狄利克雷分布

根据多项分布的形式 $\operatorname{Mult}\left(m_1, m_2, \ldots, m_K \mid \boldsymbol{\mu}, N\right)=\left(\begin{array}{c}N \\ m_1 m_2 \ldots m_K\end{array}\right) \prod_{k=1}^K \mu_k^{m_k}$
，所以 dirichlet 的 $\mu$ 应该是 $\mu k$ 的指数形式

• $p(\boldsymbol{\mu} \mid \boldsymbol{\alpha}) \propto \prod_{k=1}^K \mu_k^{\alpha_k-1}$

  • $0 \leqslant \mu_k \leqslant 1$ and $\sum_k \mu_k=1$

  • $\alpha_1, \ldots, \alpha_K$ 是分布的参数

  • $\boldsymbol{\alpha}$ denotes $\left(\alpha_1, \ldots, \alpha_K\right)^{\mathrm{T}}$

  • 满足这一约束的全体 ${\mu}$ 构成一个 K-1 维度的[[单纯形]] simplex

$\operatorname{Dir}(\boldsymbol{\mu} \mid \boldsymbol{\alpha})=\frac{\Gamma\left(\alpha_0\right)}{\Gamma\left(\alpha_1\right) \cdots \Gamma\left(\alpha_K\right)} \prod_{k=1}^K \mu_k^{\alpha_k-1}$

• $\alpha_0=\sum_{k=1}^K \alpha_k$