Aumann's famous Agreeing to Disagree Theorem states that if a group of agents share a common prior, update their beliefs by Bayesian conditioning based on private information, and have common knowledge of their posterior beliefs regarding some event, these posteriors must be identical. There is an elegant generalization of this theorem by Monderer and Samet, later refined by Neeman: if a group of agents share a common prior, update their beliefs using Bayesian conditioning on private information, and have common p-belief of their posteriors, these posteriors must be close (i.e., they cannot differ by more than 1 - p). Here, common p-belief generalizes the concept of common knowledge to probabilistic beliefs: agents commonly p-believe an event E if everyone believes E to at least degree p, everyone believes to at least degree p that everyone believes E to at least degree p, and so on. This paper further extends the Monderer-Samet-Neeman Agreement Theorem from classical probability measures to plausibility measures -- a very general framework introduced by Halpern that unifies many formal models of belief. To facilitate this extension, we provide a new proof of the Monderer-Samet-Neeman theorem in the classical setting. Building upon both the original proof and our new proof, we offer two different generalizations of the theorem to plausibility-based structures. We then apply these generalized results to several non-classical belief models, including conditional probability structures and lexicographic probability structures. Moreover, we show that whenever our generalized theorems do not apply, the Monderer-Samet-Neeman Agreement Theorem fails. These findings suggest that our results successfully identify the minimal conditions required for a belief model to satisfy the Monderer-Samet-Neeman Agreement Theorem.

The standard theory of model-free reinforcement learning assumes that the environment dynamics are stationary and that agents are decoupled from their environment, such that policies are treated as being separate from the world they inhabit. This leads to theoretical challenges in the multi-agent setting where the non-stationarity induced by the learning of other agents demands prospective learning based on prediction models. To accurately model other agents, an agent must account for the fact that those other agents are, in turn, forming beliefs about it to predict its future behavior, motivating agents to model themselves as part of the environment. Here, building upon foundational work on universal artificial intelligence (AIXI), we introduce a mathematical framework for prospective learning and embedded agency centered on self-prediction, where Bayesian RL agents predict both future perceptual inputs and their own actions, and must therefore resolve epistemic uncertainty about themselves as part of the universe they inhabit. We show that in multi-agent settings, self-prediction enables agents to reason about others running similar algorithms, leading to new game-theoretic solution concepts and novel forms of cooperation unattainable by classical decoupled agents. Moreover, we extend the theory of AIXI, and study universally intelligent embedded agents which start from a Solomonoff prior. We show that these idealized agents can form consistent mutual predictions and achieve infinite-order theory of mind, potentially setting a gold standard for embedded multi-agent learning.

Abstract-- Effective collective decision-making in swarm robotics often requires balancing exploration, communication and individual uncertainty estimation, especially in hazardous environments where direct measurements are limited or costly. We propose a decentralized Bayesian framework that enables a swarm of simple robots to identify the safer of two areas, each characterized by an unknown rate of hazardous events governed by a Poisson process. Robots employ a conjugate prior to gradually predict the times between events and derive confidence estimates to adapt their behavior . Our simulation results show that the robot swarm consistently chooses the correct area while reducing exposure to hazardous events by being sample-efficient. Compared to baseline heuristics, our proposed approach shows better performance in terms of safety and speed of convergence. The proposed scenario has potential to extend the current set of benchmarks in collective decision-making and our method has applications in adaptive risk-aware sampling and exploration in hazardous, dynamic environments. Collective decision-making under uncertainty is a fundamental challenge in multi-robot systems, including domains such as collective perception, environment classification, and spatial consensus [1]-[4]. Decentralized systems (e.g., robot swarms) operate under strict limitations on sensing, communication, and memory. Instead of sharing/storing complete observation histories, robots must maintain compact model representations of their knowledge. It is crucial to develop efficient strategies for collective decision-making, especially when observations are sparse, noisy [5], and gathered from stochastic processes [6]. This is typically characterized as a best-of-n problem [3], [7].

Offline inverse reinforcement learning (IRL) aims to recover a reward function that explains expert behavior using only fixed demonstration data, without any additional online interaction. We propose BiCQL-ML, a policy-free offline IRL algorithm that jointly optimizes a reward function and a conservative Q-function in a bi-level framework, thereby avoiding explicit policy learning. The method alternates between (i) learning a conservative Q-function via Conservative Q-Learning (CQL) under the current reward, and (ii) updating the reward parameters to maximize the expected Q-values of expert actions while suppressing over-generalization to out-of-distribution actions. This procedure can be viewed as maximum likelihood estimation under a soft value matching principle. We provide theoretical guarantees that BiCQL-ML converges to a reward function under which the expert policy is soft-optimal. Empirically, we show on standard offline RL benchmarks that BiCQL-ML improves both reward recovery and downstream policy performance compared to existing offline IRL baselines.

Counterfactual explanations enhance the actionable interpretability of machine learning models by identifying the minimal changes required to achieve a desired outcome of the model. However, existing methods often ignore the complex dependencies in real-world datasets, leading to unrealistic or impractical modifications. Motivated by cybersecurity applications in the email marketing domain, we propose a method for generating Diverse, Actionable, and kNowledge-Constrained Explanations (DANCE), which incorporates feature dependencies and causal constraints to ensure plausibility and real-world feasibility of counterfactuals. Our method learns linear and nonlinear constraints from data or integrates expert-provided dependency graphs, ensuring counterfactuals are plausible and actionable. By maintaining consistency with feature relationships, the method produces explanations that align with real-world constraints. Additionally, it balances plausibility, diversity, and sparsity, effectively addressing key limitations in existing algorithms. The work is developed based on a real-life case study with Freshmail, the largest email marketing company in Poland and supported by a joint R&D project Sendguard. Furthermore, we provide an extensive evaluation using 140 public datasets, which highlights its ability to generate meaningful, domain-relevant counterfactuals that outperform other existing approaches based on widely used metrics. The source code for reproduction of the results can be found in a GitHub repository we provide.

Large Language Models (LLMs) have demonstrated strong performance across a wide range of tasks, yet they still struggle with complex mathematical reasoning, a challenge fundamentally rooted in deep structural dependencies. To address this challenge, we propose \textbf{CA}usal \textbf{MA}thematician (\textbf{CAMA}), a two-stage causal framework that equips LLMs with explicit, reusable mathematical structure. In the learning stage, CAMA first constructs the \textbf{M}athematical \textbf{C}ausal \textbf{G}raph (\textbf{MCG}), a high-level representation of solution strategies, by combining LLM priors with causal discovery algorithms applied to a corpus of question-solution pairs. The resulting MCG encodes essential knowledge points and their causal dependencies. To better align the graph with downstream reasoning tasks, CAMA further refines the MCG through iterative feedback derived from a selected subset of the question-solution pairs. In the reasoning stage, given a new question, CAMA dynamically extracts a task-relevant subgraph from the MCG, conditioned on both the question content and the LLM's intermediate reasoning trace. This subgraph, which encodes the most pertinent knowledge points and their causal dependencies, is then injected back into the LLM to guide its reasoning process. Empirical results on real-world datasets show that CAMA significantly improves LLM performance on challenging mathematical problems. Furthermore, our experiments demonstrate that structured guidance consistently outperforms unstructured alternatives, and that incorporating asymmetric causal relationships yields greater improvements than using symmetric associations alone.

V ariational inference (VI) is a cornerstone of modern Bayesian learning, enabling approximate inference in complex models that would otherwise be intractable. However, its formulation depends on expectations and divergences defined through high-dimensional integrals, often rendering analytical treatment impossible and necessitating heavy reliance on approximate learning and inference techniques. Possibility theory, an imprecise probability framework, allows to directly model epistemic uncertainty instead of leveraging subjective probabilities. While this framework provides robustness and interpretability under sparse or imprecise information, adapting VI to the possibilistic setting requires rethinking core concepts such as entropy and divergence, which presuppose additivity. In this work, we develop a principled formulation of possibilistic variational inference and apply it to a special class of exponential-family functions, highlighting parallels with their probabilistic counterparts and revealing the distinctive mathematical structures of possibility theory.

In humans and other animals, category learning enhances discrimination between stimuli close to the category boundary. This phenomenon, called categorical perception, was also empirically observed in artificial neural networks trained on classification tasks. In previous modeling works based on neuroscience data, we show that this expansion/compression is a necessary outcome of efficient learning. Here we extend our theoretical framework to artificial networks. We show that minimizing the Bayes cost (mean of the cross-entropy loss) implies maximizing the mutual information between the set of categories and the neural activities prior to the decision layer. Considering structured data with an underlying feature space of small dimension, we show that maximizing the mutual information implies (i) finding an appropriate projection space, and, (ii) building a neural representation with the appropriate metric. The latter is based on a Fisher information matrix measuring the sensitivity of the neural activity to changes in the projection space. Optimal learning makes this neural Fisher information follow a category-specific Fisher information, measuring the sensitivity of the category membership. Category learning thus induces an expansion of neural space near decision boundaries. We characterize the properties of the categorical Fisher information, showing that its eigenvectors give the most discriminant directions at each point of the projection space. We find that, unexpectedly, its maxima are in general not exactly at, but near, the class boundaries. Considering toy models and the MNIST dataset, we numerically illustrate how after learning the two Fisher information matrices match, and essentially align with the category boundaries. Finally, we relate our approach to the Information Bottleneck one, and we exhibit a bias-variance decomposition of the Bayes cost, of interest on its own.

Higher-order networks, naturally described as hypergraphs, are essential for modeling real-world systems involving interactions among three or more entities. Stochastic block models offer a principled framework for characterizing mesoscale organization, yet their extension to hypergraphs involves a trade-off between expressive power and computational complexity. A recent simplification, a single-order model, mitigates this complexity by assuming a single affinity pattern governs interactions of all orders. This universal assumption, however, may overlook order-dependent structural details. Here, we propose a framework that relaxes this assumption by introducing a multi-order block structure, in which different affinity patterns govern distinct subsets of interaction orders. Our framework is based on a multi-order stochastic block model and searches for the optimal partition of the set of interaction orders that maximizes out-of-sample hyperlink prediction performance. Analyzing a diverse range of real-world networks, we find that multi-order block structures are prevalent. Accounting for them not only yields better predictive performance over the single-order model but also uncovers sharper, more interpretable mesoscale organization. Our findings reveal that order-dependent mechanisms are a key feature of the mesoscale organization of real-world higher-order networks.

The increasing availability of data and advancements in computational intelligence have accelerated the adoption of data-driven methods (DDMs) in product development. However, their integration into product development remains fragmented. This fragmentation stems from uncertainty, particularly the lack of clarity on what types of DDMs to use and when to employ them across the product development lifecycle. To address this, a necessary first step is to investigate the usage of DDM in engineering design by identifying which methods are being used, at which development stages, and for what application. This paper presents a PRISMA systematic literature review. The V-model as a product development framework was adopted and simplified into four stages: system design, system implementation, system integration, and validation. A structured search across Scopus, Web of Science, and IEEE Xplore (2014--2024) retrieved 1{,}689 records. After screening, 114 publications underwent full-text analysis. Findings show that machine learning (ML) and statistical methods dominate current practice, whereas deep learning (DL), though still less common, exhibits a clear upward trend in adoption. Additionally, supervised learning, clustering, regression analysis, and surrogate modeling are prevalent in design, implementation, and integration system stages but contributions to validation remain limited. Key challenges in existing applications include limited model interpretability, poor cross-stage traceability, and insufficient validation under real-world conditions. Additionally, it highlights key limitations and opportunities such as the need for interpretable hybrid models. This review is a first step toward design-stage guidelines; a follow-up synthesis should map computer science algorithms to engineering design problems and activities.