The hazard rate curve of the numerical control machine tool is a bathtub curve. The change point between the early failure period and the random failure period of the curve is difficult to obtain with a small data sample; thus, a Bayesian method is proposed. A method to build the prior distributions of the Weibull parameters is developed, which integrates the multi-source prior information of the target numerical control machine tool and the reference numerical control machine tool. The Markov chain Monte Carlo method is adopted to calculate the estimators of the Weibull parameters corresponding to each failure, which solves the problem of the absence of an analytical solution. The total working time of the numerical control machine tool when the estimator of the shape parameter is equal to 1 is estimated by taking the estimator of the shape parameter as the function of time. As a result, the change point and the early failure period are obtained. Comparison result shows that the result obtained through an existing change point solving method with a large dataset is close to the result generated through the proposed method with a small dataset. The change point and the early failure period obtained with the proposed method can be used to guide the early failure test and to design a rational maintenance strategy, which are of vital engineering significance.