the Dirichlet distribution

In the course, since we are still introducing some concepts of dependent distributions, we will talk about the Dirichlet distribution, which is a distribution over the simplex of Let denote the Gamma distribution with density (on

Let denote independent random variables, with Then where

has a Dirichlet distribution with parameter

Note that has a distribution in the simplex of,

and has density

We will write

The density for different values of can be visualized below, e.g., with some kind of symmetry,
or and, below
and finally, below,

Note that marginal distributions are also Dirichlet, in the sense that if


if, and if, then‘s have Beta distributions,

See Devroye (1986) section XI.4, or Frigyik, Kapila & Gupta (2010) .This distribution might also be called multivariate Beta distribution. In R, this function can be used as follows

> library(MCMCpack)
> alpha=c(2,2,5)
> x=seq(0,1,by=.05)
> vx=rep(x,length(x))
> vy=rep(x,each=length(x))
> vz=1-x-vy
> V=cbind(vx,vy,vz)
> D=ddirichlet(V, alpha)
> persp(x,x,matrix(D,length(x),length(x))

(to plot the density, as figures above). Note that we will come back on that distribution later on so-called Liouville copulas (see also Gupta & Richards (1986)).