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)).