Continuous Multivariate distributions

Dirichlet distribution

  • Story. The Dirichlet distribution is a generalization of the Beta distribution. It is a probability distribution describing probabilities of outcomes. Instead of describing probability of one of two outcomes of a Bernoulli trial, like the Beta distribution does, it describes probability of \(K−1\) of \(K\) outcomes. The Beta distribution is the special case of \(K=2\).
  • Parameters. The parameters are \(α_1,α_2,…α_K\), all strictly positive, defined analogously to \(α\) and \(β\) of the Beta distribution.
  • Support. The Dirichlet distribution has support on the interval [0, 1] such that \(\sum_{i=1}^K y_i = 1\).
  • Probability density function.

    \(\begin{align} f(\boldsymbol{\theta};\boldsymbol{\alpha}) = \frac{1}{B(\boldsymbol{\alpha})}\,\prod_{i=1}^K y_i^{\alpha_i-1} \end{align} \)

    where

    \(\begin{align}
    B(\boldsymbol{\alpha}) = \frac{\prod_{i=1}^K\Gamma(\alpha_i)}{\Gamma\left(\sum_{i=1}^K \alpha_i\right)}
    \end{align}\)

    is the multivariate Beta function.
  • Related distributions.
    • The special case where \(K=2\) is a Beta distribution with parameters \(α=α_1\) and \(β=α_2\).