statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...

, the inverse Dirichlet distribution is a derivation of the

matrix variate Dirichlet distribution In statistics, the matrix variate Dirichlet distribution is a generalization of the matrix variate beta distribution and of the Dirichlet distribution. Suppose U_1,\ldots,U_r are p\times p positive definite matrices with I_p-\sum_^rU_i also positi ...

. It is related to the

inverse Wishart distribution In statistics, the inverse Wishart distribution, also called the inverted Wishart distribution, is a probability distribution defined on real-valued positive-definite matrices. In Bayesian statistics it is used as the conjugate prior for the co ...

. Suppose

U_1,\ldots,U_r

are

p\times p

positive definite matrices In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a co ...

with a

\left(U_1,\ldots,U_r\right)\sim D_p\left(a_1,\ldots,a_r;a_\right)

. Then

X_i=^,i=1,\ldots,r

have an inverse Dirichlet distribution, written

\left(X_1,\ldots,X_r\right)\sim \operatorname\left(a_1,\ldots,a_r;a_\right)

. Their joint

probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...

is given by :

\left\^
\prod_^r
\det\left(X_i\right)^\det\left(I_p-\sum_^r^\right)^

References

A. K. Gupta and D. K. Nagar 1999. "Matrix variate distributions". Chapman and Hall. {{statistics-stub Probability distributions