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?
 A: 
$$ is much more significant to the variance than $$

The Sobol indices belong to a specific category of Sensitivity Analysis called Global Sensitivity Analysis (GSA). GSA computes sensitivity indices by taking into account for the model $f(X,Y)$ and the probability distributions of the input random variables $X$ and $Y$. The above statement focuses solely on the model and does not account for the input distributions.
Example
Let us shift the mean of the distributions and compute the Sobol indices (numerically) once again for scenario (3).
Case 0: $X, Y \sim \mathcal{N}(0,1)$
As expected: $S_X \approx 0.00 , S_Y \approx 0.33 $ and $S_{XY} \approx 0.66$.
Case 1: $X, Y \sim \mathcal{N}(1,1)$
We obtain: $S_X \approx 0.37 , S_Y \approx 0.25 $ and $S_{XY} \approx 0.37$.
Case 2: $X, Y \sim \mathcal{N}(100,1)$
We obtain: $S_X \approx 0.79 , S_Y \approx 0.20 $ and $S_{XY} \approx 0$.
Intuition
For Case 0: $X, Y \sim \mathcal{N}(0,1)$, most of the probability mass ($\sim 68 \%$) of the random variables is between $[-1, 1]$. Therefore $X^2$ will be smaller than $Y$ "most" of the time. Since the values of $X$ are small, the effect of their variance is also small. As the probability mass shifts to higher values in Case 1 and Case 2, the values of $X$ increase and their variance naturally has a larger contribution to the overall variance of $f(X,Y)$ and we see that the first order index $S_X$ increases.
