R 2 -equitability is satisfiable

Kinney and Atwal (1) make excellent points about mutual information, the maximal information coefficient (2, 3), and “equitability.” One of their central claims, however, is that, “No nontrivial dependence measure can satisfy R 2 R2-equitability.” We argue that this is the result of a poorly constructed definition, which we quote:“A dependence measure D [ X ; Y ] D[X;Y] is R 2 R2-equitable if and only if, when evaluated on a joint probability distribution p ( X , Y ) p(X,Y) that corresponds to a noisy functional relationship between two real random variables X and Y, the following relation holds: D [ X ; Y ] = g ( R 2 [ f ( X ) ; Y ] ) . D[X;Y]=g(R2[f(X);Y]).Here, g is a function that does not depend on p ( X , Y ) p(X,Y) and f is the function defining the noisy functional relationship, i.e., Y = f ( X ) + η … Y=f(X)+η …