  When training a predictive model over medical data, the goal is sometimes to
gain insights about a certain disease. In such cases, it is common to use
feature importance as a tool to highlight significant factors contributing to
that disease. As there are many existing methods for computing feature
importance scores, understanding their relative merits is not trivial. Further,
the diversity of scenarios in which they are used lead to different
expectations from the feature importance scores. While it is common to make the
distinction between local scores that focus on individual predictions and
global scores that look at the contribution of a feature to the model, another
important division distinguishes model scenarios, in which the goal is to
understand predictions of a given model from natural scenarios, in which the
goal is to understand a phenomenon such as a disease. We develop a set of
axioms that represent the properties expected from a feature importance
function in the natural scenario and prove that there exists only one function
that satisfies all of them, the Marginal Contribution Feature Importance (MCI).
We analyze this function for its theoretical and empirical properties and
compare it to other feature importance scores. While our focus is the natural
scenario, we suggest that our axiomatic approach could be carried out in other
scenarios too.
