Probability Mass Exclusions and the Directed Components of Pointwise Mutual Information
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The pointwise mutual information quantifies the mutual information between events x and y from random variable X and Y. This article considers the pointwise mutual information in a directed sense, examining precisely how an event y provides information about x via probability mass exclusions. Two distinct types of exclusions are identified---namely informative and misinformative exclusions. Then, inspired by Fano's postulates for the pointwise mutual information, three postulates are proposed that aim to decompose the pointwise mutual information into a non-negative informational component associated with the informative exclusions, and a non-positive informational component associated with the misinformative exclusions. This leads to a novel derivation of a familiar decomposition of the pointwise mutual information into its entropic components. The paper concludes by discussing the relevance of considering information in terms of probability mass exclusions to the ongoing effort to decompose multivariate information.
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