Dirichlet distribution

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, hence its alternative name of multivariate beta distribution (MBD). Dirichlet distributions are commonly used as prior distributions in Bayesian statistics, and in fact, the Dirichlet distribution is the conjugate prior of the categorical distribution and multinomial distribution.

Dirichlet distribution

Probability density function
Parameters	${\displaystyle K\geq 2}$ number of categories (integer) ${\displaystyle {\boldsymbol {\alpha }}=(\alpha _{1},\ldots ,\alpha _{K})}$ concentration parameters, where ${\displaystyle \alpha _{i}>0}$
Support	${\displaystyle x_{1},\ldots ,x_{K}}$ 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 ${\displaystyle \mathrm {B} ({\boldsymbol {\alpha }})={\frac {\prod _{i=1}^{K}\Gamma (\alpha _{i})}{\Gamma {\bigl (}\alpha _{0}{\bigr )}}}}$ where ${\displaystyle \alpha _{0}=\sum _{i=1}^{K}\alpha _{i}}$
Mean	${\displaystyle \operatorname {E} [X_{i}]={\frac {\alpha _{i}}{\alpha _{0}}}}$ ${\displaystyle \operatorname {E} [\ln X_{i}]=\psi (\alpha _{i})-\psi (\alpha _{0})}$ (where ${\displaystyle \psi }$ is the digamma function)
Mode	${\displaystyle x_{i}={\frac {\alpha _{i}-1}{\alpha _{0}-K}},\quad \alpha _{i}>1.}$
Variance	${\displaystyle \operatorname {Var} [X_{i}]={\frac {{\tilde {\alpha }}_{i}(1-{\tilde {\alpha }}_{i})}{\alpha _{0}+1}},}$ ${\displaystyle \operatorname {Cov} [X_{i},X_{j}]={\frac {\delta _{ij}\,{\tilde {\alpha }}_{i}-{\tilde {\alpha }}_{i}{\tilde {\alpha }}_{j}}{\alpha _{0}+1}}}$ where ${\displaystyle {\tilde {\alpha }}_{i}={\frac {\alpha _{i}}{\alpha _{0}}}}$, and ${\displaystyle \delta _{ij}}$ is the Kronecker delta
Entropy	${\displaystyle H(X)=\log \mathrm {B} ({\boldsymbol {\alpha }})}$${\displaystyle +(\alpha _{0}-K)\psi (\alpha _{0})-}$${\displaystyle \sum _{j=1}^{K}(\alpha _{j}-1)\psi (\alpha _{j})}$ with ${\displaystyle \alpha _{0}}$ defined as for variance, above; and ${\displaystyle \psi }$ is the digamma function
Method of Moments	${\displaystyle \alpha _{i}=E[X_{i}]\left({\frac {E[X_{j}](1-E[X_{j}])}{V[X_{j}]}}-1\right)}$ where ${\displaystyle j}$ is any index, possibly ${\displaystyle i}$ itself

The infinite-dimensional generalization of the Dirichlet distribution is the Dirichlet process.