Inference

Description

These functions provide the density and random number generation for the multivariate normal distribution, given the precision parameterization.

Usage

dmvnp(x, mu, Omega, log=FALSE) 
rmvnp(n=1, mu, Omega)

Arguments

x	This is data or parameters in the form of a vector of length k or a matrix with k columns.
n	This is the number of random draws.
mu	This is mean vector mu with length k or matrix with k columns.
Omega	This is the k x k precision matrix Omega.
log	Logical. If log=TRUE, then the logarithm of the density is returned.

Details

• Application: Continuous Multivariate

• Density: p(theta) = (2*pi)^(-p/2) * |Omega|^(1/2) * exp(-(1/2)*(theta-mu)^t Omega (theta-mu))

• Inventor: Unknown (to me, anyway)

• Notation 1: theta ~ MVN(mu, Omega^(-1))

• Notation 2: theta ~ N[k](mu, Omega^(-1))

• Notation 3: p(theta) = MVN(theta | mu, Omega^(-1))

• Notation 4: p(theta) = N[k](theta | mu, Omega^(-1))

• Parameter 1: location vector mu

• Parameter 2: positive-definite k x k precision matrix Omega

• Mean: E(theta) = mu

• Variance: var(theta) = Omega^(-1)

• Mode: mode(theta) = mu

The multivariate normal distribution, or multivariate Gaussian distribution, is a multidimensional extension of the one-dimensional or univariate normal (or Gaussian) distribution. It is usually parameterized with mean and a covariance matrix, or in Bayesian inference, with mean and a precision matrix, where the precision matrix is the matrix inverse of the covariance matrix. These functions provide the precision parameterization for convenience and familiarity. It is easier to calculate a multivariate normal density with the precision parameterization, because a matrix inversion can be avoided.

A random vector is considered to be multivariate normally distributed if every linear combination of its components has a univariate normal distribution. This distribution has a mean parameter vector mu of length k and a k x k precision matrix Omega, which must be positive-definite.

The conjugate prior of the mean vector is another multivariate normal distribution. The conjugate prior of the precision matrix is the Wishart distribution (see dwishart).

When applicable, the alternative Cholesky parameterization should be preferred. For more information, see dmvnpc.

For models where the dependent variable, Y, is specified to be distributed multivariate normal given the model, the Mardia test (see plot.demonoid.ppc, plot.laplace.ppc, or plot.pmc.ppc) may be used to test the residuals.

Value

dmvnp gives the density and rmvnp generates random deviates.

Author(s)

Statisticat, LLC. [e-mail]

See Also

dmvn, dmvnc, dmvnpc, dnorm, dnormp, dnormv, dwishart, plot.demonoid.ppc, plot.laplace.ppc, and plot.pmc.ppc.

Examples

library(LaplacesDemon)
x <- dmvnp(c(1,2,3), c(0,1,2), diag(3))
X <- rmvnp(1000, c(0,1,2), diag(3))
joint.density.plot(X[,1], X[,2], color=TRUE)

LaplacesDemon documentation built on July 9, 2021, 5:07 p.m.