State estimation or filtering serves as a fundamental task to enable intelligent decision-making in applications such as autonomous vehicles, robotics, healthcare monitoring, smart grids, intelligent transportation, and predictive maintenance. Standard filtering assumes prior knowledge of noise statistics to extract latent system states from noisy sensor data. However, real-world scenarios involve abnormalities like outliers, biases, drifts, and missing observations with unknown or partially known statistics, limiting conventional approaches. This thesis presents novel robust nonlinear filtering methods to mitigate these challenges. Based on insights from our filtering proposals, we extend the formulations to offline estimation/learning setups and propose smoothing extensions. Our methods leverage Bayesian inference frameworks, employing both deterministic and stochastic approximation techniques including Variational Inference (VI) and Particle Filters/Sequential Monte Carlo (SMC). We also study theoretical estimation limits using Bayesian Cramér-Rao bounds (BCRBs) in the context of measurement abnormalities. To validate the performance gains of the proposed methods, we perform simulations and experiments in scenarios including target tracking, indoor localization, 3D point cloud registration, mesh registration, and pose graph optimization. The fundamental nature of the work makes it useful in diverse applications, with possible future extensions toward developing outlier-robust machine learning pipelines, learning system dynamics from anomalous data, and addressing challenges in generative AI where standard diffusion models struggle with outliers, imbalanced datasets, and mode collapse.

Deep learning vision systems excel at pattern recognition yet falter when inputs are noisy or the model must explain its own confidence. Fuzzy inference, with its graded memberships and rule transparency, offers a remedy, while parameterized quantum circuits can embed features in richly entangled Hilbert spaces with striking parameter efficiency. Bridging these ideas, this study introduces a innovative Highly Quantized Fuzzy Neural Network (HQFNN) that realises the entire fuzzy pipeline inside a shallow quantum circuit and couples the resulting quantum signal to a lightweight CNN feature extractor. Each image feature is first mapped to a single qubit membership state through repeated angle reuploading. Then a compact rule layer refines these amplitudes, and a clustered CNOT defuzzifier collapses them into one crisp value that is fused with classical features before classification. Evaluated on standard image benchmarks, HQFNN consistently surpasses classical, fuzzy enhanced and quantum only baselines while using several orders of magnitude fewer trainable weights, and its accuracy degrades only marginally under simulated depolarizing and amplitude damping noise, evidence of intrinsic robustness. Gate count analysis further shows that circuit depth grows sublinearly with input dimension, confirming the model's practicality for larger images. These results position the model as a compact, interpretable and noise tolerant alternative to conventional vision backbones and provide a template for future quantum native fuzzy learning frameworks.

--We introduce PRISM (Predictive Reasoning in Sequential Medicine), a transformer-based architecture designed to model the sequential progression of clinical decision-making processes. Unlike traditional approaches that rely on isolated diagnostic classification, PRISM frames clinical trajectories as tokenized sequences of events -- including diagnostic tests, laboratory results, and diagnoses -- and learns to predict the most probable next steps in the patient diagnostic journey. Leveraging a large custom clinical vocabulary and an autoregressive training objective, PRISM demonstrates the ability to capture complex dependencies across longitudinal patient timelines. Experimental results show substantial improvements over random baselines in next-token prediction tasks, with generated sequences reflecting realistic diagnostic pathways, laboratory result progressions, and clinician ordering behaviors. These findings highlight the feasibility of applying generative language modeling techniques to structured medical event data, enabling applications in clinical decision support, simulation, and education. PRISM establishes a foundation for future advancements in sequence-based healthcare modeling, bridging the gap between machine learning architectures and real-world diagnostic reasoning. Accurate and timely clinical decision-making is fundamental to high-quality patient care.

Over the last few years, probabilistic machine-learned weather prediction models have begun to rival physics-based numerical weather prediction (NWP) systems in skill (Kochkov et al., 2024; Price et al., 2023; Lang et al., 2024c,b). AIFS-CRPS (Lang et al., 2024b) is based on the machined-learned weather forecasting model AIFS (Lang et al., 2024a), developed at the European Centre for Medium-Range Weather Forecasts (ECMWF). AIFS-CRPS produces skilful predictions by directly optimising a score based on a proper scoring rule, the almost fair continuous ranked probability score (afCRPS). The model learns to shape Gaussian noise to represent uncertainty in the atmospheric state and achieves ensemble forecast skill that is competitive with, or superior to, the physics-based IFS ensemble (Molteni et al., 1996; Leutbecher and Palmer, 2008; Lang et al., 2021, 2023) at ECMWF. The afCRPS loss function used in AIFS-CRPS is computed point-wise on the full output field. However, atmospheric processes are inherently multi-scale, and different scales contribute to a different degree to the loss function.

Deterministic embeddings learned by contrastive learning (CL) methods such as SimCLR and SupCon achieve state-of-the-art performance but lack a principled mechanism for uncertainty quantification. We propose Variational Contrastive Learning (VCL), a decoder-free framework that maximizes the evidence lower bound (ELBO) by interpreting the InfoNCE loss as a surrogate reconstruction term and adding a KL divergence regularizer to a uniform prior on the unit hypersphere. We model the approximate posterior $q_θ(z|x)$ as a projected normal distribution, enabling the sampling of probabilistic embeddings. Our two instantiations--VSimCLR and VSupCon--replace deterministic embeddings with samples from $q_θ(z|x)$ and incorporate a normalized KL term into the loss. Experiments on multiple benchmarks demonstrate that VCL mitigates dimensional collapse, enhances mutual information with class labels, and matches or outperforms deterministic baselines in classification accuracy, all the while providing meaningful uncertainty estimates through the posterior model. VCL thus equips contrastive learning with a probabilistic foundation, serving as a new basis for contrastive approaches.

We propose a meta-learning method for learning from multiple noisy annotators. In many applications such as crowdsourcing services, labels for supervised learning are given by multiple annotators. Since the annotators have different skills or biases, given labels can be noisy. To learn accurate classifiers, existing methods require many noisy annotated data. However, sufficient data might be unavailable in practice. To overcome the lack of data, the proposed method uses labeled data obtained in different but related tasks. The proposed method embeds each example in tasks to a latent space by using a neural network and constructs a probabilistic model for learning a task-specific classifier while estimating annotators' abilities on the latent space. This neural network is meta-learned to improve the expected test classification performance when the classifier is adapted to a given small amount of annotated data. This classifier adaptation is performed by maximizing the posterior probability via the expectation-maximization (EM) algorithm. Since each step in the EM algorithm is easily computed as a closed-form and is differentiable, the proposed method can efficiently backpropagate the loss through the EM algorithm to meta-learn the neural network. We show the effectiveness of our method with real-world datasets with synthetic noise and real-world crowdsourcing datasets.

Density regression models allow a comprehensive understanding of data by modeling the complete conditional probability distribution. While flexible estimation approaches such as normalizing flows (NF) work particularly well in multiple dimensions, interpreting the input-output relationship of such models is often difficult, due to the black-box character of deep learning models. In contrast, existing statistical methods for multivariate outcomes such as multivariate conditional transformation models (MCTM) are restricted in flexibility and are often not expressive enough to represent complex multivariate probability distributions. In this paper, we combine MCTM with state-of-the-art and autoregressive NF to leverage the transparency of MCTM for modeling interpretable feature effects on the marginal distributions in the first step and the flexibility of neural-network-based NF techniques to account for complex and non-linear relationships in the joint data distribution. We demonstrate our method's versatility in various numerical experiments and compare it with MCTM and other NF models on both simulated and real-world data.

Diffusion bridges are a promising class of deep-learning methods for sampling from unnormalized distributions. Recent works show that the Log Variance (LV) loss consistently outperforms the reverse Kullback-Leibler (rKL) loss when using the reparametrization trick to compute rKL-gradients. While the on-policy LV loss yields identical gradients to the rKL loss when combined with the log-derivative trick for diffusion samplers with non-learnable forward processes, this equivalence does not hold for diffusion bridges or when diffusion coefficients are learned. Based on this insight we argue that for diffusion bridges the LV loss does not represent an optimization objective that can be motivated like the rKL loss via the data processing inequality. Our analysis shows that employing the rKL loss with the log-derivative trick (rKL-LD) does not only avoid these conceptual problems but also consistently outperforms the LV loss. Experimental results with different types of diffusion bridges on challenging benchmarks show that samplers trained with the rKL-LD loss achieve better performance. From a practical perspective we find that rKL-LD requires significantly less hyperparameter optimization and yields more stable training behavior.

Although large language models (LLMs) are becoming increasingly capable of solving challenging real-world tasks, accurately quantifying their uncertainty remains a critical open problem, which limits their applicability in high-stakes domains. This challenge is further compounded by the closed-source, black-box nature of many state-of-the-art LLMs. Moreover, LLM-based systems can be highly sensitive to the prompts that bind them together, which often require significant manual tuning (i.e., prompt engineering). In this work, we address these challenges by viewing LLM-based systems through a Bayesian lens. We interpret prompts as textual parameters in a statistical model, allowing us to use a small training dataset to perform Bayesian inference over these prompts. This novel perspective enables principled uncertainty quantification over both the model's textual parameters and its downstream predictions, while also incorporating prior beliefs about these parameters expressed in free-form text. To perform Bayesian inference, a difficult problem even for well-studied data modalities, we introduce Metropolis-Hastings through LLM Proposals (MHLP), a novel Markov chain Monte Carlo (MCMC) algorithm that combines prompt optimization techniques with standard MCMC methods. MHLP is a turnkey modification to existing LLM pipelines, including those that rely exclusively on closed-source models. Empirically, we demonstrate that our method yields improvements in both predictive accuracy and uncertainty quantification (UQ) on a range of LLM benchmarks and UQ tasks. More broadly, our work demonstrates a viable path for incorporating methods from the rich Bayesian literature into the era of LLMs, paving the way for more reliable and calibrated LLM-based systems.

Off-policy learning serves as the primary framework for learning optimal policies from logged interactions collected under a static behavior policy. In this work, we investigate the more practical and flexible setting of adaptive off-policy learning, where policies are iteratively refined and re-deployed to collect higher-quality data. Building on the success of PAC-Bayesian learning with Logarithmic Smoothing (LS) in static settings, we extend this framework to the adaptive scenario using tools from online PAC-Bayesian theory. Furthermore, we demonstrate that a principled adjustment to the LS estimator naturally accommodates multiple rounds of deployment and yields faster convergence rates under mild conditions. Our method matches the performance of leading offline approaches in static settings, and significantly outperforms them when intermediate policy deployments are allowed. Empirical evaluations across diverse scenarios highlight both the advantages of adaptive data collection and the strength of the PAC-Bayesian formulation.