Dirichlet-multinomial distribution

Dirichlet-Multinomial
Notation
Parameters	number of trials (positive integer);
Support	;
PMF
Mean
Variance	;
MGF	; with;
CF	; with;
PGF	; 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 ${\boldsymbol {\alpha }}$ , 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.

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Dirichlet-Multinomial
Notation	$\mathrm {DirMult} (n,{\boldsymbol {\alpha }})$
Parameters	$n>0$ number of trials (positive integer) $\alpha _{1},\ldots ,\alpha _{K}>0,\alpha _{0}=\sum \alpha _{k}$
Support	$x_{i}\in \{0,\dots ,n\}$ $\Sigma x_{i}=n\!,i=1...K$
PMF	${\frac {\Gamma \left(\alpha _{0}\right)\Gamma \left(n+1\right)}{\Gamma \left(n+\alpha _{0}\right)}}\prod _{k=1}^{K}{\frac {\Gamma (x_{k}+\alpha _{k})}{\Gamma (\alpha _{k})\Gamma \left(x_{k}+1\right)}}$
Mean	$\operatorname {E} (X_{i})=n{\frac {\alpha _{i}}{\alpha _{0}}}$
Variance	$\operatorname {Var} (X_{i})=n{\frac {\alpha _{i}}{\alpha _{0}}}\left(1-{\frac {\alpha _{i}}{\alpha _{0}}}\right)\left({\frac {n+\alpha _{0}}{1+\alpha _{0}}}\right)$ $\textstyle {\mathrm {Cov} }(X_{i},X_{k})=-n{\frac {\alpha _{i}}{\alpha _{0}}}{\frac {\alpha _{k}}{\alpha _{0}}}\left({\frac {n+\alpha _{0}}{1+\alpha _{0}}}\right)~~(i\neq k)$
MGF	$\operatorname {E} (\prod \limits _{k=1}^{K}{e}^{t_{k}\cdot x_{k}})={\frac {\Gamma (\alpha _{0})\Gamma (n+1)}{\Gamma (n+\alpha _{0})}}\cdot D_{n}({\boldsymbol {\alpha }},(e^{t_{1}},...,e^{t_{K}}))$ with $D_{n}={\frac {1}{n}}\sum \limits _{u=1}^{n}\left[\left(\sum \limits _{k=1}^{K}\alpha _{k}\cdot {e}^{t_{k}\cdot u}\right)D_{n-u}\right],D_{0}=1$
CF	$\operatorname {E} (\prod \limits _{k=1}^{K}{e}^{it_{k}\cdot x_{k}})={\frac {\Gamma (\alpha _{0})\Gamma (n+1)}{\Gamma (n+\alpha _{0})}}\cdot D_{n}({\boldsymbol {\alpha }},(e^{it_{1}},...,e^{it_{K}}))$ with $D_{n}={\frac {1}{n}}\sum \limits _{u=1}^{n}\left[\left(\sum \limits _{k=1}^{K}\alpha _{k}\cdot {e}^{it_{k}\cdot u}\right)D_{n-u}\right],D_{0}=1$
PGF	$\operatorname {E} (\prod \limits _{k=1}^{K}{z_{k}}^{x_{k}})={\frac {\Gamma (\alpha _{0})\Gamma (n+1)}{\Gamma (n+\alpha _{0})}}\cdot D_{n}({\boldsymbol {\alpha }},\mathbf {z} )$ with $D_{n}={\frac {1}{n}}\sum \limits _{u=1}^{n}\left[\left(\sum \limits _{k=1}^{K}\alpha _{k}\cdot {z_{k}}^{u}\right)D_{n-u}\right],D_{0}=1$