Variable importance is central to scientific studies, including the social sciences and causal inference, healthcare, and in other domains. However, current notions of variable importance are often tied to a specific predictive model. This is problematic: what if there were multiple well-performing predictive models, and a specific variable is important to some of them and not to others? In that case, we may not be able to tell from a single well-performing model whether a variable is always important in predicting the outcome. Rather than depending on variable importance for a single predictive model, we would like to explore variable importance for all approximately-equally-accurate predictive models. This work introduces the concept of a variable importance cloud, which maps every variable to its importance for every good predictive model. We show properties of the variable importance cloud and draw connections other areas of statistics. We introduce variable importance diagrams as a projection of the variable importance cloud into two dimensions for visualization purposes. Experiments with criminal justice and marketing data illustrate how variables can change dramatically in importance for approximately-equally-accurate predictive models.
@cite_5: 
@cite_14: This work proposes a way to align statistical modeling with decision making. We provide a method that propagates the uncertainty in predictive modeling to the uncertainty in operational cost, where operational cost is the amount spent by the practitioner in solving the problem. The method allows us to explore the range of operational costs associated with the set of reasonable statistical models, so as to provide a useful way for practitioners to understand uncertainty. To do this, the operational cost is cast as a regularization term in a learning algorithm's objective function, allowing either an optimistic or pessimistic view of possible costs, depending on the regularization parameter. From another perspective, if we have prior knowledge about the operational cost, for instance that it should be low, this knowledge can help to restrict the hypothesis space, and can help with generalization. We provide a theoretical generalization bound for this scenario. We also show that learning with operational costs is related to robust optimization.
@cite_4: Model selection consistency in the high-dimensional regression setting can be achieved only if strong assumptions are fulfilled. We therefore suggest to pursue a different goal, which we call a minimal class of models. The minimal class of models includes models that are similar in their prediction accuracy but not necessarily in their elements. We suggest a random search algorithm to reveal candidate models. The algorithm implements simulated annealing while using a score for each predictor that we suggest to derive using a combination of the lasso and the elastic net. The utility of using a minimal class of models is demonstrated in the analysis of two data sets.
Finally, the idea that multiple models can explain the observed dataset well, namely the Rashomon effect ( @cite_5 ), is discussed in other works that are not related to variable importance. For example, @cite_14 implement robust optimization to find a decision rule that works well under a uncertainty set of possible situations that are consistent with the historical data. The fields of robust optimization and chance-constrained programming use uncertainty sets to make decisions; these uncertainty sets could arise from the Rashomon set. @cite_4 analyze a model selection problem in high-dimensional regression and identify a minimal class of models that rely on different subsets of variables but perform almost equally well in terms of prediction accuracy. In some sense, they are exploring the Rashomon set with random search algorithms rather than fully characterizing it.