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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?

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