Causal models provide rich descriptions of complex systems as sets of mechanisms by which each variable is influenced by its direct causes. They support reasoning about manipulating parts of the system and thus hold promise for addressing some of the open challenges of artificial intelligence (AI), such as planning, transferring knowledge in changing environments, or robustness to distribution shifts. However, a key obstacle to more widespread use of causal models in AI is the requirement that the relevant variables be specified a priori, which is typically not the case for the high-dimensional, unstructured data processed by modern AI systems. At the same time, machine learning (ML) has proven quite successful at automatically extracting useful and compact representations of such complex data. Causal representation learning (CRL) aims to combine the core strengths of ML and causality by learning representations in the form of latent variables endowed with causal model semantics. In this thesis, we study and present new results for different CRL settings. A central theme is the question of identifiability: Given infinite data, when are representations satisfying the same learning objective guaranteed to be equivalent? This is an important prerequisite for CRL, as it formally characterises if and when a learning task is, at least in principle, feasible. Since learning causal models, even without a representation learning component, is notoriously difficult, we require additional assumptions on the model class or rich data beyond the classical i.i.d. setting. By partially characterising identifiability for different settings, this thesis investigates what is possible for CRL without direct supervision, and thus contributes to its theoretical foundations. Ideally, the developed insights can help inform data collection practices or inspire the design of new practical estimation methods.

We introduce a new method to jointly reduce the dimension of the input and output space of a high-dimensional function. Choosing a reduced input subspace influences which output subspace is relevant and vice versa. Conventional methods focus on reducing either the input or output space, even though both are often reduced simultaneously in practice. Our coupled approach naturally supports goal-oriented dimension reduction, where either an input or output quantity of interest is prescribed. We consider, in particular, goal-oriented sensor placement and goal-oriented sensitivity analysis, which can be viewed as dimension reduction where the most important output or, respectively, input components are chosen. Both applications present difficult combinatorial optimization problems with expensive objectives such as the expected information gain and Sobol indices. By optimizing gradient-based bounds, we can determine the most informative sensors and most sensitive parameters as the largest diagonal entries of some diagnostic matrices, thus bypassing the combinatorial optimization and objective evaluation.

Missing Not at Random (MNAR) and nonnormal data are challenging to handle. Traditional missing data analytical techniques such as full information maximum likelihood estimation (FIML) may fail with nonnormal data as they are built on normal distribution assumptions. Two-Stage Robust Estimation (TSRE) does manage nonnormal data, but both FIML and TSRE are less explored in longitudinal studies under MNAR conditions with nonnormal distributions. Unlike traditional statistical approaches, machine learning approaches do not require distributional assumptions about the data. More importantly, they have shown promise for MNAR data; however, their application in longitudinal studies, addressing both Missing at Random (MAR) and MNAR scenarios, is also underexplored. This study utilizes Monte Carlo simulations to assess and compare the effectiveness of six analytical techniques for missing data within the growth curve modeling framework. These techniques include traditional approaches like FIML and TSRE, machine learning approaches by single imputation (K-Nearest Neighbors and missForest), and machine learning approaches by multiple imputation (micecart and miceForest). We investigate the influence of sample size, missing data rate, missing data mechanism, and data distribution on the accuracy and efficiency of model estimation. Our findings indicate that FIML is most effective for MNAR data among the tested approaches. TSRE excels in handling MAR data, while missForest is only advantageous in limited conditions with a combination of very skewed distributions, very large sample sizes (e.g., n larger than 1000), and low missing data rates.

Recently proposed quasi-Bayesian (QB) methods initiated a new era in Bayesian computation by directly constructing the Bayesian predictive distribution through recursion, removing the need for expensive computations involved in sampling the Bayesian posterior distribution. This has proved to be data-efficient for univariate predictions, but extensions to multiple dimensions rely on a conditional decomposition resulting from predefined assumptions on the kernel of the Dirichlet Process Mixture Model, which is the implicit nonparametric model used. Here, we propose a different way to extend Quasi-Bayesian prediction to high dimensions through the use of Sklar's theorem by decomposing the predictive distribution into one-dimensional predictive marginals and a high-dimensional copula. Thus, we use the efficient recursive QB construction for the one-dimensional marginals and model the dependence using highly expressive vine copulas. Further, we tune hyperparameters using robust divergences (eg. energy score) and show that our proposed Quasi-Bayesian Vine (QB-Vine) is a fully non-parametric density estimator with \emph{an analytical form} and convergence rate independent of the dimension of data in some situations. Our experiments illustrate that the QB-Vine is appropriate for high dimensional distributions ($\sim$64), needs very few samples to train ($\sim$200) and outperforms state-of-the-art methods with analytical forms for density estimation and supervised tasks by a considerable margin.

The need for regression models to predict circular values arises in many scientific fields. In this work we explore a family of expressive and interpretable distributions over circle-valued random functions related to Gaussian processes targeting two Euclidean dimensions conditioned on the unit circle. The resulting probability model has connections with continuous spin models in statistical physics. Moreover, its density is very simple and has maximum-entropy, unlike previous Gaussian process-based approaches, which use wrapping or radial marginalization. For posterior inference, we introduce a new Stratonovich-like augmentation that lends itself to fast Markov Chain Monte Carlo sampling. We argue that transductive learning in these models favors a Bayesian approach to the parameters. We present experiments applying this model to the prediction of (i) wind directions and (ii) the percentage of the running gait cycle as a function of joint angles.

The critical inquiry pervading the realm of Philosophy, and perhaps extending its influence across all Humanities disciplines, revolves around the intricacies of morality and normativity. Surprisingly, in recent years, this thematic thread has woven its way into an unexpected domain, one not conventionally associated with pondering "what ought to be": the field of artificial intelligence (AI) research. Central to morality and AI, we find "alignment", a problem related to the challenges of expressing human goals and values in a manner that artificial systems can follow without leading to unwanted adversarial effects. More explicitly and with our current paradigm of AI development in mind, we can think of alignment as teaching human values to non-anthropomorphic entities trained through opaque, gradient-based learning techniques. This work addresses alignment as a technical-philosophical problem that requires solid philosophical foundations and practical implementations that bring normative theory to AI system development. To accomplish this, we propose two sets of necessary and sufficient conditions that, we argue, should be considered in any alignment process. While necessary conditions serve as metaphysical and metaethical roots that pertain to the permissibility of alignment, sufficient conditions establish a blueprint for aligning AI systems under a learning-based paradigm. After laying such foundations, we present implementations of this approach by using state-of-the-art techniques and methods for aligning general-purpose language systems. We call this framework Dynamic Normativity. Its central thesis is that any alignment process under a learning paradigm that cannot fulfill its necessary and sufficient conditions will fail in producing aligned systems.

Approximate model counting is the task of approximating the number of solutions to an input Boolean formula. The state-of-the-art approximate model counter for formulas in conjunctive normal form (CNF), ApproxMC, provides a scalable means of obtaining model counts with probably approximately correct (PAC)-style guarantees. Nevertheless, the validity of ApproxMC's approximation relies on a careful theoretical analysis of its randomized algorithm and the correctness of its highly optimized implementation, especially the latter's stateful interactions with an incremental CNF satisfiability solver capable of natively handling parity (XOR) constraints. We present the first certification framework for approximate model counting with formally verified guarantees on the quality of its output approximation. Our approach combines: (i) a static, once-off, formal proof of the algorithm's PAC guarantee in the Isabelle/HOL proof assistant; and (ii) dynamic, per-run, verification of ApproxMC's calls to an external CNF-XOR solver using proof certificates. We detail our general approach to establish a rigorous connection between these two parts of the verification, including our blueprint for turning the formalized, randomized algorithm into a verified proof checker, and our design of proof certificates for both ApproxMC and its internal CNF-XOR solving steps. Experimentally, we show that certificate generation adds little overhead to an approximate counter implementation, and that our certificate checker is able to fully certify $84.7\%$ of instances with generated certificates when given the same time and memory limits as the counter.

Variational Inference (VI) is a commonly used technique for approximate Bayesian inference and uncertainty estimation in deep learning models, yet it comes at a computational cost, as it doubles the number of trainable parameters to represent uncertainty. This rapidly becomes challenging in high-dimensional settings and motivates the use of alternative techniques for inference, such as Monte Carlo Dropout (MCD) or Spectral-normalized Neural Gaussian Process (SNGP). However, such methods have seen little adoption in survival analysis, and VI remains the prevalent approach for training probabilistic neural networks. In this paper, we investigate how to train deep probabilistic survival models in large datasets without introducing additional overhead in model complexity. To achieve this, we adopt three probabilistic approaches, namely VI, MCD, and SNGP, and evaluate them in terms of their prediction performance, calibration performance, and model complexity. In the context of probabilistic survival analysis, we investigate whether non-VI techniques can offer comparable or possibly improved prediction performance and uncertainty calibration compared to VI. In the MIMIC-IV dataset, we find that MCD aligns with VI in terms of the concordance index (0.748 vs. 0.743) and mean absolute error (254.9 vs. 254.7) using hinge loss, while providing C-calibrated uncertainty estimates. Moreover, our SNGP implementation provides D-calibrated survival functions in all datasets compared to VI (4/4 vs. 2/4, respectively). Our work encourages the use of techniques alternative to VI for survival analysis in high-dimensional datasets, where computational efficiency and overhead are of concern.

To make accurate predictions, understand mechanisms, and design interventions in systems of many variables, we wish to learn causal graphs from large scale data. Unfortunately the space of all possible causal graphs is enormous so scalably and accurately searching for the best fit to the data is a challenge. In principle we could substantially decrease the search space, or learn the graph entirely, by testing the conditional independence of variables. However, deciding if two variables are adjacent in a causal graph may require an exponential number of tests. Here we build a scalable and flexible method to evaluate if two variables are adjacent in a causal graph, the Differentiable Adjacency Test (DAT). DAT replaces an exponential number of tests with a provably equivalent relaxed problem. It then solves this problem by training two neural networks. We build a graph learning method based on DAT, DAT-Graph, that can also learn from data with interventions. DAT-Graph can learn graphs of 1000 variables with state of the art accuracy. Using the graph learned by DAT-Graph, we also build models that make much more accurate predictions of the effects of interventions on large scale RNA sequencing data.

In recent years, predictive maintenance (PMx) has gained prominence for its potential to enhance efficiency, automation, accuracy, and cost-effectiveness while reducing human involvement. Importantly, PMx has evolved in tandem with digital advancements, such as Big Data and the Internet of Things (IOT). These technological strides have enabled Artificial Intelligence (AI) to revolutionize PMx processes, with increasing capacities for real-time automation of monitoring, analysis, and prediction tasks. However, PMx still faces challenges such as poor explainability and sample inefficiency in data-driven methods and high complexity in physics-based models, hindering broader adoption. This paper posits that Digital Twins (DTs) can be integrated into PMx to overcome these challenges, paving the way for more automated PMx applications across various stakeholders. Despite their potential, current DTs have not fully matured to bridge existing gaps. Our paper provides a comprehensive roadmap for DT evolution, addressing current limitations to foster large-scale automated PMx progression. We structure our approach in three stages: First, we reference prior work where we identified and defined the Information Requirements (IRs) and Functional Requirements (FRs) for PMx, forming the blueprint for a unified framework. Second, we conduct a literature review to assess current DT applications integrating these IRs and FRs, revealing standardized DT models and tools that support automated PMx. Lastly, we highlight gaps in current DT implementations, particularly those IRs and FRs not fully supported, and outline the necessary components for a comprehensive, automated PMx system. Our paper concludes with research directions aimed at seamlessly integrating DTs into the PMx paradigm to achieve this ambitious vision.