# dist.Multivariate.Normal: Multivariate Normal Distribution In LaplacesDemon: Complete Environment for Bayesian Inference

## Description

These functions provide the density and random number generation for the multivariate normal distribution.

## Usage

 ```1 2``` ```dmvn(x, mu, Sigma, log=FALSE) rmvn(n=1, mu, Sigma) ```

## 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. `Sigma` This is the k x k covariance matrix Sigma. `log` Logical. If `log=TRUE`, then the logarithm of the density is returned.

## Details

• Application: Continuous Multivariate

• Density: p(theta) = (1/((2*pi)^(k/2)*|Sigma|^(1/2))) * exp(-(1/2)*(theta-mu)'*Sigma^(-1)*(theta-mu))

• Inventors: Robert Adrain (1808), Pierre-Simon Laplace (1812), and Francis Galton (1885)

• Notation 1: theta ~ MVN(mu, Sigma)

• Notation 2: theta ~ N[k](mu, Sigma)

• Notation 3: p(theta) = MVN(theta | mu, Sigma)

• Notation 4: p(theta) = N[k](theta | mu, Sigma)

• Parameter 1: location vector mu

• Parameter 2: positive-definite k x k covariance matrix Sigma

• Mean: E(theta) = mu

• Variance: var(theta) = Sigma

• 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. 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 covariance matrix Sigma, which must be positive-definite.

The conjugate prior of the mean vector is another multivariate normal distribution. The conjugate prior of the covariance matrix is the inverse Wishart distribution (see `dinvwishart`).

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

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

`dmvn` gives the density and `rmvn` generates random deviates.

## Author(s)

Statisticat, LLC. [email protected]

`dinvwishart`, `dmatrixnorm`, `dmvnc`, `dmvnp`, `dnorm`, `dnormp`, `dnormv`, `plot.demonoid.ppc`, `plot.laplace.ppc`, and `plot.pmc.ppc`.
 ```1 2 3 4``` ```library(LaplacesDemon) x <- dmvn(c(1,2,3), c(0,1,2), diag(3)) X <- rmvn(1000, c(0,1,2), diag(3)) joint.density.plot(X[,1], X[,2], color=TRUE) ```