Editing Dirichlet distribution (section)

===Inequality===
Probability density function <math>f \left(x_1,\ldots, x_{K-1}; \alpha_1,\ldots, \alpha_K \right)</math> plays a key role in a multifunctional inequality which implies various bounds for the Dirichlet distribution.<ref>{{cite journal | last1=Grinshpan | first1=A. Z. | title=An inequality for multiple convolutions with respect to Dirichlet probability measure | doi=10.1016/j.aam.2016.08.001 | year=2017 | 
journal=Advances in Applied Mathematics | volume=82 | issue=1 | pages=102–119 | doi-access=free }}</ref>

Another inequality relates the moment-generating function of the Dirichlet distribution to the convex conjugate of the scaled reversed Kullback-Leibler divergence:<ref>{{cite arXiv | last1=Perrault| first1=P. | title=A New Bound on the Cumulant Generating Function of Dirichlet Processes |eprint=2409.18621 | year=2024| class=math.PR }} Theorem 3.3</ref>

<math display=block>
\log \operatorname{E}\left(\exp{\sum_{i=1}^K s_i X_i } \right) 
\leq 
\sup_p \sum_{i=1}^K \left(p_i s_i - \alpha_i\log\left(\frac{\alpha_i}{\alpha_0 p_i} \right)\right),
</math>
where the supremum is taken over {{mvar|p}} spanning the {{math|(''K'' − 1)}}-simplex.