Prostate cancer is commonly treated with androgen deprivation therapy (ADT), yet most patients eventually develop resistance. Intermittent ADT (IADT) has been proposed to reduce treatment burden and delay resistance by cycling therapy on and off. However, substantial interpatient variability in prostate-specific antigen (PSA) dynamics complicates prediction of individual treatment outcomes. We develop a real-time Bayesian framework to infer patient-specific PSA kinetics and estimate the probability of resistance during IADT. The proposed approach incorporates longitudinal PSA measurements through Bayesian filtering, enabling continuous updating of the posterior distribution of key parameters governing tumor dynamics and PSA production. This allows real-time prediction of treatment response without relying on rigid parameter assumptions. Applied to clinical PSA data from patients undergoing IADT, the model successfully captures individual PSA trajectories and quantifies uncertainty in both parameter estimates and forecasts. Posterior-based predictions identify early signals of emerging resistance and provide individualized risk estimates for upcoming treatment cycles. These results demonstrate that Bayesian inference of PSA dynamics offers a quantitative and clinically interpretable framework for predicting resistance and supporting personalized treatment strategies in prostate cancer.