Marginalization -- summing a function over all assignments to a subset of its inputs -- is a fundamental computational problem with applications from probabilistic inference to formal verification. Despite its computational hardness in general, there exist many classes of functions (e.g., probabilistic models) for which marginalization remains tractable, and they can be commonly expressed by polynomial size arithmetic circuits computing multilinear polynomials. This raises the question, can all functions with polynomial time marginalization algorithms be succinctly expressed by such circuits? We give a negative answer, exhibiting simple functions with tractable marginalization yet no efficient representation by known models, assuming $\textsf{FP}\neq\#\textsf{P}$ (an assumption implied by $\textsf{P} \neq \textsf{NP}$). To this end, we identify a hierarchy of complexity classes corresponding to stronger forms of marginalization, all of which are efficiently computable on the known circuit models. We conclude with a completeness result, showing that whenever there is an efficient real RAM performing virtual evidence marginalization for a function, then there are small circuits for that function's multilinear representation.

Foundation medical segmentation models, with MedSAM being the most popular, have achieved promising performance across organs and lesions. However, MedSAM still suffers from compromised performance on specific lesions with intricate structures and appearance, as well as bounding box prompt-induced perturbations. Although current test-time adaptation (TTA) methods for medical image segmentation may tackle this issue, partial (e.g., batch normalization) or whole parametric updates restrict their effectiveness due to limited update signals or catastrophic forgetting in large models. Meanwhile, these approaches ignore the computational complexity during adaptation, which is particularly significant for modern foundation models. To this end, our theoretical analyses reveal that directly refining image embeddings is feasible to approach the same goal as parametric updates under the MedSAM architecture, which enables us to realize high computational efficiency and segmentation performance without the risk of catastrophic forgetting. Under this framework, we propose to encourage maximizing factorized conditional probabilities of the posterior prediction probability using a proposed distribution-approximated latent conditional random field loss combined with an entropy minimization loss. Experiments show that we achieve about 3\% Dice score improvements across three datasets while reducing computational complexity by over 7 times.

Colors are omnipresent in today's world and play a vital role in how humans perceive and interact with their surroundings. However, it is challenging for computers to imitate human color perception. This paper introduces the Human Perception-Based Fuzzy Color Model, COLIBRI (Color Linguistic-Based Representation and Interpretation), designed to bridge the gap between computational color representations and human visual perception. The proposed model uses fuzzy sets and logic to create a framework for color categorization. Using a three-phase experimental approach, the study first identifies distinguishable color stimuli for hue, saturation, and intensity through preliminary experiments, followed by a large-scale human categorization survey involving more than 1000 human subjects. The resulting data are used to extract fuzzy partitions and generate membership functions that reflect real-world perceptual uncertainty. The model incorporates a mechanism for adaptation that allows refinement based on feedback and contextual changes. Comparative evaluations demonstrate the model's alignment with human perception compared to traditional color models, such as RGB, HSV, and LAB. To the best of our knowledge, no previous research has documented the construction of a model for color attribute specification based on a sample of this size or a comparable sample of the human population (n = 2496). Our findings are significant for fields such as design, artificial intelligence, marketing, and human-computer interaction, where perceptually relevant color representation is critical.

Neurosymbolic AI focuses on integrating learning and reasoning, in particular, on unifying logical and neural representations. Despite the existence of an alphabet soup of neurosymbolic AI systems, the field is lacking a generally accepted formal definition of what neurosymbolic models and inference really are. We introduce a formal definition for neurosymbolic AI that makes abstraction of its key ingredients. More specifically, we define neurosymbolic inference as the computation of an integral over a product of a logical and a belief function. We show that our neurosymbolic AI definition makes abstraction of key representative neurosymbolic AI systems.

In the study of complex dynamical systems, understanding and accurately modeling the underlying physical processes is crucial for predicting system behavior and designing effective interventions. Yet real-world systems exhibit pronounced input (or system) variability and are observed through noisy, limited data conditions that confound traditional discovery methods that assume fixed-coefficient deterministic models. In this work, we theorize that accounting for system variability together with measurement noise is the key to consistently discover the governing equations underlying dynamical systems. As such, we introduce a stochastic inverse physics-discovery (SIP) framework that treats the unknown coefficients as random variables and infers their posterior distribution by minimizing the Kullback-Leibler divergence between the push-forward of the posterior samples and the empirical data distribution. Benchmarks on four canonical problems -- the Lotka-Volterra predator-prey system (multi- and single-trajectory), the historical Hudson Bay lynx-hare data, the chaotic Lorenz attractor, and fluid infiltration in porous media using low- and high-viscosity liquids -- show that SIP consistently identifies the correct equations and lowers coefficient root-mean-square error by an average of 82\% relative to the Sparse Identification of Nonlinear Dynamics (SINDy) approach and its Bayesian variant. The resulting posterior distributions yield 95\% credible intervals that closely track the observed trajectories, providing interpretable models with quantified uncertainty. SIP thus provides a robust, data-efficient approach for consistent physics discovery in noisy, variable, and data-limited settings.

Stochastic simulators exhibit intrinsic stochasticity due to unobservable, uncontrollable, or unmodeled input variables, resulting in random outputs even at fixed input conditions. Such simulators are common across various scientific disciplines; however, emulating their entire conditional probability distribution is challenging, as it is a task traditional deterministic surrogate modeling techniques are not designed for. Additionally, accurately characterizing the response distribution can require prohibitively large datasets, especially for computationally expensive high-fidelity (HF) simulators. When lower-fidelity (LF) stochastic simulators are available, they can enhance limited HF information within a multifidelity surrogate modeling (MFSM) framework. While MFSM techniques are well-established for deterministic settings, constructing multifidelity emulators to predict the full conditional response distribution of stochastic simulators remains a challenge. In this paper, we propose multifidelity generalized lambda models (MF-GLaMs) to efficiently emulate the conditional response distribution of HF stochastic simulators by exploiting data from LF stochastic simulators. Our approach builds upon the generalized lambda model (GLaM), which represents the conditional distribution at each input by a flexible, four-parameter generalized lambda distribution. MF-GLaMs are non-intrusive, requiring no access to the internal stochasticity of the simulators nor multiple replications of the same input values. We demonstrate the efficacy of MF-GLaM through synthetic examples of increasing complexity and a realistic earthquake application. Results show that MF-GLaMs can achieve improved accuracy at the same cost as single-fidelity GLaMs, or comparable performance at significantly reduced cost.

Diffusion models have quickly become some of the most popular and powerful generative models for high-dimensional data. The key insight that enabled their development was the realization that access to the score -- the gradient of the log-density at different noise levels -- allows for sampling from data distributions by solving a reverse-time stochastic differential equation (SDE) via forward discretization, and that popular denoisers allow for unbiased estimators of this score. In this paper, we demonstrate that an alternative, backward discretization of these SDEs, using proximal maps in place of the score, leads to theoretical and practical benefits. We leverage recent results in proximal matching to learn proximal operators of the log-density and, with them, develop Proximal Diffusion Models (ProxDM). Theoretically, we prove that $\widetilde{O}(d/\sqrt{\varepsilon})$ steps suffice for the resulting discretization to generate an $\varepsilon$-accurate distribution w.r.t. the KL divergence. Empirically, we show that two variants of ProxDM achieve significantly faster convergence within just a few sampling steps compared to conventional score-matching methods.

Data privacy is a central concern in many applications involving ranking from incomplete and noisy pairwise comparisons, such as recommendation systems, educational assessments, and opinion surveys on sensitive topics. In this work, we propose differentially private algorithms for ranking based on pairwise comparisons. Specifically, we develop and analyze ranking methods under two privacy notions: edge differential privacy, which protects the confidentiality of individual comparison outcomes, and individual differential privacy, which safeguards potentially many comparisons contributed by a single individual. Our algorithms--including a perturbed maximum likelihood estimator and a noisy count-based method--are shown to achieve minimax optimal rates of convergence under the respective privacy constraints. We further demonstrate the practical effectiveness of our methods through experiments on both simulated and real-world data.

We present the Context-Content Uncertainty Principle (CCUP), a unified framework that models cognition as the directed flow of information between high-entropy context and low-entropy content. Inference emerges as a cycle of bidirectional interactions, bottom-up contextual disambiguation paired with top-down content reconstruction, which resolves the Information Bottleneck in Optimal Transport (iBOT). Implemented via Rao-Blackwellized variational entropy minimization, CCUP steers representations toward minimal joint uncertainty while preserving inferential directionality. Local cycle completion underpins temporal bootstrapping, chaining simulations to refine memory, and spatial bootstrapping, enabling compositional hierarchical inference. We prove a Delta Convergence Theorem showing that recursive entropy minimization yields delta-like attractors in latent space, stabilizing perceptual schemas and motor plans. Temporal bootstrapping through perception-action loops and sleep-wake consolidation further transforms episodic traces into semantic knowledge. Extending CCUP, each hierarchical level performs delta-seeded inference: low-entropy content seeds diffuse outward along goal-constrained paths shaped by top-down priors and external context, confining inference to task-relevant manifolds and circumventing the curse of dimensionality. Building on this, we propose that language emerges as a symbolic transport system, externalizing latent content to synchronize inference cycles across individuals. Together, these results establish iBOT as a foundational principle of information flow in both individual cognition and collective intelligence, positioning recursive inference as the structured conduit through which minds adapt, align, and extend.

Continual learning is an online paradigm where a learner continually accumulates knowledge from different tasks encountered over sequential time steps. Importantly, the learner is required to extend and update its knowledge without forgetting about the learning experience acquired from the past, and while avoiding the need to retrain from scratch. Given its sequential nature and its resemblance to the way humans think, continual learning offers an opportunity to address several challenges which currently stand in the way of widening the range of applicability of deep models to further real-world problems. The continual need to update the learner with data arriving sequentially strikes inherent congruence between continual learning and Bayesian inference which provides a principal platform to keep updating the prior beliefs of a model given new data, without completely forgetting the knowledge acquired from the old data. This survey inspects different settings of Bayesian continual learning, namely task-incremental learning and class-incremental learning. We begin by discussing definitions of continual learning along with its Bayesian setting, as well as the links with related fields, such as domain adaptation, transfer learning and meta-learning. Afterwards, we introduce a taxonomy offering a comprehensive categorization of algorithms belonging to the Bayesian continual learning paradigm. Meanwhile, we analyze the state-of-the-art while zooming in on some of the most prominent Bayesian continual learning algorithms to date. Furthermore, we shed some light on links between continual learning and developmental psychology, and correspondingly introduce analogies between both fields. We follow that with a discussion of current challenges, and finally conclude with potential areas for future research on Bayesian continual learning.