Bayesian Probabilistic approaches in the health monitoring of civil engineering structures has gained remarkable interest during past decades. When compared to the available Operational Modal Analysis (OMA) methods, Bayesian Operational Modal Analysis (BAYOMA) determines a probabilistic range with a most probable value and uncertainty instead of a certain result. For this reason, the most important difference of BAYOMA lies in its capability of uncertainty quantification. Therefore, the modal parameters of a measured structure can be determined based on a probabilistic logic according to various cases (for example single measurement setup, well separated and/or closely spaced modes, multiple measurement setups). Further, the finite element model of the investigated structure can also be updated by a Bayesian approach incorporated with modal identification procedure. Some efficient BAYOMA methods such as Bayesian Spectral Density Approach (BSDA) and Bayesian Fast Fourier Transform Approach (BFFTA) have been presented by various researchers during the past two decades. Despite their efficient and fast solution procedure, the available methods have some critical issues that need to be solved. Most of these problems especially lie in the quantification of posterior uncertainties and some special cases arise in closely spaced modes and/or multiple setup measurement cases. In the literature, solutions for the aforementioned problems have been also presented by various researchers. In the light of the accumulated knowledge in the literature, this study presents a computational framework for BAYOMA and Bayesian Model Updating (BMU). In addition to some improvements to the available methods, new and alternative approaches are presented for BAYOMA and BMU. According to the results, it is seen that the quality of identified modal parameters and updated finite element models increases significantly by the proposed computational procedure.