Bayesian Neural Networks

This thesis investigates how Bayesian principles can deepen our understanding of modern deep learning systems. While neural networks achieve remarkable predictive performance, their ability to generalize and to quantify uncertainty remains only partly understood. This thesis approaches this challenge from both methodological and theoretical angles: unifying Bayesian inference, function-space modeling, and large-deviation theory under a common probabilistic perspective. On the methodological side, the thesis introduces the Deep Variational Implicit Process (DVIP), a scalable Bayesian framework that extends implicit processes to deep architectures. Complementing this, two post-hoc methods -- the Variational Linearized Laplace Approximation (VaLLA) and the Fixed-Mean Gaussian Process (FMGP) -- are proposed to equip pretrained deterministic networks with calibrated uncertainty estimates. The theoretical contributions focus on one of the central open questions in modern machine learning: why do large, over-parameterized neural networks generalize so well? To address this, the thesis develops a unified probabilistic framework that connects three key mechanisms -- diversity, smoothness, and stochasticity -- within the language of PAC-Bayesian and large-deviation theory.

Vessel trajectory prediction from Automatic Identification System (AIS) data is essential for maritime situational awareness, yet it remains challenging due to irregular sampling, missing reports, and complex dynamics. Beyond accurate point forecasts, maritime applications also demand well-calibrated uncertainty estimates for reliable decision-making. Bayesian Neural Ordinary Differential Equations (ODEs) offer a principled framework for continuous-time trajectory modeling with uncertainty quantification by placing a prior over the neural vector field parameters. However, the commonly used isotropic Gaussian weight prior fails to encode informative structural properties of vessel dynamics, such as smoothness and locality. Existing function-space Bayesian neural network methods address this limitation for static mappings, but do not transfer directly to Neural ODEs, where the primary quantity of interest is the trajectory rather than the vector field itself. In principle, one could place a Gaussian process (GP) prior directly over ODE solutions, but this requires propagating distributions through a nonlinear ODE solver, which is analytically intractable. To address this challenge, we adopt a practical approach that imposes a GP-kernel-based prior directly on the vector field evaluated at a finite set of measurement points. Specifically, we augment the standard weight-space variational objective with a kernel-based regularizer that penalizes deviations of the vector field from the structure implied by a GP prior. To handle long and irregular AIS trajectories, we further combine this function-space regularization with probabilistic multiple shooting, which decouples inference across temporal segments while maintaining global consistency.

A central aim of deep learning theory is to characterize how neural networks make predictions in the regime of simultaneously large model and training set size. Since the limits of diverging number of model parameters and dataset size do not commute it is not clear a priori what limits exist. In this work, we shed new light on these questions by studying Bayesian inference in deep non-linear MLPs in the regime where the number of training samples ($P$), the input dimension ($N_0$), the hidden layer width ($N$), and the number of hidden layers ($L$) can all be large. We build on the Neural Covariance SDE (Li et al., 2022) to analyze predictive posteriors in the regime where $LP/N\inΘ(1)$, playing the role of an effective network depth. Our framework covers both smooth and ReLU activation functions and applies to arbitrary temperature. We find to first order in $LP/N$ a simple criterion for which data generating processes benefit from depth in the sense that larger $LP/N$ increases the Bayesian model evidence. We also give a novel derivation of a prior result from the physics literature that at least to first order in $LP/N$, the Bayesian predictive posterior is remarkably simple and is simply equivalent to that of a data-dependent kernel method.

Always-on edge systems must keep learning as conditions change under tight compute budgets and must detect unreliable predictions. Bayesian binary neural networks are attractive in this setting, but mean-field Bernoulli posteriors can saturate on long non-stationary streams, wiping out epistemic uncertainty and freezing plasticity. We propose BiMU, derived from a bounded-memory variational objective that balances stability, plasticity, and forgetting. BiMU combines a data term with controlled relaxation toward the prior and an uncertainty-dependent step size that prevents saturation and sustains informative uncertainty. This non-degenerate posterior enables fully online, buffer-free active querying via Monte Carlo disagreement, reducing label queries and backpropagation updates under imbalance. BiMU sustains learning and strong OOD detection on 1000-tasks Permuted-MNIST, and on OpenLORIS-Object achieves up to 32$\times$ label/update savings at matched accuracy under class imbalance and feature compression.

The scaling limit where both the size of the training set $P$ and the width $N$ of a deep neural network grow at the same rate, the so-called proportional-width regime, has been intensely studied for shallow, single-hidden-layer networks. However, extending these non-perturbative results from shallow architectures to deep non-linear networks has proven very challenging. Here we present an effective approximate approach to predict the generalization performance of Bayesian multi-layer perceptrons (MLPs) of fixed depth $L$ on arbitrary high-dimensional data. We propose an equivalent Wishart Ansatz to capture the dominant stochastic fluctuations of the hierarchical empirical kernels of MLPs. This allows us to perform a large deviation analysis for the partition function of MLPs in the proportional limit, expressed in terms of a renormalized NNGP kernel. In this description, even strong representation learning in the proportional limit is encoded in at most $L$ scalar order parameters, determined self-consistently. Extending the approach to convolutional architectures (CNNs), we identify a hierarchical local kernel renormalization mechanism, which allows to quantify more complex data-dependent transformations of the large-width kernel in CNNs due to finite-width effects. We test our effective theory against sampling experiments from the Bayesian posterior of finite deep neural networks with depths $L \sim O(10)$ and $P\sim O(10^3)$ on classic benchmark datasets, finding overall very good agreement together with two distinct types of systematic deviations.

Gaussian processes (GPs) provide a principled Bayesian framework for uncertainty estimation, but their computational complexity severely limits scalability to large datasets. We propose SIKA-GP, which accelerates GP inference using sparse inducing kernel approximations based on a dyadic ordered template basis, incurring only ${O}(\log M)$ complexity dependence on the number of inducing points. Our approach constructs compact and expressive kernel representations from sparsely activated bases, enabling efficient tensorized GPU computation and seamless integration with modern large-scale models. SIKA-GP can be naturally embedded into Bayesian neural networks (BNNs) with sparse activations, yielding significant speedups in both training and inference without sacrificing predictive performance. The method naturally extends to deep feature learning, addressing the scalability challenges introduced by deep architectures and high-dimensional feature representations. Empirical results on vision and transformer-based language benchmarks demonstrate that our approach consistently delivers fast and accurate GP models, providing a principled path toward scalable kernel learning.

Epistemic uncertainty is often viewed as a reducible uncertainty that vanishes with increasing data. This perspective implicitly assumes parameter identifiability and equates epistemic uncertainty with predictive variability. In overparametrized neural networks, however, model parameters are typically non-identifiable due to symmetries and redundant representations. As a consequence, substantial parameter uncertainty can persist even when the underlying function is fully identified. In this work, we analyze epistemic uncertainty through the lens of non-identifiability and characterize both discrete and continuous sources of residual uncertainty. Focusing on one-hidden-layer ReLU networks, we thoroughly analyze the resulting posterior structure and validate our theoretical insights through empirical studies.

Advanced driver assistance systems (ADAS) play an important role in modern automotive intelligence, significantly enhancing vehicle safety and stability. The performance of ADAS critically relies on accurate and reliable vehicle state estimation, particularly from vehicle dynamic sensors. Among these signals, wheel load is a key variable for chassis control and safety-critical functions, yet it remains difficult to estimate robustly due to complex suspension geometry, nonlinear dynamics, and measurement noise. To address this issue, we propose DBPnet, a Bayesian physics-informed neural network (PINN) with a physics-aware embedding module inspired by damper characteristics. First, this paper presents a suspension linkage-level modeling (SLLM) approach that constructs a nonlinear instantaneous dynamic model by explicitly considering the complex geometric structure of the suspension. Building upon SLLM, Bayesian inference is integrated into the PINN to effectively cope with noise and uncertainty in the vehicle chassis system, thereby improving the model's robustness. Then, a physics-informed loss function is employed to ensure consistency with fundamental physical principles, while the damper characteristics-inspired embedding module extracts temporal variation features of input signals and incorporates them into each layer of the PINN, ensuring that physical observations guide the neural network without being constrained by fixed physical models. Extensive evaluations on high-fidelity simulations and real-world experiments demonstrate that our DBPnet consistently achieves lower RMSE and MaxError than baseline methods. These results highlight the potential of our DBPnet to advance wheel load estimation and contribute to the development of more reliable ADAS actuator functions.

Annotator disagreement in emotion classification reflects ambiguity intrinsic to emotion concepts and is essential for predictor-quality assessment in subjective NLP. Yet no prior work integrates soft-label learning with Bayesian deep learning to evaluate uncertainty along axes including annotator-distribution fidelity. We train a linear head on a frozen RoBERTa via cyclical stochastic gradient Markov chain Monte Carlo (cSG-MCMC), targeting the empirical annotator distribution with a soft-label objective under a five-axis evaluation. On the 28-emotion GoEmotions benchmark, the proposed method outperforms Monte Carlo Dropout and Deep Ensemble simultaneously on three axes -- Jensen-Shannon divergence (JSD) to the annotator distribution, Spearman correlation between per-emotion aleatoric uncertainty and disagreement, and selective-prediction Area Under the Risk-Coverage Curve (AURC) and Area Under the ROC Curve (AUROC) -- showing independent axes are jointly attainable from one posterior. Post-hoc temperature scaling exhibits a bidirectional effect, establishing hard-label calibration and annotator-JSD as independent dimensions and motivating joint reporting as an honest protocol.

Machine learning methods provide a methodological innovation that can help screen for cardiovascular disease through noninvasive and readily available measurement modalities. Recent investments in using electrocardiogram (ECG) data to screen for structural heart disease (SHD) are one example, where ECGs provide a low-cost, available modality for screening. This has led to the EchoNext dataset, a paired ECG-echocardiogram data repository for testing new methods of SHD detection. However, relatively few studies have investigated how more probabilistic classification through Bayesian inference may improve uncertainty quantification in this setting. Moreover, few studies have considered how triage systems can be developed to alleviate healthcare bottlenecks, such as the review of data from underserved, rural clinics by expert sonographers for SHD assessment. In this study, we leverage existing ECG-echocardiogram data to compare frequentist and Bayesian neural network classifiers. We show that the Bayesian approach is comparable or better than frequentist methods in SHD classification, and that they have a more robust uncertainty quantification attached to them. We provide an example of how this uncertainty-aware classification scheme can be used for screening SHD, providing a proof-of-concept for how machine learning can help with triage in getting individuals expert sonographer input when SHD is highly likely or measurements are highly uncertain.

The practical adoption of sampling-based inference (SAI) in Bayesian neural networks (BNNs) remains limited, partly due to persistent misconceptions about the feasibility and efficiency of sampling. This position paper argues that SAI has achieved computational parity with optimization-based methods and is at the verge of superseding such methods for effective and efficient inference in BNNs. This development should be in the interest of the whole community, promoting BNNs as a principled paradigm with its long-standing yet unfulfilled promise of providing principled uncertainty quantification for neural networks. SAI can even do more -- yielding superior prediction performance through model averaging, serving as the foundation for a plethora of possible downstream tasks, and providing crucial insights into the landscape of BNNs. In order to make such a change happen and unfold the potential of sampling, overcoming current misconceptions is a necessary first step. The next step is to realign research efforts toward addressing remaining challenges in SAI. In particular, the community must focus on two core problems: sufficient exploration of the posterior landscape and high-fidelity distillation of posterior samples for efficient downstream inference. By addressing conceptual and practical obstacles, we can unlock the full potential of SAI and establish it as a central tool in Bayesian deep learning.

Post-hoc explainable AI (XAI) methods typically produce deterministic attribution maps, whereas Bayesian neural networks (BNNs) induce a distribution over explanations. Capturing the variability of this distribution is important for uncertainty-aware decision-making. This paper formalises the explanation distribution as the push-forward measure of the BNN posterior through any Lipschitz-continuous attribution operator. It further proposes the uncertainty-aware relevance attribution operator (UA-RAO), a general family of operators that summarises the explanation distribution using the mean, variance, coefficient of variation, quantiles, and set-theoretic aggregation measures. Theoretical support is provided through Monte Carlo accessibility and Wasserstein approximation bounds. The framework is evaluated on a 15-class power quality disturbance (PQD) classification benchmark, comparing three BNN approximations paired with three attribution operators using relevance mass accuracy and intersection-over-union as localisation metrics. Results show that deep ensembles with the mean UA-RAO improve localisation over the deterministic baseline, while other UA-RAO summaries reveal uncertainty patterns absent from point-estimate attributions. Qualitative results on measured signals further suggest that these patterns generalise beyond the synthetic training distribution. The framework is domain-agnostic and can be applied to any BNN paired with a Lipschitz-continuous attribution operator.

Federated Bayesian neural networks require fixing a prior on the model parameters together with a likelihood. Eliciting meaningful priors on the weight space of modern overparameterized models is notoriously difficult, and misspecification of either component can severely degrade accuracy and calibration. Motivated by the rapid progress of predictive models such as large language models, the martingale posterior, also known as predictive Bayes, replaces the prior--likelihood pair with a predictive distribution and recovers parameter uncertainty by repeatedly drawing predictive samples and refitting the model. A direct federated implementation, however, would require clients to share the local data sets. This letter proposes {federated martingale posterior} (FMP) sampling, a one-shot embarrassingly parallel protocol in which each client uploads a small set of trainable data embeddings and the server runs the predictive sampler centrally. Experiments on MNIST, CIFAR-10, and CIFAR-100 show that FMP closely matches the centralized counterpart and significantly improves calibration over consensus-style baselines.

The minima of modern neural network loss functions are typically not isolated, rather they form connected components of reparameterization invariant solutions on the training data. Analytically characterizing these solutions is a hard problem, but sampling approaches are feasible. By construction, existing methods either spread over low-loss regions, and thus do not sample reparameterization invariant solutions exactly, or are inherently local, which limits exploration of other minima valleys. We propose sampling such reparameterization invariant models using a dynamical system based on kinetic energy, subject to a gravitational pull and a friction term that dissipates energy from the system. Our proposed sampler, DiMS, is guaranteed to sample exactly from the minimum level sets and depends on physically motivated hyperparameters which allows control over the exploration capabilities of the sampler. We consider uncertainty quantification in Bayesian inference as the motivating problem and observe improved performance compared to previously proposed approaches.

Approximate Bayesian inference typically revolves around computing the posterior parameter distribution. In practice, however, the main object of interest is often a model's predictions rather than its parameters. In this work, we propose to bypass the parameter posterior and focus directly on approximating the posterior predictive distribution. We achieve this by drawing inspiration from self-training within self-supervised and semi-supervised learning. Essentially, we quantify a Bayesian model's predictive uncertainty by refitting on self-predicted data. The idea is strikingly simple: If a model assigns high likelihood to self-predicted data, these predictions are of low uncertainty, and vice versa. This yields a deterministic, sampling-free approximation of the posterior predictive. The modular structure of our Self-Supervised Laplace Approximation (SSLA) further allows us to plug in different prior specifications, enabling classical Bayesian sensitivity (w.r.t. prior choice) analysis. In order to bypass expensive refitting, we further introduce an approximate version of SSLA, called ASSLA. We study (A)SSLA both theoretically and empirically in regression models ranging from Bayesian linear models to Bayesian neural networks. Across a wide array of regression tasks with simulated and real-world datasets, our methods outperform classical Laplace approximations in predictive calibration while remaining computationally efficient.

We study posterior contraction rates for sparse Bayesian Kolmogorov-Arnold networks (KANs) over anisotropic Besov spaces, providing a statistical foundation of KANs from a Bayesian point of view. We show that sparse Bayesian KANs equipped with spike-and-slab-type sparsity priors attain the near-minimax posterior contraction. In particular, the contraction rate depends on the intrinsic anisotropic smoothness of the underlying function. Moreover, by placing a hyperprior on a single model-size parameter, the resulting posterior adapts to unknown anisotropic smoothness and still achieves the corresponding near-minimax rate. A distinctive feature of our results, compared with those for standard sparse MLP-based models, is that the KAN depth can be kept fixed: owing to the flexibility of learnable spline edge functions, the required approximation complexity is controlled through the network width, spline-grid range and size, and parameter sparsity. Our analysis develops theoretical tools tailored to sparse spline-edge architectures, including approximation and complexity bounds for Bayesian KANs. We then extend to compositional Besov spaces and show that the contraction rates depend on layerwise smoothness and effective dimension of the underlying compositional structure, thereby effectively avoiding the curse of dimensionality. Together, the developed tools and findings advance the theoretical understanding of Bayesian neural networks and provide rigorous statistical foundations for KANs.

The EU Artificial Intelligence Act (Regulation 2024/1689), fully applicable to high-risk systems from August 2026, creates urgent demand for AI architectures that are simultaneously trustworthy, transparent, and feasible to deploy on resource-constrained edge devices. Brain-like neural networks built on the Bayesian Confidence Propagation Neural Network (BCPNN) formalism have re-emerged as a credible alternative to backpropagation-driven deep learning. They deliver state-of-the-art unsupervised representation learning, neuromorphic-friendly sparsity, and existing FPGA implementations that target edge deployment. Despite this momentum, no systematic framework exists for explaining BCPNN decisions -- a gap the present paper fills. We argue that BCPNN is, in the sense of Rudin's interpretable-by-design agenda, an inherently transparent model whose architectural primitives map directly onto established explainable-AI (XAI) families. We make four contributions. First, we propose the first XAI taxonomy for BCPNN. It maps weights, biases, hypercolumn posteriors, structural-plasticity usage scores, attractor dynamics, and input-reconstruction populations onto attribution, prototype, concept, counterfactual, and mechanistic explanation modalities. Second, we introduce sixteen architecture-level explanation primitives (P1--P16), several without analogue in standard ANNs. We provide closed-form algorithms for computing each from quantities the model already maintains. Third, we introduce five design-time Configuration-as-Explanation primitives (Config-P1 to Config-P5) that treat BCPNN hyperparameter choices as an auditable pre-deployment explanation artifact. Fourth, we sketch a roadmap for integration into industrial IoT deployments and discuss EU AI Act alignment, edge feasibility, and Industry 5.0 implications.

Although the Laplace approximation offers a simple route to uncertainty quantification in deep neural networks, its reliance on inverting large Hessian matrices has motivated a range of computationally feasible low-dimensional or sparse approximations. A prominent class of such methods - sub-network Laplace approximations, constructs surrogates by restricting attention to a small subset of parameters. Existing approaches in this family typically rely on diagonal, layer-wise, or other architectural heuristics for subset selection, which ignore cross-parameter interactions and lack formal optimality guarantees. In this paper, we provide a rigorous theoretical analysis of the sub-network Laplace paradigm. We prove that all sub-network Laplace methods systematically underestimate the predictive variance of the full Laplace posterior, and that this bias decreases monotonically as the retained sub-matrix expands. Leveraging this insight, we propose two principled, analytically grounded sub-network Hessian approximations: Gradient-Laplace selects parameters with the largest average squared gradients of the model output with respect to the parameters over a reference dataset; while Greedy-Laplace iteratively refines this selection by accounting for off-diagonal interactions in the precision matrix. We establish theoretical guarantees characterizing their optimality properties and show that Gradient-Laplace provably outperforms existing heuristic approaches. Extensive numerical studies across diverse settings indicate that these methods perform strongly relative to existing benchmarks.

Bayesian neural networks are typically trained against the evidence lower bound (ELBO), whose Jensen gap closes only when the variational posterior is exact. We instead train by local consistency: gradient descent on the Bethe free energy, driving the belief at every factor toward agreement with its neighbours rather than placing a loss on the output. The resulting objective scores each observation by its own predictive density: a strictly proper rule whose optimum is the true conditional, for any likelihood with a tractable predictive convolution. Instantiated with a Gaussian last layer over a deterministic backbone, exact inference appears as one known corner: the neural-linear marginal likelihood. That corner is evidence-optimal; the shared-cavity, free-routed interior is predictive-optimal, improving NLL and calibration over it. This instance, SCROLL (Shared-Cavity fRee-rOuting Last-Layer), is a single-pass Bayesian neural network: batchable, any-likelihood, and implicitly empirical-Bayes-prior precision, observation noise, covariance, and backbone fit in one gradient pass. Prior work enters the interior only through the ELBO and its Jensen gap, even in this conjugate setting. At a single training run and forward pass per architecture-where the validation-tuned conventional references cross-validate $λ$ and ensembles pay $5$-$50\times$ at inference-a fixed SCROLL variant is best-or-tied on NLL and calibration on 7/8 UCI regression benchmarks, and best on 4/5 across three large tabular datasets (up to 515k examples) and two frozen text/vision embeddings.

Low-Rank Adaptation (LoRA) enables parameter-efficient fine-tuning of large models by decomposing weight updates into low-rank matrices, significantly reducing storage and computational overhead. While effective, standard LoRA lacks mechanisms for uncertainty quantification, leading to overconfident and poorly calibrated models. Bayesian variants of LoRA address this limitation, but at the cost of a significantly increased number of trainable parameters, partially offsetting the original efficiency gains. Additionally, these models are harder to train and may suffer from unstable convergence. In this work, we propose a novel framework for parameter-efficient Bayesian fine-tuning, demonstrating that effective uncertainty quantification can be achieved in very low-dimensional parameter spaces. The proposed method achieves strong performance with improved calibration and generalization while maintaining computational efficiency. Our empirical findings show that, with the appropriate projection of the weight space uncertainty can be effectively modeled in a low-dimensional space, and weight covariances exhibit low ranks.

Label noise remains a critical bottleneck for the generalization of supervised deep learning models, particularly when errors are structured rather than random. Standard robust training methods often fail in the presence of such semantically proximal classification errors. This work presents an architecture-agnostic Lipschitz-constant Bayesian header that can be integrated into feature extractors such as vision transformers, yielding the bi-Lipschitz-constrained Bayesian Vision Transformer (LipB-ViT). In contrast to conventional Bayesian layers, our approach enforces spectral normalization on both the mean and log-variance of the variational weights, which promotes calibrated predictive uncertainty and mitigates noise amplification. We further propose a novel metric to jointly capture uncertainty and confidence across misclassification rates, as well as an adaptive arithmetic-mean fusion scheme that combines feature-space proximity with predictive uncertainty to detect corrupted labels outperforming the state of the art k-nearest neighbor based identification methods by more than 7% reaching a recall of more than 0.93 at 15% semantically misclassified labels. Although computational costs increase due to Monte Carlo sampling, the method offers plug-and-play compatibility with pre-trained backbones and consistent hyperparameters across domains, suggesting strong utility for high-stakes applications with variable annotation reliability. The stabilized confidence estimates serve as the foundation for an analysis pipeline that jointly assesses dataset quality and label noise, yielding a second novel metric for their combined quantification. Lastly, we systematically evaluate LipB-ViT under both structured (adversarial) and unstructured noise at inference time, demonstrating its robustness in realistic high-noise and attack scenarios. We compare its performance against baseline methods.

Visual affordances identify regions in an image with potential interactions, offering a novel paradigm for scene understanding. Recognizing affordances allows autonomous robots to act more naturally, could enhance human-robot interactions, enrich augmented reality systems, and benefit prosthetic vision devices. Accurate and localized prediction of affordance regions, rather than general saliency maps is crucial for these applications. We present a model for instance segmentation of affordances by adopting sample-based and ensembles approaches for uncertainty estimation. We extend an attention-based architecture for our novel task, showing with detailed ablation experiments the effects of each component. By comparing the distribution of these different detections, we extract pixel-wise epistemic and aleatoric variances at both the semantic and spatial levels. In addition, we propose a novel measure called Probability-based Mask Quality, which enables a comprehensive analysis of semantic and spatial variations in a probabilistic instance segmentation model. Our results show that the global consensus of multiple sub-networks of Bayesian models improve deterministic networks due to a better mask refinement and generalization. This fact, joined with the more powerful features extracted by attention-based mechanisms, represent an improvement of +7.4 p.p on the $F_β^w$ score in the challenging IIT-Aff dataset. Bayesian models are also better calibrated, producing less overconfident probabilities and with a better uncertainty estimation. Qualitative results show that aleatoric variance appears in the contour of the objects, while the epistemic variance is observed in visual challenging pixels, adding interpretability to the neural network.

Accurate state estimation of nonlinear dynamical systems is fundamental to modern aerospace operations across air, sea, and space domains. Online tracking of adversarial unmanned aerial vehicles (UAVs) is especially challenging due to agile nonlinear motion, noisy and sparse sensor measurements, and unknown control inputs; conditions that violate key assumptions of classical Kalman filter variants and degrade estimation performance. Neural networks (NNs) can learn complex nonlinear relationships from data, but lack principled uncertainty quantification, which is critical for state estimation tasks where confidence bounds drive downstream decisions. We address this with Bayesian Neural Networks (BNNs), which model uncertainty through distributions over network weights and produce predictive means and uncertainties via Monte Carlo sampling. Building on this, we propose the Bayesian Neural Kalman Filter (BNKF): a hybrid framework coupling a trained BNN with a Kalman correction step for robust online UAV state estimation. Unlike related neural Kalman approaches, BNKF produces full state predictions and incorporates Bayesian uncertainty directly into covariance propagation, improving robustness under high noise conditions. We evaluate BNKF under varying radar noise levels and sampling rates using synthetic nonlinear UAV flight data. Five fold cross validation demonstrates that BNKF outperforms Extended and Unscented Kalman Filters in accuracy, precision, and truth containment under degraded sensing. An ensemble variant (BNKFe) further improves precision in high-noise edge cases at a slight accuracy tradeoff. Runtime analysis confirms minimal inference overhead, supporting real-time deployment feasibility.

Machine Learning is becoming more prevalent in science and engineering, but many approaches do not provide meaningful uncertainty estimates and predictions may also violate known physical knowledge. We propose a Bayesian framework to embed linear relationships across inputs and outputs into the learning process, whilst characterizing full predictive uncertainty over both the model parameters and the domain knowledge. We evaluated our method on learning the single particle battery model subject to voltage and energy balances, showing its ability to provide reduced credible intervals and constraint violations compared to standard Bayesian neural networks based on variational inference.

Low-Rank Adaptation (LoRA) has become the standard for fine-tuning large pre-trained models at reduced computational cost. However, its low-rank point-estimate updates limit expressiveness, leave a persistent gap relative to full fine-tuning accuracy, and provide no built-in uncertainty quantification, limiting its applicability in settings where reliability matters as much as accuracy. We introduce BaLoRA, a Bayesian extension of LoRA with a novel input-adaptive Bayesian parameterization of LoRA matrices that adds minimal parameters and compute. Surprisingly, not only does the Bayesian extension yield well-calibrated uncertainty estimates, but the adaptive noise injection underlying our approach also significantly improves prediction accuracy, narrowing the gap with full fine-tuning across both natural language reasoning and vision tasks. When applied to band gap prediction in metal-organic frameworks, BaLoRA produces zero-shot test-time uncertainty estimates that correlate more strongly with model error than a trained ensemble of LoRA models, and improve monotonically with compute without sacrificing accuracy.

In limited-data settings, a single endpoint mean of an evaluation metric such as the Continuous Ranked Probability Score (CRPS) is itself a random variable, yet it is routinely reported as if it were a stable property of the method. We study when this practice fails. Using 50 independent repetitions across six regression datasets, we show that CRPS variance trajectories differ substantially across methods and are not always well described by a smooth power-law decay. Methods with a learned heteroscedastic variance head, namely MAP and Deep Ensembles, can develop pronounced, reproducible variance peaks at intermediate training sizes on real datasets, whereas MC Dropout and Bayes by Backprop typically show smooth variance contraction. These peaks have direct practical consequences: at the variance peak on Seoul Bike, the relative RMSE of a single-seed MAP estimate reaches 93.6%, and the probability of falling within $\pm 10\%$ of the repeated-run mean drops to 5.9%. We show that local CRPS variance provides a direct signal of single-seed estimation error, with Spearman correlations above 0.96 on every real dataset. Power-law fit quality and monotonicity together provide compact method-level summaries of trajectory regularity. Finally, replacing the standard heteroscedastic objective with $β$-NLL substantially reduces the irregular behavior, consistent with the view that the heteroscedastic training objective contributes to the instability. Practitioners should report trajectory summaries alongside endpoint means and concentrate repeated evaluation in high-variance regions.

Standard evaluations of Bayesian deep learning methods assume that metric estimates are reliable, but we show this assumption fails under data scarcity. Method rankings are not only unreliable at small $n$, but also dataset-dependent in ways that point estimates cannot reveal: the same method comparison yields $P(\mathrm{MCD} \prec \mathrm{Ensemble}) = 1.000$ at $n = 50$ on one dataset and remains below $0.95$ even at $n = 500$ on another. Across the datasets we consider, no universal sample size threshold exists, which is precisely why dataset-specific posterior inference is necessary. To address this, we use a Bayesian hierarchical model with method-specific variances to treat evaluation metrics as random variables across data realizations, and we use a predictive Minimum Detectable Difference curve to assess whether an observed gap would be detectable at a given training size. Across six Bayesian deep learning methods and five regression datasets, our results show that uncertainty-aware evaluation is necessary in low-data settings, because current evidence for method superiority and predictive detectability at the same training size can diverge substantially. Our framework provides practitioners with principled tools to determine whether their evaluation data is sufficient before drawing conclusions about method superiority.

Complex-Valued Neural Networks (CVNNs) have significant advantages in handling tasks that involve complex numbers. However, existing CVNNs are unable to quantify predictive uncertainty. We propose, for the first time, dropout-based Bayesian Complex-Valued Neural Networks (BayesCVNNs) to enable uncertainty quantification for complex-valued applications, exhibiting broad applicability and efficiency for hardware implementation due to modularity. Furthermore, as the dual-part nature of complex values significantly broadens the design space and enables novel configurations based on layer-mixing and part-mixing, we introduce an automated search approach to effectively identify optimal configurations for both real and imaginary components. To facilitate deployment, we present a framework that generates customized FPGA-based accelerators for BayesCVNNs, leveraging a set of optimized building blocks. Experiments demonstrate the best configuration can be effectively found via the automated search, attaining higher performance with lower hardware costs compared with manually crafted models. The optimized accelerators achieve approximately 4.5x and 13x speedups on different models with less than 10% power consumption compared to GPU implementations, and outperform existing work in both algorithm and hardware aspects. Our code is publicly available at: https://github.com/zehuanzhang/BayesCVNN.git.

Spiking Neural Networks (SNNs) are naturally suited for speech processing tasks due to their specific dynamics, which allows them to handle temporal data. However, the threshold-based generation of spikes in SNNs intuitively causes an angular or irregular predictive landscape. We explore the effect of using the Bayesian learning approach for the weights on the irregular predictive landscape. For the surrogate-gradient SNNs, we also explore the application of the Improved Variational Online Newton (IVON) approach, which is an efficient variational approach. The performance of the proposed approach is evaluated on the Heidelberg Digits and Speech Commands datasets. The hypothesis is that the Bayesian approach will result in a smoother and more regular predictive landscape, given the angular nature of the deterministic predictive landscape. The experimental evaluation of the proposed approach shows improved performance on the negative log-likelihood and Brier score. Furthermore, the proposed approach has resulted in a smoother and more regular predictive landscape compared to the deterministic approach, based on the one-dimensional slices of the weight space

Machine learning models, and deep neural networks in particular, are increasingly deployed in risk-sensitive domains such as healthcare, environmental forecasting, and finance, where reliable quantification of predictive uncertainty is essential. However, many uncertainty quantification (UQ) methods remain difficult to apply due to their substantial computational cost. Sampling-based Bayesian learning approaches, such as Bayesian neural networks (BNNs), are particularly expensive since drawing and evaluating multiple parameter samples rapidly exhausts memory and compute resources. These constraints have limited the accessibility and exploration of Bayesian techniques thus far. To address these challenges, we introduce sampling parallelism, a simple yet powerful parallelization strategy that targets the primary bottleneck of sampling-based Bayesian learning: the samples themselves. By distributing sample evaluations across multiple GPUs, our method reduces memory pressure and training time without requiring architectural changes or extensive hyperparameter tuning. We detail the methodology and evaluate its performance on a few example tasks and architectures, comparing against distributed data parallelism (DDP) as a baseline. We further demonstrate that sampling parallelism is complementary to existing strategies by implementing a hybrid approach that combines sample and data parallelism. Our experiments show near-perfect scaling when the sample number is scaled proportionally to the computational resources, confirming that sample evaluations parallelize cleanly. Although DDP achieves better raw speedups under scaling with constant workload, sampling parallelism has a notable advantage: by applying independent stochastic augmentations to the same batch on each GPU, it increases augmentation diversity and thus reduces the number of epochs required for convergence.