Pointwise Information Decomposition Using the Specificity and Ambiguity Lattices

What are the distinct ways in which a set of predictor variables can provide information about a target variable? When does a variable provide unique information, when do variables share redundant information, and when do variables combine to provide complementary information? The redundancy lattice from the partial information decomposition of Williams and Beer provided a promising glimpse at the answer to these questions; however, this structure was constructed using a much criticised measure of redundant information. Despite much research effort, no satisfactory replacement measure has been proposed. This paper takes a different approach, applying the axiomatic derivation of the redundancy lattice to a single realisation from the set of variables. In order to do this, one must overcome the difficulties associated with signed pointwise mutual information. This is done by applying the decomposition separately to the non-negative entropic components of the pointwise mutual information, which we refer to as the specificity and the ambiguity. Then, based upon an operational interpretation of redundancy, measures of redundant specificity and ambiguity are defined. It is shown that the decomposed specificity and ambiguity can be recombined to yield the sought-after information decomposition. The decomposition is applied to canonical examples from the literature and its various properties are discussed. In particular, the pointwise decomposition using specificity and ambiguity satisfies a chain rule over target variables, which provides new insights into interpreting the well-known two-bit copy example.