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Dirichlet distribution
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===Bayesian models=== Dirichlet distributions are most commonly used as the [[prior distribution]] of [[categorical distribution|categorical variable]]s or [[multinomial distribution|multinomial variable]]s in Bayesian [[mixture model]]s and other [[hierarchical Bayesian model]]s. (In many fields, such as in [[natural language processing]], categorical variables are often imprecisely called "multinomial variables". Such a usage is unlikely to cause confusion, just as when [[Bernoulli distribution]]s and [[binomial distribution]]s are commonly conflated.) Inference over hierarchical Bayesian models is often done using [[Gibbs sampling]], and in such a case, instances of the Dirichlet distribution are typically [[Marginal distribution|marginalized out]] of the model by integrating out the Dirichlet [[random variable]]. This causes the various categorical variables drawn from the same Dirichlet random variable to become correlated, and the joint distribution over them assumes a [[Dirichlet-multinomial distribution]], conditioned on the hyperparameters of the Dirichlet distribution (the [[concentration parameter]]s). One of the reasons for doing this is that Gibbs sampling of the [[Dirichlet-multinomial distribution]] is extremely easy; see that article for more information.
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