A new paradigm for global sensitivity analysis

It is well-known that Sobol indices, which count among the most popular sensitivity indices, are based on the Sobol decomposition. Here we challenge this construction by redefining Sobol indices without the Sobol decomposition. In fact, we show that Sobol indices are a particular instance of a more general concept which we call sensitivity measures. A sensitivity measure of a system taking inputs and returning outputs is a set function that is null at a subset of inputs if and only if, with probability one, the output actually does not depend on those inputs. A sensitivity measure evaluated at the whole set of inputs represents the uncertainty about the output. We show that measuring sensitivity to a particular subset is akin to measuring the expected output’s uncertainty conditionally on the fact that the inputs belonging to that subset have been fixed to random values. By considering all of the possible combinations of inputs, sensitivity measures induce an implicit symmetric factorial experiment with two levels, the factorial effects of which can be calculated. This new paradigm generalizes many known sensitivity indices, can create new ones, and defines interaction effects independently of the choice of the sensitivity measure. No assumption about the distribution of the inputs is required.

💡 Research Summary

The paper proposes a fundamentally new framework for global sensitivity analysis (GSA) that does not rely on the classical Sobol decomposition. Traditional Sobol indices are built on a variance‑based ANOVA‑type decomposition that requires the input variables to be statistically independent. The authors introduce the concept of a “sensitivity measure” τ, a set‑function defined on all subsets A of the input index set D, with the property that τ(A)=0 if and only if the model output f(X) does not depend on the variables in A (they call such subsets “super‑flous”). τ(∅)=0 and τ(D) quantifies the total uncertainty of the output.

The key construction uses a “Dirac test” functional φ on probability measures: φ(Q)≥0 and φ(Q)=0 iff Q is a Dirac measure. For each subset A, the conditional distribution Q_A = Pr{f(X)∈· | X_{D\A}} is fed into φ, and the sensitivity measure is defined as τ(A)=E