Maximum entropy characterizations of the multivariate Liouville distributions

A random vector X = ( X 1 , X 2 , … , X n ) with positive components has a Liouville distribution with parameter θ = ( θ 1 , θ 2 , … , θ n ) if its joint probability density function is proportional to h ( ∑ i = 1 n x i ) ∏ i = 1 n x i θ i - 1 , θ i > 0 [R.D. Gupta, D.S.P. Richards, Multivariate Liouville distributions, J. Multivariate Anal. 23 (1987) 233–256]. Examples include correlated gamma variables, Dirichlet and inverted Dirichlet distributions. We derive appropriate constraints which establish the maximum entropy characterization of the Liouville distributions among all multivariate distributions. Matrix analogs of the Liouville distributions are considered. Some interesting results related to I-projection from a Liouville distribution are presented.