R: Multivariate Normal Distribution: Precision Parameterization

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 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. software@bayesian-inference.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)

[Package LaplacesDemon version 16.1.1 Index]