A Bayesian piecewise fitting method for estimating probability distributions of performance functions

The probability distribution of the performance function plays an important role in many fields. However, it is challenging to obtain this distribution because of the difficulty in capturing the tails on both sides, particularly for high-dimensional problems. To estimate the probability distribution of the performance function efficiently and accurately, this study proposes a piecewise fitting method based on the simulation-based Bayesian post-processing method. The method first divides the whole distribution into the main body and the left and right tail distributions. Subsequently, the samples for the main body are generated by a randomized Sobol sequence, while the samples for the left and right tails are produced through Markov chain Monte Carlo sampling. Thereafter, the shifted generalized lognormal distribution model is applied to reconstruct the main body distribution, and the truncated shifted generalized lognormal distribution is used to fit the tail distributions. Finally, the overall distribution is obtained, and the shape parameters of the distribution model are determined using Bayesian estimation methods. The efficiency and accuracy of the proposed method are demonstrated through four numerical examples, including a simple toy example and cases involving strongly nonlinear, implicit, high-dimensional performance functions.