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What are the distinct ways in which a set of predictor variables can provideinformation about a target variable? When does a variable provide uniqueinformation, when do variables share redundant information, and when dovariables combine synergistically to provide complementary information? Theredundancy lattice from the partial information decomposition of Williams andBeer provided a promising glimpse at the answer to these questions. However,this structure was constructed using a much criticised measure of redundantinformation, and despite sustained research, no completely satisfactoryreplacement measure has been proposed. In this paper, we take a differentapproach, applying the axiomatic derivation of the redundancy lattice to asingle realisation from a set of discrete variables. To overcome the difficultyassociated with signed pointwise mutual information, we apply thisdecomposition separately to the unsigned entropic components of pointwisemutual information which we refer to as the specificity and ambiguity. Thisyields a separate redundancy lattice for each component. Then based upon anoperational interpretation of redundancy, we define measures of redundantspecificity and ambiguity enabling us to evaluate the partial information atomsin each lattice. These atoms can be recombined to yield the sought-aftermultivariate information decomposition. We apply this framework to canonicalexamples from the literature and discuss the results and the various propertiesof the decomposition. In particular, the pointwise decomposition usingspecificity and ambiguity satisfies a chain rule over target variables, whichprovides new insights into the so-called two-bit-copy example.

Pointwise Partial Information Decomposition using the Specificity and Ambiguity Lattices