Dirichlet 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

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,^[1] hence its alternative name of multivariate beta distribution (MBD).^[2] 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.

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