Multivariate normal (MVN) probabilities arise in myriad applications, but they are analytically intractable and need to be evaluated via Monte-Carlo-based numerical integration. For the state-of-the-art minimax exponential tilting (MET) method, we show that the complexity of each of its components can be greatly reduced through an integrand parameterization that utilizes the sparse inverse Cholesky factor produced by the Vecchia approximation, whose approximation error is often negligible relative to the Monte-Carlo error. Based on this idea, we derive algorithms that can estimate MVN probabilities and sample from truncated MVN distributions in linear time (and that are easily parallelizable) at the same convergence or acceptance rate as MET, whose complexity is cubic in the dimension of the MVN probability. We showcase the advantages of our methods relative to existing approaches using several simulated examples. We also analyze a groundwater-contamination dataset with over twenty thousand censored measurements to demonstrate the scalability of our method for partially censored Gaussian-process models.

翻译：多元正态（MVN）概率出现在无数应用中，但其解析形式难以处理，需要通过基于蒙特卡洛的数值积分进行评估。针对最先进的极小极大指数倾斜（MET）方法，我们证明其各组成部分的复杂度可通过一种被积函数参数化方法大幅降低，该方法利用了Vecchia近似产生的稀疏逆Cholesky因子——其近似误差相对于蒙特卡洛误差通常可忽略不计。基于这一思想，我们推导出能在与MET相同的收敛率或接受率下，以线性时间复杂度（且易于并行化）估计MVN概率并从截断MVN分布中采样的算法，而MET的复杂度与MVN概率维度的立方成正比。我们通过多个模拟示例展示了所提方法相对于现有方法的优势。此外，我们分析了一个包含两万多个截尾测量值的地下水污染数据集，以证明我们的方法在部分截尾高斯过程模型中的可扩展性。