Information Theory and Measure Theory - Multivariate Mutual Information

Multivariate Mutual Information

Certain extensions to the definitions of Shannon's basic measures of information are necessary to deal with the σ-algebra generated by the sets that would be associated to three or more arbitrary random variables. (See Reza pp. 106–108 for an informal but rather complete discussion.) Namely needs to be defined in the obvious way as the entropy of a joint distribution, and a multivariate mutual information defined in a suitable manner so that we can set:

in order to define the (signed) measure over the whole σ-algebra. There is no single universally accepted definition for the mutivariate mutual information, but the one that corresponds here to the measure of a set intersection is due to Fano (Srinivasa). The definition is recursive. As a base case the mutual information of a single random variable is defined to be its entropy: . Then for we set

where the conditional mutual information is defined as

The first step in the recursion yields Shannon's definition It is interesting to note that the mutual information (same as interaction information but for a change in sign) of three or more random variables can be negative as well as positive: Let X and Y be two independent fair coin flips, and let Z be their exclusive or. Then bit.

Many other variations are possible for three or more random variables: for example, is the mutual information of the joint distribution of X and Y relative to Z, and can be interpreted as Many more complicated expressions can be built this way, and still have meaning, e.g. or