  In this paper, we present a new explainability formalism designed to shed
light on how each input variable of a test set impacts the predictions of
machine learning models. Hence, we propose a group explainability formalism for
trained machine learning decision rules, based on their response to the
variability of the input variables distribution. In order to emphasize the
impact of each input variable, this formalism uses an information theory
framework that quantifies the influence of all input-output observations based
on entropic projections. This is thus the first unified and model agnostic
formalism enabling data scientists to interpret the dependence between the
input variables, their impact on the prediction errors, and their influence on
the output predictions. Convergence rates of the entropic projections are
provided in the large sample case. Most importantly, we prove that computing an
explanation in our framework has a low algorithmic complexity, making it
scalable to real-life large datasets. We illustrate our strategy by explaining
complex decision rules learned by using XGBoost, Random Forest or Deep Neural
Network classifiers on various datasets such as Adult Income, MNIST, CelebA,
Boston Housing, Iris, as well as synthetic ones. We finally make clear its
differences with the explainability strategies LIME and SHAP, that are based on
single observations. Results can be reproduced by using the freely distributed
Python toolbox https://gems-ai.aniti.fr/.
