DirichletAnteile

DirichletAnteile, or Dirichlet shares, denotes the components of a random vector that lies on the probability simplex. If X = (X1, ..., XK) follows a Dirichlet distribution with parameters α = (α1, ..., αK) with αi > 0, then each Xi is nonnegative and the sum of all components equals 1. The vector X is commonly described as consisting of DirichletAnteile.

The Dirichlet distribution Dirichlet(α) is a multivariate generalization of the Beta distribution. Its probability density on the simplex is proportional to the product over i of Xi^(αi − 1). The parameter α controls the concentration and the expected shares among the components. Let α0 be the sum of all αi.

Key properties: The expected value of each DirichletAnteil is E[Xi] = αi / α0. The variance is Var[Xi] = αi(α0 − αi) / (α0^2(α0 + 1)). The covariance between distinct components is Cov[Xi, Xj] = − αi αj / (α0^2(α0 + 1)) for i ≠ j. The components are negatively correlated and, except in special cases, not independent. When K = 2, the Dirichlet distribution reduces to a Beta distribution for X1 (or X2).

Common uses and significance: DirichletAnteile appear as the conjugate prior for multinomial likelihoods in Bayesian statistics, enabling tractable posterior updates. They are central to mixture models and topic models such as latent Dirichlet allocation, where Dirichlet priors govern distributions over categories, topics, or components. The term emphasizes the view of the vector as a set of shares of a whole, rather than as independent random variables.

Historically, the Dirichlet distribution is named after Johann Peter Gustav Lejeune Dirichlet, with the component shares often discussed in German texts as DirichletAnteile.