Structural health monitoring (SHM) is extensively employed to ensure the integrity of structures and infrastructures. However, the data-to-decision process is often highly uncertain, primarily due to insufficient information about the monitored structure and the limitations of numerical modeling. To address these challenges, recent advances in artificial intelligence offer promising solutions for handling various sources of uncertainty. This study introduces a Bayesian Neural Network (BNN) designed to identify both undamaged and damaged states in a structure over time, utilizing data from diverse instrumentation sources. Unlike traditional neural networks, the BNN is particularly effective with small datasets and provides probabilistic outputs, yielding mean and standard deviation estimates that quantify prediction uncertainty. As more data become available, the BNN is naturally designed to improve the prediction of the network outputs in a Bayesian fashion. As an illustrative case study, the model is applied to a two-span simply supported prestressed girder bridge. The SHM system within the FEM is represented by virtual measurement points along the girder, where both dynamic and static response characteristics are monitored under various simulated damage scenarios, including changes in concrete properties and supports stiffness. The BNN is trained using the sensor data as inputs, with values of girder segments stiffness as target variables, thereby addressing the solution of the inverse problem. Results of a parametric investigation show that the trained BNN is capable of identifying the most probable damage configuration over time, giving an estimate of the confidence level of the prediction, using synthetic SHM system data.