Methods for probability updating, of which Bayesian conditionalization is the most well-known and widely used, are modeling tools that aim to represent the process of modifying an initial epistemic state, typically represented by a prior probability function P, which is adjusted in light of new information. Notably, updating methods and conditional sentences seem to intuitively share a deep connection, as is evident in the case of conditionalization. The present work contributes to this line of research and aims at shedding new light on the relationship between updating methods and conditional connectives. Departing from previous literature that often focused on a specific type of conditional or a particular updating method, our goal is to prove general results concerning the connection between conditionals and their probabilities. This will allow us to characterize the probabilities of certain conditional connectives and to understand what class of updating procedures can be represented using specific conditional connectives. Broadly, we adopt a general perspective that encompasses a large class of conditionals and a wide range of updating methods, enabling us to prove some general results concerning their interrelation.

We present BKP, a user-friendly and extensible R package that implements the Beta Kernel Process (BKP) -- a fully nonparametric and computationally efficient framework for modeling spatially varying binomial probabilities. The BKP model combines localized kernel-weighted likelihoods with conjugate beta priors, resulting in closed-form posterior inference without requiring latent variable augmentation or intensive MCMC sampling. The package supports binary and aggregated binomial responses, allows flexible choices of kernel functions and prior specification, and provides loss-based kernel hyperparameter tuning procedures. In addition, BKP extends naturally to the Dirichlet Kernel Process (DKP) for modeling spatially varying multinomial or compositional data. To our knowledge, this is the first publicly available R package for implementing BKP-based methods. We illustrate the use of BKP through several synthetic and real-world datasets, highlighting its interpretability, accuracy, and scalability. The package aims to facilitate practical application and future methodological development of kernel-based beta modeling in statistics and machine learning.

Within the formal setting of the Lockean thesis, an agent belief set is defined in terms of degrees of confidence and these are described in probabilistic terms. This approach is of established interest, notwithstanding some limitations that make its use troublesome in some contexts, like, for instance, in belief change theory. Precisely, Lockean belief sets are not generally closed under (classical) logical deduction. The aim of the present paper is twofold: on one side we provide two characterizations of those belief sets that are closed under classical logic deduction, and on the other we propose an approach to probabilistic update that allows us for a minimal revision of those beliefs, i.e., a revision obtained by making the fewest possible changes to the existing belief set while still accommodating the new information. In particular, we show how we can deductively close a belief set via a minimal revision.

There has been much recent interest in two-sided markets and dynamics thereof. In a rather a general discrete-time feedback model, which we show conditions that assure that for each agent, there exists the limit of a long-run average allocation of a resource to the agent, which is independent of any initial conditions. We call this property the unique ergodicity. Our model encompasses two-sided markets and more complicated interconnections of workers and customers, such as in a supply chain. It allows for non-linearity of the response functions of market participants. Finally, it allows for uncertainty in the response of market participants by considering a set of the possible responses to either price or other signals and a measure to sample from these.

The beta-negative binomial process (BNBP), an integer-valued stochastic process, is employed to partition a count vector into a latent random count matrix. As the marginal probability distribution of the BNBP that governs the exchangeable random partitions of grouped data has not yet been developed, current inference for the BNBP has to truncate the number of atoms of the beta process. This paper introduces an exchangeable partition probability function to explicitly describe how the BNBP clusters the data points of each group into a random number of exchangeable partitions, which are shared across all the groups. A fully collapsed Gibbs sampler is developed for the BNBP, leading to a novel nonparametric Bayesian topic model that is distinct from existing ones, with simple implementation, fast convergence, good mixing, and state-of-the-art predictive performance.

In this work, we study probability functions associated with Gaussian mixture models. Our primary focus is on extending the use of spherical radial decomposition for multivariate Gaussian random vectors to the context of Gaussian mixture models, which are not inherently spherical but only conditionally so. Specifically, the conditional probability distribution, given a random parameter of the random vector, follows a Gaussian distribution, allowing us to apply Bayesian analysis tools to the probability function. This assumption, together with spherical radial decomposition for Gaussian random vectors, enables us to represent the probability function as an integral over the Euclidean sphere. Using this representation, we establish sufficient conditions to ensure the differentiability of the probability function and provide and integral representation of its gradient. Furthermore, leveraging the Bayesian decomposition, we approximate the probability function using random sampling over the parameter space and the Euclidean sphere. Finally, we present numerical examples that illustrate the advantages of this approach over classical approximations based on random vector sampling.

Scientists often search for phenomena of interest while exploring new environments. Autonomous vehicles are deployed to explore such areas where human-operated vehicles would be costly or dangerous. Online control of autonomous vehicles for information-gathering is called adaptive sampling and can be framed as a POMDP that uses information gain as its principal objective. While prior work focuses largely on single-agent scenarios, this paper confronts challenges unique to multi-agent adaptive sampling, such as avoiding redundant observations, preventing vehicle collision, and facilitating path planning under limited communication. We start with Multi-Agent Path Finding (MAPF) methods, which address collision avoidance by decomposing the MAPF problem into a series of single-agent path planning problems. We then present information-driven MAPF which addresses multi-agent information gain under limited communication. First, we introduce an admissible heuristic that relaxes mutual information gain to an additive function that can be evaluated as a set of independent single agent path planning problems. Second, we extend our approach to a distributed system that is robust to limited communication. When all agents are in range, the group plans jointly to maximize information. When some agents move out of range, communicating subgroups are formed and the subgroups plan independently. Since redundant observations are less likely when vehicles are far apart, this approach only incurs a small loss in information gain, resulting in an approach that gracefully transitions from full to partial communication. We evaluate our method against other adaptive sampling strategies across various scenarios, including real-world robotic applications. Our method was able to locate up to 200% more unique phenomena in certain scenarios, and each agent located its first unique phenomenon faster by up to 50%.

Forward-only learning algorithms have recently gained attention as alternatives to gradient backpropagation, replacing the backward step of this latter solver with an additional contrastive forward pass. Among these approaches, the so-called Forward-Forward Algorithm (FFA) has been shown to achieve competitive levels of performance in terms of generalization and complexity. Networks trained using FFA learn to contrastively maximize a layer-wise defined goodness score when presented with real data (denoted as positive samples) and to minimize it when processing synthetic data (corr. negative samples). However, this algorithm still faces weaknesses that negatively affect the model accuracy and training stability, primarily due to a gradient imbalance between positive and negative samples. To overcome this issue, in this work we propose a novel implementation of the FFA algorithm, denoted as Polar-FFA, which extends the original formulation by introducing a neural division (\emph{polarization}) between positive and negative instances. Neurons in each of these groups aim to maximize their goodness when presented with their respective data type, thereby creating a symmetric gradient behavior. To empirically gauge the improved learning capabilities of our proposed Polar-FFA, we perform several systematic experiments using different activation and goodness functions over image classification datasets. Our results demonstrate that Polar-FFA outperforms FFA in terms of accuracy and convergence speed. Furthermore, its lower reliance on hyperparameters reduces the need for hyperparameter tuning to guarantee optimal generalization capabilities, thereby allowing for a broader range of neural network configurations.

Advances in neural computation have predominantly relied on the gradient backpropagation algorithm (BP). However, the recent shift towards non-stationary data modeling has highlighted the limitations of this heuristic, exposing that its adaptation capabilities are far from those seen in biological brains. Unlike BP, where weight updates are computed through a reverse error propagation path, Hebbian learning dynamics provide synaptic updates using only information within the layer itself. This has spurred interest in biologically plausible learning algorithms, hypothesized to overcome BP's shortcomings. In this context, Hinton recently introduced the Forward-Forward Algorithm (FFA), which employs local learning rules for each layer and has empirically proven its efficacy in multiple data modeling tasks. In this work we argue that when employing a squared Euclidean norm as a goodness function driving the local learning, the resulting FFA is equivalent to a neo-Hebbian Learning Rule. To verify this result, we compare the training behavior of FFA in analog networks with its Hebbian adaptation in spiking neural networks. Our experiments demonstrate that both versions of FFA produce similar accuracy and latent distributions. The findings herein reported provide empirical evidence linking biological learning rules with currently used training algorithms, thus paving the way towards extrapolating the positive outcomes from FFA to Hebbian learning rules. Simultaneously, our results imply that analog networks trained under FFA could be directly applied to neuromorphic computing, leading to reduced energy usage and increased computational speed.

The beta-negative binomial process (BNBP), an integer-valued stochastic process, is employed to partition a count vector into a latent random count matrix. As the marginal probability distribution of the BNBP that governs the exchangeable random partitions of grouped data has not yet been developed, current inference for the BNBP has to truncate the number of atoms of the beta process. This paper introduces an exchangeable partition probability function to explicitly describe how the BNBP clusters the data points of each group into a random number of exchangeable partitions, which are shared across all the groups. A fully collapsed Gibbs sampler is developed for the BNBP, leading to a novel nonparametric Bayesian topic model that is distinct from existing ones, with simple implementation, fast convergence, good mixing, and state-of-the-art predictive performance.