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 GNU Scientific Library Reference Manual - Third Edition (v1.12) by M. Galassi, J. Davies, J. Theiler, B. Gough, G. Jungman, P. Alken, M. Booth, F. RossiPaperback (6"x9"), 592 pages, 60 figuresISBN 0954612078RRP £24.95 (\$39.95)

## 19.27 The Dirichlet Distribution

Function: void gsl_ran_dirichlet (const gsl_rng * r, size_t K, const double alpha[], double theta[])
This function returns an array of K random variates from a Dirichlet distribution of order K-1. The distribution function is
p(\theta_1, ..., \theta_K) d\theta_1 ... d\theta_K =
(1/Z) \prod_{i=1}^K \theta_i^{\alpha_i - 1} \delta(1 -\sum_{i=1}^K \theta_i) d\theta_1 ... d\theta_K


for theta_i >= 0 and alpha_i >= 0. The delta function ensures that \sum \theta_i = 1. The normalization factor Z is

Z = {\prod_{i=1}^K \Gamma(\alpha_i)} / {\Gamma( \sum_{i=1}^K \alpha_i)}


The random variates are generated by sampling K values from gamma distributions with parameters a=alpha_i, b=1, and renormalizing. See A.M. Law, W.D. Kelton, Simulation Modeling and Analysis (1991).

Function: double gsl_ran_dirichlet_pdf (size_t K, const double alpha[], const double theta[])
This function computes the probability density p(\theta_1, ... , \theta_K) at theta[K] for a Dirichlet distribution with parameters alpha[K], using the formula given above.
Function: double gsl_ran_dirichlet_lnpdf (size_t K, const double alpha[], const double theta[])
This function computes the logarithm of the probability density p(\theta_1, ... , \theta_K) for a Dirichlet distribution with parameters alpha[K].
 ISBN 0954612078 GNU Scientific Library Reference Manual - Third Edition (v1.12) See the print edition