  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.
