  Variable importance is central to scientific studies, including the social
sciences and causal inference, healthcare, and 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
to 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, marketing data, and
image classification tasks illustrate how variables can change dramatically in
importance for approximately-equally-accurate predictive models
