Uncertainty-Aware Symbolic Regression through Bayesian Support Selection

Abstract: The Sure Independence Screening and Sparsifying Operator (SISSO) framework is a powerful symbolic-regression method for extracting compact and interpretable descriptors from large nonlinear feature spaces. However, standard SISSO is deterministic: it returns a single descriptor and point prediction, without quantifying uncertainty in descriptor selection, regression coefficients, or predictions. Here we introduce a probabilistic extension in which the sure independence screening (SIS) stage is kept deterministic to preserve scalability, while the sparsifying operator (SO) stage is reformulated as Bayesian inference over the SIS-screened support space. The resulting deterministic-SIS/Bayesian-SO framework yields posterior probabilities for competing descriptor supports, feature-inclusion probabilities, Bayesian-model-averaged predictions, and predictive credible intervals, while recovering the deterministic SO descriptor of standard SISSO in the maximum-a-posteriori limit. Applied to an $X_2YZ$ Heusler-alloy magnetic-moment dataset, the approach gives modest improvements in five-fold cross-validation RMSE and near-nominal empirical coverage of the 95$\%$ predictive intervals. More importantly, the posterior exposes competing, physically related symbolic descriptor families that would appear artificially unique in a deterministic analysis. These results suggest that deterministic-SIS/Bayesian-SO can be used as an uncertainty-aware diagnostic extension of SISSO: a tool for assessing descriptor confidence, stability, and non-uniqueness in small-data materials regression problems.