# Intuition for Sobol Indices

I have recently been learning about Sobol indices and have found them quite informative. However, as I have explored them I have encoutnered situations where I have found them counter intuitive and in fact more misleading than helpful.

I am lead to assume that this must be a misconception on my part, so perhaps somebody could shed some light on this for me?

For our considerations let $$X,Y \sim \mathcal N(0,1)$$ be standard normally distribution and consider a model $$Z = f(X,Y)$$.

Scenario (1): Let $$f(X,Y) = aX+bY$$, then the Sobol indices are $$S_X = \frac{a^2}{a^2+b^2}$$, $$S_Y = \frac{b^2}{a^2+b^2}$$ and $$S_{XY} = 0$$, which makes a lot of sense each of them contributing a fraction of the variance proportional to their own variance.

Scenario (2): Let $$f(X,Y) = XY$$, then the Sobol indices are $$S_X = S_Y = 0$$ and $$S_{XY} = 1$$, which again makes sense. All variation is contributed by their combination.

Scenario (3): Let $$f(X,Y) = X^2Y$$. Now it gets counter-intuitive to me. Here the Sobol indices are $$S_X = 0$$, $$S_Y = \frac13$$ and $$S_{XY} = \frac23$$. I can compute the indices, no problem, but the way I see it $$X$$ is much more significant to the variance than $$Y$$ here, which is not clear from the indices at all. In fact, they kind of seem to tell the opposite story.

Of course, $$S_X = 0$$ in (3) is due to it being proportional to $$\mathbb E_Y[f(X,Y)] = 0$$, so $$f(X,Y) = X^kY$$ for all $$k > 0$$ has $$S_X = 0$$.

What exactly am I supposed to take away from scenario (3)? And if this is kind of an artifact of the setup, how am I supposed to identify such an issue in another setting where the model $$f$$ is way more obscure? or straight up black box?