Detecting Functional Dependencies via Symmetric Delta

Given a set of N variables and a set of M samples representing information on each of the variables, how do we find functional relationships between the variables? MIST tackles the problem by using a divide-and-conquer approach: while considering the set of all variable tuples as the search space, MIST divides that search space among parallel threads, and then conquers it by computing dependency measures for each tuple.
Symmetric Delta (Galas et al, 2020; Galas et al, 2014 (link)) is the measure used in a MIST search. The Symmetric Delta is a novel symmetric measure of functional dependence (it is symmetric under exchange of variables) constructed from joint entropies. Joint entropies between variables (using the Shannon entropy (Shannon and Weaver, 1949 ) defined as the expectation of the logarithm of each element of the joint probability distribution: $H\left(X\right)=-{\sum }_{i}P\left(x_i\right)log\left(P\left(x_i\right)\right)$ ). For variable tuples of size T there are N choose T tuples in the search space. Thus, the problem is reduced to computing joint probability distributions for a very large number of variable tuples.
Dependence between two variables X and Y can be directly measured with mutual information $I\left(X,Y\right)$ , defined as
$$I\left(X,Y\right)=H\left(X\right)+H\left(Y\right)-H\left(X,Y\right),$$
where $H\left(X\right)$ and $H\left(Y\right)$ are single entropies of variables X and Y and $H\left(X,Y\right)$ is their joint entropy.
A general dependence among three variables, X, Y, and Z, can be measured with symmetric delta. $\overline{\Delta }\left(X,Y,Z\right)$ . To see clearly the definition of symmetric delta, we need to introduce interaction information, which is a multivariable generalization of mutual information (McGill, 1954 ), defined for three variables as
$$I\left(X,Y,Z\right)=I\left(X,Y\right)-I\left(X,Y|Z\right).$$
Given interaction information, differential interaction information $\Delta$ is defined as a difference between values of successive interaction information arising from adding a variable:
$$\begin{array}{cc}{\Delta }_X& =I\left(X,Y,Z\right)-I\left(Y,Z\right),\\ {\Delta }_Y& =I\left(X,Y,Z\right)-I\left(X,Z\right),\\ {\Delta }_Z& =I\left(X,Y,Z\right)-I\left(X,Y\right).\\ \end{array}$$
Here ${\Delta }_X$ is called asymmetric delta for the target variable X. To detect a fully synergistic dependence among a set of variables, we want a single measure, which is symmetric. Consequently, we defined a general measure $\overline{\Delta }$ , called symmetric delta (or simply delta), by multiplying the $\Delta$ 's with all possible choices of the target variable:
$$\overline{\Delta }\left(X,Y,Z\right)={\Delta }_X\cdot {\Delta }_Y\cdot {\Delta }_Z.$$
The critical property of this delta measure is that it is zero whenever any of the three variables is independent of the others. It is important to note that the absolute values of the delta measure indicate the degree to which the corresponding variables are collectively interdependent.