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Exploring the cloud of variable importance for the set of all good models

A preprint version of the article is available at arXiv.


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 but not to others? In that case, we cannot tell from a single well-performing model if a variable is always important, sometimes important, never important or perhaps only important when another variable is not important. Ideally, we would like to explore variable importance for all approximately equally accurate predictive models within the same model class. In this way, we can understand the importance of a variable in the context of other variables, and for many good 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.

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Fig. 1: The VIC for uncorrelated variables ρ12 = 0, ρ1Y = 0.4 and ρ2Y = 0.5.
Fig. 2: The VIC for correlated variables.
Fig. 3: Tuning the parameters (r, M).
Fig. 4: VID for recidivism using logistic regression.
Fig. 5: The VID for image classification.
Fig. 6: Visualizing representative models.

Data availability

The datasets analysed in this paper are publicly available. The COMPAS dataset is available in Propublica’s repository7. The image dataset of dogs and cats is available at ImageNet8. The image we use in Fig. 6 is provided in the Supplementary Information. Our experiment on the in-vehicle coupon recommendation dataset in the Supplementary Information uses data from ref. 21. Source data are provided with this paper.

Code availability

The code we use in our paper can be downloaded from ref. 22.


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Author information

Authors and Affiliations



Both authors contributed to the conception, analytics and writing of the study. The experiments were conducted by J.D. The code was designed by J.D.

Corresponding author

Correspondence to Jiayun Dong.

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Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature Machine Intelligence thanks Professor Kristian Kersting and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data

Extended Data Fig. 1 Logistic loss function.

Logistic loss.

Extended Data Fig. 2 VID of decision tree models for the recidivism experiment.

VID for Recidivism: decision trees. This is the projective of the VIC onto the space spanned by the four variables of interest: age, race, prior criminal history and gender. Unlike Fig. 4, the VIC is generated by the Rashomon set that consists of the all the good decision trees instead of logistic regression models. However, the diagrams should be interpreted in the same way as before.

Supplementary information

Supplementary Information

Supplementary experiment.

Source data

Source Data Fig. 1

Source data for our experiment.

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Dong, J., Rudin, C. Exploring the cloud of variable importance for the set of all good models. Nat Mach Intell 2, 810–824 (2020).

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