LaplacesDemon (version 16.1.0)

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

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 \(k \times k\) precision matrix \(\Omega\).

log

Logical. If log=TRUE, then the logarithm of the density is returned.

Value

dmvnp gives the density and rmvnp generates random deviates.

Details

• Application: Continuous Multivariate

• Density: \(p(\theta) = (2\pi)^{-p/2} |\Omega|^{1/2} \exp(-\frac{1}{2} (\theta-\mu)^T \Omega (\theta-\mu))\)

• Inventor: Unknown (to me, anyway)

• Notation 1: \(\theta \sim \mathcal{MVN}(\mu, \Omega^{-1})\)

• Notation 2: \(\theta \sim \mathcal{N}_k(\mu, \Omega^{-1})\)

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

• Notation 4: \(p(\theta) = \mathcal{N}_k(\theta | \mu, \Omega^{-1})\)

• Parameter 1: location vector \(\mu\)

• Parameter 2: positive-definite \(k \times k\) 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 \(k \times k\) 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.

See Also

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

Examples

# NOT RUN {
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)
# }