Bayesian Neural Networks for Nonlinear Regression: Posterior Inference, Uncertainty Quantification and Scalability

Bayesian neural networks provide a probabilistic framework for nonlinear regression by combining the expressive flexibility of neural networks with principled uncertainty quantification. However, practical implementation remains challenging because posterior inference is computationally demanding and often requires approximate methods. This study presents a comparative analysis of several inference approaches for Bayesian neural-network regression, including Hamiltonian Monte Carlo, the No- U-Turn Sampler and variational inference, with emphasis on predictive uncertainty, posterior calibration and computational efficiency. The results show that different inference strategies often achieve comparable predictive accuracy, whereas substantially larger differences emerge in uncertainty quantification and scalability. Sampling-based approaches provide more reliable posterior characterization and better-calibrated predictive uncertainty, particularly under complex noise structures, but incur substantially higher computational cost. In contrast, variational inference offers competitive predictive performance together with markedly improved computational efficiency, although it may underestimate posterior uncertainty in more challenging settings. Overall, the findings suggest that the primary practical benefit of Bayesian neural networks lies in reliable and interpretable uncertainty quantification rather than solely improved point prediction. The choice of inference strategy should therefore balance posterior fidelity, uncertainty calibration, and computational scalability according to