Summary: In competing risks problem, a subset of risks is needed more attention for inferential purposes. In the objective Bayesian paradigm, reference priors enable to achieve such inferential objectives. In this article, the Marshall-Olkin bivariate Weibull distribution is considered to model the competing risks data. In the availability of partial information for some of the parameters, the reference priors are derived as per the importance of the parameters. The Dirichlet prior is taken as a conditional subjective prior and the marginal reference prior has been derived. Also, the propriety of the resulting posterior density has been proved. The Bayesian estimates of the parameters are obtained under squared error and linear-exponential loss functions. Further, the derived reference prior is used for the computation of Bayes factors or posterior odds in testing the hypothesis that the competing risks are identical. The performance of established Bayesian estimators is illustrated using the Diabetic Retinopathy Study (DRS) and Prostate Cancer data sets. Finally, the model compatibility is done for the considered data sets under Bayesian Paradigm.