Dirichlet distribution
| Notation | $\mathbf{p}\sim\operatorname{Dir}(\mathbf{\alpha})$ |
| $\operatorname{B}(\mathbf{\alpha}) = \frac{\prod_{i=1}^K \Gamma(\alpha_i)}{\Gamma\bigl(\sum_{i=1}^K \alpha_i\bigr)} \prod_{i=1}^K x_i^{\alpha_i - 1} $ | |
| cdf | |
| EV | $\operatorname{E}[X_i] = \frac{\alpha_i}{\sum_k \alpha_k}$ |
| Var | $\mathrm{Var}[X_i] = \frac{\alpha_i (\alpha_0-\alpha_i)}{\alpha_0^2 (\alpha_0+1)}$, where
$\alpha_0 = \sum_{i=1}^K\alpha_i$.
$\mathrm{Cov}[X_i,X_j] = \frac{- \alpha_i \alpha_j}{\alpha_0^2 (\alpha_0+1)}~~(i\neq j)$ |
| Char. function | |
| Entropy | |
| Exponential family |
$\mathbf{\alpha}=(\alpha_1, \alpha_2, \dots, \alpha_K)$,
$\alpha_i>0, \sum_i^K\alpha_i=1$