On ordered beta distribution and the generalized incomplete beta function

Motivated by applications in Bayesian analysis we introduce a multidimensional beta distribution in an ordered simplex. We study properties of this distribution and connect them with the generalized incomplete beta function. This function is crucial in applications of multidimensional beta distribution, thus we present two efficient numerical algorithms for computing the generalized incomplete beta function, one based on Taylor series expansion and another based on Chebyshev polynomials.