Dirichlet Distribution¶
Dirichlet Distribution¶
Dirichlet Distribution is an extension of the Beta distribution to multiple random variables and is also called Multivariate Beta Distribution (MBD). For \(K \geq 2\) it is defined as
\[
\begin{aligned}
f(x_{1}, \ldots, x_{K}, \alpha_{1}, \ldots, \alpha_{K}) = \frac{1}{B(\boldsymbol{\alpha})} \prod_{i=1}^{K} x_{i}^{\alpha_{i} - 1}\newline
\text{with} \quad \sum_{i=1}^{K} x_{i} = 1, \quad x_{i} \geq 0 \; \forall \; i=1,\ldots,K\newline
B(\boldsymbol{\alpha}) = \frac{\prod_{i=1}^{K}\Gamma(\alpha_{i})}{\Gamma \bigg(\sum_{i=1}^{K} \alpha_{i} \bigg)} \quad \text{with} \; \boldsymbol{\alpha} = (\alpha_{1}, \ldots, \alpha_{K})
\end{aligned}
\]