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<title>The Dirichlet Distribution - GNU Scientific Library -- Reference Manual</title>
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<h3 class="section">20.27 The Dirichlet Distribution</h3>
<div class="defun">
— Function: void <b>gsl_ran_dirichlet</b> (<var>const gsl_rng * r, size_t K, const double alpha</var>[]<var>, double theta</var>[])<var><a name="index-gsl_005fran_005fdirichlet-1853"></a></var><br>
<blockquote><p><a name="index-Dirichlet-distribution-1854"></a>This function returns an array of <var>K</var> random variates from a Dirichlet
distribution of order <var>K</var>-1. The distribution function is
<pre class="example"> 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
</pre>
<p class="noindent">for <!-- {$\theta_i \ge 0$} -->
theta_i >= 0
and <!-- {$\alpha_i > 0$} -->
alpha_i > 0. The delta function ensures that \sum \theta_i = 1.
The normalization factor Z is
<pre class="example"> Z = {\prod_{i=1}^K \Gamma(\alpha_i)} / {\Gamma( \sum_{i=1}^K \alpha_i)}
</pre>
<p>The random variates are generated by sampling <var>K</var> values
from gamma distributions with parameters
<!-- {$a=\alpha_i$, $b=1$} -->
a=alpha_i, b=1,
and renormalizing.
See A.M. Law, W.D. Kelton, <cite>Simulation Modeling and Analysis</cite> (1991).
</p></blockquote></div>
<div class="defun">
— Function: double <b>gsl_ran_dirichlet_pdf</b> (<var>size_t K, const double alpha</var>[]<var>, const double theta</var>[])<var><a name="index-gsl_005fran_005fdirichlet_005fpdf-1855"></a></var><br>
<blockquote><p>This function computes the probability density
<!-- {$p(\theta_1, \ldots , \theta_K)$} -->
p(\theta_1, ... , \theta_K)
at <var>theta</var>[<var>K</var>] for a Dirichlet distribution with parameters
<var>alpha</var>[<var>K</var>], using the formula given above.
</p></blockquote></div>
<div class="defun">
— Function: double <b>gsl_ran_dirichlet_lnpdf</b> (<var>size_t K, const double alpha</var>[]<var>, const double theta</var>[])<var><a name="index-gsl_005fran_005fdirichlet_005flnpdf-1856"></a></var><br>
<blockquote><p>This function computes the logarithm of the probability density
<!-- {$p(\theta_1, \ldots , \theta_K)$} -->
p(\theta_1, ... , \theta_K)
for a Dirichlet distribution with parameters
<var>alpha</var>[<var>K</var>].
</p></blockquote></div>
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