Deep ensembles provide the most reliable uncertainty estimates in deep learning, but their cost grows linearly with the number of members. Implicit ensembles lower this cost by sharing a single backbone across members. Member diversity is a primary determinant of ensemble quality, yet no implicit ensemble can shape it during training; existing methods fix it at initialisation or build it into the architecture. We introduce $σ$N-Ens, a normalisation-based implicit ensemble that treats each member as a task in a multi-task architecture and modulates the shared backbone through sigmoid-bounded scalers. We also introduce a softmax-temperature regulariser, which shapes the equilibrium level of sharing between members and traces the accuracy-calibration frontier. Because only normalisation layers are replicated, the mechanism can wrap convolutional and transformer backbones alike, also allowing pretrained models to be adapted through a short fine-tune. We frame the epistemic uncertainty such an ensemble expresses as modulation uncertainty, and explain why its calibration holds under input corruption, and why its out-of-distribution detection is weaker. Our method is evaluated across ResNets and transformers on CIFAR-10/100, ImageNet and SST-2. $σ$N-Ens matches or outperforms deep ensembles at a fraction of their parameter cost, scales with ensemble size where partitioning methods collapse, and maintains calibration under distribution shift.
Sina Aghaee Dabaghan Fard, Marie Maros, Jaesung Leecs.LG stat.ME
We introduce an efficient Bayesian deep ensemble method for predictive regression designed to enhance interpretability while maintaining competitive predictive performance and computational efficiency. Our method combines the statistical rigor of Bayesian inference with the scalability of deep ensembles, providing calibrated uncertainty estimates that enable its use not only for standalone prediction but also as a component within broader learning systems. To achieve these goals, our work relies on three key design components: (i) low-dimensional ensemble representation: predictions are expressed as a combination of a small number of trained neural predictors, enabling scalable inference whose cost depends on ensemble size rather than dataset size; (ii) closed-form Bayesian aggregation: ensemble predictions are combined using Bayesian linear regression, yielding interpretable posterior weights and calibrated uncertainty without approximate inference; and (iii) Independent ensemble training: multiple neural networks are trained separately, producing diverse predictive representations that improve robustness and uncertainty calibration. Empirical results on standard regression benchmarks demonstrate that the proposed approach achieves competitive predictive performance while maintaining reliable uncertainty estimates across settings.
Training data for machine learning is routinely collected by a selection process the model never sees: loans are observed only when granted, outcomes only when a test was ordered. The standard fixes -- importance weighting, covariate-shift correction, MAR imputation -- assume selection is ignorable given observables. Econometrics solved the harder case in 1979: Heckman's two-equation model jointly fits a probit selection equation and an outcome equation linked through correlated errors, and the inverse-Mills-ratio term corrects for selection on unobservables, where importance weighting is structurally helpless. We instantiate this for deep epistemic uncertainty: a deep outcome network, a linear selection head, and a joint bivariate-normal likelihood over all units, ensembled for predictive variance. In a controlled generator where sampling probability depends on an unobservable correlated (rho up to 0.9) with the outcome noise, deep ensembles, MC dropout, and GP baselines are overconfident exactly where data was avoided: coverage of nominal-90% intervals falls to 64.4% at rho=0.9, and importance weighting with oracle propensities does not fix it (43.1%) -- reweighting corrects the covariate distribution, not the conditional bias E[y|x,selected] != E[y|x]. The Heckman correction restores coverage (88.9%) when the selection equation has an instrument -- a variable affecting selection but not the outcome -- and degrades measurably without one (40.3%); we chart this honesty curve rather than hide it. On real tabular data with induced MNAR selection, the corrected intervals are the best-calibrated (lowest region-ECE) non-oracle method in selected-against regions; baselines matching its raw coverage do so only by over-widening everywhere. Our estimators reproduce classic Stata output to seven digits. We state which identification regime a practitioner is in, and release the code.