  There is broad agreement in the literature that explanation methods should be
faithful to the model that they explain, but faithfulness remains a rather
vague term. We revisit faithfulness in the context of continuous data and
propose two formal definitions of faithfulness for feature attribution methods.
Qualitative faithfulness demands that scores reflect the true qualitative
effect (positive vs. negative) of the feature on the model and quanitative
faithfulness that the magnitude of scores reflect the true quantitative effect.
We discuss under which conditions these requirements can be satisfied to which
extent (local vs global). As an application of the conceptual idea, we look at
differentiable classifiers over continuous data and characterize
Gradient-scores as follows: every qualitatively faithful feature attribution
method is qualitatively equivalent to Gradient-scores. Furthermore, if an
attribution method is quantitatively faithful in the sense that changes of the
output of the classifier are proportional to the scores of features, then it is
either equivalent to gradient-scoring or it is based on an inferior
approximation of the classifier. To illustrate the practical relevance of the
theory, we experimentally demonstrate that popular attribution methods can fail
to give faithful explanations in the setting where the data is continuous and
the classifier differentiable.
