# dist.Multivariate.Normal.Precision.Cholesky: Multivariate Normal Distribution: Precision-Cholesky... 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-Cholesky parameterization.

## Usage

 ```1 2``` ```dmvnpc(x, mu, U, log=FALSE) rmvnpc(n=1, mu, U) ```

## 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. `U` This is the k x k upper-triangular of the precision matrix that is Cholesky factor U of 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-Cholesky 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. The precision matrix is replaced with an upper-triangular k x k matrix that is Cholesky factor U, as per the `chol` function for Cholesky decomposition.

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.

In practice, U is fully unconstrained for proposals when its diagonal is log-transformed. The diagonal is exponentiated after a proposal and before other calculations. Overall, Cholesky parameterization is faster than the traditional parameterization. Compared with `dmvnp`, `dmvnpc` must additionally matrix-multiply the Cholesky back to the covariance matrix, but it does not have to check for or correct the precision matrix to positive-definiteness, which overall is slower. Compared with `rmvnp`, `rmvnpc` is faster because the Cholesky decomposition has already been performed.

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

`dmvnpc` gives the density and `rmvnpc` generates random deviates.

## Author(s)

Statisticat, LLC. [email protected]

## See Also

`chol`, `dmvn`, `dmvnc`, `dmvnp`, `dnorm`, `dnormp`, `dnormv`, `dwishartc`, `plot.demonoid.ppc`, `plot.laplace.ppc`, and `plot.pmc.ppc`.

## Examples

 ```1 2 3 4 5 6``` ```library(LaplacesDemon) Omega <- diag(3) U <- chol(Omega) x <- dmvnpc(c(1,2,3), c(0,1,2), U) X <- rmvnpc(1000, c(0,1,2), U) joint.density.plot(X[,1], X[,2], color=TRUE) ```

LaplacesDemon documentation built on May 29, 2017, 11:54 p.m.