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Dirichlet distribution

In probability and statistics, the Dirichlet distribution (after Johann Peter Gustav Lejeune Dirichlet) is a continuous multivariate probability distribution. The Dirichlet distribution is the multivariate generalization of the beta distribution. It is the conjugate prior of the multinomial distribution in Bayesian statistics.

Contents

1 Specification of the Dirichlet distribution

2 Probability density function

3 Random number generation

4 Related topics

Specification of the Dirichlet distribution

Probability density function

The probability density function of the Dirichlet distribution of order K is the following function of a K-dimensional vector x = (x₁, ..., x_K) with x_i ≥ 0:

Failed to parse (unknown function \propto): f(x) \propto \prod_{i=1}^K x_i^{\alpha_i - 1} \;\delta(0, 1 -\sum_{i=1}^K x_i)

where α = (α₁, ..., α_K) is a parameter vector with α_i ≥ 0. The Kronecker delta δ ensures that the density is zero unless

$\sum_{i=1}^K x_i = 1\,\!$ .

The normalizing constant is the multinomial beta function, which is expressed in terms of the gamma function:

$\frac{\prod_{i=1}^K \Gamma(\alpha_i)}{\Gamma(\sum_{i=1}^K \alpha_i)} = \mathrm{B}(\alpha)$ .

The density can therefore be written as the function g given by

$g(x) = \frac{1}{\mathrm{B}(\alpha)} \prod_{i=1}^K x_i^{\alpha_i-1}$

where the domain of g are the K-dimensional vectors x over the nonnegative reals with |x|₁ = 1.

Let $\alpha_0 = \sum_{i=1}^K\alpha_i$ . Then the means of the random variables $x_1, \ldots, x_K$ are $\frac{\alpha_i}{\alpha_0}$ , respectively. The variances are $\frac{\alpha_i (\alpha_0-\alpha_i)}{\alpha_0^2 (\alpha_0+1)}$ , respectively.

Random number generation

One way to generate a random draw of $x_1, \ldots, x_K$ from the Dirichlet is to draw independent random variables $y_1, \ldots, y_k$ from the gamma distribution with density $\frac{e^{-y} y^{\alpha_i-1}}{\Gamma (\alpha_i)}$ , then set $x_i = y_i/\sum_{j=1}^K y_j$ .

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