Editing Dirichlet distribution (section)

===Probability density function===
[[Image:LogDirichletDensity-alpha 0.3 to alpha 2.0.gif|thumb|right|250px|Illustrating how the log of the density function changes when {{math|1=''K'' = 3}} as we change the vector {{math|'''α'''}} from {{math|1='''α''' = (0.3, 0.3, 0.3)}} to {{math|(2.0, 2.0, 2.0)}}, keeping all the individual <math>\alpha_i</math>'s equal to each other.]]

The Dirichlet distribution of order {{math|''K'' ≥ 2}} with parameters {{math|''α''{{sub|1}}, ..., ''α''{{sub|''K''}} > 0}} has a [[probability density function]] with respect to [[Lebesgue measure]] on the [[Euclidean space]] {{math|'''R'''{{isup|''K''−1}}}} given by

<math display=block>f \left(x_1,\ldots, x_{K}; \alpha_1,\ldots, \alpha_K \right) = \frac{1}{\mathrm{B}(\boldsymbol\alpha)} \prod_{i=1}^K x_i^{\alpha_i - 1}</math>
where <math>\{x_k\}_{k=1}^{k=K}</math> belong to the standard <math>K-1</math> [[simplex]], or in other words:
<math display=block>\sum_{i=1}^{K} x_i = 1 \mbox{ and } x_i \in \left[0,1\right] \mbox{ for all } i \in \{1,\dots,K\}\,.</math>

The [[normalizing constant]] is the multivariate [[beta function]], which can be expressed in terms of the [[gamma function]]:

<math display=block>\mathrm{B}(\boldsymbol\alpha) = \frac{\prod\limits_{i=1}^K \Gamma(\alpha_i)}{\Gamma\left(\sum\limits_{i=1}^K \alpha_i\right)},\qquad\boldsymbol{\alpha}=(\alpha_1,\ldots,\alpha_K).</math>