Probability theory has become the predominant framework for quantifying uncertainty across scientific and engineering disciplines, with a particular focus on measurement and control systems. However, the widespread reliance on simple Gaussian assumptions--particularly in control theory, manufacturing, and measurement systems--can result in incomplete representations and multistage lossy approximations of complex phenomena, including inaccurate propagation of uncertainty through multi stage processes. This work proposes a comprehensive yet computationally tractable framework for representing and propagating quantitative attributes arising in measurement systems using Probability Density Functions (PDFs). Recognizing the constraints imposed by finite memory in software systems, we advocate for the use of Gaussian Mixture Models (GMMs), a principled extension of the familiar Gaussian framework, as they are universal approximators of PDFs whose complexity can be tuned to trade off approximation accuracy against memory and computation. From both mathematical and computational perspectives, GMMs enable high performance and, in many cases, closed form solutions of essential operations in control and measurement. The paper presents practical applications within manufacturing and measurement contexts especially circular factory, demonstrating how the GMMs framework supports accurate representation and propagation of measurement uncertainty and offers improved accuracy--compared to the traditional Gaussian framework--while keeping the computations tractable.

Our approach is purely based on 2D convolutions. Nevertheless, it3 outperforms or performs comparably to many more costly 3D models. We thank the reviewers for pointing out some related (or missing) references. The12 Timeception layers involve group convolutions at different time scales while our TAM layers only use depthwise13 convolution. As a result, the Timeception has significantly more parameters than the TAM (10% vs. 0.1% of the14 totalmodelparameters).

In appendix, we Reward OptimizationWepropose .

The code for our experiments is available at https://github.com/AndyShih12/HyperSPN. To examine the merits of HyperSPNs as discussed in Section 3, we construct a hand-crafted dataset to test the three types of models described in Figure 4: SPN-Large, SPN-Small, and HyperSPN. The hand-crafted dataset is procedurally generated with 256 binary variables and 10000 instances, broken into train/valid/test splits at 70/10/20%. The generation procedure is designed such that the correlation between variable i and j is dependent on the path length between leaves i and j of a complete binary tree over the 256 variables. The exact details can be found in our code.

Usage Application The model is intended for open-vocabulary object detection.Known Caveats

Major challenges in scaling self-training are the choice of label space, pseudo-annotation filtering, and training efficiency.