dist_dirichlet: The Dirichlet distribution

In distributional: Vectorised Probability Distributions

The Dirichlet distribution

Description

The Dirichlet distribution is a multivariate generalisation of the Beta distribution. It is the conjugate prior of the Categorical and Multinomial distributions, and describes a probability distribution over the (k-1)-simplex — the set of k-dimensional vectors whose components are non-negative and sum to one.

Usage

dist_dirichlet(alpha)

Arguments

alpha

A list of positive numeric concentration vectors.

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_dirichlet.html

In the following, let \mathbf{X} = (X_1, \ldots, X_k) be a Dirichlet random variable with concentration parameter alpha = \boldsymbol{\alpha} = (\alpha_1, \ldots, \alpha_k), where each \alpha_i > 0.

Support: \mathbf{x} on the (k-1)-simplex, i.e. x_i \geq 0 and \sum_{i=1}^k x_i = 1.

Mean: E(X_i) = \frac{\alpha_i}{\alpha_0} where \alpha_0 = \sum_{i=1}^k \alpha_i.

Variance:

\mathrm{Var}(X_i) = \frac{\alpha_i(\alpha_0 - \alpha_i)}{\alpha_0^2(\alpha_0 + 1)}

Covariance:

\mathrm{Cov}(X_i, X_j) = \frac{-\alpha_i \alpha_j}{\alpha_0^2(\alpha_0 + 1)}, \quad i \neq j

Probability density function (p.d.f):

f(\mathbf{x}) = \frac{1}{B(\boldsymbol{\alpha})} \prod_{i=1}^k x_i^{\alpha_i - 1}

where B(\boldsymbol{\alpha}) = \frac{\prod_{i=1}^k \Gamma(\alpha_i)}{\Gamma(\alpha_0)} is the multivariate Beta function.

See Also

LaplacesDemon::ddirichlet(), LaplacesDemon::rdirichlet()

Examples

dist <- dist_dirichlet(alpha = list(c(2, 5, 3)))
dist


mean(dist)
variance(dist)
support(dist)
generate(dist, 10)

density(dist, cbind(0.2, 0.5, 0.3))
density(dist, cbind(0.2, 0.5, 0.3), log = TRUE)

distributional documentation built on June 11, 2026, 9:07 a.m.