Learning about the causal structure of the world is a fundamental problem for human cognition. Causal models and especially causal learning have proved to be difficult for large pretrained models using standard techniques of deep learning. In contrast, cognitive scientists have applied advances in our formal understanding of causation in computer science, particularly within the Causal Bayes Net formalism, to understand human causal learning. In the very different tradition of reinforcement learning, researchers have described an intrinsic reward signal called "empowerment" which maximizes mutual information between actions and their outcomes. "Empowerment" may be an important bridge between classical Bayesian causal learning and reinforcement learning and may help to characterize causal learning in humans and enable it in machines. If an agent learns an accurate causal world model, they will necessarily increase their empowerment, and increasing empowerment will lead to a more accurate causal world model. Empowerment may also explain distinctive features of childrens causal learning, as well as providing a more tractable computational account of how that learning is possible. In an empirical study, we systematically test how children and adults use cues to empowerment to infer causal relations, and design effective causal interventions.

Self supervised learning (SSL) is a machine learning paradigm where models learn to understand the underlying structure of data without explicit supervision from labeled samples. The acquired representations from SSL have demonstrated useful for many downstream tasks including clustering, and linear classification, etc. To ensure smoothness of the representation space, most SSL methods rely on the ability to generate pairs of observations that are similar to a given instance. However, generating these pairs may be challenging for many types of data. Moreover, these methods lack consideration of uncertainty quantification and can perform poorly in out-of-sample prediction settings. To address these limitations, we propose Gaussian process self supervised learning (GPSSL), a novel approach that utilizes Gaussian processes (GP) models on representation learning. GP priors are imposed on the representations, and we obtain a generalized Bayesian posterior minimizing a loss function that encourages informative representations. The covariance function inherent in GPs naturally pulls representations of similar units together, serving as an alternative to using explicitly defined positive samples. We show that GPSSL is closely related to both kernel PCA and VICReg, a popular neural network-based SSL method, but unlike both allows for posterior uncertainties that can be propagated to downstream tasks. Experiments on various datasets, considering classification and regression tasks, demonstrate that GPSSL outperforms traditional methods in terms of accuracy, uncertainty quantification, and error control.

Moralisation and Triangulation are transformations allowing to switch between different ways of factoring a probability distribution into a graphical model. Moralisation allows to view a Bayesian network (a directed model) as a Markov network (an undirected model), whereas triangulation addresses the opposite direction. We present a categorical framework where these transformations are modelled as functors between a category of Bayesian networks and one of Markov networks. The two kinds of network (the objects of these categories) are themselves represented as functors from a `syntax' domain to a `semantics' codomain. Notably, moralisation and triangulation can be defined inductively on such syntax via functor pre-composition. Moreover, while moralisation is fully syntactic, triangulation relies on semantics. This leads to a discussion of the variable elimination algorithm, reinterpreted here as a functor in its own right, that splits the triangulation procedure in two: one purely syntactic, the other purely semantic. This approach introduces a functorial perspective into the theory of probabilistic graphical models, which highlights the distinctions between syntactic and semantic modifications.

Inverse problems arise across scientific and engineering domains, where the goal is to infer hidden parameters or physical fields from indirect and noisy observations. Classical approaches, such as variational regularization and Bayesian inference, provide well established theoretical foundations for handling ill posedness. However, these methods often become computationally restrictive in high dimensional settings or when the forward model is governed by complex physics. Physics Informed Neural Networks (PINNs) have recently emerged as a promising framework for solving inverse problems by embedding physical laws directly into the training process of neural networks. In this paper, we introduce a new perspective on the Bayesian Physics Informed Neural Network (BPINN) framework, extending classical PINNs by explicitly incorporating training data generation, modeling and measurement uncertainties through Bayesian prior modeling and doing inference with the posterior laws. Also, as we focus on the inverse problems, we call this method BPINN-IP, and we show that the standard PINN formulation naturally appears as its special case corresponding to the Maximum A Posteriori (MAP) estimate. This unified formulation allows simultaneous exploitation of physical constraints, prior knowledge, and data-driven inference, while enabling uncertainty quantification through posterior distributions. To demonstrate the effectiveness of the proposed framework, we consider inverse problems arising in infrared image processing, including deconvolution and super-resolution, and present results on both simulated and real industrial data.

This study develops a hierarchical Bayesian framework that integrates expert domain knowledge to quantify player-specific effects in expected goals (xG) estimation, addressing a limitation of standard models that treat all players as identical finishers. Using 9,970 shots from StatsBomb's 2015-16 data and Football Manager 2017 ratings, we combine Bayesian logistic regression with informed priors to stabilise player-level estimates, especially for players with few shots. The hierarchical model reduces posterior uncertainty relative to weak priors and achieves strong external validity: hierarchical and baseline predictions correlate at R2 = 0.75, while an XGBoost benchmark validated against StatsBomb xG reaches R2 = 0.833. The model uncovers interpretable specialisation profiles, including one-on-one finishing (Aguero, Suarez, Belotti, Immobile, Martial), long-range shooting (Pogba), and first-touch execution (Insigne, Salah, Gameiro). It also identifies latent ability in underperforming players such as Immobile and Belotti. The framework supports counterfactual "what-if" analysis by reallocating shots between players under identical contexts. Case studies show that Sansone would generate +2.2 xG from Berardi's chances, driven largely by high-pressure situations, while Vardy-Giroud substitutions reveal strong asymmetry: replacing Vardy with Giroud results in a large decline (about -7 xG), whereas the reverse substitution has only a small effect (about -1 xG). This work provides an uncertainty-aware tool for player evaluation, recruitment, and tactical planning, and offers a general approach for domains where individual skill and contextual factors jointly shape performance.

The idea of using Formal Verification tools with large language models (LLMs) has enabled scaling software verification beyond manual workflows. However, current methods remain unreliable. Without a solid theoretical footing, the refinement process can wander; sometimes it settles, sometimes it loops back, and sometimes it breaks away from any stable trajectory. This work bridges this critical gap by developing an LLM-Verifier Convergence Theorem, providing the first formal framework with provable guarantees for termination and convergence. We model the interaction between the LLM and the verifier as a discrete-time Markov Chain, with state transitions determined by a key parameter: the error-reduction probability ($\delta$). The procedure reaching the Verified state almost surely demonstrates that the program terminates for any $\delta > 0$, with an expected iteration count bounded by $\mathbb{E}[n] \leq 4/\delta$. We then stress-tested this prediction in an extensive empirical campaign comprising more than 90,000 trials. The empirical results match the theory with striking consistency. Every single run reached verification, and the convergence factor clustered tightly around $C_f\approx$ 1.0. Consequently, the bound mirrors the system's actual behavior. The evidence is sufficiently robust to support dividing the workflow into three distinct operating zones: marginal, practical, and high-performance. Consequently, we establish the design thresholds with absolute confidence. Together, the theoretical guarantee and the experimental evidence provide a clearer architectural foundation for LLM-assisted verification. Heuristic tuning no longer has to be carried out by the system. Engineers gain a framework that supports predictable resource planning and performance budgeting, precisely what is needed before deploying these pipelines into safety-critical software environments.

We seek to clarify the concept of active inference by disentangling it from the Free Energy Principle. We show how the optimizations that need to be carried out in order to implement active inference in discrete state spaces can be formulated as constrained divergence minimization problems which can be solved by standard mean field methods that do not appeal to the idea of expected free energy. When it is used to model perception, the perception/action divergence criterion that we propose coincides with variational free energy. When it is used to model action, it differs from an expected free energy functional by an entropy regularizer.

Investment planning in power utilities, such as generation and transmission expansion, requires decade-long forecasts under profound uncertainty. Forecasting of energy mix and energy use decades ahead is nontrivial. Classical approaches focus on generating a finite number of scenarios (modeled as a mixture of Diracs in statistical theory terms), which limits insight into scenario-specific volatility and hinders robust decision-making. We propose an alternative using tractable probabilistic models (TPMs), particularly sum-product networks (SPNs). These models enable exact, scalable inference of key quantities such as scenario likelihoods, marginals, and conditional probabilities, supporting robust scenario expansion and risk assessment. This framework enables direct embedding of chance-constrained optimization into investment planning, enforcing safety or reliability with prescribed confidence levels. TPMs allow both scenario analysis and volatility quantification by compactly representing high-dimensional uncertainties. We demonstrate the approach's effectiveness through a representative power system planning case study, illustrating computational and reliability advantages over traditional scenario-based models.

Generative flow networks are able to sample, via sequential construction, high-reward, complex objects according to a reward function. However, such reward functions are often estimated approximately from noisy data, leading to epistemic uncertainty in the learnt policy. We present an approach to quantify this uncertainty by constructing a surrogate model composed of a polynomial chaos expansion, fit on a small ensemble of trained flow networks. This model learns the relationship between reward functions, parametrised in a low-dimensional space, and the probability distributions over actions at each step along a trajectory of the flow network. The surrogate model can then be used for inexpensive Monte Carlo sampling to estimate the uncertainty in the policy given uncertain rewards. We illustrate the performance of our approach on a discrete and continuous grid-world, symbolic regression, and a Bayesian structure learning task.

Estimating mutual information (MI) is a fundamental task in data science and machine learning. Existing estimators mainly rely on either highly flexible models (e.g., neural networks), which require large amounts of data, or overly simplified models (e.g., Gaussian copula), which fail to capture complex distributions. Drawing upon recent vector copula theory, we propose a principled interpolation between these two extremes to achieve a better trade-off between complexity and capacity. Experiments on state-of-the-art synthetic benchmarks and real-world data with diverse modalities demonstrate the advantages of the proposed estimator.

We explore the use of Active Inference (AIF) as a computational user model for spatial pointing, a key problem in Human-Computer Interaction (HCI). We present an AIF agent with continuous state, action, and observation spaces, performing one-dimensional mouse pointing and clicking. We use a simple underlying dynamic system to model the mouse cursor dynamics with realistic perceptual delay. In contrast to previous optimal feedback control-based models, the agent's actions are selected by minimizing Expected Free Energy, solely based on preference distributions over percepts, such as observing clicking a button correctly. Our results show that the agent creates plausible pointing movements and clicks when the cursor is over the target, with similar end-point variance to human users. In contrast to other models of pointing, we incorporate fully probabilistic, predictive delay compensation into the agent. The agent shows distinct behaviour for differing target difficulties without the need to retune system parameters, as done in other approaches. We discuss the simulation results and emphasize the challenges in identifying the correct configuration of an AIF agent interacting with continuous systems.

We present the design of an autoregressive active inference agent in the form of message passing on a factor graph. Expected free energy is derived and distributed across a planning graph. The proposed agent is validated on a robot navigation task, demonstrating exploration and exploitation in a continuous-valued observation space with bounded continuous-valued actions. Compared to a classical optimal controller, the agent modulates action based on predictive uncertainty, arriving later but with a better model of the robot's dynamics.