Methods to integrate multinormals and compute classification measures
Authors: Abhranil Das, Wilson S Geisler
Organizations: Department of Physics, The University of Texas at Austin · Center for Perceptual Systems, The University of Texas at Austin · Center for Theoretical and Computational Neuroscience, The University of Texas at Austin · Department of Psychology, The University of Texas at Austin
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) ways to scale the discriminability of two distributions, (v) dimension reduction and visualizations for such problems, and (vi) 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.
We present four new mathematical methods, two exact and two approximate, along with open-source software, to compute the cdf, pdf and inverse cdf of the generalized chi-square distribution. Some methods are geared for speed, while others are designed to be accurate far into the tails, using which we can also measure large values of the discriminability index d′ between multivariate normal distributions. We compare the accuracy and speed of these and previous methods, characterize their advantages and limitations, and identify the best methods to use in different cases.
We propose Differences in Detection (DnD), an intuitive method to compare two object detection models. Based on the same matching algorithm, it complements the standard metrics of mean Average Precision (mAP) and TIDE error analysis with the ability to compare two models directly. More specifically, we calculate the intersection of ground truth labels that are recognized by both models, followed by the corresponding difference sets and the complement set of ground truth labels that are missed by both models. The resulting comparison is more direct and intuitive than a comparison of independent summary statistics. It reveals individual and shared mistakes and becomes particularly interesting when combined with error types. In this case, the differences in detection errors can be analyzed naturally in a standard confusion matrix. While valuable in itself, we believe that one of the best applications of DnD is to guide explainability methods such as ODAM towards metric-relevant examples, grounded in structured subsets. The code for our method is available here: https://github.com/JohannesTheo/differences-in-detection
Johannes Theodoridis, Johannes Maucher, Andreas Schilling
Implicit neural representations (INRs) offer compact encoding of volumes, but as lossy approximators, inevitably have prediction errors. We consider INRs that can simultaneously encode relative error scales by predicting distributions using tools from uncertainty estimation. Typically, uncertainty estimation relies on computationally expensive approaches or on predefined parametric assumptions about the predictive distribution (e.g., Gaussian). In this study, we propose a lightweight method that reformulates regression-based INR training as a classification task by discretizing continuous targets into bins, enabling flexible distribution modeling to capture complex multimodal behaviors. We analyze the trade-off between regression and classification for INR training and demonstrate that the classification setting tends to achieve high reconstruction quality and competitive error awareness through uncertainty estimation, compared to regression-based approaches.