Bayesian Residual Cascade for Uncertainty Quantification in Deep Regression

Abstract

Uncertainty quantification in deep regression is still dominated by post-hoc sampling or layer-wise Bayesian parameterization, while the residual pathways that govern information flow across depth usually remain deterministic. This paper proposes the Bayesian Residual Cascade (BRC), a deep regression architecture that treats residual connections themselves as stochastic regulators of uncertainty propagation. BRC integrates three coupled mechanisms: Beta-distributed residual weights for inter-stage variance modulation, a Bayesian variance-thresholded intra-stage adaptive refinement loop that repeatedly applies the Bayesian block until the estimated variance falls below a stage-specific tolerance threshold defined by a validation-selected schedule, and a cascade uncertainty aggregation mechanism with sample-dependent intra-stage refinement. The model is trained end-to-end with a composite ELBO objective and evaluated on five benchmark datasets, mainly covering structured and geospatial regression tasks, with one time-series forecasting benchmark used to examine behavior under sequential dependence. On structured regression tasks, BRC consistently improves calibration over MC Dropout, Variational Bayesian Neural Network, and Deep Ensemble, achieving ECE values of 0.047 on Energy Efficiency and 0.081 on California Housing while maintaining competitive $R^{2}$ and prediction interval coverage close to or above the nominal 95% target. On Power Plant, BRC shows stable but conservative interval coverage, with PICP remaining around 99.5% and showing smaller variability than sampling-based baselines. Although the advantage becomes weaker on time-series and small-sample tasks, the overall results indicate that the current cascade mechanism is particularly effective for medium-to-large regression problems dominated by feature-level interactions. By combining robust confidence calibration with sample-dependent refinement, BRC offers an architecture-integrated foundation for trustworthy deployment in safety-critical and resource-constrained applications such as structural health monitoring, energy management, and geospatial risk assessment.