# Sobol Indices for Discrete Variables

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 ?

• May be this reference is very useful Sobol I (1993) Sensitivity estimates for non linear mathematical models. Mathematical Modelling and Computational Experiments 1:407–414. and this book Roger Ghanem,David Higdon,Houman Owhadi ," Handbook of Uncertainty Quantification ", Elsevier. Nov 27 '17 at 12:30