Bayesian model updating (BMU) is frequently used in Structural Health Monitoring to investigate the structure’s dynamic behavior under various operational and environmental loadings for decision-making, e.g., to determine whether maintenance is required. Data collected by sensors are used to update the prior of some physics-based model’s latent parameters to yield the posterior. The choice of prior may significantly affect posterior predictions and subsequent decision-making, especially under the typical case in engineering applications of little informative data. Therefore, understanding how the choice of prior affects the posterior prediction is of great interest. In this article, a robust Bayesian inference technique evaluates the optimal and worst-case prior in the vicinity of a chosen nominal prior and their corresponding posteriors. This technique derives an interacting Wasserstein gradient flow that minimizes and maximizes/minimizes the KL divergence between the posterior and the approximation to the posterior, with respect to the approximation to the posterior and the prior. Two numerical case studies are used to showcase the proposed algorithm: a double-banana-posterior and a double-beam structure. Optimal and worst-case priors are modeled by specifying an ambiguity set containing any distribution at a statistical distance to the nominal prior, less than or equal to the radius. The resulting posteriors may be used to yield the lower and upper bounds on subsequent calculations of an engineering metric (e.g., failure probability) used for decision-making. If the metric used for decision-making is not sensitive to the resulting posteriors, it may be assumed that decisions taken are robust to prior uncertainty.