Bayesian operational modal analysis of structures with tuned mass damper

•A Bayesian approach is developed for identifying the modal parameters of the TMD and primary structure.•The proposed approach accounts for the non-classically damped modal dynamics.•A parameter scheme is proposed to make the problem globally identifiable.•Efficient algorithms are developed to compute the MPV and quantify the identification uncertainty.•The proposed theory and algorithms are verified based on synthetic data and applied to field data.