Local vs global importance

Let’s assume a simple model described as the interaction of $X^1$ and $X^1$ in values from 0 to 1 $[0,1]$.

$$f(x^1, x^2) = x^1 * x^2$$

• Globally both variables are equally important, because the model is symmetrical.

• But if the instance we want to explain $\underline{x}_*$ has $x^1 = 0$ and $x^2 = 1$. Then the importance of $X^1$ is larger than $X^2$:

  • $h^1_{x_*}(z) = z$ as $x^2 = 1$ for any value of $z$.
  • $h^2_{x_*}(z) = 0$ as $x^1 = 0$ for any value of $z$.