In the era of Explainable Artificial Intelligence, there is a renewed focus on single trees for their ease of interpretation. This paper introduces Simultaneous Latent Budget Trees, a probabilistic machine learning framework for classification trees in the presence of a stratification factor such as a temporal, spatial, or demographic variable, acting as a control variable or potential confounder. Standard tree growth procedures are not designed to optimize a conditional split rule. A model-based split rule is proposed in which child nodes are interpreted as latent components of a simultaneous mixture model, such as the Simultaneous Latent Budget Model and its constrained versions, fitted to the parent node. Mixing parameters drive the observations, differently for each group, to the child nodes whereas latent budgets parameters update the response classes profile of each level of the control variable. Parameters are estimated by least squares considering a neural network perspective of the model. An informative tree structure can be interactively visualized with interpretation aids on the node and the paths, including visual pruning and decision tree selection procedure. Suitable measures are proposed to handle an unbalanced response class distribution. The proposed methodology is applied to investigate gender-related differences in disease progression of Amyotrophic Lateral Sclerosis. The SLBT library with the various tree-based algorithms is available in the linked GitHub repository.

Modeling and forecasting nonlinear dynamics under distribution shifts is essential for robust decision-making in real-world systems. In this work, we propose MetaKoopman, a Bayesian meta-learning framework for modeling nonlinear dynamics through linear latent representations. MetaKoopman learns a Matrix Normal-Inverse Wishart (MNIW) prior over the Koopman operator, enabling closed-form Bayesian updates conditioned on recent trajectory segments. Moreover, it provides a closed-form posterior predictive distribution over future state trajectories, capturing both epistemic and aleatoric uncertainty in the learned dynamics. We evaluate MetaKoopman on a full-scale autonomous truck and trailer system across a wide range of adverse winter scenarios--including snow, ice, and mixed-friction conditions--as well as in simulated control tasks with diverse distribution shifts. MetaKoopman consistently outperforms prior approaches in multi-step prediction accuracy, uncertainty calibration and robustness to distributional shifts. Field experiments further demonstrate its effectiveness in dynamically feasible motion planning, particularly during evasive maneuvers and operation at the limits of traction.

The Poisson Generalized Linear Model (GLM) is a foundational tool for analyzing neural spike train data. However, standard implementations rely on discretizing spike times into binned count data, limiting temporal resolution and scalability. Here, we develop Monte Carlo (MC) methods and polynomial approximations (PA) to the continuous-time analog of these models, and show them to be advantageous over their discrete-time counterparts. Further, we propose using a set of exponentially scaled Laguerre polynomials as an orthogonal temporal basis, which improves filter identification and yields closed-form integral solutions under the polynomial approximation. Applied to both synthetic and real spike-time data from rodent hippocampus, our methods demonstrate superior accuracy and scalability compared to traditional binned GLMs, enabling functional connectivity inference in large-scale neural recordings that are temporally precise on the order of synaptic dynamical timescales and in agreement with known anatomical properties of hippocampal subregions. We provide open-source implementations of both MC and PA estimators, optimized for GPU acceleration, to facilitate adoption in the neuroscience community1.

In standard autoregressive generation, an LLM predicts the next-token distribution, samples a discrete token, and then discards the distribution, passing only the sampled token as new input. To preserve this distribution's rich information, we propose Mixture of Inputs (MOI), a training-free method for autoregressive generation. After generating a token following the standard paradigm, we construct a new input that blends the generated discrete token with the previously discarded token distribution. Specifically, we employ a Bayesian estimation method that treats the token distribution as the prior, the sampled token as the observation, and replaces the conventional one-hot vector with the continuous posterior expectation as the new model input. MOI allows the model to maintain a richer internal representation throughout the generation process, resulting in improved text quality and reasoning capabilities. On mathematical reasoning, code generation, and PhDlevel QA tasks, MOI consistently improves performance across multiple models including QwQ-32B, Nemotron-Super-49B, Gemma-3-27B, and DAPO-Qwen32B, with no additional training and negligible computational overhead.

Planning under opponent uncertainty is a fundamental challenge in multi-agent environments, where an agent must act while inferring the hidden policies of its opponents. Existing type-based methods rely on manually defined behavior classes and struggle to scale, while model-free approaches are sample-inefficient and lack a principled way to incorporate uncertainty into planning. We propose Quantized Opponent Models (QOM), which learn a compact catalog of opponent types via a quantized autoencoder and maintain a Bayesian belief over these types online. This posterior supports both a belief-weighted meta-policy and a Monte-Carlo planning algorithm that directly integrates uncertainty, enabling real-time belief updates and focused exploration. Experiments show that QOM achieves superior performance with lower search cost, offering a tractable and effective solution for belief-aware planning.

Understanding representational similarity between neural recordings and computational models is essential for neuroscience, yet remains challenging to measure reliably due to the constraints on the number of neurons that can be recorded simultaneously. In this work, we apply tools from Random Matrix Theory to investigate how such limitations affect similarity measures, focusing on Centered Kernel Alignment (CKA) and Canonical Correlation Analysis (CCA). We propose an analytical framework for representational similarity analysis that relates measured similarities to the spectral properties of the underlying representations. We demonstrate that neural similarities are systematically underestimated under finite neuron sampling, mainly due to eigenvector delocalization. Moreover, for power-law population spectra, we show that the number of localized eigenvectors scales as the square root of the number of recorded neurons, providing a simple rule of thumb for practitioners. To overcome sampling bias, we introduce a denoising method to infer population-level similarity, enabling accurate analysis even with small neuron samples. Theoretical predictions are validated on synthetic and real datasets, offering practical strategies for interpreting neural data under finite sampling constraints.

Markov chain Monte Carlo (MCMC) methods are foundational algorithms for Bayesian inference and probabilistic modeling. However, most MCMC algorithms are inherently sequential and their time complexity scales linearly with the sequence length. Previous work on adapting MCMC to modern hardware has therefore focused on running many independent chains in parallel. Here, we take an alternative approach: we propose algorithms to evaluate MCMC samplers in parallel across the chain length. To do this, we build on recent methods for parallel evaluation of nonlinear recursions that formulate the state sequence as a solution to a fixed-point problem and solve for the fixed-point using a parallel form of Newton's method. We show how this approach can be used to parallelize Gibbs, Metropolis-adjusted Langevin, and Hamiltonian Monte Carlo sampling across the sequence length. In several examples, we demonstrate the simulation of up to hundreds of thousands of MCMC samples with only tens of parallel Newton iterations. Additionally, we develop two new parallel quasi-Newton methods to evaluate nonlinear recursions with lower memory costs and reduced runtime. We find that the proposed parallel algorithms accelerate MCMC sampling across multiple examples, in some cases by more than an order of magnitude compared to sequential evaluation.

Gene regulatory network inference (GRNI) aims to discover how genes causally regulate each other from gene expression data. It is well-known that statistical dependencies in observed data do not necessarily imply causation, as spurious dependencies may arise from latent confounders, such as non-coding RNAs. Numerous GRNI methods have thus been proposed to address this confounding issue. However, dependencies may also result from selection-only cells satisfying certain survival or inclusion criteria are observed-while these selection-induced spurious dependencies are frequently overlooked in gene expression data analyses. In this work, we show that such selection is ubiquitous and, when ignored or conflated with true regulations, can lead to flawed causal interpretation and misguided intervention recommendations. To address this challenge, a fundamental question arises: can we distinguish dependencies due to regulation, confounding, and crucially, selection? We show that gene perturbations offer a simple yet effective answer: selection-induced dependencies are symmetric under perturbation, while those from regulation or confounding are not. Building on this motivation, we propose GISL (Gene regulatory network Inference in the presence of Selection bias and Latent confounders), a principled algorithm that leverages perturbation data to uncover both true gene regulatory relations and non-regulatory mechanisms of selection and confounding up to the equivalence class. Experiments on synthetic and real-world gene expression data demonstrate the effectiveness of our method.

Shuffled regression is the problem of learning regression functions from shuffled data where the correspondence between the input features and target response is unknown. This paper proposes a probabilistic model for shuffled regression called Gaussian Process Shuffled Regression (GPSR). By introducing Gaussian processes as a prior of regression functions in function space via the kernel function, GPSR can express a wide variety of functions in a nonparametric manner while quantifying the uncertainty of the prediction. By adopting the Bayesian evidence maximization framework and a theoretical analysis of the connection between the marginal likelihood/predictive distribution of GPSR and that of standard Gaussian process regression (GPR), we derive an easy-to-implement inference algorithm for GPSR that iteratively applies GPR and updates the input-output correspondence. To reduce computation costs and obtain closed-form solutions for correspondence updates, we also develop a sparse approximate variant of GPSR using its weight space formulation, which can be seen as Bayesian shuffled linear regression with random Fourier features. Experiments on benchmark datasets confirm the effectiveness of our GPSR proposal.

In pattern classification, efficient uncertainty reasoning plays a critical role, particularly in real-time applications involving noisy data, ambiguous class boundaries, or overlapping categories. Leveraging the advanced computational power of quantum computing, an Adaptive Quantum Circuit for Dempster's Rule of Combination (AQC-DRC) is proposed to address efficient classification under uncertain environments. The AQC-DRC is developed within the framework of quantum evidence theory (QET) and facilitates decision-making based on quantum basic probability and plausibility levels, which is a generalized Bayesian inference method. The AQC-DRC provides a deterministic computation of DRC, ensuring that quantum fusion outcomes in uncertain pattern classification are exactly aligned with those of the classical method, while simultaneously achieving exponential reductions in the computational complexity of evidence combination and significantly improving fusion efficiency. It is founded that the quantum basic probability amplitude function in QET, as a generalized quantum probability amplitude, can be naturally utilized to express the quantum amplitude encoding. In addition, the quantum basic probability in QET, as a generalized quantum probability, naturally forms a quantum basic probability distribution and can be used to represent quantum measurement outcomes for quantum basic probability level decision-making. Furthermore, the quantum plausibility function in QET also can be naturally used to express the quantum measurement outcomes for quantum plausibility level decision-making. These findings enrich the physical understanding of quantum amplitude encoding and quantum measurement outcomes, offering broad application prospects for representing and processing uncertain knowledge in pattern classification.