Polynomial chaos expansions (PCE) are widely used for uncertainty quantification (UQ) tasks, particularly in the applied mathematics community. However, PCE has received comparatively less attention in the statistics literature, and fully Bayesian formulations remain rare--especially with implementations in R. Motivated by the success of adaptive Bayesian machine learning models such as BART, BASS, and BPPR, we develop a new fully Bayesian adaptive PCE method with an efficient and accessible R implementation: khaos. Our approach includes a novel proposal distribution that enables data-driven interaction selection, and supports a modified g-prior tailored to PCE structure. Through simulation studies and real-world UQ applications, we demonstrate that Bayesian adaptive PCE provides competitive performance for surrogate modeling, global sensitivity analysis, and ordinal regression tasks.

Physics-Informed Machine Learning (PIML) has successfully integrated mechanistic understanding into machine learning, particularly in domains governed by well-known physical laws. This success has motivated efforts to apply PIML to biology, a field rich in dynamical systems but shaped by different constraints. Biological modeling, however, presents unique challenges: multi-faceted and uncertain prior knowledge, heterogeneous and noisy data, partial observability, and complex, high-dimensional networks. In this position paper, we argue that these challenges should not be seen as obstacles to PIML, but as catalysts for its evolution. We propose Biology-Informed Machine Learning (BIML): a principled extension of PIML that retains its structural grounding while adapting to the practical realities of biology. Rather than replacing PIML, BIML retools its methods to operate under softer, probabilistic forms of prior knowledge. We outline four foundational pillars as a roadmap for this transition: uncertainty quantification, contextualization, constrained latent structure inference, and scalability. Foundation Models and Large Language Models will be key enablers, bridging human expertise with computational modeling. We conclude with concrete recommendations to build the BIML ecosystem and channel PIML-inspired innovation toward challenges of high scientific and societal relevance.

Regression tasks, notably in safety-critical domains, require proper uncertainty quantification, yet the literature remains largely classification-focused. In this light, we introduce a family of measures for total, aleatoric, and epistemic uncertainty based on proper scoring rules, with a particular emphasis on kernel scores. The framework unifies several well-known measures and provides a principled recipe for designing new ones whose behavior, such as tail sensitivity, robustness, and out-of-distribution responsiveness, is governed by the choice of kernel. We prove explicit correspondences between kernel-score characteristics and downstream behavior, yielding concrete design guidelines for task-specific measures. Extensive experiments demonstrate that these measures are effective in downstream tasks and reveal clear trade-offs among instantiations, including robustness and out-of-distribution detection performance.

Bayesian inference of Bayesian network structures is often performed by sampling directed acyclic graphs along an appropriately constructed Markov chain. We present two techniques to improve sampling. First, we give an efficient implementation of basic moves, which add, delete, or reverse a single arc. Second, we expedite summing over parent sets, an expensive task required for more sophisticated moves: we devise a preprocessing method to prune possible parent sets so as to approximately preserve the sums. Our empirical study shows that our techniques can yield substantial efficiency gains compared to previous methods.

Causal discovery aims to recover graphs that represent causal relations among given variables from observations, and new methods are constantly being proposed. Increasingly, the community raises questions about how much progress is made, because properly evaluating discovered graphs remains notoriously difficult, particularly under latent confounding. We propose a graph distance measure for acyclic directed mixed graphs (AD-MGs) based on the downstream task of cause-effect estimation under unobserved confounding. Our approach uses identification via fixing and a symbolic verifier to quantify how graph differences distort cause-effect esti-mands for different treatment-outcome pairs. We analyze the behavior of the measure under different graph perturbations and compare it against existing distance metrics.

Modeling complex, non-linear, and uncertain relationships between input and output variables remains a central challenge in modern statistical learning and artificial intelligence. Traditional neural networks, trained via point estimation, have demonstrated remarkable success in a variety of domains but inherently provide deterministic predictions - that is, single-valued outputs without accompanying measures of uncertainty. This limitation becomes critical in domains characterized by limited, noisy, or ambiguous data, such as medicine, climate science, or finance, where quantifying uncertainty is as important as producing accurate predictions (Gal & Ghahramani, 2016; Kendall & Gal, 2017; Abdar et al., 2021). Bayesian Neural Networks (BNNs) provide a probabilistic extension of standard neural networks by treating weights and biases as random variables endowed with prior distributions (MacKay, 1992; Neal, 2012). Through Bayes' theorem, BNNs infer a posterior distribution over weights, allowing predictions to reflect epistemic uncertainty - the uncertainty arising from limited data and model knowledge. However, the exact posterior is analytically intractable for deep models, motivating approximate inference methods such as variational inference (Graves, 2011; Blundell et al., 2015) and Monte Carlo dropout (Gal & Ghahramani, 2016). Despite their appeal, these approaches may yield biased or overconfident posteriors due to restrictive variational families (Hern andez-Lobato & Adams, 2015a; Osband et al., 2023), often resulting in over-smoothed predictive distributions. An alternative paradigm for probabilistic modeling is the Mixture Density Network (MDN), introduced by Bridle (1990) and developed further by Jacobs et al. (1991).

This thesis delves into the world of non-invasive electrophysiological brain signals like electroencephalography (EEG) and magnetoencephalography (MEG), focusing on modelling and decoding such data. The research aims to investigate what happens in the brain when we perceive visual stimuli or engage in covert speech (inner speech) and enhance the decoding performance of such stimuli. The thesis is divided into two main sections, methodological and experimental work. A central concern in both sections is the large variability present in electrophysiological recordings, whether it be within-subject or between-subject variability, and to a certain extent between-dataset variability. In the methodological sections, we explore the potential of deep learning for brain decoding. We present advancements in decoding visual stimuli using linear models at the individual subject level. We then explore how deep learning techniques can be employed for group decoding, introducing new methods to deal with between-subject variability. Finally, we also explores novel forecasting models of MEG data based on convolutional and Transformer-based architectures. In particular, Transformer-based models demonstrate superior capabilities in generating signals that closely match real brain data, thereby enhancing the accuracy and reliability of modelling the brain's electrophysiology. In the experimental section, we present a unique dataset containing high-trial inner speech EEG, MEG, and preliminary optically pumped magnetometer (OPM) data. Our aim is to investigate different types of inner speech and push decoding performance by collecting a high number of trials and sessions from a few participants. However, the decoding results are found to be mostly negative, underscoring the difficulty of decoding inner speech.

The heterogeneity of multimodal data leads to inconsistencies and imbalance, allowing a dominant modality to steer gradient updates. Existing solutions mainly focus on optimization- or data-based strategies but rarely exploit the information inherent in multimodal imbalance or conduct its quantitative analysis. To address this gap, we propose a novel quantitative analysis framework for Multimodal Imbalance and design a sample-level adaptive loss function. We define the Modality Gap as the Softmax score difference between modalities for the correct class and model its distribution using a bimodal Gaussian Mixture Model(GMM), representing balanced and imbalanced samples. Using Bayes' theorem, we estimate each sample's posterior probability of belonging to these two groups. Based on this, our adaptive loss (1) minimizes the overall Modality Gap, (2) aligns imbalanced samples with balanced ones, and (3) adaptively penalizes each according to its imbalance degree. A two-stage training strategy-warm-up and adaptive phases,yields state-of-the-art performance on CREMA-D (80.65%), AVE (70.40%), and KineticSound (72.42%). Fine-tuning with high-quality samples identified by the GMM further improves results, highlighting their value for effective multimodal fusion.

Quantifying the uncertainty in the output of a neural network is essential for deployment in scientific or engineering applications where decisions must be made under limited or noisy data. Bayesian neural networks (BNNs) provide a framework for this purpose by constructing a Bayesian posterior distribution over the network parameters. However, the prior, which is of key importance in any Bayesian setting, is rarely meaningful for BNNs. This is because the complexity of the input-to-output map of a BNN makes it difficult to understand how certain distributions enforce any interpretable constraint on the output space. Gaussian processes (GPs), on the other hand, are often preferred in uncertainty quantification tasks due to their interpretability. The drawback is that GPs are limited to small datasets without advanced techniques, which often rely on the covariance kernel having a specific structure. To address these challenges, we introduce a new class of priors for BNNs, called Mercer priors, such that the resulting BNN has samples which approximate that of a specified GP. The method works by defining a prior directly over the network parameters from the Mercer representation of the covariance kernel, and does not rely on the network having a specific structure. In doing so, we can exploit the scalability of BNNs in a meaningful Bayesian way.

Variational mean field approximations tend to struggle with contemporary overparametrized deep neural networks. Where a Bayesian treatment is usually associated with high-quality predictions and uncertainties, the practical reality has been the opposite, with unstable training, poor predictive power, and subpar calibration. Building upon recent work on reparametrizations of neural networks, we propose a simple variational family that considers two independent linear subspaces of the parameter space. These represent functional changes inside and outside the support of training data. This allows us to build a fully-correlated approximate posterior reflecting the overparametrization that tunes easy-to-interpret hyperparameters. We develop scalable numerical routines that maximize the associated evidence lower bound (ELBO) and sample from the approximate posterior. Empirically, we observe state-of-the-art performance across tasks, models, and datasets compared to a wide array of baseline methods. Our results show that approximate Bayesian inference applied to deep neural networks is far from a lost cause when constructing inference mechanisms that reflect the geometry of reparametrizations.