# 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 then 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)).

## 2 thoughts on “the Dirichlet distribution”

1. Andreas Holzhammer says:

Sounds really interesting! Could you please fix the pix?

2. alekk says:

great post!

Just wondering how you created those nice animated gifs?

Alekk.

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