Bayesian active learning methods have emerged for structural reliability analysis and shown more attractive features than existing active learning methods. However, it remains a challenge to actively learn the failure probability by fully exploiting its posterior statistics. In this study, a novel Bayesian active learning method termed ‘Parallel Bayesian Probabilistic Integration’ (PBPI) is proposed for structural reliability analysis, especially when involving small failure probabilities. A pseudo posterior variance of the failure probability is first heuristically proposed for providing a pragmatic uncertainty measure over the failure probability. The variance amplified importance sampling is modified in a sequential manner to allow the estimations of posterior mean and pseudo posterior variance with a large sample population. A learning function derived from the pseudo posterior variance and a stopping criterion associated with the pseudo posterior coefficient of variance of the failure probability are then presented to enable active learning. In addition, a new adaptive multi-point selection method is developed to identify multiple sample points at each iteration without the need to predefine the number, thereby allowing parallel computing. The effectiveness of the proposed PBPI method is verified by investigating four numerical examples, including a turbine blade structural model and a transmission tower structure. Results indicate that the proposed method is capable of estimating small failure probabilities with superior accuracy and efficiency over several other existing active learning reliability methods.

贝叶斯主动学习方法已经出现，用于结构可靠性分析，并显示出比现有主动学习方法更有吸引力的特征。然而，通过充分利用其后验统计来主动学习失败概率仍然是一个挑战。在本研究中，提出了一种称为“并行贝叶斯概率积分”（PBPI）的新型贝叶斯主动学习方法，用于结构可靠性分析，特别是在涉及小故障概率时。首先启发式地提出了故障概率的伪后验方差，以提供故障概率的实用不确定性度量。以顺序方式修改方差放大重要性采样，以允许对大样本群体的后验均值和伪后验方差进行估计。然后提出从伪后验方差导出的学习函数和与故障概率的伪后验方差系数相关联的停止标准以实现主动学习。此外，还开发了一种新的自适应多点选择方法，可以在每次迭代时识别多个样本点，而无需预先定义数量，从而允许并行计算。通过研究四个数值实例（包括涡轮叶片结构模型和输电塔结构），验证了所提出的 PBPI 方法的有效性。结果表明，与其他几种现有的主动学习可靠性方法相比，所提出的方法能够以更高的准确性和效率来估计小故障概率。