There is a notion of 'feature importance,' which especially comes up in decision trees and random forests, giving a notion of how much a particular feature contributes to the overall prediction. It seems like combining predictive power with feature importance would be an interesting alternate route to demonstrating important correlations. (For example, maybe a model predicts lung cancer with 90% precision, and 'is_smoker' has a 80% feature importance.) Of course, these importances depend a lot on the other features used by the model! If you include a lot of junk features and/or exclude other important features, the importance of your pet feature will shoot up.