Dirichlet distribution

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\(\operatorname {Dir} ({\boldsymbol {\alpha }})\) is a family of continuous multivariate probability distributions parameterized by a vector \(\boldsymbol {\alpha }\) of positive reals.

\(\alpha_K\) called concentration parameters (also known as scaling parameter)

\(\displaystyle f\left(x_{1},\ldots ,x_{K-1};\alpha _{1},\ldots ,\alpha _{K}\right)={\frac {1}{\mathrm {B} ({\boldsymbol {\alpha }})}}\prod _{i=1}^{K}x_{i}^{\alpha _{i}-1},\) where \(\displaystyle x_{K}=1-\sum \limits _{i=1}^{K-1}x_{i}\)
\(\displaystyle \lbrace {x_{1},\cdots ,x_{K}\rbrace }\) belongs to the standard simplex.

The marginal distributions are beta distributions:
\(\displaystyle X_{i}\sim \operatorname {Beta} (\alpha _{i},\alpha _{0}-\alpha _{i}).\)

binomial과 beta distribution의 관계가 multinomial과 dirichlet distribution의 관계와 같다.