Univariate and multivariate normal probability distributions are widely used when modeling decisions under uncertainty. Computing the performance of such models requires integrating these distributions over specific domains, which can vary widely across models. Besides some special cases, there exist no general analytical expressions, standard numerical methods or software for these integrals. Here we present mathematical results and open-source software that provide (i) the probability in any domain of a normal in any dimensions with any parameters, (ii) the probability density, cumulative distribution, and inverse cumulative distribution of any function of a normal vector, (iii) the classification errors among any number of normal distributions, the Bayes-optimal discriminability index and relation to the operating characteristic, (iv) dimension reduction and visualizations for such problems, and (v) tests for how reliably these methods may be used on given data. We demonstrate these tools with vision research applications of detecting occluding objects in natural scenes, and detecting camouflage.

翻译：单变量与多变量正态概率分布广泛应用于不确定性条件下的决策建模中。计算此类模型的性能需要对这些分布在特定域上进行积分，而不同模型的积分域可能差异显著。除少数特例外，此类积分通常缺乏通用的解析表达式、标准数值方法或软件工具。本文提出数学结果与开源软件，可实现：(i) 任意维度、任意参数正态分布在任意域内的概率计算；(ii) 正态向量任意函数的概率密度、累积分布及逆累积分布计算；(iii) 任意数量正态分布间的分类误差、贝叶斯最优可分性指数及其与操作特征的关系；(iv) 此类问题的降维与可视化方法；(v) 在给定数据上使用这些方法的可靠性检验。我们通过自然场景中遮挡物体检测与伪装检测的视觉研究应用，展示了这些工具的效能。