Sobol Indices for Discrete Variables

Question

It is well known that the Sobol sensitivity indices are derived from the ANOVA decomposition. Most proofs that I read concerning this subject assume that the original model is of the form $$ Y=f(X_1,X_2,\cdots, X_n)$$ Where $Y$ is the dependent variable and $X_i's$ are the independent variables having continuous uniform distribution. Then accordingly the first order Sobol indices are defined as $$ S_i=\frac{\operatorname{var}(E[Y|X_i])}{\operatorname{var}(Y)}$$

My Question is: Suppose our model $ Y=f(X_1,X_2,\cdots, X_n)$ has only discrete variables, i.e. the $X_i$'s take only discrete values, can we apply the same formula to find the Sobol indices in this case ?

Answer 1

Sobol sensitivity indices can be applied to discrete input random variables without any change.

Their definition is not dependent on the distribution of input variables. It could even be stochastic processes, for example.

Here a paper where it is stated that we can use Sobol with discrete inputs (table 1): onlinelibrary.wiley.com/doi/10.1002/psp4.6/pdf

And here a paper in which it is actually done on a small example (section 5): informs-sim.org/wsc97papers/0261.PDF

Answer 2

How about when the dependent variable Y is a discrete response rather than a continuous response, say for a classification model where Y is either 0 or 1. Can we still use the Sobol's method to gain insights into model sensitivities?

Answer 3

It depends if X is categorical of integer. If it is integer with a notion of larger and smaller values, then traditional methods should work. Otherwise, you should directly compute the sobol indices by holding ove categorical value constant at a time.