Understanding how individual metro usage evolves over multi-year horizons is essential for transit planning and passenger retention. However, existing approaches typically characterize mobility patterns as static clusters or short-term variability, leaving the lifecycle dynamics of transit participation underexplored. This study proposes a state-based lifecycle modeling framework that integrates Hidden Semi-Markov Models (HSMM) with discrete-time survival analysis to characterize the evolution of individual metro mobility. The HSMM infers latent mobility states with explicit duration distributions and a transition matrix governing regime changes, while the survival component models exit and re-entry events via state-dependent hazard functions conditioned on mobility-state trajectories and behavioral history. Applied to four years of smart card data from the Shanghai metro system (2021-2024), the framework enables the identification of interpretable mobility states, the characterization of transition dynamics, and the quantification of state-dependent exit and re-entry processes. The analysis reveals five robust mobility states with a directional transition hierarchy centered on an occasional-usage gateway state, and fundamentally different temporal mechanisms governing disengagement and return: exit hazard is state-dependent but duration-independent, whereas re-entry hazard decays sharply with inactivity length. These findings provide a methodological foundation for lifecycle-oriented mobility analysis and practical guidance for transit operators to identify at-risk users and time retention interventions.

This paper builds a hierarchy of explicit, non-asymptotic tail bounds for the supremum of the Kostlan--Shub--Smale (KSS) random field on the sphere, and applies it to two problems: Spiked Tensor PCA and the landscape of the spherical $k$-spin model. For Tensor PCA, we study the non-asymptotic statistical limits of estimating a rank-$R$ symmetric signal tensor of order~$k\ge 3$ and dimension~$d\ge 3$ from a single Gaussian observation at signal-to-noise ratio~$λ$, through the \emph{profile maximum likelihood estimator}, the MLE restricted to normalized rank-$R$ tensors of coherence at least~$κ$. Our analysis uses a single reduction: a deterministic geometric inequality (the Tube Method) and a rank-reduction step bound the estimation error by the supremum of the canonical KSS field, which the Kac--Rice formula turns into a Gaussian integral against the expected absolute characteristic polynomial of a shifted Gaussian Orthogonal Ensemble, controlled in turn by the four explicit tail bounds of our hierarchy (three from a Mehta--Fyodorov representation, one from a Ben Arous--Dembo--Guionnet large deviation). The same reduction yields two results, each with explicit constants. For estimation, a finite-$(k,d)$ error bound recovers the asymptotically optimal rate~$\sqrt{d\log k}$ of Perry, Wein and Bandeira, with explicit dependence on the rank~$R$ and the coherence~$κ$. For the landscape, a two-sided non-asymptotic bracketing of the annealed complexity of the spherical $k$-spin Hamiltonian recovers the Auffinger--Ben Arous--Černý complexity function in the high-dimensional limit.

Matrix factorization is a fundamental method in statistics and machine learning for inferring and summarizing structure in multivariate data. Modern data sets often come with "side information" of various forms (images, text, graphs) that can be leveraged to improve estimation of the underlying structure. However, existing methods that leverage side information are limited in the types of data they can incorporate, and they assume specific parametric models. Here, we introduce a novel method for this problem, covariate-moderated empirical Bayes matrix factorization (cEBMF).

With the advancing reasoning capabilities of Large Language Models (LLMs), they are increasingly employed for complex evaluation tasks, such as grading student responses, verifying factual claims, and comparing competing answers. Leveraging multiple LLMs as automated judges can enhance robustness and accuracy by aggregating diverse perspectives, yet existing approaches often rely on static and simple aggregation methods, such as majority voting, which may produce incorrect judgments despite correct individual assessments. We propose a novel multiagent debate framework where LLMs collaboratively reason and iteratively refine judgments, formalizing this process mathematically and proving its advantages over static ensembles. To ensure computational efficiency, we introduce a stability detection mechanism using a time-varying Beta-Binomial mixture model (a mixture of two Beta-Binomial distributions) that tracks judge consensus dynamics and applies adaptive stopping via Kolmogorov-Smirnov testing. Experiments across diverse benchmarks and models demonstrate significant improvements in judgment accuracy over majority voting while maintaining computational efficiency.

A fundamental challenge in probabilistic modeling is to balance expressivity and inference efficiency. Tractable probabilistic models (TPMs) aim to directly address this tradeoff by imposing constraints that guarantee efficient inference of certain queries while maintaining expressivity. In particular, probabilistic circuits (PCs) provide a unifying framework for many TPMs, by characterizing families of models as circuits satisfying different structural properties. Because the complexity of inference on PCs is a function of the circuit size, understanding the size requirements of different families of PCs is fundamental in mapping the trade-off between tractability and expressive efficiency. However, the study of expressive efficiency of circuits are often concerned with exact representations, which may not align with model learning, where we look to approximate the underlying data distribution closely by some distance measure.

Dynamic pricing algorithms typically assume continuous price variables, which may not reflect real-world scenarios where prices are often discrete. This paper demonstrates that leveraging discrete price information within a semi-parametric model can substantially improve performance, depending on the size of the support set of the price variable relative to the time horizon. Specifically, we propose a novel semi-parametric contextual dynamic pricing algorithm, namely BayesCoxCP, based on a Bayesian approach to the Cox proportional hazards model. Our theoretical analysis establishes high-probability regret bounds that adapt to the sparsity level γ, proving that our algorithm achieves a regret upper bound of eO(T(1+γ)/2 + dT) for γ < 1/3 and eO(T2/3 + dT) for γ 1/3, where γ represents the sparsity of the price grid relative to the time horizon T. Through numerical experiments, we demonstrate that our proposed algorithm significantly outperforms an existing method, particularly in scenarios with sparse discrete price points.

Estimating the uncertainty of responses from Large Language Models (LLMs) remains a critical challenge. While recent Bayesian methods have demonstrated effectiveness in quantifying uncertainty through low-rank weight updates, they typically require complex fine-tuning or post-training procedures. In this paper, we propose Training-Free Bayesianization (TFB), a simple yet theoretically grounded framework that efficiently transforms trained low-rank adapters into Bayesian ones without additional training. TFBsystematically searches for the maximally acceptable level of variance in the weight posterior, constrained within a family of low-rank isotropic Gaussian distributions. Our theoretical analysis shows that under mild conditions, this search process is equivalent to KL-regularized variational optimization, a generalized form of variational inference. Through comprehensive experiments, we show that TFB achieves superior uncertainty estimation and generalization compared to existing methods while eliminating the need for complex Bayesianization training procedures.

AlphaZero has achieved remarkable success in complex decision-making problems through self-play and neural network training. However, its self-play process remains inefficient due to limited exploration of high-uncertainty positions, the overlooked runner-up decisions in Monte Carlo Tree Search (MCTS), and high variance in value labels. To address these challenges, we propose and evaluate uncertainty-guided exploration by branching from high-uncertainty positions using our proposed Label Change Rate (LCR) metric, which is further refined by a Bayesian inference framework. Our proposed approach leverages runner-up MCTS decisions to create multiple variations, and ensembles value labels across these variations to reduce variance. We investigate three key design parameters for our branching strategy: where to branch, how many variations to branch, and which move to play in the new branch. Our empirical findings indicate that branching with 10 variations per game provides the best performance-exploration balance. Overall, our end-to-end results show an improved sample efficiency over the baseline by 58.5% on 9x9 Go in the early stage of training and by 47.3% on 19x19 Go in the late stage of training.

Generative models that maximize model likelihood have gained traction in many practical settings. Among them, perturbation-based approaches underpin many state-of-the-art likelihood estimation models, yet they often face slow convergence and limited theoretical understanding. In this paper, we derive a tighter likelihood bound for noise-driven models to improve both the accuracy and efficiency of maximum likelihood learning. Our key insight extends the classical Kullback-Leibler (KL) divergence-Fisher information relationship to arbitrary noise perturbations, going beyond the Gaussian assumption and enabling structured noise distributions. This formulation allows flexible use of randomized noise distributions that naturally account for sensor artifacts, quantization effects, and data distribution smoothing, while remaining compatible with standard diffusion training. Treating the diffusion process as a Gaussian channel, we further express the mismatched entropy between data and model, showing that the proposed objective upper-bounds the negative log-likelihood (NLL). In experiments, our models achieve competitive NLL on CIFAR-10 and state-of-the-art results on ImageNet across multiple resolutions, all without data augmentation, and the framework extends naturally to discrete data.

Expanding on neural operators, we propose a novel framework for stochastic process learning across arbitrary domains. In particular, we develop operator flow matching (OFM) for learning stochastic process priors on function spaces. OFM provides the probability density of the values of any collection of points and enables mathematically tractable functional regression at new points with mean and density estimation. Our method outperforms state-of-the-art models in stochastic process learning, functional regression, and prior learning.