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：多元正态概率在众多应用中频繁出现，但解析计算困难，需通过基于蒙特卡洛的数值积分进行评估。针对当前最先进的极小极大指数倾斜方法，我们证明通过利用Vecchia近似产生的稀疏逆Cholesky因子对被积函数进行参数化，可大幅降低其各组件的复杂度，且该近似误差相对于蒙特卡洛误差通常可忽略不计。基于此思路，我们推导出能在线性时间内（且易于并行化）估计多元正态概率并从截断多元正态分布中采样的算法，同时保持与极小极大指数倾法相同的收敛率或接受率，而后者复杂度随多元正态概率维度的立方增长。通过多个模拟示例，我们展示了所提方法相对于现有方法的优势。我们还分析了一个包含两万余条删失实测数据的地下水污染数据集，验证了该方法在部分删失高斯过程模型中的可扩展性。