An extended prior distribution-based Bayes formula method for cumulative time-dependent failure probability function

Since the cumulative time-dependent failure probability function (CTFPF) can provide the time-dependent failure probability (TFP) with respect to distribution parameters and upper bound of time interval (UBTI), estimating CTFPF can provide great convenience for solving time-dependent reliability-based design optimization. However, the existing direct Monte Carlo simulation method (MCS) for estimating CTFPF is time-consuming. Therefore, this paper proposes an extended prior distribution-based Bayes formula method (EPD-Bayes) to improve the efficiency and accuracy of estimating CTFPF. The EPD-Bayes adopts the Bayes formula to transform the focus of estimating CTFPF into efficiently estimating the time-dependent failure domain under different UBTI. Then, a first failure instant (FFI) learning function combined with adaptive candidate sample pool reduction technology (ACSPRT) is established to efficiently obtain the time-dependent failure domain under different UBTI. At the meanwhile, to avoid the boundary effect of kernel density estimation method (KDE) in estimating the conditional probability density function (PDF) of distribution parameters, an extended prior distribution is proposed to improve the accuracy of estimating the conditional PDF at the boundary of distribution parameter space. The results of three examples verify the advantage of the proposed EPD-Bayes.