Papers

We study distribution-free risk control for model routing, motivated by large language model reasoning. We formalize pointwise conditional efficiency under a probably approximately correct guarantee and show that it forces a nearly impossible router: at almost every input where the fast model exceeds the target loss, the algorithm must route to the expert with probability at least one minus the prescribed error level. We therefore introduce a restricted conditional formulation based on a prespecified family of conditioning sets, together with an explicit router. The proposed router achieves finite-sample conditional validity and, under separation and margin conditions, near-oracle expert usage. The main insight is that the level of conditioning determines whether distribution-free reliability can coexist with computational savings: pointwise control is too strong, whereas structured setwise control remains feasible.

Research increasingly relies on computational methods to analyze experimental data and predict molecular properties. Current approaches often require researchers to use a variety of tools for statistical analysis and machine learning, creating workflow inefficiencies. We present an integrated platform that combines classical statistical methods with Random Forest classification for comprehensive data analysis that can be used in the biological sciences. The platform implements automated hyperparameter optimization, feature importance analysis, and a suite of statistical tests including t tests, ANOVA, and Pearson correlation analysis. Our methodology addresses the gap between traditional statistical software, modern machine learning frameworks and biology, by providing a unified interface accessible to researchers without extensive programming experience. The system achieves this through automatic data preprocessing, categorical encoding, and adaptive model configuration based on dataset characteristics. Initial testing protocols are designed to evaluate classification accuracy across diverse chemical datasets with varying feature distributions. This work demonstrates that integrating statistical rigor with machine learning interpretability can accelerate biological discovery workflows while maintaining methodological soundness. The platform's modular architecture enables future extensions to additional machine learning algorithms and statistical procedures relevant to bioinformatics.

This paper develops a spatial-dynamic framework to analyze the theoretical and quantitative effects of a Land Value Tax (LVT) on urban land markets, capital accumulation, and spatial redistribution. Building upon the Georgist distinction between produced value and unearned rent, the model departs from the static equilibrium tradition by introducing an explicit diffusion process for land values and a local investment dynamic governed by profitability thresholds. Land value $V (x, y, t)$ and built capital $K(x, y, t)$evolve over a two-dimensional urban domain according to coupled nonlinear partial differential equations, incorporating local productivity $A(x, y)$, centrality effects $\mu(x, y)$, depreciation $\delta$, and fiscal pressure $\tau$ . Analytical characterization of the steady states reveals a transcritical bifurcation in the parameter $\tau$ , separating inactive (low-investment) and active (self-sustaining) spatial regimes. The equilibrium pair $(V ^*, K^*)$ is shown to exist only when the effective decay rate $\alpha = r + \tau - \mu(x, y)$ exceeds a profitability threshold $\theta = \kappa + \delta / I_0$, and becomes locally unstable beyond this boundary. The introduction of diffusion, $D_V \Delta V$, stabilizes spatial dynamics and generates continuous gradients of land value and capital intensity, mitigating speculative clustering while preserving productive incentives. Numerical simulations confirm these analytical properties and display the emergence of spatially heterogeneous steady states driven by urban centrality and local productivity. The model also quantifies key aggregate outcomes, including dynamic tax revenues, adjusted capital-to-land ratios, and net present values under spatial heterogeneity and temporal discounting. Sensitivity analyses demonstrate that the main qualitative mechanisms-critical activation, spatial recomposition, and bifurcation structure-remain robust under alternative spatial profiles $(A, \mu)$, discretization schemes, and moderate differentiation of the tax rate $\tau (x, y)$. From an economic perspective, the results clarify the dual nature of the LVT: while it erodes unproductive rents and speculative land holding, its dynamic incidence on built capital depends on local profitability and financing constraints. The taxation parameter $\tau$ thus acts as a control variable in a nonlinear spatial system, shaping transitions between rent-driven and production-driven equilibria. Within a critical range around $\tau_c$, the LVT functions as an efficient spatial reallocation operator-reducing inequality in land values and investment density without impairing aggregate productivity. Beyond this range, excessive taxation induces systemic contraction and investment stagnation. Overall, this research bridges static urban tax theory with dynamic spatial economics by formalizing how a land-based fiscal instrument can reshape the geography of value creation through endogenous diffusion and nonlinear feedback. The framework provides a foundation for future extensions involving stochastic shocks, adaptive policy feedbacks, or endogenous public investment, offering a unified quantitative perspective on the dynamic efficiency and spatial equity of land value taxation.

Recent theoretical work established the unsupervised identifiability of quantized factors under any diffeomorphism. The theory assumes that quantization thresholds correspond to axis-aligned discontinuities in the probability density of the latent factors. By constraining a learned map to have a density with axis-aligned discontinuities, we can recover the quantization of the factors. However, translating this high-level principle into an effective practical criterion remains challenging, especially under nonlinear maps. Here, we develop a criterion for unsupervised disentanglement by encouraging axis-aligned discontinuities. Discontinuities manifest as sharp changes in the estimated density of factors and form what we call cliffs. Following the definition of independent discontinuities from the theory, we encourage the location of the cliffs along a factor to be independent of the values of the other factors. We show that our method, Cliff, outperforms the baselines on all disentanglement benchmarks, demonstrating its effectiveness in unsupervised disentanglement.

Bogdan et al. established a new criterion to determine the existence of a maximum likelihood estimator in discrete exponential families. It uses the notion of the set of uniqueness, which allows one to apply the problem to the Ising model from statistical mechanics. We propose a full characterization of the existence of the MLE for the Ising model among the level sets used in related combinatorial problems. We then establish new bounds for the size of the smallest set of uniqueness for the products of Rademacher functions.

We study nonparametric regression with covariates $X$ and outcome $Y$ under random unbiased perturbations (RUPs) of the conditional distribution $Y|X$, where the marginal distribution of covariates, $P^X$, remains fixed but the conditional law, $P^{Y|X}$, varies randomly across datasets. Unlike adversarial distribution shift frameworks that yield conservative worst-case guarantees, RUPs induce dataset-level variance inflation rather than systematic bias. We provide examples of RUPs and show that this distributional uncertainty reduces the effective sample size to $n_{\mathrm{eff}} = n/(1 + n \tau)$, where $\tau\in [0,1]$ quantifies the perturbation strength. For local polynomial estimators, we derive an extended bias-variance decomposition that includes a distributional variance term with the same bandwidth scaling as classical sampling variance. This leads to a modified bandwidth selection principle: when distributional uncertainty dominates sampling uncertainty ($\tau \gg 1/n$), optimal bandwidths scale as $\tau^{1/(2\beta+1)}$ rather than the usual $n^{-1/(2\beta+1)}$, where $\beta$ indicates the smoothness of the function class considered. We also establish matching minimax lower bounds showing that there exists an RUP for which this effective sample size $n_{\mathrm{eff}}$ is fundamental. Our results demonstrate that random dataset-level perturbations create a distinct mode of uncertainty that affects both practical tuning and fundamental statistical limits.

We study streaming data with categorical features where the vocabulary of categorical feature values is changing and can even grow unboundedly over time. Feature hashing is commonly used as a pre-processing step to map these categorical values into a feature space of fixed size before learning their embeddings. While these methods have been developed and evaluated for offline or batch settings, in this paper we consider online settings. We show that deterministic embeddings are sensitive to the arrival order of categories and suffer from forgetting in online learning, leading to performance deterioration. To mitigate this issue, we propose a probabilistic hash embedding (PHE) model that treats hash embeddings as stochastic and applies Bayesian online learning to learn incrementally from data. Based on the structure of PHE, we derive a scalable inference algorithm to learn model parameters and infer/update the posteriors of hash embeddings and other latent variables. Our algorithm (i) can handle an evolving vocabulary of categorical items, (ii) is adaptive to new items without forgetting old items, (iii) is implementable with a bounded set of parameters that does not grow with the number of distinct observed values on the stream, and (iv) is invariant to the item arrival order. Experiments in classification, sequence modeling, and recommendation systems in online learning setups demonstrate the superior performance of PHE while maintaining high memory efficiency (consumes as low as 2~4 memory of a one-hot embedding table). Supplementary materials are at https://github.com/aodongli/probabilistic-hash-embeddings

This paper argues that DNNs implement a computational Occam's razor – finding the `simplest' algorithm that fits the data – and that this could explain their incredible and wide-ranging success over more traditional statistical methods. We start with the discovery that the set of real-valued function $f$ that can be $\epsilon$-approximated with a binary circuit of size at most $c\epsilon^{-\gamma}$ becomes convex in the `Harder than Monte Carlo' (HTMC) regime, when $\gamma>2$, allowing for the definition of a HTMC norm on functions. In parallel one can define a complexity measure on the parameters of a ResNets (a weighted $\ell_1$ norm of the parameters), which induce a `ResNet norm' on functions. The HTMC and ResNet norms can then be related by an almost matching sandwich bound. Thus minimizing this ResNet norm is equivalent to finding a circuit that fits the data with an almost minimal number of nodes (within a power of 2 of being optimal). ResNets thus appear as an alternative model for computation of real functions, better adapted to the HTMC regime and its convexity.

Across many disciplines, causal inference often relies on randomized experiments with binary outcomes. In such experiments, often analysts are interested in testing Fisher's sharp null hypothesis, i.e., the treatment has zero effect on the outcome for every study subject. Sometimes the outcomes are sensitive and must be kept confidential, for example, when they comprise physical or mental health measurements. Releasing test statistics or p-values computed with the confidential outcomes can leak information about the individuals in the study. Those responsible for sharing the analysis results may wish to bound this information leakage, which they can do by ensuring the released outputs satisfy differential privacy. In this article, we develop several differentially private tests of Fisher's sharp null for binary outcomes. Specifically, we consider direct perturbation approaches that inject calibrated noise into test statistics or p-values, as well as a Bayesian denoising framework that explicitly models the privacy mechanism. We further develop decision-making procedures under privacy constraints, including a Bayes risk-optimal rule and a frequentist-calibrated significance test. Through theoretical results, simulation studies, and an application to the ADAPTABLE clinical trial, we demonstrate that our methods can achieve valid and interpretable causal inference while ensuring the differential privacy guarantee.

Vertical Federated Learning (VFL) often suffers from client-wise missingness, where entire feature blocks from some clients are unobserved, and conventional approaches are vulnerable to privacy leakage. We propose a Gaussian copulabased framework for VFL data privatization under missingness constraints, which requires no prior specification of downstream analysis tasks and imposes no restriction on the number of analyses. To privately estimate copula parameters, we introduce a debiased randomized response mechanism for correlation matrix estimation from perturbed ranks, together with a nonparametric privatized marginal estimation that yields consistent CDFs even under MAR. The proposed methods comprise VCDS for MCAR data, EVCDS for MAR data, and IEVCDS, which iteratively refines copula parameters to mitigate MAR-induced bias. Notably, EVCDS and IEVCDS also apply under MCAR, and the framework accommodates mixed data types, including discrete variables. Theoretically, we introduce the notion of Vertical Distributed Attribute Differential Privacy (VDADP), tailored to the VFL setting, establish corresponding privacy and utility guarantees, and investigate the utility of privatized data for GLM coefficient estimation and variable selection. We further establish asymptotic properties including estimation and variable selection consistency for VFL-GLMs. Extensive simulations and a real-data application demonstrate the effectiveness of the proposed framework.

State resetting is a fundamental but often overlooked capability of simulators. It supports sample-based planning by allowing resets to previously encountered simulation states, and enables calibration of simulators using real data by resetting to states observed in real-system traces. While often taken for granted, state resetting in complex simulators can be nontrivial: when the simulator comes with latent variables (states), state resetting requires sampling from the posterior over the latent state given the observable history, a.k.a. the belief state (Silver and Veness, 2010). While exact sampling is often infeasible, many approximate belief-state samplers can be constructed, raising the question of how to select among them using only sampling access to the simulator. In this paper, we show that this problem reduces to a general conditional distribution-selection task and develop a new algorithm and analysis under sampling-only access. Building on this reduction, the belief-state selection problem admits two different formulations: latent state-based selection, which directly targets the conditional distribution of the latent state, and observation-based selection, which targets the induced distribution over the observation. Interestingly, these formulations differ in how their guarantees interact with the downstream roll-out methods: perhaps surprisingly, observation-based selection may fail under the most natural roll-out method (which we call Single-Reset) but enjoys guarantees under the less conventional alternative (which we call Repeated-Reset). Together with discussion on issues such as distribution shift and the choice of sampling policies, our paper reveals a rich landscape of algorithmic choices, theoretical nuances, and open questions, in this seemingly simple problem.

Closure modeling - the statistical modeling of missing dynamics in the natural sciences and engineering - is a growing and active area of research. Existing methods for closure modeling are often computationally prohibitive, lack uncertainty quantification, or require noise-free observations of the temporal derivatives over the system state. We propose a novel, computationally efficient approach for the modeling and estimation of closure terms over the spatiotemporal domain that provides uncertainty quantification and is effective even when the observations of the system state are sparse or contain moderate levels of noise. The efficacy of our approach is demonstrated in both one and two spatial dimensions through numerical experiments using the Fisher-KPP reaction-diffusion equation and the advection-diffusion equation as exemplars.

Feature selection remains difficult in modern high-dimensional settings, and established methods such as Boruta and Recursive Feature Elimination are either computationally costly or lack a statistically justified stopping criterion for their importance scores. A common heuristic adds random noise features and retains any predictor ranking above the strongest one, but this rule is purely ad hoc. We introduce a method that keeps the noise-augmentation idea while grounding it in theory: each feature's importance is tested against the maximum noise importance using a non-parametric bootstrap hypothesis test, with statistical derivations supporting the algorithm's design. On controlled simulations, the method recovers true signal more consistently than Boruta and Knockoff-based procedures; on diverse real-world datasets, it outperforms Boruta, RFE, and Extra Trees. The result is a robust, principled selector that yields reliable inference, improved prediction, and efficient computation.

In incomplete financial markets, pricing and hedging European options lack a unique no-arbitrage solution due to unhedgeable risks. This paper introduces a constrained deep learning approach to determine option prices and hedging strategies that minimize the Profit and Loss (P&L) distribution around zero. We employ a single neural network to represent the option price function, with its gradient serving as the hedging strategy, optimized via a loss function enforcing the self-financing portfolio condition. A key challenge arises from the non-smooth nature of option payoffs (e.g., vanilla calls are non-differentiable at-the-money, while digital options are discontinuous), which conflicts with the inherent smoothness of standard neural networks. To address this, we compare unconstrained networks against constrained architectures that explicitly embed the terminal payoff condition, drawing inspiration from PDE-solving techniques. Our framework assumes two tradable assets: the underlying and a liquid call option capturing volatility dynamics. Numerical experiments evaluate the method on simple options with varying non-smoothness, the exotic Equinox option, and scenarios with market jumps for robustness. Results demonstrate superior P&L distributions, highlighting the efficacy of constrained networks in handling realistic payoffs. This work advances machine learning applications in quantitative finance by integrating boundary constraints, offering a practical tool for pricing and hedging in incomplete markets.

In cluster-randomized trials (CRTs), entire clusters of individuals are randomized to treatment, and outcomes within a cluster are typically correlated. While frequentist approaches are standard practice for CRT analysis, Bayesian methods have emerged as a strong alternative. Previous work has investigated the use of Bayesian hierarchical models for continuous, binary, and count outcomes in CRTs, but these approaches focus on model-based treatment effect coefficients as the target estimands, which may have ambiguous interpretation under model misspecification and informative cluster size. In this article, we introduce a calibrated Bayesian procedure for estimand-aligned analysis of CRTs even in the presence of potentially misspecified models. We propose estimators targeting both the cluster-average treatment effect (cluster-ATE) and individual-average treatment effect (individual-ATE), particularly in scenarios with informative cluster sizes. We additionally explore strategies for summarizing the posterior samples that can achieve the frequentist coverage even under working model misspecification. We provide simulation evidence to demonstrate the model-robustness property of the proposed estimators in CRTs, and further investigate the impact of covariate adjustment as well as the use of more flexible Bayesian nonparametric working models in the CRT context.

Inferring who infected whom in an outbreak is essential for characterising transmission dynamics and guiding public health interventions. However, this task is challenging due to limited surveillance data and the complexity of immunological and social interactions. Instead of a single definitive transmission tree, epidemiologists often consider multiple plausible trees forming epidemic forests. Various inference methods and assumptions can yield different epidemic forests, yet no formal test exists to assess whether these differences are statistically significant. We propose such a framework using a chi-square test and permutational multivariate analysis of variance (PERMANOVA). We assessed each method's ability to distinguish simulated epidemic forests generated under different offspring distributions. While both methods achieved perfect specificity for forests with 100+ trees, PERMANOVA consistently outperformed the chi-square test in sensitivity across all epidemic and forest sizes. Implemented in the R package mixtree, we provide the first statistical framework to robustly compare epidemic forests.

We develop a data-driven approach for runtime safety monitoring in flight testing, where pilots perform maneuvers on aircraft with uncertain parameters. Because safety violations can arise unexpectedly as a result of these uncertainties, pilots need clear, preemptive criteria to abort the maneuver in advance of safety violation. To solve this problem, we use offline stochastic trajectory simulation to learn a calibrated statistical model of the short-term safety risk facing pilots. We use flight testing as a motivating example for data-driven learning/monitoring of safety due to its inherent safety risk, uncertainty, and human-interaction. However, our approach consists of three broadly-applicable components: a model to predict future state from recent observations, a nearest neighbor model to classify the safety of the predicted state, and classifier calibration via conformal prediction. We evaluate our method on a flight dynamics model with uncertain parameters, demonstrating its ability to reliably identify unsafe scenarios, match theoretical guarantees, and outperform baseline approaches in preemptive classification of risk.

This short note contains a simple argument that allows us to go from fixed-time to any-time bounds for the concentration of matrix products. The result presented here is motivated by the analysis of Oja's algorithms.

Time-to-event models are commonly used to study associations between risk factors and disease outcomes in the setting of electronic health records (EHR). In recent years, focus has intensified on social determinants of health, highlighting the need for methods that account for patients' locations. We propose a Bayesian approach for introducing point-referenced spatial effects into a competing risks proportional hazards model. Our method leverages Gaussian process (GP) priors for spatially varying intercept and slope. To improve computational efficiency under a large number of spatial locations, we implemented a Hilbert space low-rank approximation of the GP. We modeled the baseline hazard curves as piecewise constant, and introduced a novel multiplicative gamma process prior to induce shrinkage and smoothing. A loss-based clustering method was then used on the spatial random effects to identify high-risk regions. We demonstrate the utility of this method through simulation and a real-world analysis of EHR data from Duke Hospital to study readmission risk of elderly patients with upper extremity fractures. Our results showed that the proposed method improved inference efficiency and provided valuable insights for downstream policy decisions.

We study the optimization of functions with $n>2$ arguments that have a representation as a sum of several functions that have only $2$ of the $n$ arguments each, termed sums of bivariates, on finite domains. The complexity of optimizing sums of bivariates is shown to be NP-equivalent and it is shown that there exists free lunch in the optimization of sums of bivariates. Based on measure-valued extensions of the objective function, so-called relaxations, $\ell^2$-approximation, and entropy-regularization, we derive several tractable problem formulations solvable with linear programming, coordinate ascent as well as with closed-form solutions. The limits of applying tractable versions of such relaxations to sums of bivariates are investigated using general results for reconstructing measures from their bivariate marginals. Experiments in which the derived algorithms are applied to random functions, vertex coloring, and signal reconstruction problems provide insights into qualitatively different function classes that can be modeled as sums of bivariates.

We introduce and analyse active learning markets as a way to purchase labels, in situations where analysts aim to acquire additional data to improve model fitting, or to better train models for predictive analytics applications. This comes in contrast to the many proposals that already exist to purchase features and examples. By originally formalising the market clearing as an optimisation problem, we integrate budget constraints and improvement thresholds into the label acquisition process. We focus on a single-buyer-multiple-seller setup and propose the use of two active learning strategies (variance based and query-by-committee based), paired with distinct pricing mechanisms. They are compared to benchmark baselines including random sampling and a greedy knapsack heuristic. The proposed strategies are validated on real-world datasets from two critical application domains: real estate pricing and energy forecasting. Results demonstrate the robustness of our approach, consistently achieving superior performance with fewer labels acquired compared to conventional methods. Our proposal comprises an easy-to-implement practical solution for optimising data acquisition in resource-constrained environments.

Variational Bayes methods are popular due to their computational efficiency and adaptability to diverse applications. In specifying the variational family, mean-field classes are commonly used, which enables efficient algorithms such as coordinate ascent variational inference (CAVI) but fails to capture parameter dependence and typically underestimates uncertainty. In this work, we introduce a variational bagging approach that integrates a bagging procedure with variational Bayes, resulting in a bagged variational posterior for improved inference. We establish strong theoretical guarantees, including posterior contraction rates for general models and a Bernstein-von Mises (BVM) type theorem that ensures valid uncertainty quantification. Notably, our results show that even when using a mean-field variational family, our approach can recover off-diagonal elements of the limiting covariance structure and provide proper uncertainty quantification. In addition, variational bagging is robust to model misspecification, with covariance structures matching those of the target covariance. We illustrate our variational bagging method in numerical studies through applications to parametric models, finite mixture models, deep neural networks, and variational autoencoders (VAEs).

In this paper, we design $MC^2$ algorithms for Mixed Integer and Linear Programming. By expressing a constrained optimisation as one of simulation from a Boltzmann distribution, we reformulate integer and linear programming as Monte Carlo optimisation problems. The key insight is that solving these optimisation problems requires the ability to simulate from truncated distributions, namely multivariate exponentials and Gaussians. Efficient simulation can be achieved using the algorithms of Kent and Davis. We demonstrate our methodology on portfolio optimisation and the classical farmer problem from stochastic programming. Finally, we conclude with directions for future research.

The abundance of fine-grained spatio-temporal data, such as traffic sensor networks, offers vast opportunities for scientific discovery. However, inferring causal relationships from such observational data remains challenging, particularly due to unobserved confounders that are specific to units (e.g., geographical locations) yet influence outcomes over time. Most existing methods for spatio-temporal causal inference assume that all confounders are observed, an assumption that is often violated in practice. In this paper, we introduce Spatio-Temporal Hierarchical Causal Models (ST-HCMs), a novel graphical framework that extends hierarchical causal modeling to the spatio-temporal domain. At the core of our approach is the Spatio-Temporal Collapse Theorem, which shows that a complex ST-HCM converges to a simpler flat causal model as the amount of subunit data increases. This theoretical result enables a general procedure for causal identification, allowing ST-HCMs to recover causal effects even in the presence of unobserved, time-invariant unit-level confounders, a scenario where standard non-hierarchical models fail. We validate the effectiveness of our framework on both synthetic and real-world datasets, demonstrating its potential for robust causal inference in complex dynamic systems.

Generative modeling is typically framed as learning mapping rules, but from an observer's perspective without access to these rules, the task becomes disentangling the geometric support from the probability distribution. We propose that continuum percolation is uniquely suited to this support analysis, as the sampling process effectively projects high-dimensional density estimation onto a geometric counting problem on the support. In this work, we establish a rigorous correspondence between the topological phase transitions of random geometric graphs and the underlying data manifold in high-dimensional space. By analyzing the relationship between our proposed Percolation Shift metric and FID, we show that this metric captures structural pathologies, such as implicit mode collapse, where standard statistical metrics fail. Finally, we translate this topological phenomenon into a differentiable loss function that guides training. Experimental results confirm that this approach not only prevents manifold shrinkage but also fosters a form of synergistic improvement, where topological stability becomes a prerequisite for sustained high fidelity in both static generation and sequential decision making.

This study investigates the determinants of formal reporting to law enforcement among victims undertaking a pathway out of violence in Anti-Violence Centers (AVCs) of the Apulia region (Southern Italy). Micro-level data are obtained by integrating the official survey on AVCs conducted by the Italian National Institute of Statistics with the regional administrative monitoring on their service users. We employ a Bayesian multilevel logistic regression model with municipality-level random effects to explore individual and contextual factors impacting on the reporting outcome while accounting for the hierarchical data structure. At individual level, the analysis reveals significant differences in formal reporting depending on the relationship with the perpetrator, the nature and severity of violence, the presence of minor children and age. Among the contextual variables, the study highlights the role of specific AVC service provison, of territorial anti-violence networks as well as geographical disparities among provinces and major municipalities. This work adds on the value of integrating multiple data sources for an effective characterization and understanding of the complex and delicate dynamics of formal reporting to police by women victims of violence and informs evidence-based policies to concretely strengthen targeted anti-violence interventions at individual and territorial levels.ling in capturing reporting dynamics and informing policy interventions.

Understanding complex dependencies and extrapolating beyond observations are key challenges in modeling environmental space-time extremes. To address this, we introduce a simplifying approach that projects a wide range of multivariate exceedance problems onto a univariate peaks-over-threshold problem. In this framework, an estimator is computed by minimizing the $L_2$-distance between the empirical distribution function of the data and the theoretical distribution of the model. Asymptotic properties of this estimator are derived and validated in a simulation study. We evaluated our estimator in the EVA (2025) conference Data Challenge as part of Team Bochum's submission. The challenge provided precipitation data from four runs of LENS2, an ensemble of long-term weather simulations, on a $5 \times 5$ grid of locations centered at the grid point closest to Asheville, NC. Our estimator achieved a top-three rank in two of six competitive categories and won the overall preliminary challenge against ten competing teams.

Modeling nonlinear systems with Volterra series is challenging because the number of kernel coefficients grows exponentially with the model order. This work introduces Bayesian Tensor Network Volterra kernel machines (BTN-V), extending the Bayesian Tensor Network framework to Volterra system identification. BTN-V represents Volterra kernels using canonical polyadic decomposition, reducing model complexity from O(I^D) to O(DIR). By treating all tensor components and hyperparameters as random variables, BTN-V provides predictive uncertainty estimation at no additional computational cost. Sparsity-inducing hierarchical priors enable automatic rank determination and the learning of fading-memory behavior directly from data, improving interpretability and preventing overfitting. Empirical results demonstrate competitive accuracy, enhanced uncertainty quantification, and reduced computational cost.

We propose a method to approximate continuous-time, continuous-state stochastic processes by a discrete-time Markov chain defined on a nonuniform grid. Our method provides exact moment matching for processes whose first and second moments are linear functions of time. In particular, we show that, under certain conditions, the transition probabilities of a Markov chain can be chosen so that its first two moments match prescribed linear functions of time. These conditions depend on the grid points of the Markov chain and the coefficients of the linear mean and variance functions. Our proof relies on two recurrence relations for the expectation and variance across time. This approach enables simulation-based numerical analysis of continuous processes while preserving their key characteristics. We illustrate its efficacy by approximating continuous processes describing heat diffusion and geometric Brownian motion (GBM). For heat diffusion, we show that the heat profile at a set of points can be investigated by embedding those points inside the nonuniform grid of our Markov chain. For GBM, numerical simulations demonstrate that our approach, combined with suitable nonuniform grids, yields accurate approximations, with consistently small empirical Wasserstein-1 distances at long time horizons.

Decision-focused learning (DFL) trains predictive models to optimize downstream decisions rather than prediction accuracy alone. While recent studies have extended this paradigm to online settings with streaming data, existing online DFL methods generally maintain a point estimate, while their gradient-based updates require either a differentiable optimization layer or a problem-specific surrogate loss. Consequently, they can be unstable under limited data and difficult to apply across heterogeneous optimization problems. We introduce Bayesian Online Contextual Optimization (BOCO), a framework that maintains a decision-focused posterior over model parameters. BOCO aggregates the resulting predictions when prescribing decisions, thereby accounting for parameter uncertainty. To track this posterior in evolving environments, we develop two particle-based inference algorithms: a sequential Monte Carlo sampler for nondifferentiable problems and a function-space Stein variational gradient descent algorithm for differentiable problems. Across both real-world tasks, BOCO reduces running mean regret and temporal regret variability relative to two frequentist online DFL baselines. At the full horizon, relative to the best frequentist baseline, BOCO reduces running mean regret and temporal regret variability by 6.9% and 6.3% on knapsack and by 49.3% and 38.1% on energy scheduling, respectively. The gains are larger early in the data stream, a pattern consistent with a benefit from accounting for parameter uncertainty when data are limited.