We explore the use of conformal prediction to provide statistical uncertainty guarantees for runway detection in vision-based landing systems (VLS). Using fine-tuned YOLOv5 and YOLOv6 models on aerial imagery, we apply conformal prediction to quantify localization reliability under user-defined risk levels. We also introduce Conformal mean Average Precision (C-mAP), a novel metric aligning object detection performance with conformal guarantees. Our results show that conformal prediction can improve the reliability of runway detection by quantifying uncertainty in a statistically sound way, increasing safety on-board and paving the way for certification of ML system in the aerospace domain.

Asymmetric, non-Gaussian probability distributions are often observed in the analysis of natural and engineering datasets. The lognormal distribution is a standard model for data with skewed frequency histograms and fat tails. However, the lognormal law severely restricts the asymptotic dependence of the probability density and the hazard function for high values. Herein we present a family of three-parameter non-Gaussian probability density functions that are based on generalized kappa-exponential and kappa-logarithm functions and investigate its mathematical properties. These kappa-lognormal densities represent continuous deformations of the lognormal with lighter right tails, controlled by the parameter kappa. In addition, bimodal distributions are obtained for certain parameter combinations. We derive closed-form analytic expressions for the main statistical functions of the kappa-lognormal distribution. For the moments, we derive bounds that are based on hypergeometric functions as well as series expansions. Explicit expressions for the gradient and Hessian of the negative log-likelihood are obtained to facilitate numerical maximum-likelihood estimates of the kappa-lognormal parameters from data. We also formulate a joint probability density function for kappa-lognormal stochastic processes by applying Jacobi's multivariate theorem to a latent Gaussian process. Estimation of the kappa-lognormal distribution based on synthetic and real data is explored. Furthermore, we investigate applications of kappa-lognormal processes with different covariance kernels in time series forecasting and spatial interpolation using warped Gaussian process regression. Our results are of practical interest for modeling skewed distributions in various scientific and engineering fields.

In electromyogram (EMG)-based motion recognition, a subject-specific classifier is typically trained with sufficient labeled data. However, this process demands extensive data collection over extended periods, burdening the subject. To address this, utilizing information from pre-training on multiple subjects for the training of the target subject could be beneficial. This paper proposes an inter-subject variance transfer learning method based on a Bayesian approach. This method is founded on the simple hypothesis that while the means of EMG features vary greatly across subjects, their variances may exhibit similar patterns. Our approach transfers variance information, acquired through pre-training on multiple source subjects, to a target subject within a Bayesian updating framework, thereby allowing accurate classification using limited target calibration data. A coefficient was also introduced to adjust the amount of information transferred for efficient transfer learning. Experimental evaluations using two EMG datasets demonstrated the effectiveness of our variance transfer strategy and its superiority compared to existing methods.

In statistical learning, models are classified as regular or singular depending on whether the mapping from parameters to probability distributions is injective. Most models with hierarchical structures or latent variables are singular, for which conventional criteria such as the Akaike Information Criterion and the Bayesian Information Criterion are inapplicable due to the breakdown of normal approximations for the likelihood and posterior. To address this, the Widely Applicable Information Criterion (WAIC) and the Widely Applicable Bayesian Information Criterion (WBIC) have been proposed. Since WAIC and WBIC are computed using posterior distributions at different temperature settings, separate posterior sampling is generally required. In this paper, we theoretically derive an asymptotic equation that links WAIC and WBIC, despite their dependence on different posteriors. This equation yields an asymptotically unbiased expression of WAIC in terms of the posterior distribution used for WBIC. The result clarifies the structural relationship between these criteria within the framework of singular learning theory, and deepens understanding of their asymptotic behavior. This theoretical contribution provides a foundation for future developments in the computational efficiency of model selection in singular models.

We revisit the classical problem of Bayesian ensembles and address the challenge of learning optimal combinations of Bayesian models in an online, continual learning setting. To this end, we reinterpret existing approaches such as Bayesian model averaging (BMA) and Bayesian stacking through a novel empirical Bayes lens, shedding new light on the limitations and pathologies of BMA. Further motivated by insights from online optimization, we propose Online Bayesian Stacking (OBS), a method that optimizes the log-score over predictive distributions to adaptively combine Bayesian models. A key contribution of our work is establishing a novel connection between OBS and portfolio selection, bridging Bayesian ensemble learning with a rich, well-studied theoretical framework that offers efficient algorithms and extensive regret analysis. We further clarify the relationship between OBS and online BMA, showing that they optimize related but distinct cost functions. Through theoretical analysis and empirical evaluation, we identify scenarios where OBS outperforms online BMA and provide principled guidance on when practitioners should prefer one approach over the other.

Causal inference determines cause-and-effect relationships between variables and has broad applications across disciplines. Traditional time-series methods often reveal causal links only in a time-averaged sense, while ensemble-based information transfer approaches detect the time evolution of short-term causal relationships but are typically limited to low-dimensional systems. In this paper, a new causal inference framework, called assimilative causal inference (ACI), is developed. Fundamentally different from the state-of-the-art methods, ACI uses a dynamical system and a single realization of a subset of the state variables to identify instantaneous causal relationships and the dynamic evolution of the associated causal influence range (CIR). Instead of quantifying how causes influence effects as done traditionally, ACI solves an inverse problem via Bayesian data assimilation, thus tracing causes backward from observed effects with an implicit Bayesian hypothesis. Causality is determined by assessing whether incorporating the information of the effect variables reduces the uncertainty in recovering the potential cause variables. ACI has several desirable features. First, it captures the dynamic interplay of variables, where their roles as causes and effects can shift repeatedly over time. Second, a mathematically justified objective criterion determines the CIR without empirical thresholds. Third, ACI is scalable to high-dimensional problems by leveraging computationally efficient Bayesian data assimilation techniques. Finally, ACI applies to short time series and incomplete datasets. Notably, ACI does not require observations of candidate causes, which is a key advantage since potential drivers are often unknown or unmeasured. The effectiveness of ACI is demonstrated by complex dynamical systems showcasing intermittency and extreme events.

Accurately estimating the informativeness of individual samples in a dataset is an important objective in deep learning, as it can guide sample selection, which can improve model efficiency and accuracy by removing redundant or potentially harmful samples. We propose Laplace Sample Information (LSI) measure of sample informativeness grounded in information theory widely applicable across model architectures and learning settings. LSI leverages a Bayesian approximation to the weight posterior and the KL divergence to measure the change in the parameter distribution induced by a sample of interest from the dataset. We experimentally show that LSI is effective in ordering the data with respect to typicality, detecting mislabeled samples, measuring class-wise informativeness, and assessing dataset difficulty. We demonstrate these capabilities of LSI on image and text data in supervised and unsupervised settings. Moreover, we show that LSI can be computed efficiently through probes and transfers well to the training of large models.

Ensuring the reliability and safety of machine learning models in open-world deployment is a central challenge in AI safety. This thesis develops both algorithmic and theoretical foundations to address key reliability issues arising from distributional uncertainty and unknown classes, from standard neural networks to modern foundation models like large language models (LLMs). Traditional learning paradigms, such as empirical risk minimization (ERM), assume no distribution shift between training and inference, often leading to overconfident predictions on out-of-distribution (OOD) inputs. This thesis introduces novel frameworks that jointly optimize for in-distribution accuracy and reliability to unseen data. A core contribution is the development of an unknown-aware learning framework that enables models to recognize and handle novel inputs without labeled OOD data. We propose new outlier synthesis methods, VOS, NPOS, and DREAM-OOD, to generate informative unknowns during training. Building on this, we present SAL, a theoretical and algorithmic framework that leverages unlabeled in-the-wild data to enhance OOD detection under realistic deployment conditions. These methods demonstrate that abundant unlabeled data can be harnessed to recognize and adapt to unforeseen inputs, providing formal reliability guarantees. The thesis also extends reliable learning to foundation models. We develop HaloScope for hallucination detection in LLMs, MLLMGuard for defending against malicious prompts in multimodal models, and data cleaning methods to denoise human feedback used for better alignment. These tools target failure modes that threaten the safety of large-scale models in deployment. Overall, these contributions promote unknown-aware learning as a new paradigm, and we hope it can advance the reliability of AI systems with minimal human efforts.

This work introduces a new method called scalable Bayesian Monte Carlo (SBMC). The model interpolates between a point estimator and the posterior, and the algorithm is a parallel implementation of a consistent (asymptotically unbiased) Bayesian deep learning algorithm: sequential Monte Carlo (SMC) or Markov chain Monte Carlo (MCMC). The method is motivated theoretically, and its utility is demonstrated on practical examples: MNIST, CIFAR, IMDb. A systematic numerical study reveals that parallel implementations of SMC and MCMC are comparable to serial implementations in terms of performance and total cost, and they achieve accuracy at or beyond the state-of-the-art (SOTA) methods like deep ensembles at convergence, along with substantially improved uncertainty quantification (UQ)--in particular, epistemic UQ. But even parallel implementations are expensive, with an irreducible time barrier much larger than the cost of the MAP estimator. Compressing time further leads to rapid degradation of accuracy, whereas UQ remains valuable. By anchoring to a point estimator we can recover accuracy, while retaining valuable UQ, ultimately delivering strong performance across metrics for a cost comparable to the SOTA.

We investigate the convergence properties of popular data-augmentation samplers for Bayesian probit regression. Leveraging recent results on Gibbs samplers for log-concave targets, we provide simple and explicit non-asymptotic bounds on the associated mixing times (in Kullback-Leibler divergence). The bounds depend explicitly on the design matrix and the prior precision, while they hold uniformly over the vector of responses. We specialize the results for different regimes of statistical interest, when both the number of data points $n$ and parameters $p$ are large: in particular we identify scenarios where the mixing times remain bounded as $n,p\to\infty$, and ones where they do not. The results are shown to be tight (in the worst case with respect to the responses) and provide guidance on choices of prior distributions that provably lead to fast mixing. An empirical analysis based on coupling techniques suggests that the bounds are effective in predicting practically observed behaviours.