> [!tldr] > A **Dirichlet distribution** $D$ with parameters $\alpha=(\alpha_{1},\dots,\alpha_{k})$ is a continuous distribution in $[0,1]^{k}$, a generalization of the [[Beta Distribution|beta distribution]] (special case of $k=2$). > > It can be thought of as the distribution of some pmf $\theta=(\theta_{1},\dots,\theta_{k})$, so its support is the set $S=\left\{ \theta \in [0,1]^{k} ~|~ \sum\nolimits_{j}\theta_{j} =1 \right\}.$Moreover, it is conjugate with the multinomial distribution. If $\theta \sim D(\alpha)$, then its density is $f(\theta)\propto \prod_{j=1}^{k}\theta_{j}^{\alpha_{j}-1},$and with the normalizing constant: $f(\theta)=\frac{\Gamma( \sum_{j}\alpha_{j} )}{\prod_{j}\Gamma(\alpha_{j})}\prod_{j=1}^{k} \theta_{j}^{\alpha_{j}-1}.$Note that $\sum_{j}\theta_{j}=1$, so in fact there are only $k-1$ degrees of freedom in $\theta_{j}$.