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 exists no general analytical expression, standard numerical method 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, distribution, and percentage points of any function of a normal vector, (iii) the error matrix that measures classification performance amongst any number of normal distributions, and the optimal discriminability index, (iv) dimension reduction and visualizations for such problems, and (v) tests for how reliably these methods can be used on given data. We demonstrate these tools with vision research applications of detecting occluding objects in natural scenes, and detecting camouflage.