Dirichelet Distribution

Dirichlet Distribution

A Dirichlet distribution is a continuous distribution on discrete probability vectors. It is the multivariate generalization of Beta Distribution.

• Parameter
  • $\alpha \in {\mathbb{R}}_{++}^V$
• PDF
  • $p\left(x\mid \alpha \right)=\frac{\alpha \left({\sum }_v{\alpha }_v\right)}{{\prod }_{v=1}^V\alpha \left({\alpha }_v\right)}{\prod }_{v=1}^V{x}_v^{{\alpha }_v-1}$
• Mean
  • $\left({\alpha }_i/{\sum }_v{\alpha }_v\right)$
• Variance
  • $\mathrm{Cov}(x_i,x_j)=\frac{{\delta }_{ij}{\tilde{\alpha }}_i-{\tilde{\alpha }}_i{\tilde{\alpha }}_j}{{\sum }_v{\alpha }_v+1}$, where ${\tilde{\alpha }}_i={\alpha }_i/\left({\sum }_v{\alpha }_v\right)$

With a larger $\alpha$, $x$ will be uniform.