One of the main challenges for interpreting black-box models is the ability to uniquely decompose square-integrable functions of non-independent random inputs into a sum of functions of every possible subset of variables. However, dealing with dependencies among inputs can be complicated. We propose a novel framework to study this problem, linking three domains of mathematics: probability theory, functional analysis, and combinatorics. We show that, under two reasonable assumptions on the inputs (non-perfect functional dependence and non-degenerate stochastic dependence), it is always possible to decompose such a function uniquely. This generalizes the well-known Hoeffding decomposition. The elements of this decomposition can be expressed using oblique projections and allow for novel interpretability indices for evaluation and variance decomposition purposes. The properties of these novel indices are studied and discussed. This generalization offers a path towards a more precise uncertainty quantification, which can benefit sensitivity analysis and interpretability studies whenever the inputs are dependent. This decomposition is illustrated analytically, and the challenges for adopting these results in practice are discussed.

翻译：解释黑箱模型的主要挑战之一在于，能够将非独立随机输入的平方可积函数唯一地分解为每个可能变量子集函数之和。然而，处理输入间的依赖关系可能颇为复杂。我们提出了一种新的框架来研究这一问题，该框架连接了概率论、泛函分析和组合学三个数学领域。我们证明，在关于输入的两个合理假设（非完美函数依赖和非退化随机依赖）下，总是可能唯一地分解此类函数。这推广了著名的Hoeffding分解。该分解的各个分量可使用斜投影表示，并可用于评估和方差分解的新型可解释性指标。我们研究并讨论了这些新指标的性质。这一推广为实现更精确的不确定性量化提供了途径，当输入存在依赖时，这有助于敏感性分析和可解释性研究。我们通过分析实例说明了这一分解，并讨论了在实践中应用这些结果所面临的挑战。