# Efficiency of Representations

(, , Definition 2) measures the efficiency of a representation in causal language. Specifically, given a sample with representation \(Z=z\) and label \(Y=y\), the efficiency of \(Z\) for label \(Y\) is the Probability of Necessity (PN) of \(\mathbb{I}(Z=z)\) for \(\mathbb{I}(Y=y)\), denoted \(PN_{Z=z,Y=y}=\mathbb{P}(Y(Z\neq z)\neq y|Z= z, Y=y)\).

For e.g. is the label \(Y\) denotes whether a dog is in the image or not, and we have 2 features corresponding to the presence of a dog's face and the presence of 4 legs, using both of these representations is inefficient since given a dog's face, the presence of 4 legs is redundant, since the dog's face is a necessary cause of the label.