Dual total correlation


In information theory, dual total correlation, information rate, excess entropy, or binding information is one of several known non-negative generalizations of mutual information. While total correlation is bounded by the sum entropies of the n elements, the dual total correlation is bounded by the joint-entropy of the n elements. Although well behaved, dual total correlation has received much less attention than the total correlation. A measure known as "TSE-complexity" defines a continuum between the total correlation and dual total correlation.

Definition

For a set of n random variables, the dual total correlation is given by
where is the joint entropy of the variable set and is the conditional entropy of variable, given the rest.

Normalized

The dual total correlation normalized between is simply the dual total correlation divided by its maximum value,

Bounds

Dual total correlation is non-negative and bounded above by the joint entropy.
Secondly, Dual total correlation has a close relationship with total correlation,. In particular,

Relation to other quantities

In measure theoretic terms, by the definition of dual total correlation:
which is equal to the union of the pairwise mutual informations:

History

Han originally defined the dual total correlation as,
However Abdallah and Plumbley showed its equivalence to the easier-to-understand form of the joint entropy minus the sum of conditional entropies via the following: