dist.Multivariate.Normal.Precision {LaplacesDemon}  R Documentation 
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

n 
This is the number of random draws. 
mu 
This is mean vector 
Omega 
This is the 
log 
Logical. If 
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: positivedefinite
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 onedimensional 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 positivedefinite.
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. software@bayesianinference.com
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)