A method to integrate and classify normal distributions

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 where these integrals are easy to calculate, 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 error matrix for classification amongst any number of normal distributions, and the Bayes-optimal discriminability index, (iv) dimension reduction and visualizations for such problems, and (v) tests for how reliable are the methods on given data. We demonstrate these tools with vision research applications of detecting occluding objects in natural scenes, and detecting camouflage.