This is the appendix of the paper " Quantification of Uncertainty with Adversarial Models ". It consists of three sections.

Quantifying uncertainty is important for actionable predictions in real-world applications.

Accurate EV power estimation underpins range prediction and energy management, yet practitioners need both point accuracy and trustworthy uncertainty. We propose an anchored-ensemble Long Short-Term Memory (LSTM) with a Student-t likelihood that jointly captures epistemic (model) and aleatoric (data) uncertainty. Anchoring imposes a Gaussian weight prior (MAP training), yielding posterior-like diversity without test-time sampling, while the t-head provides heavy-tailed robustness and closed-form prediction intervals. Using vehicle-kinematic time series (e.g., speed, motor RPM), our model attains strong accuracy: RMSE 3.36 +/- 1.10, MAE 2.21 +/- 0.89, R-squared = 0.93 +/- 0.02, explained variance 0.93 +/- 0.02, and delivers well-calibrated uncertainty bands with near-nominal coverage. Against competitive baselines (Student-t MC dropout; quantile regression with/without anchoring), our method matches or improves log-scores while producing sharper intervals at the same coverage. Crucially for real-time deployment, inference is a single deterministic pass per ensemble member (or a weight-averaged collapse), eliminating Monte Carlo latency. The result is a compact, theoretically grounded estimator that couples accuracy, calibration, and systems efficiency, enabling reliable range estimation and decision-making for production EV energy management.

Despite remarkable progress in recent years, vision language models (VLMs) remain prone to overconfidence and hallucinations on tasks such as Visual Question Answering (VQA) and Visual Reasoning. Bayesian methods can potentially improve reliability by helping models selectively predict, that is, models respond only when they are sufficiently confident. Unfortunately, Bayesian methods are often assumed to be costly and ineffective for large models, and so far there exists little evidence to show otherwise, especially for multimodal applications. Here, we show the effectiveness and competitive edge of variational Bayes for selective prediction in VQA for the first time. We build on recent advances in variational methods for deep learning and propose an extension called "Variational VQA". This method improves calibration and yields significant gains for selective prediction on VQA and Visual Reasoning, particularly when the error tolerance is low ($\leq 1\%$). Often, just one posterior sample can yield more reliable answers than those obtained by models trained with AdamW. In addition, we propose a new risk-averse selector that outperforms standard sample averaging by considering the variance of predictions. Overall, we present compelling evidence that variational learning is a viable option to make large VLMs safer and more trustworthy.

Accurate brain tumor segmentation from MRI is vital for diagnosis and treatment planning. Although Monte Carlo (MC) Dropout is widely used to estimate model uncertainty, its effectiveness in identifying segmentation errors -- especially near tumor boundaries -- remains unclear. This study empirically examines the relationship between MC Dropout--based uncertainty and segmentation error in 2D brain tumor MRI segmentation using a U-Net trained under four augmentation settings: none, horizontal flip, rotation, and scaling. Uncertainty was computed from 50 stochastic forward passes and correlated with pixel-wise errors using Pearson and Spearman coefficients. Results show weak global correlations ($r \approx 0.30$--$0.38$) and negligible boundary correlations ($|r| < 0.05$). Although differences across augmentations were statistically significant ($p < 0.001$), they lacked practical relevance. These findings suggest that MC Dropout uncertainty provides limited cues for boundary error localization, underscoring the need for alternative or hybrid uncertainty estimation methods in medical image segmentation.

Estimating accurate and well-calibrated predictive uncertainty is important for enhancing the reliability of computer vision models, especially in safety-critical applications like traffic scene perception. While ensemble methods are commonly used to quantify uncertainty by combining multiple models, a mixture of experts (MoE) offers an efficient alternative by leveraging a gating network to dynamically weight expert predictions based on the input. Building on the promising use of MoEs for semantic segmentation in our previous works, we show that well-calibrated predictive uncertainty estimates can be extracted from MoEs without architectural modifications. W e investigate three methods to extract predictive uncertainty estimates: predictive entropy, mutual information, and expert variance. W e evaluate these methods for an MoE with two experts trained on a semanti-cal split of the A2D2 dataset. Our results show that MoEs yield more reliable uncertainty estimates than ensembles in terms of conditional correctness metrics under out-of-distribution (OOD) data. Additionally, we evaluate routing uncertainty computed via gate entropy and find that simple gating mechanisms lead to better calibration of routing uncertainty estimates than more complex classwise gates. Finally, our experiments on the Cityscapes dataset suggest that increasing the number of experts can further enhance uncertainty calibration. Our code is available at https: // github.com/

This figure compares (a) a spaghetti plot of ensemble members, (b) a circular tube, and (c) our uncertainty tube for visualizing model uncertainty. Previous methods face challenges such as visual clutter (a) or the assumption of symmetric uncertainty (a, b), but our uncertainty tube (c), constructed using superellipses, provides a more accurate visualization of asymmetric uncertainty. Its superelliptical shape distinctly improves the visualization of the uncertainty orientation and its evolution along trajectories, as highlighted in the boxes. The visualization is further enhanced with a color palette that uses gray for low uncertainty, blue for large asymmetric uncertainty, and yellow for large symmetric uncertainty. Predicting particle trajectories with neural networks (NNs) has substantially enhanced many scientific and engineering domains. However, effectively quantifying and visualizing the inherent uncertainty in predictions remains challenging. Without an understanding of the uncertainty, the reliability of NN models in applications where trustworthiness is paramount is significantly compromised. This paper introduces the uncertainty tube, a novel, computationally efficient visualization method designed to represent this uncertainty in NN-derived particle paths. By integrating well-established uncertainty quantification techniques, such as Deep Ensembles, Monte Carlo Dropout (MC Dropout), and Stochastic Weight Averaging-Gaussian (SW AG), we demonstrate the practical utility of the uncertainty tube, showcasing its application on both synthetic and simulation datasets. Understanding and analyzing flow field data is fundamental for numerous scientific and engineering disciplines, including fluid dynamics, atmospheric science, and material processing. Traditional computational fluid dynamics (CFD) simulations are often computationally intensive, a limitation that has led researchers to explore more efficient paradigms. This exploration has given rise to neural networks (NNs) as a transformative tool in this domain, driven by their capacity to overcome these computational bottlenecks. Notably, recent work, such as Han et al. [26, 27], leverages NNs to learn Lagrangian-based flow maps, enabling efficient and robust particle tracing in time-varying fields. These data-driven models demonstrate remarkable accuracy and speed, making them increasingly indispensable for accelerating discovery and design cycles in fluid dynamics. Despite these advancements, a significant challenge remains in providing a comprehensive understanding of the confidence associated with NN predictions in flow fields.

This study evaluated the probability and uncertainty forecasts of five recently proposed Bayesian deep learning methods relative to a deterministic residual neural network (ResNet) baseline for 0-1 h convective initiation (CI) nowcasting using GOES-16 satellite infrared observations. Uncertainty was assessed by how well probabilistic forecasts were calibrated and how well uncertainty separated forecasts with large and small errors. Most of the Bayesian deep learning methods produced probabilistic forecasts that outperformed the deterministic ResNet, with one, the initial-weights ensemble + Monte Carlo (MC) dropout, an ensemble of deterministic ResNets with different initial weights to start training and dropout activated during inference, producing the most skillful and well-calibrated forecasts. The initial-weights ensemble + MC dropout benefited from generating multiple solutions that more thoroughly sampled the hypothesis space. The Bayesian ResNet ensemble was the only one that performed worse than the deterministic ResNet at longer lead times, likely due to the challenge of optimizing a larger number of parameters. To address this issue, the Bayesian-MOPED (MOdel Priors with Empirical Bayes using Deep neural network) ResNet ensemble was adopted, and it enhanced forecast skill by constraining the hypothesis search near the deterministic ResNet hypothesis. All Bayesian methods demonstrated well-calibrated uncertainty and effectively separated cases with large and small errors. In case studies, the initial-weights ensemble + MC dropout demonstrated better forecast skill than the Bayesian-MOPED ensemble and the deterministic ResNet on selected CI events in clear-sky regions. However, the initial-weights ensemble + MC dropout exhibited poorer generalization in clear-sky and anvil cloud regions without CI occurrence compared to the deterministic ResNet and Bayesian-MOPED ensemble.