Seismic risk assessment entails frequently a large number of nonlinear time-history analyses. The use of high-fidelity Finite Element Models (FEMs) in this context, though supports high accuracy predictions, involves a substantial computational burden. This has incentivized researchers to explore the use of calibrated Reduced Order Models (ROMs) to replace the computational expensive FEMs, with the calibration process established so that the ROM can match the desired FEM characteristics. Recent studies have explicitly examined this calibration for hysteretic, multi-degree-of-freedom ROMs utilizing FEM time-history response data. This calibration has been posed as a least-squares optimization problem for the configurable ROM parameters, considering the discrepancy between the ROM and FEM responses for a selected number of excitations, for different output quantities of interest (QoIs). This work extends these efforts to examine a Bayesian calibration framework of ROMs for earthquake engineering applications. The entire posterior distribution of the ROM parameters is identified, instead of solely the point estimate that the current least-squares optimization provides. This facilitates a quantification of the uncertainty in the calibrated ROM parameters. For formulating the likelihood function different assumptions are discussed, impacting, ultimately, how the different earthquake excitations and QoIs are effectively combined. Comparisons to the objective function utilized in the established least-squares approach are drawn. Furthermore, a model class selection is advocated to compare different candidate ROMs in order to choose the most appropriate one given the available data. Finally, a hierarchical Bayesian inference approach is examined, treating independently the ROM calibration for each earthquake excitation, but unifying the ROM parameters under a common prior, updated using data from all excitations. Different challenges are discussed for implementing such a multilevel Bayesian framework for the current application. The illustrative implementation considers two different structures, corresponding to different materials and heights, with the FEMs developed in OpenSees. The different calibration approaches are evaluated by utilizing the calibrated ROMs for seismic risk assessment for different seismicity scenarios.