In statistics, the inverse Dirichlet distribution is a derivation of the matrix variate Dirichlet distribution. It is related to the inverse Wishart distribution. Suppose U 1 , … , U r {\displaystyle U_{1},\ldots ,U_{r}} are p × p {\displaystyle p\times p} positive definite matrices with a matrix variate Dirichlet distribution, ( U 1 , … , U r ) ∼ D p ( a 1 , … , a r ; a r + 1 ) {\displaystyle \left(U_{1},\ldots ,U_{r}\right)\sim D_{p}\left(a_{1},\ldots ,a_{r};a_{r+1}\right)} . Then X i = U i − 1 , i = 1 , … , r {\displaystyle X_{i}={U_{i}}^{-1},i=1,\ldots ,r} have an inverse Dirichlet distribution, written ( X 1 , … , X r ) ∼ ID ( a 1 , … , a r ; a r + 1 ) {\displaystyle \left(X_{1},\ldots ,X_{r}\right)\sim \operatorname {ID} \left(a_{1},\ldots ,a_{r};a_{r+1}\right)} . Their joint probability density function is given by
{ β p ( a 1 , … , a r ; a r + 1 ) } − 1 ∏ i = 1 r det ( X i ) − a i − ( p + 1 ) / 2 det ( I p − ∑ i = 1 r X i − 1 ) a r + 1 − ( p + 1 ) / 2 {\displaystyle \left\{\beta _{p}\left(a_{1},\ldots ,a_{r};a_{r+1}\right)\right\}^{-1}\prod _{i=1}^{r}\det \left(X_{i}\right)^{-a_{i}-(p+1)/2}\det \left(I_{p}-\sum _{i=1}^{r}{X_{i}}^{-1}\right)^{a_{r+1}-(p+1)/2}}
References A. K. Gupta and D. K. Nagar 1999. "Matrix variate distributions". Chapman and Hall.