Dirichlet distribution

Dirichlet Distribution

Probability density function
Parameters	${\displaystyle K\geq 2}$ number of categories (integer) ${\displaystyle \alpha _{1},\cdots ,\alpha _{K}}$ concentration parameters, where ${\displaystyle \alpha _{i}>0}$
Support	${\displaystyle x_{1},\cdots ,x_{K}}$ where ${\displaystyle x_{i}\in (0,1)}$ and ${\displaystyle \sum _{i=1}^{K}x_{i}=1}$
PDF	${\displaystyle {\frac {1}{\mathrm {B} ({\boldsymbol {\alpha }})}}\prod _{i=1}^{K}x_{i}^{\alpha _{i}-1}}$ where ${\displaystyle \mathrm {B} ({\boldsymbol {\alpha }})={\frac {\prod _{i=1}^{K}\Gamma (\alpha _{i})}{\Gamma {\bigl (}\sum _{i=1}^{K}\alpha _{i}{\bigr )}}}}$ where ${\displaystyle {\boldsymbol {\alpha }}=(\alpha _{1},\ldots ,\alpha _{K})}$
Mean	${\displaystyle \operatorname {E} [X_{i}]={\frac {\alpha _{i}}{\sum _{k}\alpha _{k}}}}$ ${\displaystyle \operatorname {E} [\ln X_{i}]=\psi (\alpha _{i})-\psi (\textstyle \sum _{k}\alpha _{k})}$ (see digamma function)
Mode	${\displaystyle x_{i}={\frac {\alpha _{i}-1}{\sum _{k=1}^{K}\alpha _{k}-K}},\quad \alpha _{i}>1.}$
Variance	${\displaystyle \mathrm {Var} [X_{i}]={\frac {\alpha _{i}(\alpha _{0}-\alpha _{i})}{\alpha _{0}^{2}(\alpha _{0}+1)}},}$ where ${\displaystyle \alpha _{0}=\sum _{i=1}^{K}\alpha _{i}}$ ${\displaystyle \mathrm {Cov} [X_{i},X_{j}]={\frac {-\alpha _{i}\alpha _{j}}{\alpha _{0}^{2}(\alpha _{0}+1)}}~~(i\neq j)}$
Entropy	${\displaystyle H(X)=\log \mathrm {B} (\alpha )-(K-\alpha _{0})\psi (\alpha _{0})-\sum _{j=1}^{K}(\alpha _{j}-1)\psi (\alpha _{j})}$ with ${\displaystyle \alpha _{0}}$ defined as for variance, above.

In probability and statistics, the Dirichlet distribution (after Peter Gustav Lejeune Dirichlet), often denoted ${\displaystyle \operatorname {Dir} ({\boldsymbol {\alpha }})}$, is a family of continuous multivariate probability distributions parameterized by a vector ${\displaystyle {\boldsymbol {\alpha }}}$ of positive reals. It is a multivariate generalization of the beta distribution. Dirichlet distributions are very often used as prior distributions in Bayesian statistics, and in fact the Dirichlet distribution is the conjugate prior of the categorical distribution and multinomial distribution.