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A distribution over categorical probability vectors, conjugate to multinomial.
DESCRIPTION
The Dirichlet distribution is a multivariate distribution over probability vectors on the simplex. Parameterized by positive concentration parameters, it generalizes the Beta distribution to multiple categories. It is conjugate to the Multinomial and Categorical likelihoods, enabling tractable Bayesian updates. Applications include topic models (LDA), mixture weights, and compositional data. Its concentration controls sparsity versus uniformity in category probabilities.
ALTERNATIVE DEFINITIONS
Wikipedia
In probability and statistics, the Dirichlet distribution is a family of continuous multivariate probability distributions parameterized by a vector of positive reals.
Stresses multivariate nature and positive parameter vector.
Wolfram MathWorld
The Dirichlet distribution is a multivariate generalization of the beta distribution whose support is the probability simplex.
Emphasizes relationship to Beta and support on the simplex.
RELATED TERMS
Multinomial Distribution Categorical Distribution Beta Distribution Bayesian Inference Simplex Concentration Parameter Topic Modeling Latent Dirichlet Allocation Conjugate Prior
COMPETING TERMS
Logistic-Normal Distribution
KNOWLEDGE AREAS
TARGET AUDIENCE
EXTERNAL SOURCES
- Wikipedia - Dirichlet distribution
- Statlect - Definition and properties on simplex
- Penn State STAT 414 - Course notes on Dirichlet
- CRAN MCMCpack - Dirichlet background and usage in R
TOP QUESTIONS
Why is the Dirichlet Distribution used in topic modeling?
It provides a conjugate prior over document-topic and topic-word distributions, enabling efficient inference in LDA and related models.
Category: Explanation
How do concentration parameters affect sparsity?
Lower values (<1) encourage sparse probability vectors; higher values (>1) push toward uniformity across categories.
Category: Explanation
What is an alternative to the Dirichlet prior?
The Logistic-Normal allows correlated components on the simplex and different tail behavior.
Category: Alternatives
How is the Dirichlet Distribution parameterized?
By a vector α = (α1,…,αK) of positive reals; the mean is α/α0, where α0 is the sum of components.
Category: Explanation
How do I sample from a Dirichlet?
Draw independent Gamma(αk,1) variates and normalize by their sum to get a probability vector.
Category: How-To
What’s a common mistake with Dirichlet modeling?
Using it for compositional data with strong subcomponent correlations; Logistic-Normal may be more appropriate.
Category: Mistakes
Is Dirichlet conjugate to the Multinomial?
Yes, yielding a Dirichlet posterior after observing category counts from a Multinomial or Categorical model.
Category: Explanation
How to set a symmetric Dirichlet prior?
Use αk = α0/K for all k. Choose α0 to control overall concentration.
Category: How-To
Where else is Dirichlet applied?
In mixture modeling (component weights), Bayesian bandits (probability vectors), and probabilistic graphical models.
Category: Example