In Classical Bayesian approach, estimation of lifetime data usually is dealing with exact information. However, in real world, some informations about an underlying system might be imprecise and represented in the form of vague quantities. In these situations, we need to generalize classical methods to vague environment for studying and analyzing the systems of interest.\\\\\\\\ \\\\indent In this paper, we propose the Bayesian estimation of failure rate and mean time to failure based on vague set theory in the case of complete and censored data sets. To employ the Bayesian approach, model parameters are assumed to be vague random variables with vague prior distributions. This approach will be used to induce the vague Bayes estimate of failure rate and mean time to failure by introducing and applying a theorem called \\\"Resolution Identity\\\" for vague sets. In order to evaluate the membership degrees of vague Bayesian estimate for these quantities, a computational procedure is investigated. In the proposed method, the original problem is transformed into a nonlinear programming problem which is then divided into eight subproblems to simplifying computations.

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