Doctor Rashomon and the UNIVERSE of Madness: Variable Importance with Unobserved Confounding and the Rashomon Effect

TL;DR

UNIVERSE introduces a theoretically grounded framework to bound VI in the presence of unobserved confounding and the Rashomon effect by leveraging Rashomon sets extended to unobserved features and finite-sample corrections. The approach yields high-probability bounds on the VI of the true conditional mean function $g^*$ and, via a VI-drift parameter, accounts for distributional shifts induced by unobserved variables. It provides finite-sample guarantees and demonstrates tight, useful bounds on VI through semi-synthetic experiments across multiple datasets and a FICO credit-risk case study. The results show that accounting for model uncertainty, VI estimation uncertainty, and drift is essential to achieve reliable bounds, with practical implications for model interpretation and data collection decisions. The framework is general across model classes and VI metrics, with future work aimed at scaling to complex models and broader problem domains.

Abstract

Variable importance (VI) methods are often used for hypothesis generation, feature selection, and scientific validation. In the standard VI pipeline, an analyst estimates VI for a single predictive model with only the observed features. However, the importance of a feature depends heavily on which other variables are included in the model, and essential variables are often omitted from observational datasets. Moreover, the VI estimated for one model is often not the same as the VI estimated for another equally-good model - a phenomenon known as the Rashomon Effect. We address these gaps by introducing UNobservables and Inference for Variable importancE using Rashomon SEts (UNIVERSE). Our approach adapts Rashomon sets - the sets of near-optimal models in a dataset - to produce bounds on the true VI even with missing features. We theoretically guarantee the robustness of our approach, show strong performance on semi-synthetic simulations, and demonstrate its utility in a credit risk task.

Figures (10)

• Figure 1: Variable importance intervals for two variables from the FICO dataset, as more variables are observed. When we estimate the range of variable importance fisher2019all with only one observed feature -- % Trades w/ Balance -- this feature seems important, with a confidence interval that does not overlap with 0. However, as we introduce more variables into the analysis pipeline, this variable's confidence intervals overlap with 0, no longer yielding significant results. In contrast, External Risk Estimate -- the third feature incorporated into the analysis -- remains significant even after controlling for other observed variables. Thus, observing additional information changes the conclusion a practitioner would draw about % Trades w/ Balance.
• Figure 2: An overview of our framework showing the expansion of the Rashomon set as we account for more sources of error. The ${\color{gold} \blacklozenge}$ represents the true importance of variable $X_1$ having observed all necessary features. We consider three key factors: the Rashomon Effect, unobserved variables $U$, and finite sample errors. When adjusting for $U$, we expand which models are considered as part of the Rashomon set, potentially widening our variable importance interval. Adjusting for finite sample considerations both expands the Rashomon set and expands the range of variable importance values for each model, further widening our intervals.
• Figure 3: Verifying Theorem \ref{['thm:smart_eps']} in finite sample datasets. We compute the proportion of 100 random draws of the each dataset in which Rashomon sets estimated with the Rashomon threshold adjusting for finite sample biases as in Theorem \ref{['thm:smart_eps']} (in blue) and without any adjustment (in black) captures each $f_{u}$ for each setting. The target coverage rate is $\geq 0.9$, with $\delta=0.1$. Across all sample sizes and datasets, omitting finite sample adjustments yields Rashomon sets that leave out necessary models. In contrast, our adjustment yields the target coverage rate, verifying the theorem holds. We use the estimated Rashomon set size as our upper bound on the size of $S^*$.
• Figure 4: Verifying Theorem \ref{['thm:vi_coverage_uncond']}. We achieve the specified coverage rate of $\geq0.8$ only when adjusting for (i) model uncertainty via Theorem \ref{['thm:smart_eps']} and (ii) variable importance estimation uncertainty (in gold). Adjusting for model uncertainty alone (in blue) is not sufficient. For each setting, we compute the proportion of 100 experiments where our variable importance bounds capture the true variable importance for all submodels $f_u \in S^*$, averaged over variables. We use an estimate of the true Rashomon set size as our upper bound on the size of $S^*$.
• Figure 5: Verifying Theorem \ref{['thm:optmodel_coverage']}. We consistently achieve the specified coverage rate of $\geq 0.8$ only when we account for (i) model uncertainty, (ii) variable importance uncertainty, and (iii) VI drift. Each bar measures the proportion of 100 experiments in which our bounds capture the true variable importance for the true model $g^*$. Plots are colored such that blue only accounts for finite sample model uncertainty as in Theorem \ref{['thm:smart_eps']}, gold also adjusts for uncertainty in estimating subtractive model reliance (MR) at the model-level as in Theorem \ref{['thm:vi_coverage_uncond']}, and red adjusts for the previous two and distribution shifts induced by omitted variables \ref{['thm:optmodel_coverage']}. All three adjustments are necessary to achieve the target coverage rate of $\geq 0.8$, with $\delta=\gamma=0.1$.

Theorems & Definitions (22)

• Proposition 1
• Theorem 1
• Corollary 1
• Proposition 2
• Theorem 2
• Theorem 3
• Proposition 1
• proof
• Theorem 1: Recovering the Population Rashomon Set
• proof