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
Image:VennInfo3Var.svg|thumb|256px|right|Venn diagram of information theoretic measures for three variables x, y, and z. The dual total correlation is represented by the union of the three mutual informations and is shown in the diagram by the yellow, magenta, cyan, and gray regions.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,Relationship with Total Correlation
Dual total correlation is non-negative and bounded above by the joint entropy.Secondly, Dual total correlation has a close relationship with total correlation,, and can be written in terms of differences between the total correlation of the whole, and all subsets of size :
where and
Furthermore, the total correlation and dual total correlation are related by the following bounds:
Finally, the difference between the total correlation and the dual total correlation defines a novel measure of higher-order information-sharing: the O-information:
The O-information or redundant interactions (in which case, and have found multiple applications in neuroscience.
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: