Bayesian R-LayerNorm: Uncertainty-Aware Adaptive Normalization—Revised Version

This paper introduces Bayesian R-LayerNorm, a normalization layer that extends the previously proposed R-LayerNorm with uncertainty quantification. Building upon R-LayerNorm, we draw connections to statistical field theory, renormalization group methods, and infor-mation geometry to motivate the design. The method incorporates uncertainty estimation through a stable ψ-function, enabling adaptive noise suppression based on local entropy esti-mates. We provide theoretical analysis of numerical stability, gradient stability, and training convergence under standard assumptions. A key practical contribution is the integration of uncertainty quantification directly into the normalization operation, providing confidence estimates for each normalized activation without additional cost. The method adapts to local noise, varying normalization strength spatially based on estimated noise levels. The implementation is simple, adding only two learnable parameters per layer, and serves as a drop-in replacement for existing normalization layers. Due to computational constraints (Kaggle P100 GPU, limited epochs), we evaluate Bayesian R-LayerNorm on CIFAR-10-C using 50 training epochs and 3 random seeds. Under these limitations, it achieves average accuracy gains of +0.49% over standard LayerNorm across four common corruptions, with the largest improvement of +0.74% on shot noise. While these gains are modest, they are consistent across seeds. The method requires mini-mal computational overhead ( 10%) and we provide complete open-source implementation. We further show that the learned λ parameters offer interpretability, revealing which layers adapt most strongly to different corruptions. The framework suggests promising directions for trustworthy normalization in safety-critical applications where uncertainty matters alongside accuracy.