Linear models for the conditional distribution

Our aim is to introduce a regression model for the conditional probablity distributions, of observable random variables as a function of regressors. The approach linear models in Bayes spaces. Bayes spaces are a generalisation of the Aitchison Simplex, which can be seen as the space of all distributions with support 1,...,n to probablity distributions on arbitrary measure spaces. Compositional linear models on the Aitchison simplex turn out to be equivalent to multivariate logistic regression if written in alr representation. Then, likelihood theory for generalized linear models is used to estimate the regression coefficients based on observed realisation of the random variable instead of observed compositions as in compositional regression. This idea is generalized towards Bayes spaces, generating a technique allowing to define and estimate models for the conditional distribution of an observed quantity as a Bayes-space-linear function of regressors.