Bayesian updating plays an important role in reducing epistemic uncertainty and making more reliable predictions of the structural failure probability. In this context, it should be noted that the posterior failure probability conditional on the updated uncertain parameters becomes a random variable itself. Hence, characterizing the statistical properties of the posterior failure probability is important, yet challenging task for risk-based decision-making. In this study, an efficient framework is proposed to fully characterize the statistical properties of the posterior failure probability. The framework is based on the concept of Bayesian updating and keeps the effect of aleatory and epistemic uncertainty separated. To improve the efficiency of the proposed framework, a weighted sparse grid numerical integration is suggested to evaluate the first three raw moments of the corresponding posterior reliability index. This enables the reuse of evaluation results stemming from previous analyses. In addition, the proposed framework employs the shifted lognormal distribution to approximate the probability distribution of the posterior reliability index, from which the mean, quantile, and even the distribution of the posterior failure probability can be easily obtained in closed form. Four examples illustrate the efficiency and accuracy of the proposed method, and results generated with Markov Chain Monte Carlo combined with plain Monte Carlo simulation are employed as a reference.

中文翻译：

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用于表征后验失效概率的有效贝叶斯更新框架


贝叶斯更新在减少认知不确定性和对结构失效概率进行更可靠的预测方面发挥着重要作用。在这种情况下，应该注意的是，以更新的不确定参数为条件的后验失效概率本身变成了随机变量。因此，表征后验失效概率的统计特性对于基于风险的决策来说非常重要，但也是一项具有挑战性的任务。在本研究中，提出了一个有效的框架来充分表征后验失效概率的统计特性。该框架基于贝叶斯更新的概念，并将偶然和认知不确定性的影响分开。为了提高所提出框架的效率，建议使用加权稀疏网格数值积分来评估相应后验可靠性指数的前三个原始矩。这使得可以重复使用先前分析的评估结果。此外，所提出的框架采用平移对数正态分布来近似后验可靠性指标的概率分布，从中可以很容易地以封闭形式获得后验失效概率的均值、分位数甚至分布。四个例子说明了该方法的效率和准确性，并采用马尔可夫链蒙特卡罗结合普通蒙特卡罗模拟产生的结果作为参考。