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Applications such as the analysis of microbiome data have led to renewedinterest in statistical methods for compositional data, i.e., multivariate datain the form of probability vectors that contain relative proportions. Inparticular, there is considerable interest in modeling interactions among suchrelative proportions. To this end we propose a class of exponential familymodels that accommodate general patterns of pairwise interaction while beingsupported on the probability simplex. Special cases include the family ofDirichlet distributions as well as Aitchison's additive logistic normaldistributions. Generally, the distributions we consider have a density thatfeatures a difficult to compute normalizing constant. To circumvent this issue,we design effective estimation methods based on generalized versions of scorematching. A high-dimensional analysis of our estimation methods shows that thesimplex domain is handled as efficiently as previously studied full-dimensionaldomains.

Interaction Models and Generalized Score Matching for Compositional Data