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 \(z\).