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The process personnel always seek the opportunity to improve the processes. One of the essential steps for process improvement is to quickly recognize the starting time or the change point of a process disturbance. The proposed approach combines the CUSUM control chart with the Bayesian estimation technique. We show that the control chart has some information about the change point and this information can be used to make an informative prior. Two Bayes estimators corresponding to the informative and a non informative prior along with MLE, frequentist approach, are considered. Their mean square error of estimators, are compared through a series of simulations. The results show that the Bayes estimator with the informative prior is more accurate and more precise for almost all values of the shift in the process mean compared to Bayes estimator with a non-informative prior and MLE.