In probability and statistics, the Dirichlet distribution (after Peter Gustav Lejeune Dirichlet), often denoted , is a family of continuous multivariate probability distributions parameterized by a vector of positive reals. It is a multivariate generalization of the beta distribution,[1] hence its alternative name of multivariate beta distribution (MBD).[2] Dirichlet distributions are commonly used as prior distributions in Bayesian statistics, and in fact, the Dirichlet distribution is the conjugate prior of the categorical distribution and multinomial distribution.
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Dirichlet distribution
Probability density function |
Parameters |
number of categories (integer) concentration parameters, where |
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Support |
where and |
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PDF |
where where |
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Mean |
(where is the digamma function) |
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Mode |
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Variance |
where , and is the Kronecker delta |
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Entropy |
with defined as for variance, above; and is the digamma function |
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Method of Moments |
where is any index, possibly itself |
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The infinite-dimensional generalization of the Dirichlet distribution is the Dirichlet process.