Bits and pieces: understanding information decomposition from part-whole relationships and formal logic

Partial information decomposition (PID) seeks to decompose the multivariate mutual information that a set of source variables contains about a target variable into basic pieces, the so-called ‘atoms of information’. Each atom describes a distinct way in which the sources may contain information about the target. For instance, some information may be contained uniquely in a particular source, some information may be shared by multiple sources and some information may only become accessible synergistically if multiple sources are combined. In this paper, we show that the entire theory of PID can be derived, firstly, from considerations of part-whole relationships between information atoms and mutual information terms, and secondly, based on a hierarchy of logical constraints describing how a given information atom can be accessed. In this way, the idea of a PID is developed on the basis of two of the most elementary relationships in nature: the part-whole relationship and the relation of logical implication. This unifying perspective provides insights into pressing questions in the field such as the possibility of constructing a PID based on concepts other than redundant information in the general n-sources case. Additionally, it admits of a particularly accessible exposition of PID theory.