Editing Dirichlet distribution (section)

{{short description|Probability distribution}}
{{Probability distribution |
  name       =Dirichlet distribution|
  type       =density|
  pdf_image  =Dirichlet.pdf|
  cdf_image  =|
  parameters =<math>K \geq 2</math> number of categories ([[integer]])<br /><math>\boldsymbol\alpha=(\alpha_1,\ldots,\alpha_K)</math> [[concentration parameter]]s, where <math>\alpha_i > 0</math>|
  support    =<math>x_1, \ldots, x_K</math> where <math>x_i \in [0,1]</math> and <math>\sum_{i=1}^K x_i = 1</math> <br /> (i.e. a <math>K-1</math> [[simplex]])|
  pdf        =<math>\frac{1}{\mathrm{B}(\boldsymbol\alpha)} \prod_{i=1}^K x_i^{\alpha_i - 1} </math><br />where <math>\mathrm{B}(\boldsymbol\alpha) = \frac{\prod_{i=1}^K \Gamma(\alpha_i)}{\Gamma\bigl(\alpha_0\bigr)}</math><br />where <math>\alpha_0 = \sum_{i=1}^K\alpha_i</math>|
  cdf        =|
  mean       =<math>\operatorname{E}[X_i] = \frac{\alpha_i}{\alpha_0}</math><br /><math> \operatorname{E}[\ln X_i] = \psi(\alpha_i)-\psi(\alpha_0)</math><br />(where <math>\psi</math> is the [[digamma function]])|
  median     =|
  mode       =<math>x_i = \frac{\alpha_i - 1}{\alpha_0 - K}, \quad \alpha_i > 1. </math>|
  variance   =<math>\operatorname{Var}[X_i] = \frac{\tilde{\alpha}_i(1-\tilde{\alpha}_i)}{\alpha_0+1},</math> <math>\operatorname{Cov}[X_i,X_j] = \frac{\delta_{ij}\,\tilde{\alpha}_i-\tilde{\alpha}_i \tilde{\alpha}_j}{\alpha_0+1}</math> <br/>where <math>\tilde{\alpha}_i = \frac{\alpha_i}{\alpha_0}</math>, and <math>\delta_{ij}</math> is the [[Kronecker delta]] |
  skewness   =|
  kurtosis   =|
  entropy    = <math> H(X) = \log \mathrm{B}(\boldsymbol\alpha)</math><math> + (\alpha_0-K)\psi(\alpha_0) -</math><math> \sum_{j=1}^K (\alpha_j-1)\psi(\alpha_j) </math><br/>with <math>\alpha_0</math> defined as for variance, above; and <math>\psi</math> is the [[digamma function]]|
moments = <math> \alpha_i = E[X_i]\left(\frac{E[X_j](1 - E[X_j])}{V[X_j]} - 1 \right)</math> where {{mvar|j}} is any index, possibly {{mvar|i}} itself
}}

In [[probability]] and [[statistics]], the '''Dirichlet distribution''' (after [[Peter Gustav Lejeune Dirichlet]]), often denoted <math>\operatorname{Dir}(\boldsymbol\alpha)</math>, is a family of [[Continuous probability distribution|continuous]] [[multivariate random variable|multivariate]] [[probability distribution]]s parameterized by a vector {{math|'''α'''}} of positive [[real number|reals]]. It is a multivariate generalization of the [[beta distribution]],<ref name=KBJ>{{cite book|author1=S. Kotz |author2=N. Balakrishnan |author3=N. L. Johnson |title= Continuous Multivariate Distributions. Volume 1: Models and Applications|year=2000| publisher=Wiley|location= New York|isbn=978-0-471-18387-7}} (Chapter 49: Dirichlet and Inverted Dirichlet Distributions)</ref> hence its alternative name of '''multivariate beta distribution''' ('''MBD''').<ref>{{Cite journal |jstor = 2238036|title = Multivariate Beta Distributions and Independence Properties of the Wishart Distribution|journal = The Annals of Mathematical Statistics|volume = 35|issue = 1|pages = 261–269|last1 = Olkin|first1 = Ingram|last2 = Rubin|first2 = Herman|year = 1964|doi=10.1214/aoms/1177703748|doi-access = free}}</ref>  Dirichlet distributions are commonly used as [[prior distribution]]s 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]]''.