Dirichlet-multinomial distribution

withwithIn 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 {oldsymbol {alpha }}} , and an observation drawn from a multinomial distribution with probability vector p and number of trials n. The compounding corresponds to a Pólya urn scheme. It is frequently encountered in Bayesian statistics, empirical Bayes methods and classical statistics as an overdispersed multinomial distribution. 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 {oldsymbol {alpha }}} , and an observation drawn from a multinomial distribution with probability vector p and number of trials n. The compounding corresponds to a Pólya urn scheme. It is frequently encountered in Bayesian statistics, 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. The Dirichlet distribution is a conjugate distribution to the multinomial distribution. This fact leads to an analytically tractable compound distribution.For a random vector of category counts x = ( x 1 , … , x K ) {displaystyle mathbf {x} =(x_{1},dots ,x_{K})} , distributed according to a multinomial distribution, the marginal distribution is obtained by integrating on the distribution for p which can be thought of as a random vector following a Dirichlet distribution: