In Structural Health Monitoring (SHM), ensuring the integrity of infrastructure faces notable challenges due to uncertainties stemming from limited information about the structural properties to be used in numerical modeling. Leveraging recent advancements in artificial intelligence, this study introduces a Bayesian Neural Network (BNN) tailored for identifying both undamaged and damaged states in structures over time by integrating multi-source data from various monitoring instruments. The BNN framework, unlike traditional neural networks, excels in handling smaller datasets while providing probabilistic outputs, including mean and standard deviation estimates, which quantify prediction uncertainty. A key advantage of the BNN is its adaptability allowing it to incorporate new data from experimental campaigns or ambient vibration tests to refine predictions in a Bayesian manner as more data becomes available. For this study, a prestressed concrete box-girder bridge with vertically prestressed internal joints is used as an illustrative case. A Finite Element Model (FEM) of the bridge is built and calibrated using Ambient Vibration Test data. Measurement points along the bridge girder in the FEM represent the SHM system, where both dynamic features and static responses are monitored under simulated damage scenarios. The BNN is trained with sensor data as inputs, aiming to predicting the variations in girder stiffness, to address the inverse problem of damage identification and quantification. Parametric investigations demonstrate that the BNN effectively identifies probable damage configurations over time and provides confidence levels for each prediction. Importantly, the Bayesian approach, included in the training process, enables continuous updates, enhancing predictive accuracy with ongoing data from field experiments or additional campaigns, thereby improving the robustness of SHM-driven decisions over the structure's lifecycle.