We consider the sample efficient estimation of failure probabilities from expensive oracle evaluations of a limit state function via importance sampling (IS). In contrast to conventional ``two stage'' approaches, which first train a surrogate model for the limit state and then construct an IS proposal to estimate failure probability using separate oracle evaluations, we propose a \emph{single stage} approach where a Gaussian process surrogate and a surrogate for the optimal (zero-variance) IS density are trained from shared evaluations of the oracle, making better use of a limited budget. With such an approach, small failure probabilities can be learned with relatively few oracle evaluations. We propose \emph{kernel density estimation adaptive importance sampling} (\texttt{KDE-AIS}), which combines Gaussian process surrogates with kernel density estimation to adaptively construct the IS proposal density, leading to sample efficient estimation of failure probabilities. We show that \texttt{KDE-AIS} density asymptotically converges to the optimal zero-variance IS density in total variation. Empirically, \texttt{KDE-AIS} enables accurate and sample efficient estimation of failure probabilities compared to the state of the art, including previous work on Gaussian process based adaptive importance sampling.

翻译：本文关注通过重要性采样，从极限状态函数的高耗时代价预言机评估中高效估计失效概率的问题。与传统的“两阶段”方法（即先训练极限状态的代理模型，再构建重要性采样建议分布，并利用独立的预言机评估来估计失效概率）不同，我们提出了一种“单阶段”方法，其中高斯过程代理模型和最优（零方差）重要性采样密度的代理模型通过共享预言机评估进行训练，从而更有效地利用有限预算。采用这种方法，可以用相对较少的预言机评估学习到小的失效概率。我们提出了核密度估计自适应重要性采样（\texttt{KDE-AIS}），该方法将高斯过程代理模型与核密度估计相结合，自适应地构建重要性采样建议密度，从而实现失效概率的高效样本估计。我们证明了\texttt{KDE-AIS}密度在总变差距离上渐近收敛于最优零方差重要性采样密度。实验表明，与现有最先进方法（包括基于高斯过程的自适应重要性采样的先前工作）相比，\texttt{KDE-AIS}能够实现准确且样本高效的失效概率估计。