E ( ∏ k = 1 K e i t k ⋅ x k ) = Γ ( α 0 ) Γ ( n + 1 ) Γ ( n + α 0 ) ⋅ D n ( α , ( e i t 1 , . . . , e i t K ) ) {\displaystyle \operatorname {E} (\prod \limits _{k=1}^{K}{e}^{it_{k}\cdot x_{k}})={\frac {\Gamma (\alpha _{0})\Gamma (n+1)}{\Gamma (n+\alpha _{0})}}\cdot D_{n}({\boldsymbol {\alpha }},(e^{it_{1}},...,e^{it_{K}}))} with
E ( ∏ k = 1 K z k x k ) = Γ ( α 0 ) Γ ( n + 1 ) Γ ( n + α 0 ) ⋅ D n ( α , z ) {\displaystyle \operatorname {E} (\prod \limits _{k=1}^{K}{z_{k}}^{x_{k}})={\frac {\Gamma (\alpha _{0})\Gamma (n+1)}{\Gamma (n+\alpha _{0})}}\cdot D_{n}({\boldsymbol {\alpha }},\mathbf {z} )} with
In probability theory and statistics, the Dirichlet-multinomial distribution is a family of discrete multivariate probability distributions on a finite support of non-negative integers. It is also called the Dirichlet compound multinomial distribution (DCM) or multivariate Pólya distribution (after George Pólya). It is a compound probability distribution, where a probability vector p is drawn from a Dirichlet distribution with parameter vector α {\displaystyle {\boldsymbol {\alpha }}} , and an observation drawn from a multinomial distribution with probability vector p and number of trials n. The Dirichlet parameter vector captures the prior belief about the situation and can be seen as a pseudocount: observations of each outcome that occur before the actual data is collected. The compounding corresponds to a Pólya urn scheme. It is frequently encountered in Bayesian statistics, machine learning, empirical Bayes methods and classical statistics as an overdispersed multinomial distribution.
It reduces to the categorical distribution as a special case when n = 1. It also approximates the multinomial distribution arbitrarily well for large α. The Dirichlet-multinomial is a multivariate extension of the beta-binomial distribution, as the multinomial and Dirichlet distributions are multivariate versions of the binomial distribution and beta distributions, respectively.
number of trials
![{\displaystyle D_{n}={\frac {1}{n}}\sum \limits _{u=1}^{n}\left[\left(\sum \limits _{k=1}^{K}\alpha _{k}\cdot {e}^{t_{k}\cdot u}\right)D_{n-u}\right],D_{0}=1}](./c7ea7f49e672323660a86ce30c1c37f806f25ddb.svg)
![{\displaystyle D_{n}={\frac {1}{n}}\sum \limits _{u=1}^{n}\left[\left(\sum \limits _{k=1}^{K}\alpha _{k}\cdot {e}^{it_{k}\cdot u}\right)D_{n-u}\right],D_{0}=1}](./b6788a491edfe38f602b98e126232a3fcfcf34a7.svg)
![{\displaystyle D_{n}={\frac {1}{n}}\sum \limits _{u=1}^{n}\left[\left(\sum \limits _{k=1}^{K}\alpha _{k}\cdot {z_{k}}^{u}\right)D_{n-u}\right],D_{0}=1}](./277d5f926715737ea0da06205f90388bd570f9b3.svg)
