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Partial Information Decomposition and the Information Delta: A Geometric Unification Disentangling Non-Pairwise Information James Kunert-Graf 1*, Nikita Sakhanenko 1 and David Galas 1 1 Pacific Northwest Research Institute * Correspondence: [email protected] Version September 23, 2020 submitted to Entropy; Typeset by LATEX using class file mdpi.cls Abstract: Information theory provides robust measures of multivariable interdependence, but 1 classically does little to characterize the multivariable relationships it detects. The Partial 2 Information Decomposition (PID) characterizes the mutual information between variables by 3 decomposing it into unique, redundant, and synergistic components. This has been usefully 4 applied, particularly in neuroscience, but there is currently no generally accepted method for 5 its computation. Independently, the Information Delta framework characterizes non-pairwise 6 dependencies in genetic datasets. This framework has developed an intuitive geometric 7 interpretation for how discrete functions encode information, but lacks some important 8 generalizations. This paper shows that the PID and Delta frameworks are largely equivalent. 9 We equate their key expressions, allowing for results in one framework to apply towards open 10 questions in the other. For example, we find that the approach of Bertschinger et al. is useful for 11 the open Information Delta question of how to deal with linkage disequilibrium. We also show how 12 PID solutions can be mapped onto the space of delta measures. Using Bertschinger et al. as an 13 example solution, we identify a specific plane in delta-space on which this approach’s optimization 14 is constrained, and compute it for all possible three-variable discrete functions of a three-letter 15 alphabet. This yields a clear geometric picture of how a given solution decomposes information. 16