6.3 Probability distributions on the simplex
In most statistical analysis, probability models for random variables need to be specified. For statistical analysis of compositional data, some standard probability laws are available. The main distribution for compositional data is the normal on the simplex (Mateu-Figueras et al., 2013), also called additive logistic-normal distribution (Aitchison and Shen, 1980; Aitchison, 1982). A very popular, although rigid, distribution is the Dirichlet distribution (Aitchison, 1986; Narayanan, 1991) and its variants, such as the multivariate beta-distributions or compound distributions (Connor and Mosimann, 1969; Mosimann, 1962; Nadarajah and Kotz, 2007).
Probability distributions for random compositions are presented as pdf's. However, the expression of the pdf depends on the reference measure in the sample space, that is, how areas or volumes are measured in the sample space. When it is considered embedded in a real space, the Euclidean structure of the real space is implicitly inherited by the sample space, and the reference measure is the Lebesgue measure. Alternatively, when the sample space is the simplex with the Aitchison geometry, the reference measure should be the Aitchison measure. The adoption of these alternative measures produces different expressions of the corresponding pdf, although they represent the same probability distribution.
Before presenting the pdf expressions of the standard probability distributions, it is convenient ...