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.

翻译：众所周知，基于Sobol分解的Sobol指标是最流行的敏感性指标之一。本文通过重新定义无需Sobol分解的Sobol指标来挑战这一构建方法。事实上，我们证明Sobol指标是更广义概念——即敏感性测度——的一个特例。对于接收输入并返回输出的系统，其敏感性测度是这样一个集合函数：当且仅当输出几乎必然不依赖于某输入子集时，该函数在该子集上取零值。对全体输入集评估的敏感性测度表征了输出的不确定性。我们证明，对特定输入子集的敏感性度量，本质上等价于在给定该子集内输入被固定为随机值的条件下，对输出期望不确定性的测量。通过考虑所有可能的输入组合，敏感性测度隐式诱导出一个具有两个水平的对称因子试验，其因子效应可被计算。这一新范式不仅推广了众多已知敏感性指标，还能创建新指标，并独立于敏感性测度的选择来定义交互效应。该框架无需对输入分布做任何假设。