Learning causal structure in complex systems is a fundamental challenge across a broad range of disciplines, from traditional scientific fields to modern engineering and technology. Unlike conventional statistical methods that focus merely on correlation, the field of causal discovery primarily considers the problem of discovering the directionality and strength of causal relationships between variables, often from observational data. Thus, it has become a critical tool for researchers aiming to predict the effects of interventions on the systems, especially where controlled experimentation may be expensive, unethical, or even infeasible. Such necessities arise not only in various areas of natural science, such as epidemiology [56], public health [65], genomics [14], neuroscience [86], and climate and environmental science [60], but also in numerous domains in social science, such as psychology [50], philosophy [26], and economics [37]. Moreover, with recent advances in science and technology and the increase in size and complexity of data generation processes, causal discovery has acquired significant relevance in the fields of machine learning [63] and artificial intelligence [81, 82] through various emerging areas such as causal representation learning [64, 85], causal transfer learning [83], causal algorithmic fairness [84], and causal reinforcement learning [5]. This work focuses on learning causal structures from purely observational data within the framework of causal Bayesian networks, which are widely used to represent causal relationships among variables through directed acyclic graphs (DAGs). This is, in general, a nontrivial and difficult task due to the vast number of potential DAG structures and multiple DAGs representing the same set of conditional independence relationships. In fact, DAGs are generally identifiable only up to their corresponding Markov equivalence class, in which all DAGs encode the same conditional independencies [31].

The use of machine learning tools has given causal inference a new lease of life, enabling complex models to be used with principled causal estimators and guarantees about statistically important quantities (Wager and Athey, 2018; Chernozhukov et al., 2018; Hahn et al., 2020). To build trustworthy causal models, however, we also need to understand when these methods may be more or less reliable, or perhaps fail completely. This implies that causal inference needs a set of good benchmarking tools. Unfortunately, real-world datasets are not ideal for this task, because they cannot give us access to the ground truth other than in a few very special circumstances. In particular, they rarely provide the counterfactual outcomes we care about, and the distribution we want to evaluate often differs from the one that produced the observations. Well-designed simulations can address this discrepancy (Neal et al., 2020; Parikh et al., 2022); they allow us to choose a ground truth, stress-test new methods, compare their generalizability and stability, and expose failure modes before deployment.

Multi-class classification annotations have significantly advanced AI applications, with truth inference serving as a critical technique for aggregating noisy and biased annotations. Existing state-of-the-art methods typically model each annotator's expertise using a confusion matrix. However, these methods suffer from two widely recognized issues: 1) when most annotators label only a few tasks, or when classes are imbalanced, the estimated confusion matrices are unreliable, and 2) a single confusion matrix often remains inadequate for capturing each annotator's full expertise patterns across all tasks. To address these issues, we propose a novel confusion-matrix-based method, PTBCC (ProtoType learning-driven Bayesian Classifier Combination), to introduce a reliable and richer annotator estimation by prototype learning. Specifically, we assume that there exists a set $S$ of prototype confusion matrices, which capture the inherent expertise patterns of all annotators. Rather than a single confusion matrix, the expertise per annotator is extended as a Dirichlet prior distribution over these prototypes. This prototype learning-driven mechanism circumvents the data sparsity and class imbalance issues, ensuring a richer and more flexible characterization of annotators. Extensive experiments on 11 real-world datasets demonstrate that PTBCC achieves up to a 15% accuracy improvement in the best case, and a 3% higher average accuracy while reducing computational cost by over 90%.

Counterfactual explanations study what should have changed in order to get an alternative result, enabling end-users to understand machine learning mechanisms with counterexamples. Actionability is defined as the ability to transform the original case to be explained into a counterfactual one. We develop a method for actionable counterfactual explanations that, unlike predecessors, does not directly leverage training data. Rather, data is only used to learn a density estimator, creating a search landscape in which to apply path planning algorithms to solve the problem and masking the endogenous data, which can be sensitive or private. We put special focus on estimating the data density using Bayesian networks, demonstrating how their enhanced interpretability is useful in high-stakes scenarios in which fairness is raising concern. Using a synthetic benchmark comprised of 15 datasets, our proposal finds more actionable and simpler counterfactuals than the current state-of-the-art algorithms. We also test our algorithm with a real-world Environmental Protection Agency dataset, facilitating a more efficient and equitable study of policies to improve the quality of life in United States of America counties. Our proposal captures the interaction of variables, ensuring equity in decisions, as policies to improve certain domains of study (air, water quality, etc.) can be detrimental in others. In particular, the sociodemographic domain is often involved, where we find important variables related to the ongoing housing crisis that can potentially have a severe negative impact on communities.

The exponential growth of scientific knowledge has made the automated generation of scientific hypotheses that combine novelty, feasibility, and research value a core challenge. Existing methods based on large language models fail to systematically model the inherent in hypotheses or incorporate the closed-loop feedback mechanisms crucial for refinement. This paper proposes a multi-agent collaborative framework called HypoAgents, which for the first time integrates Bayesian reasoning with an information entropy-driven search mechanism across three stages-hypotheses generation, evidence validation, and hypotheses Refinement-to construct an iterative closed-loop simulating scientists' cognitive processes. Specifically, the framework first generates an initial set of hypotheses through diversity sampling and establishes prior beliefs based on a composite novelty-relevance-feasibility (N-R-F) score. It then employs etrieval-augmented generation (RAG) to gather external literature evidence, updating the posterior probabilities of hypotheses using Bayes' theorem. Finally, it identifies high-uncertainty hypotheses using information entropy $H = - \sum {{p_i}\log {p_i}}$ and actively refines them, guiding the iterative optimization of the hypothesis set toward higher quality and confidence. Experimental results on the ICLR 2025 conference real-world research question dataset (100 research questions) show that after 12 optimization iterations, the average ELO score of generated hypotheses improves by 116.3, surpassing the benchmark of real paper abstracts by 17.8, while the framework's overall uncertainty, as measured by Shannon entropy, decreases significantly by 0.92. This study presents an interpretable probabilistic reasoning framework for automated scientific discovery, substantially improving the quality and reliability of machine-generated research hypotheses.

This study proposes a novel perspective on multimodal deep learning for biomedical signal classification, systematically analyzing how complementary feature domains impact model performance. While fusing multiple domains often presumes enhanced accuracy, this work demonstrates that adding modalities can yield diminishing returns, as not all fusions are inherently advantageous. To validate this, five deep learning models were designed, developed, and rigorously evaluated: three unimodal (1D-CNN for time, 2D-CNN for time-frequency, and 1D-CNN-Transformer for frequency) and two multimodal (Hybrid 1, which fuses 1D-CNN and 2D-CNN; Hybrid 2, which combines 1D-CNN, 2D-CNN, and a Transformer). For ECG classification, bootstrapping and Bayesian inference revealed that Hybrid 1 consistently outperformed the 2D-CNN baseline across all metrics (p-values < 0.05, Bayesian probabilities > 0.90), confirming the synergistic complementarity of the time and time-frequency domains. Conversely, Hybrid 2's inclusion of the frequency domain offered no further improvement and sometimes a marginal decline, indicating representational redundancy; a phenomenon further substantiated by a targeted ablation study. This research redefines a fundamental principle of multimodal design in biomedical signal analysis. We demonstrate that optimal domain fusion isn't about the number of modalities, but the quality of their inherent complementarity. This paradigm-shifting concept moves beyond purely heuristic feature selection. Our novel theoretical contribution, "Complementary Feature Domains in Multimodal ECG Deep Learning," presents a mathematically quantifiable framework for identifying ideal domain combinations, demonstrating that optimal multimodal performance arises from the intrinsic information-theoretic complementarity among fused domains.

In this paper, we investigate how concept-based models (CMs) respond to out-of-distribution (OOD) inputs. CMs are interpretable neural architectures that first predict a set of high-level concepts (e.g., stripes, black) and then predict a task label from those concepts. In particular, we study the impact of concept interventions (i.e., operations where a human expert corrects a CM's mispredicted concepts at test time) on CMs' task predictions when inputs are OOD. Our analysis reveals a weakness in current state-of-the-art CMs, which we term leakage poisoning, that prevents them from properly improving their accuracy when intervened on for OOD inputs. To address this, we introduce MixCEM, a new CM that learns to dynamically exploit leaked information missing from its concepts only when this information is in-distribution. Our results across tasks with and without complete sets of concept annotations demonstrate that MixCEMs outperform strong baselines by significantly improving their accuracy for both in-distribution and OOD samples in the presence and absence of concept interventions.

Quantifying uncertainty in word embeddings is crucial for reliable inference from textual data. However, existing Bayesian methods such as Hamiltonian Monte Carlo (HMC) and mean-field variational inference (MFVI) are either computationally infeasible for large data or rely on restrictive assumptions. We propose a scalable Gibbs sampler using Polya-Gamma augmentation as well as Laplace approximation and compare them with MFVI and HMC for word embeddings. In addition, we address non-identifiability in word embeddings. Our Gibbs sampler and HMC correctly estimate uncertainties, while MFVI does not, and Laplace approximation only does so on large sample sizes, as expected. Applying the Gibbs sampler to the US Congress and the Movielens datasets, we demonstrate the feasibility on larger real data. Finally, as a result of having draws from the full posterior, we show that the posterior mean of word embeddings improves over maximum a posteriori (MAP) estimates in terms of hold-out likelihood, especially for smaller sampling sizes, further strengthening the need for posterior sampling of word embeddings.

The energy market has faced a significant structural change in the past decade. The global strife for decarbonization is encouraging the use of renewable energy sources, thus affecting the traditional supply-demand pattern, which were historically dominated by fossil fuels like coal, oil, and natural gas [18]. The growing integration of renewable energy sources into the power supply increases uncertainties in the electricity market due to intermittent nature of the sources such as wind or sunshine [57]. The volatility of the generation sources causes high price shocks and regime changes that is compromising to financial stability as well as investment strategies in the power market [58]. Particularly for countries such as Germany, where the larger percentage of electricity is produced by renewable energy sources [37], levels of sunlight and wind impact electricity generation and thus prices. This introduces, in addition to the physical problem of balancing the grid, non-stationarity to most price models, which further adds unreliability to the predictions. Accurate electricity price forecasting is crucial for efficient resource planning, financial risk management, and stabilization of the market, especially with increasing renewable energy penetration, which enables utilities, businesses, and governments to optimize planning and policy maximization while matching demand and supply. The building of an adequate prediction model, which is relatively straightforward and understandable but at the same time can reflect the market complexity and all influence factors engaged in it is not straightforward, and authors have utilized quite broadly three types of model for prediction: statistical/(probability-based) models [12], machine learning/deep learning models [42], and mixed models [30]. Precise forecasting allows the players in the market to make sound monetary policy.

Density operators, quantum generalizations of probability distributions, are gaining prominence in machine learning due to their foundational role in quantum computing. Generative modeling based on density operator models (\textbf{DOMs}) is an emerging field, but existing training algorithms -- such as those for the Quantum Boltzmann Machine -- do not scale to real-world data, such as the MNIST dataset. The Expectation-Maximization algorithm has played a fundamental role in enabling scalable training of probabilistic latent variable models on real-world datasets. \textit{In this paper, we develop an Expectation-Maximization framework to learn latent variable models defined through \textbf{DOMs} on classical hardware, with resources comparable to those used for probabilistic models, while scaling to real-world data.} However, designing such an algorithm is nontrivial due to the absence of a well-defined quantum analogue to conditional probability, which complicates the Expectation step. To overcome this, we reformulate the Expectation step as a quantum information projection (QIP) problem and show that the Petz Recovery Map provides a solution under sufficient conditions. Using this formulation, we introduce the Density Operator Expectation Maximization (DO-EM) algorithm -- an iterative Minorant-Maximization procedure that optimizes a quantum evidence lower bound. We show that the \textbf{DO-EM} algorithm ensures non-decreasing log-likelihood across iterations for a broad class of models. Finally, we present Quantum Interleaved Deep Boltzmann Machines (\textbf{QiDBMs}), a \textbf{DOM} that can be trained with the same resources as a DBM. When trained with \textbf{DO-EM} under Contrastive Divergence, a \textbf{QiDBM} outperforms larger classical DBMs in image generation on the MNIST dataset, achieving a 40--60\% reduction in the Fréchet Inception Distance.