论文标题
随机几何图中的罕见事件
Rare Events in Random Geometric Graphs
论文作者
论文摘要
这项工作介绍了和比较方法，用于估计与泊松点过程中随机几何图中边缘数量相关的稀有事实概率。在一维环境中，我们得出了与随机几何图中边缘数量相关的各种条件概率的封闭形式表达式，并在此基础上开发了条件蒙特卡洛算法，以估算稀有事实概率。与粗蒙特卡洛估计量相比，我们严格地降低了方差，并说明了一项仿真研究的改进程度。在较高的维度中，我们利用条件蒙特卡洛（Monte Carlo）来消除来自节点泊松数的随机性的估计器中的波动。最后，基于大问题理论的概念见解，我们说明使用Gibbsian Point Process进行的重要性取样可以进一步降低估计差异。
This work introduces and compares approaches for estimating rare-event probabilities related to the number of edges in the random geometric graph on a Poisson point process. In the one-dimensional setting, we derive closed-form expressions for a variety of conditional probabilities related to the number of edges in the random geometric graph and develop conditional Monte Carlo algorithms for estimating rare-event probabilities on this basis. We prove rigorously a reduction in variance when compared to the crude Monte Carlo estimators and illustrate the magnitude of the improvements in a simulation study. In higher dimensions, we leverage conditional Monte Carlo to remove the fluctuations in the estimator coming from the randomness in the Poisson number of nodes. Finally, building on conceptual insights from large-deviations theory, we illustrate that importance sampling using a Gibbsian point process can further substantially reduce the estimation variance.
