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We develop the information-theoretical concepts required to study thestatistical dependencies among three variables. Some of such dependencies arepure triple interactions, in the sense that they cannot be explained in termsof a combination of pairwise correlations. We derive bounds for tripledependencies, and characterize the shape of the joint probability distributionof three binary variables with high triple interaction. The analysis alsoallows us to quantify the amount of redundancy in the mutual informationbetween pairs of variables, and to assess whether the information between twovariables is or is not mediated by a third variable. These concepts are appliedto the analysis of written texts. We find that the probability that a givenword is found in a particular location within the text is not only modulated bythe presence or absence of other nearby words, but also, on the presence orabsence of nearby pairs of words. We identify the words enclosing the keysemantic concepts of the text, the triplets of words with high pairwise andtriple interactions, and the words that mediate the pairwise interactionsbetween other words.

Information-theoretical analysis of the statistical dependencies among three variables: Applications to written language