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In statistics, the matrix variate Dirichlet distribution is a generalization of the matrix variate beta distribution and of the Dirichlet distribution.
Suppose U 1 , … , U r {\displaystyle U_{1},\ldots ,U_{r}} are p × p {\displaystyle p\times p} positive definite matrices with I p − ∑ i = 1 r U i {\displaystyle I_{p}-\sum _{i=1}^{r}U_{i}} also positive-definite, where I p {\displaystyle I_{p}} is the p × p {\displaystyle p\times p} identity matrix. Then we say that the U i {\displaystyle U_{i}} have a matrix variate Dirichlet distribution, ∼ D p {\displaystyle \left\sim D_{p}\left} , if their joint probability density function is
where a i > / 2 , i = 1 , … , r + 1 {\displaystyle a_{i}>/2,i=1,\ldots ,r+1} and β p {\displaystyle \beta _{p}\left} is the multivariate beta function.
If we write U r + 1 = I p − ∑ i = 1 r U i {\displaystyle U_{r+1}=I_{p}-\sum _{i=1}^{r}U_{i}} then the PDF takes the simpler form