We present a Bayesian deep learning architecture with multiplicative normalizing flows for precise uncertainty quantification (UQ) at the physics event level. This method distinguishes both types of uncertainties, aleatoric and epistemic, offering nuanced insights. When applied to Deep Inelastic Scattering (DIS) events, the model extracts kinematic variables effectively, paralleling the efficacy of recent techniques, but with event-level UQ. This UQ proves essential for tasks like event filtering and can rectify errors without the ground truth. Tests using the H1 detector at HERA suggest potential applications for the future EIC, including data monitoring and anomaly detection. Notably, our model efficiently handles large samples with low inference time.