The ratio of two densities characterizes their differences. We consider learning the density ratio given i.i.d. observations from each of the two distributions. We propose additive tree models for the density ratio along with efficient algorithms for training these models using a new loss function called the balancing loss. With this loss, additive tree models for the density ratio can be trained using algorithms original designed for supervised learning. Specifically, they can be trained from both an optimization perspective that parallels tree boosting and from a (generalized) Bayesian perspective that parallels Bayesian additive regression trees (BART). For the former, we present two boosting algorithms -- one based on forward-stagewise fitting and the other based on gradient boosting, both of which produce a point estimate for the density ratio function. For the latter, we show that due to the loss function's resemblance to an exponential family kernel, the new loss can serve as a pseudo-likelihood for which conjugate priors exist, thereby enabling effective generalized Bayesian inference on the density ratio using backfitting samplers designed for BART. The resulting uncertainty quantification on the inferred density ratio is critical for applications involving high-dimensional and complex distributions in which uncertainty given limited data can often be substantial. We provide insights on the balancing loss through its close connection to the exponential loss in binary classification and to the variational form of f-divergence, in particular that of the squared Hellinger distance. Our numerical experiments demonstrate the accuracy of the proposed approach while providing unique capabilities in uncertainty quantification. We demonstrate the application of our method in a case study involving assessing the quality of generative models for microbiome compositional data.

Ensuring fairness in AI systems is critical, especially in high-stakes domains such as lending, hiring, and healthcare. This urgency is reflected in emerging global regulations that mandate fairness assessments and independent bias audits. However, procuring the necessary complete data for fairness testing remains a significant challenge. In industry settings, legal and privacy concerns restrict the collection of demographic data required to assess group disparities, and auditors face practical and cultural challenges in gaining access to data. In practice, data relevant for fairness testing is often split across separate sources: internal datasets held by institutions with predictive attributes, and external public datasets such as census data containing protected attributes, each providing only partial, marginal information. Our work seeks to leverage such available separate data to estimate model fairness when complete data is inaccessible. We propose utilising the available separate data to estimate a set of feasible joint distributions and then compute the set plausible fairness metrics. Through simulation and real experiments, we demonstrate that we can derive meaningful bounds on fairness metrics and obtain reliable estimates of the true metric. Our results demonstrate that this approach can serve as a practical and effective solution for fairness testing in real-world settings where access to complete data is restricted.

Robust information representation and its persistent maintenance are fundamental for higher cognitive functions. Existing models employ distinct neural mechanisms to separately address noise-resistant processing or information maintenance, yet a unified framework integrating both operations remains elusive -- a critical gap in understanding cortical computation. Here, we introduce a recurrent neural circuit that combines divisive normalization with self-excitation to achieve both robust encoding and stable retention of normalized inputs. Mathematical analysis shows that, for suitable parameter regimes, the system forms a continuous attractor with two key properties: (1) input-proportional stabilization during stimulus presentation; and (2) self-sustained memory states persisting after stimulus offset. We demonstrate the model's versatility in two canonical tasks: (a) noise-robust encoding in a random-dot kinematogram (RDK) paradigm; and (b) approximate Bayesian belief updating in a probabilistic Wisconsin Card Sorting Test (pWCST). This work establishes a unified mathematical framework that bridges noise suppression, working memory, and approximate Bayesian inference within a single cortical microcircuit, offering fresh insights into the brain's canonical computation and guiding the design of biologically plausible artificial neural architectures.

The dominant approach to generating from language models subject to some constraint is locally constrained decoding (LCD), incrementally sampling tokens at each time step such that the constraint is never violated. Typically, this is achieved through token masking: looping over the vocabulary and excluding non-conforming tokens. There are two important problems with this approach. (i) Evaluating the constraint on every token can be prohibitively expensive -- LM vocabularies often exceed $100,000$ tokens. (ii) LCD can distort the global distribution over strings, sampling tokens based only on local information, even if they lead down dead-end paths. This work introduces a new algorithm that addresses both these problems. First, to avoid evaluating a constraint on the full vocabulary at each step of generation, we propose an adaptive rejection sampling algorithm that typically requires orders of magnitude fewer constraint evaluations. Second, we show how this algorithm can be extended to produce low-variance, unbiased estimates of importance weights at a very small additional cost -- estimates that can be soundly used within previously proposed sequential Monte Carlo algorithms to correct for the myopic behavior of local constraint enforcement. Through extensive empirical evaluation in text-to-SQL, molecular synthesis, goal inference, pattern matching, and JSON domains, we show that our approach is superior to state-of-the-art baselines, supporting a broader class of constraints and improving both runtime and performance. Additional theoretical and empirical analyses show that our method's runtime efficiency is driven by its dynamic use of computation, scaling with the divergence between the unconstrained and constrained LM, and as a consequence, runtime improvements are greater for better models.

Molecular dynamics simulations (MDS) face challenges, including resource-heavy computations and the need to manually scan outputs to detect "interesting events," such as the formation and persistence of hydrogen bonds between atoms of different molecules. A critical research gap lies in identifying the underlying causes of hydrogen bond formation and separation -understanding which interactions or prior events contribute to their emergence over time. With this challenge in mind, we propose leveraging spatio-temporal data analytics and machine learning models to enhance the detection of these phenomena. In this paper, our approach is inspired by causal modeling and aims to identify the root cause variables of hydrogen bond formation and separation events. Specifically, we treat the separation of hydrogen bonds as an "intervention" occurring and represent the causal structure of the bonding and separation events in the MDS as graphical causal models. These causal models are built using a variational autoencoder-inspired architecture that enables us to infer causal relationships across samples with diverse underlying causal graphs while leveraging shared dynamic information. We further include a step to infer the root causes of changes in the joint distribution of the causal models. By constructing causal models that capture shifts in the conditional distributions of molecular interactions during bond formation or separation, this framework provides a novel perspective on root cause analysis in molecular dynamic systems. We validate the efficacy of our model empirically on the atomic trajectories that used MDS for chiral separation, demonstrating that we can predict many steps in the future and also find the variables driving the observed changes in the system.

Active inference is a formal approach to study cognition based on the notion that adaptive agents can be seen as engaging in a process of approximate Bayesian inference, via the minimisation of variational and expected free energies. Minimising the former provides an account of perceptual processes and learning as evidence accumulation, while minimising the latter describes how agents select their actions over time. In this way, adaptive agents are able to maximise the likelihood of preferred observations or states, given a generative model of the environment. In the literature, however, different strategies have been proposed to describe how agents can plan their future actions. While they all share the notion that some kind of expected free energy offers an appropriate way to score policies, sequences of actions, in terms of their desirability, there are different ways to consider the contribution of past motor experience to the agent's future behaviour. In some approaches, agents are assumed to know their own actions, and use such knowledge to better plan for the future. In other approaches, agents are unaware of their actions, and must infer their motor behaviour from recent observations in order to plan for the future. This difference reflects a standard point of departure in two leading frameworks in motor control based on the presence, or not, of an efference copy signal representing knowledge about an agent's own actions. In this work we compare the performances of action-aware and action-unaware agents in two navigations tasks, showing how action-unaware agents can achieve performances comparable to action-aware ones while at a severe disadvantage.

Objective: Sparse Bayesian learning provides an effective scheme to solve the high-dimensional problem in brain signal decoding. However, traditional assumptions regarding data distributions such as Gaussian and binomial are potentially inadequate to characterize the noisy signals of brain activity. Hence, this study aims to propose a robust sparse Bayesian learning framework to address noisy highdimensional brain activity decoding. Methods: Motivated by the commendable robustness of the minimum error entropy (MEE) criterion for handling complex data distributions, we proposed an MEE-based likelihood function to facilitate the accurate inference of sparse Bayesian learning in analyzing noisy brain datasets. Results: Our proposed approach was evaluated using two high-dimensional brain decoding tasks in regression and classification contexts, respectively. The experimental results showed that, our approach can realize superior decoding metrics and physiological patterns than the conventional and state-of-the-art methods. Conclusion: Utilizing the proposed MEE-based likelihood model, sparse Bayesian learning is empowered to simultaneously address the challenges of noise and high dimensionality in the brain decoding task. Significance: This work provides a powerful tool to realize robust brain decoding, advancing biomedical engineering applications such as brain-computer interface.

The Expectation Maximization (EM) algorithm is the default algorithm for inference in latent variable models. As in any other field of machine learning, applications of latent variable models to very large datasets makes the use of advanced parallel and distributed architectures mandatory. This paper introduces FedEM, which is the first extension of the EM algorithm to the federated learning context. FedEM is a new communication efficient method, which handles partial participation of local devices, and is robust to heterogeneous distributions of the datasets. To alleviate the communication bottleneck, FedEM compresses appropriately defined complete data sufficient statistics. We also develop and analyze an extension of FedEM to further incorporate a variance reduction scheme. In all cases, we derive finite-time complexity bounds for smooth non-convex problems. Numerical results are presented to support our theoretical findings, as well as an application to federated missing values imputation for biodiversity monitoring.

We believe this work brings fundamental insights into the related fields.