In graphical models, factor graphs, and more generally energy-based models, the interactions between variables are encoded by a graph, a hypergraph, or, in the most general case, a partially ordered set (poset). Inference on such probabilistic models cannot be performed exactly due to cycles in the underlying structures of interaction. Instead, one resorts to approximate variational inference by optimizing the Bethe free energy. Critical points of the Bethe free energy correspond to fixed points of the associated Belief Propagation algorithm. A full characterization of these critical points for general graphs, hypergraphs, and posets with a finite number of variables is still an open problem. We show that, for hypergraphs and posets with chains of length at most 1, changing the poset of interactions of the probabilistic model to one with the same homotopy type induces a bijection between the critical points of the associated free energy. This result extends and unifies classical results that assume specific forms of collapsibility to prove uniqueness of the critical points of the Bethe free energy.

The Internet of Things (IoT) has revolutionized various applications including agriculture, but it still faces challenges in data collection and understanding. This paper proposes a real-time framework with three additional semantic layers to help IoT devices and sensors comprehend data meaning and source. The framework consists of six layers: perception, semantic annotation, interoperability, transportation, semantic reasoning, and application, suitable for dynamic environments. Sensors collect data in the form of voltage, which is then processed by microprocessors or microcontrollers in the semantic annotation and preprocessing layer. Metadata is added to the raw data, including the purpose, ID number, and application. Two semantic algorithms are proposed in the semantic interoperability and ontologies layer: the interoperability semantic algorithm for standardizing file types and the synonym identification algorithm for identifying synonyms. In the transportation layer, raw data and metadata are sent to other IoT devices or cloud computing platforms using techniques like WiFi, Zigbee networks, Bluetooth, and mobile communication networks. A semantic reasoning layer is proposed to infer new knowledge from the existing data, using fuzzy logic, Dempster-Shafer theory, and Bayesian networks. A Graphical User Interface (GUI) is proposed in the application layer to help users communicate with and monitor IoT sensors, devices, and new knowledge inferred. This framework provides a robust solution for managing IoT data, ensuring semantic completeness, and enabling real-time knowledge inference. The integration of uncertainty reasoning methods and semantic interoperability techniques makes this framework a valuable tool for advancing IoT applications in general and in agriculture in particular.

The growing complexity of machine learning (ML) models in big data analytics, especially in domains such as environmental monitoring, highlights the critical need for interpretability and explainability to promote trust, ethical considerations, and regulatory adherence (e.g., GDPR). Traditional "black-box" models obstruct transparency, whereas post-hoc explainable AI (XAI) techniques like LIME and SHAP frequently compromise accuracy or fail to deliver inherent insights. This paper presents a novel framework that combines type-2 fuzzy sets, granular computing, and clustering to boost explainability and fairness in big data environments. When applied to the UCI Air Quality dataset, the framework effectively manages uncertainty in noisy sensor data, produces linguistic rules, and assesses fairness using silhouette scores and entropy. Key contributions encompass: (1) A type-2 fuzzy clustering approach that enhances cohesion by about 4% compared to type-1 methods (silhouette 0.365 vs. 0.349) and improves fairness (entropy 0.918); (2) Incorporation of fairness measures to mitigate biases in unsupervised scenarios; (3) A rule-based component for intrinsic XAI, achieving an average coverage of 0.65; (4) Scalable assessments showing linear runtime (roughly 0.005 seconds for sampled big data sizes). Experimental outcomes reveal superior performance relative to baselines such as DBSCAN and Agglomerative Clustering in terms of interpretability, fairness, and efficiency. Notably, the proposed method achieves a 4% improvement in silhouette score over type-1 fuzzy clustering and outperforms baselines in fairness (entropy reduction by up to 1%) and efficiency.

Large language models (LLMs) are increasingly used for decision-making tasks under uncertainty; however, their risk profiles and how they are influenced by prompting and alignment methods remain underexplored. Existing studies have primarily examined personality prompting or multi-agent interactions, leaving open the question of how post-training influences the risk behavior of LLMs. In this work, we propose a new pipeline for eliciting, steering, and modulating LLMs' risk profiles, drawing on tools from behavioral economics and finance. Using utility-theoretic models, we compare pre-trained, instruction-tuned, and RLHF-aligned LLMs, and find that while instruction-tuned models exhibit behaviors consistent with some standard utility formulations, pre-trained and RLHF-aligned models deviate more from any utility models fitted. We further evaluate modulation strategies, including prompt engineering, in-context learning, and post-training, and show that post-training provides the most stable and effective modulation of risk preference. Our findings provide insights into the risk profiles of different classes and stages of LLMs and demonstrate how post-training modulates these profiles, laying the groundwork for future research on behavioral alignment and risk-aware LLM design.

Catastrophic interference, also known as catastrophic forgetting, is a fundamental challenge in machine learning, where a trained learning model progressively loses performance on previously learned tasks when adapting to new ones. In this paper, we aim to better understand and model the catastrophic interference problem from a latent representation learning point of view, and propose a novel theoretical framework that formulates catastrophic interference as an identification problem. Our analysis demonstrates that the forgetting phenomenon can be quantified by the distance between partial-task aware (PTA) and all-task aware (ATA) setups. Building upon recent advances in identifiability theory, we prove that this distance can be minimized through identification of shared latent variables between these setups. When learning, we propose our method \ourmeos with two-stage training strategy: First, we employ maximum likelihood estimation to learn the latent representations from both PTA and ATA configurations. Subsequently, we optimize the KL divergence to identify and learn the shared latent variables. Through theoretical guarantee and empirical validations, we establish that identifying and learning these shared representations can effectively mitigate catastrophic interference in machine learning systems. Our approach provides both theoretical guarantees and practical performance improvements across both synthetic and benchmark datasets.

Abstract--Dempster-Shafer Theory (DST) provides a powerful framework for modeling uncertainty and has been widely applied to multi-attribute classification tasks. However, traditional DST - based attribute fusion-based classifiers suffer from oversimplified membership function modeling and limited exploitation of the belief structure brought by basic probability assignment (BPA), reducing their effectiveness in complex real-world scenarios. This paper presents an enhanced attribute fusion-based classifier that addresses these limitations through two key innovations. First, we adopt a selective modeling strategy that utilizes both single Gaussian and Gaussian Mixture Models (GMMs) for membership function construction, with model selection guided by cross-validation and a tailored evaluation metric. Second, we introduce a novel method to transform the possibility distribution into a BPA by combining simple BPAs derived from normalized possibility distributions, enabling a much richer and more flexible representation of uncertain information. Furthermore, we apply the belief structure-based BPA generation method to the evidential K-Nearest Neighbors (EKNN) classifier, enhancing its ability to incorporate uncertainty information into decision-making. Comprehensive experiments on benchmark datasets are conducted to evaluate the performance of the proposed attribute fusion-based classifier and the enhanced evidential K-Nearest Neighbors classifier in comparison with both evidential classifiers and conventional machine learning classifiers. The results demonstrate that the proposed classifier outperforms the best existing evidential classifier, achieving an average accuracy improvement of 4.86%, while maintaining low variance, thus confirming its superior effectiveness and robustness.

Learning about the directed acyclic graph (DAG) underlying a system's data-generating process from observational data under causal sufficiency is a fundamental causal discovery task (Pearl, 2009). Score-based algorithms address this task by assigning penalized likelihood scores to each DAG and seeking graphs whose scores are optimal. Identifiability theory asks when such score-optimal graphs identify the target graph (or its equivalence class) in the infinite-sample limit, with various results under different assumptions and scores (Chickering, 2002; Nandy et al., 2018). Exact algorithms, that are guaranteed to find a score-optimal graph, have exponential run-time and are feasible up to roughly 30 variables (Koivisto & Sood, 2004; Silander & Myllym aki, 2006). For larger graphs, local search must be employed, which evaluates neighbouring graphs to find graphs with better scores; canonical moves for this hill climbing are single edge insertions, deletions, or reversals (Heckerman et al., 1995). In the sample limit, greedy discrete search with a neighbourhood notion that respects score equivalence provably finds a graph with optimal score (Chickering, 2002). In finite samples, scores are inexact and local search may get stuck in local optima or, as we demonstrate, even find graphs with better scores than the true graph. Finite-sample performance is a practical challenge, despite the mature identifiability theory and asymptotic guarantees. Continuous optimization methods have emerged as a popular alternative.

Many machine learning algorithms rely on iterative updates of uncertainty representations, ranging from variational inference and expectation-maximization, to reinforcement learning, continual learning, and multi-agent learning. In the presence of imprecision and ambiguity, credal sets -- closed, convex sets of probability distributions -- have emerged as a popular framework for representing imprecise probabilistic beliefs. Under such imprecision, many learning problems in imprecise probabilistic machine learning (IPML) may be viewed as processes involving successive applications of update rules on credal sets. This naturally raises the question of whether this iterative process converges to stable fixed points -- or, more generally, under what conditions on the updating mechanism such fixed points exist, and whether they can be attained. We provide the first analysis of this problem and illustrate our findings using Credal Bayesian Deep Learning as a concrete example. Our work demonstrates that incorporating imprecision into the learning process not only enriches the representation of uncertainty, but also reveals structural conditions under which stability emerges, thereby offering new insights into the dynamics of iterative learning under imprecision.

A generic D-dimensional Gaussian can be conditioned or projected onto the D-1 unit sphere, thereby leading to the well-known Fisher-Bingham (FB) or Angular Gaussian (AG) distribution families, respectively. These are some of the most fundamental distributions on the sphere, yet cannot straightforwardly be written as a normalizing flow except in two special cases: the von-Mises Fisher in D=3 and the central angular Gaussian in any D. In this paper, we describe how to generalize these special cases to a family of normalizing flows that behave similarly to the full FB or AG family in any D. We call them "zoom-linear-project" (ZLP)-Fisher flows. Unlike a normal Fisher-Bingham distribution, their composition allows to gradually add complexity as needed. Furthermore, they can naturally handle conditional density estimation with target distributions that vary by orders of magnitude in scale - a setting that is important in astronomical applications but that existing flows often struggle with. A particularly useful member of the new family is the Kent analogue that can cheaply upgrade any flow in this situation to yield better performance.

Learning graphical conditional independence structures from nonlinear, continuous or mixed data is a central challenge in machine learning and the sciences, and many existing methods struggle to scale to thousands of samples or hundreds of variables. We introduce two basis-expansion tools for scalable causal discovery. First, the Basis Function BIC (BF-BIC) score uses truncated additive expansions to approximate nonlinear dependencies. BF-BIC is theoretically consistent under additive models and extends to post-nonlinear (PNL) models via an invertible reparameterization. It remains robust under moderate interactions and supports mixed data through a degenerate-Gaussian embedding for discrete variables. In simulations with fully nonlinear neural causal models (NCMs), BF-BIC outperforms kernel- and constraint-based methods (e.g., KCI, RFCI) in both accuracy and runtime. Second, the Basis Function Likelihood Ratio Test (BF-LRT) provides an approximate conditional independence test that is substantially faster than kernel tests while retaining competitive accuracy. Extensive simulations and a real-data application to Canadian wildfire risk show that, when integrated into hybrid searches, BF-based methods enable interpretable and scalable causal discovery. Implementations are available in Python, R, and Java.