Bayesian PASA: Provably Stable AdaptiveActivation with Uncertainty Quantification

The choice of activation function is a fundamental design decision in deep learning, yet most popular options like ReLU, GELU, or Swish are static and treat all inputs uniformly. This one-size-fits-all approach breaks down in the presence of noisy or corrupted data, where the optimal non-linearity should depend on the input's statistical context. In this paper, we introduce Bayesian Probabilistic Adaptive Sigmoidal Activation (Bayesian PASA), a novel activation function that dynamically adapts its behavior based on the input's uncertainty. Bayesian PASA is not just a new function, but a new paradigm. It frames activation selection as a Bayesian model averaging problem, adaptively mixing sigmoidal, linear, and noise-aware behaviors. The mixing weights are derived from a principled variational evidence lower bound (ELBO), regularized by a stable ψ-function that guarantees bounded influence from noise estimates. We provide three formal theorems proving its Lipschitz continuity, gradient stability, and convergence under standard training assumptions. On the challenging CIFAR-100 benchmark, Bayesian PASA achieves a state-of-the-art test accuracy of 76.38% , outperforming ReLU (75.68%), GELU (75.98%), and the original PASA (75.53%). On the corrupted CIFAR-10-C dataset, the full Bayesian PASA model combined with Bayesian R-LayerNorm achieves an average accuracy of 53.91% , a  + 1.87% improvement over the ReLU+LayerNorm baseline. This work provides a drop-in replacement for existing activations, offering not only improved performance but also built-in uncertainty quantification for more robust deep learning systems.