A measure of dependence is a way to assign a number (usually normalized between 0 and 1) to a couple of random variable, such that $\delta(X,Y)=0$ if and only of $X$ and $Y$ are independent, and $\delta(X,Y)=1$ as soon as there is a complete dependence between $X$ and $Y$. Rényi introduced in the paper On measures of dependence (Acta Math. Acad. Sci. Hungar. 10 1959 441--451) a set of axioms that a measure of dependence should fulfill. Among them are nonparametricity ($\delta(f(X),g(Y))=\delta(X,Y)$ as soon as $f,g$ are inversible bimeasurable functions, so that $\delta$ dos not rely on a metric, or affine structure unlike the correlation) and symmetry ($\delta(X,Y)=\delta(Y,X)$), and the "complete dependence" is defined by the existence of a relation $X=f(Y)$ of $Y=f(X)$.
I feel that the symmetry assumption is unnatural. In view of, for example, Bell inequalities, one would want to have some sort of triangle inequality, so that if $\delta(X,Y)$ and $\delta(Y,Z)$ are both very high, $\delta(X,Z)$ should be quite high too. This rules out symmetry since one can easily construct random variables such that $X=f(Y)$, $Z=g(Y)$ but $X$ and $Z$ are independent (in this example, one would want to say that $X$ depends heavily on $Y$, but that $Y$ does not depend that much on $X$).
Question: do you know a measure of dependence, or a similar tool, that is nonparametric and nonsymmetric? Does it satisfy a triangle inequality? Any reference would be useful.
