Deep Bayesian Approaches for Interpretable Nonlinear Structural System Identification

System identification, namely the inference of mathematical representations of dynamical systems from measured data, is fundamental to engineering diagnostics, prognostics, and decision-making. Bayesian filtering offers a principled framework for probabilistic state estimation but traditionally relies on prescribed transition and observation models. Recent advances in deep Bayesian filtering address this limitation by embedding learnable representations within probabilistic state-space formulations, enabling data-driven identification of unknown dynamics while retaining physical structure and uncertainty-aware inference. This tutorial provides a systematic introduction to deep Bayesian filtering for nonlinear structural system identification, focusing on the interplay between learned dynamics, physical inductive biases, and posterior inference. The Deep Markov Model serves as a data-driven reference, while the Physics-guided Deep Markov Model (PgDMM) and Neural Extended Kalman Filter (Neural EKF) illustrate complementary approaches based on amortized variational inference and differentiable Kalman filtering, respectively. Their capabilities are examined through dynamic state estimation and joint dynamic state and parameter estimation on a three-dimensional shear-frame benchmark with hysteretic Bouc-Wen connections and sparse acceleration measurements. The results highlight important distinctions between measurement-conditioned reconstruction and open-loop forecasting, demonstrate how physical structure can improve latent-state interpretability without necessarily ensuring predictive fidelity, and reveal differences in the robustness of response and parameter estimation. Practical guidelines for model construction, training, evaluation, and deployment are provided, together with an outlook on remaining challenges in uncertainty calibration, identifiability, and real-world validation.