Dirichlet

Parameters

\(n\), called \(\alpha_i \) for \(i=1,\ldots ,n\) below, with \({\Sigma _\alpha } \equiv {\alpha _1} +  \ldots + {\alpha _n}\)

Kernel

\(\prod\limits_i {x_i^{{\alpha _i} - 1}} \). If the \(\alpha \) are all 1, this is uniform on its support

Support

\(x_i \geq 0\), \(\sum\limits_i {{x_i}}  = 1\)

Mean

for component \(i\),

\({\alpha _i}/{\Sigma _\alpha }\)

Variance

for component \(i\), \(\frac{{{\alpha _i}\left( {{\Sigma _\alpha } - {\alpha _i}} \right)}}{{\Sigma _\alpha ^2 ({\Sigma _\alpha } + 1}})\). The larger the \(\alpha \), the smaller the variance

Main Uses

Priors and posteriors for parameters that measure probabilities with more than two alternatives.

Density Function

%LOGDIRICHLET(x,a) is the log density at the VECTOR x (whose elements have to sum to 1) with parameter VECTOR a.

Random Draws

%RANDIRICHLET(alpha) draws a vector of Dirichlet probabilities with the alpha as the vector of shape parameters.