We build information geometry for a partially ordered set of variables and define the orthogonal decomposition of information theoretic quantities. The natural connection between information geometry and order theory leads to efficient decomposition algorithms. This generalization of Amari's seminal work on hierarchical decomposition of probability distributions on event combinations enables us to analyze high-order statistical interactions arising in neuroscience, biology, and machine learning.

5 pages, 5 figures, accepted to the 2016 IEEE International Symposium on Information Theory (ISIT 2016)