Papers

We show that the cumulative distribution function corresponding to a kernel density estimator with optimal bandwidth lies outside any confidence interval, around the empirical distribution function, with probability tending to 1 as the sample size increases.

Grokking occurs when a model achieves high training accuracy but generalization to unseen test points happens long after that. This phenomenon was initially observed on a class of algebraic problems, such as learning modular arithmetic (Power et al., 2022). We study grokking on algebraic tasks in a class of feature learning kernels via the Recursive Feature Machine (RFM) algorithm (Radhakrishnan et al., 2024), which iteratively updates feature matrices through the Average Gradient Outer Product (AGOP) of an estimator in order to learn task-relevant features. Our main experimental finding is that generalization occurs only when a certain symmetry in the training set is broken. Furthermore, we empirically show that RFM generalizes by recovering the underlying invariance group action inherent in the data. We find that the learned feature matrices encode specific elements of the invariance group, explaining the dependence of generalization on symmetry.

Recent advances in generative networks have enabled new approaches to subsurface velocity model synthesis, offering a compelling alternative to traditional methods such as Full Waveform Inversion. However, these approaches predominantly rely on the availability of large-scale datasets of high-quality, geologically realistic subsurface velocity models, which are often difficult to obtain in practice. We introduce SAGE, a novel framework for statistically consistent proxy velocity generation from incomplete observations, specifically sparse well logs and migrated seismic images. During training, SAGE learns a proxy posterior over velocity models conditioned on both modalities (wells and seismic); at inference, it produces full-resolution velocity fields conditioned solely on migrated images, with well information implicitly encoded in the learned distribution. This enables the generation of geologically plausible and statistically accurate velocity realizations. We validate SAGE on both synthetic and field datasets, demonstrating its ability to capture complex subsurface variability under limited observational constraints. Furthermore, samples drawn from the learned proxy distribution can be leveraged to train downstream networks, supporting inversion workflows. Overall, SAGE provides a scalable and data-efficient pathway toward learning geological proxy posterior for seismic imaging and inversion. Repo link: https://github.com/slimgroup/SAGE.

Deep generative models for tabular data (GANs, diffusion models, and LLM-based generators) exhibit highly non-uniform behavior across datasets; the best-performing synthesizer family depends strongly on distributional stressors such as long-tailed marginals, high-cardinality categorical, Zipfian imbalance, and small-sample regimes. This brittleness makes practical deployment challenging, especially when users must balance competing objectives of fidelity, privacy, and utility. We study {intent-conditioned tabular synthesis selection}: given a dataset and a user intent expressed as a preference over evaluation metrics, the goal is to select a synthesizer that minimizes regret relative to an intent-specific oracle. We propose {stress profiling}, a synthesis-specific meta-feature representation that quantifies dataset difficulty along four interpretable stress dimensions, and integrate it into {SYNTHONY}, a selection framework that matches stress profiles against a calibrated capability registry of synthesizer families. Across a benchmark of 7 datasets, 10 synthesizers, and 3 intents, we demonstrate that stress-based meta-features are highly predictive of synthesizer performance: a $k$NN selector using these features achieves strong Top-1 selection accuracy, substantially outperforming zero-shot LLM selectors and random baselines. We analyze the gap between meta-feature-based and capability-based selection, identifying the hand-crafted capability registry as the primary bottleneck and motivating learned capability representations as a direction for future work.

Neuron-level firing data is believed to be governed by latent activation patterns during task completion. Analysing repeated trials of a task allows us to study these patterns, typically by averaging in-vivo neural spikes across trials. However, estimates of underlying latent activation patterns show trial-to-trial variability. Our aim is to determine whether this variation arises from observed data differences or changes in the latent activation patterns themselves. The latter would imply that current approaches overlook meaningful activation changes, necessitating adjustments in dimension reduction and downstream analysis. We propose a test that compares the eigendecompositions of two samples of functional data based on the covariance matrix of scores derived from a functional principal component analysis of the pooled data. Initially developed for independent samples, we later extend the test to paired samples, as necessary for our data. Simulation studies demonstrate its superior power compared to leading methods across various scenarios. In an experiment with 157 trials, we analyse all pairwise comparisons using a permutation approach to test the null hypothesis of shared latent activation patterns across trials. Our findings reveal trial-to-trial variation in latent activation patterns that cannot be attributed to sampling noise.

We consider Targeted Maximum Likelihood Estimation (TMLE) of weighted average treatment effects (WATEs), a class of causal estimands that reweight the covariate distribution using a specified function of the propensity score. This class includes the average treatment effect and average treatment effect on the treated, as well as various overlap-based targets. We provide a comprehensive analysis of the one-step TMLE along the universal least favorable path for such parameters. Under explicit regularity conditions on the weight function and initialization, we show that the targeting procedure is well-defined, reaches a solution of the estimating equation in finite time, and yields an asymptotically efficient estimator. In particular, convergence of the targeting dynamics and control of the second-order remainder are derived from these conditions rather than imposed as separate assumptions on the output of the algorithm.

This paper extends the Acemoglu-Restrepo task exposure framework to address the labor market effects of agentic artificial intelligence systems: autonomous AI agents capable of completing entire occupational workflows rather than discrete tasks. Unlike prior automation technologies that substitute for individual subtasks, agentic AI systems execute end-to-end workflows involving multi-step reasoning, tool invocation, and autonomous decision-making, substantially expanding occupational displacement risk beyond what existing task-level analyses capture. We introduce the Agentic Task Exposure (ATE) score, a composite measure computed algorithmically from O*NET task data using calibrated adoption parameters–not a regression estimate–incorporating AI capability scores, workflow coverage factors, and logistic adoption velocity. Applying the ATE framework across five major US technology regions (Seattle-Tacoma, San Francisco Bay Area, Austin, New York, and Boston) over a 2025-2030 horizon, we find that 93.2% of the 236 analyzed occupations across six information-intensive SOC groups (financial, legal, healthcare, healthcare support, sales, and administrative/clerical) cross the moderate-risk threshold (ATE >= 0.35) in Tier 1 regions by 2030, with credit analysts, judges, and sustainability specialists reaching ATE scores of 0.43-0.47. We simultaneously identify seventeen emerging occupational categories benefiting from reinstatement effects, concentrated in human-AI collaboration, AI governance, and domain-specific AI operations roles. Our findings carry implications for workforce transition policy, regional economic planning, and the temporal dynamics of labor market adjustment

We present a unified theoretical model relating stochastic microscopic epidemic dynamics with macroscopic non-linear population behavior. Utilizing the Doi-Peliti formalism, we model the pathogen as a gauge mediator field coupled to susceptible and infected host populations, and introduce a Reactive Immunity Field capable of spontaneous symmetry breaking. We demonstrate that the naive epidemic vacuum is destabilized by radiative loop corrections via the Coleman-Weinberg mechanism, generating a dynamic herd immunity threshold. By extracting the classical saddle-point limit of the Effective Action, we derive the macroscopic reaction-diffusion equations governing the host population. We show that integrating out the gauge mediator inherently generates a thermodynamic Free Energy dependent on the square of the susceptible density. This non-linearity produces a macroscopic spatial “Fear Drift” proportional to the magnitude of the immunity field, and a cubic shielding penalty in the effective reproductive number ($R_{eff}$). In this work, we establish a mapping between fundamental field-theoretic mechanisms and specific terms in the macroscopic behavioral equations. We demonstrate that Debye screening is physically executed by the spatial cross-diffusion fluxes driving host evacuation. Simultaneously, vacuum polarization manifests as a non-linear cubic penalty ($-S^3 I$) in the dressed reaction rate that dynamically suppresses the effective reproductive number. As a validation of our model, we apply the formalism to high-resolution spatiotemporal COVID-19 data from Germany.

Understanding vaccine effects on post-infection outcomes is critical for evaluating the full value proposition of a vaccine. However, defining appropriate causal effects on such outcomes is challenging because infection is affected by vaccination. Existing approaches focus on the Doomed stratum, individuals who would be infected regardless of vaccine receipt. For many relevant outcomes, however, this estimand understates vaccine benefit by excluding individuals whose adverse post-infection outcomes improve because vaccination prevented infection. We therefore propose causal estimands for post-infection outcomes in the Naturally Infected, individuals who would be infected in absence of vaccine. We derive bounds under minimal assumptions and give point identification results under an exclusion restriction and/or a partial principal ignorability assumption. For point-identified settings, we develop efficient one-step estimators with robustness properties under inconsistent nuisance parameter estimation. We further show under what conditions the same identification functional can be interpreted as targeting an effect among individuals exposed to a sufficiently infectious dose of the pathogen, thereby avoiding reliance on cross-world parameters and fundamentally untestable causal assumptions. Simulations show that the bounds are valid but often wide, and that the point estimators perform well when their identifying assumptions hold. In a reanalysis of a rotavirus vaccine trial, marginal and Doomed-stratum analyses showed little evidence of an effect on antibiotic use, whereas analyses targeting the Naturally Infected suggested a protective effect under principal ignorability-based assumptions.

Can classifier-based safety gates maintain reliable oversight as AI systems improve over hundreds of iterations? We provide comprehensive empirical evidence that they cannot. On a self-improving neural controller (d=240), eighteen classifier configurations – spanning MLPs, SVMs, random forests, k-NN, Bayesian classifiers, and deep networks – all fail the dual conditions for safe self-improvement. Three safe RL baselines (CPO, Lyapunov, safety shielding) also fail. Results extend to MuJoCo benchmarks (Reacher-v4 d=496, Swimmer-v4 d=1408, HalfCheetah-v4 d=1824). At controlled distribution separations up to delta_s=2.0, all classifiers still fail – including the NP-optimal test and MLPs with 100% training accuracy – demonstrating structural impossibility. We then show the impossibility is specific to classification, not to safe self-improvement itself. A Lipschitz ball verifier achieves zero false accepts across dimensions d in {84, 240, 768, 2688, 5760, 9984, 17408} using provable analytical bounds (unconditional delta=0). Ball chaining enables unbounded parameter-space traversal: on MuJoCo Reacher-v4, 10 chains yield +4.31 reward improvement with delta=0; on Qwen2.5-7B-Instruct during LoRA fine-tuning, 42 chain transitions traverse 234x the single-ball radius with zero safety violations across 200 steps. A 50-prompt oracle confirms oracle-agnosticity. Compositional per-group verification enables radii up to 37x larger than full-network balls. At d<=17408, delta=0 is unconditional; at LLM scale, conditional on estimated Lipschitz constants.

We study trajectory forecasting under squared loss for time series with weak conditional structure, using highly expressive prediction models. Building on the classical characterization of squared-loss risk minimization, we emphasize regimes in which the conditional expectation of future trajectories is effectively degenerate, leading to trivial Bayes-optimal predictors (flat for prices and zero for returns in standard financial settings). In this regime, increased model expressivity does not improve predictive accuracy but instead introduces spurious trajectory fluctuations around the optimal predictor. These fluctuations arise from the reuse of noise and result in increased prediction variance without any reduction in bias. This provides a process-level explanation for the degradation of Transformerbased forecasts on financial time series. We complement these theoretical results with numerical experiments on high-frequency EUR/USD exchange rate data, analyzing the distribution of trajectory-level forecasting errors. The results show that Transformer-based models yield larger errors than a simple linear benchmark on a large majority of forecasting windows, consistent with the variance-driven mechanism identified by the theory.

The low-rank matrix recovery problem seeks to reconstruct an unknown $n_1 \times n_2$ rank-$r$ matrix from $m$ linear measurements, where $m\ll n_1n_2$. This problem has been extensively studied over the past few decades, leading to a variety of algorithms with solid theoretical guarantees. Among these, gradient descent based non-convex methods have become particularly popular due to their computational efficiency. However, these methods typically suffer from two key limitations: a sub-optimal sample complexity of $O((n_1 + n_2)r^2)$ and an iteration complexity of $O(\kappa \log(1/\epsilon))$ to achieve $\epsilon$-accuracy, resulting in slow convergence when the target matrix is ill-conditioned. Here, $\kappa$ denotes the condition number of the unknown matrix. Recent studies show that a preconditioned variant of GD, known as scaled gradient descent (ScaledGD), can significantly reduce the iteration complexity to $O(\log(1/\epsilon))$. Nonetheless, its sample complexity remains sub-optimal at $O((n_1 + n_2)r^2)$. In contrast, a delicate virtual sequence technique demonstrates that the standard GD in the positive semidefinite (PSD) setting achieves the optimal sample complexity $O((n_1 + n_2)r)$, but converges more slowly with an iteration complexity $O(\kappa^2 \log(1/\epsilon))$. In this paper, through a more refined analysis, we show that ScaledGD achieves both the optimal sample complexity $O((n_1 + n_2)r)$ and the improved iteration complexity $O(\log(1/\epsilon))$. Notably, our results extend beyond the PSD setting to general low-rank matrix recovery problem. Numerical experiments further validate that ScaledGD accelerates convergence for ill-conditioned matrices with the optimal sampling complexity.

We propose a simple detection mechanism for the Gumbel watermarking scheme proposed by Aaronson (2022). The new mechanism is proven to be near-optimal in a problem-dependent sense among all model-agnostic watermarking schemes under the assumption that the next-token distribution is sampled i.i.d.

Reliable estimation of predictive performance is essential for spatial environmental modeling, where machine-learning models are used to generate maps from unevenly distributed observations. Standard cross-validation (CV) assumes that validation data are representative of prediction conditions across the target domain. In practice, this assumption is often violated due to preferential or clustered sampling, leading to biased performance and uncertainty estimates. We introduce a deployment-oriented validation framework based on weighted CV that aligns validation tasks with the distribution of prediction tasks across a specified domain. The framework includes importance-weighted cross-validation (IWCV) and a calibration-based approach, Target-Weighted Cross-Validation (TWCV), which uses spatially meaningful task descriptors such as environmental covariates and prediction distance. Simulation experiments show that conventional non-spatial and spatial CV strategies can exhibit substantial bias under realistic sampling designs, whereas weighted CV approaches substantially reduce this bias when validation tasks adequately cover the deployment-task space. A case study on mapping nitrogen dioxide (NO$_2$) concentrations across Germany demonstrates that standard CV can overestimate prediction error due to sampling bias, while weighted CV yields estimates more consistent with deployment conditions. The framework separates validation task generation from risk estimation and provides a practical approach for improving performance assessment in spatial prediction settings where sample distributions differ from prediction domains.

Scientists often want to explain why an outcome is different in two groups. For instance, differences in patient mortality rates across two hospitals could be due to differences in the patients themselves (covariates) or differences in medical care (outcomes given covariates). The Oaxaca–Blinder decomposition (OBD) is a standard tool to tease apart these factors. It is well known that the OBD requires choosing one of the groups as a reference, and the numerical answer can vary with the reference. To the best of our knowledge, there has been no systematic investigation into whether the choice of OBD reference can yield different substantive conclusions and how common this issue is. In the present paper, we give existence proofs in real and simulated data that the OBD references can in fact yield substantively different conclusions. Our empirical exercises find that this sensitivity is more common when the OBD is extended to more complex regression models, including a pretrained transformer. Our theoretical and empirical results together establish that these conclusion reversals are not entirely driven by model misspecification, small data, or adversarial parameter choices. Our results suggest that practitioners should always report both directions of the OBD; that modern machine learning and large datasets do not automatically resolve the conclusion reversal problem; and that further work on this problem is needed.

Hilbert's sixth problem calls for the axiomatization of physics, particularly the derivation of macroscopic statistical laws from microscopic mechanical principles. A conceptual difficulty arises in classical probability theory: in continuous spaces every individual microstate has probability zero. In this paper, we introduce a probabilistic framework based on Soft Logic and Soft Numbers in which point events possess infinitesimal Soft probabilities rather than the classical zero. We show that Soft probability can be interpreted as an infinitesimal refinement of classical probability and discuss its implications for statistical mechanics and Hilbert's sixth problem. In addition, we show rigorously how to construct a Mobius strip, based on the soft numbers, and we discuss how this Mobius strip representation with soft numbers allows for a deeper understanding of the nature and character of Hilbert's sixth problem. Inspired by the collapsing of that classical probability to zero, we suggest adding an axiom for an Infinitesimal Probability into the list of Kolmogorov's five Probability axioms. Furthermore, we suggest a probabilistic framework based on Soft Numbers for assigning values to probabilities of impossible events of a discrete random variable with realizations outside its support (which, in the ordinary probability, collapse to zero). This assignment of Soft Number values is based on an extension of the Pascal triangle to have soft zeros outside of the regular Pascal triangle (with real values) based on factorials of negative numbers.

This paper develops a penalized GMM (PGMM) framework for automatic debiased inference on functionals of nonparametric instrumental variable estimators. We derive convergence rates for the PGMM estimator and provide conditions for root-n consistency and asymptotic normality of debiased functional estimates, covering both linear and nonlinear functionals. Monte Carlo experiments on average derivative show that the PGMM-based debiased estimator performs on par with the analytical debiased estimator that uses the known closed-form Riesz representer, achieving 90-96% coverage while the plug-in estimator falls below 5%. We apply our procedure to estimate mean own-price elasticities in a semiparametric demand model for differentiated products. Simulations confirm near-nominal coverage while the plug-in severely undercovers. Applied to IRI scanner data on carbonated beverages, debiased semiparametric estimates are approximately 20% more elastic compared to the logit benchmark, and debiasing corrections are heterogeneous across products, ranging from negligible to several times the standard error.

We study various properties of $f$-divergences and Csiszár indices between two probability distributions in very general setups for the convex function $f$ and for the probability distributions. We establish general structural properties of $f$-divergences and show how they are inherited by the associated Csiszár indices, including monotonicity and invariance under suitable transformations. We also study the relationship between Csiszár indices and copula representations of random vectors. When the marginal distributions have atoms, the copula representation is not unique and the Csiszár index of the transformed vectors may increase. We build a large family of interpolating copulas which minimize the Csiszár index and thus preserve the dependence structure of the initial vector.

For events $A$ and $B$, we have \[ \mathbb{P}(A\mid B) > \mathbb{P}(A\mid \neg B) \qquad\Longleftrightarrow\qquad \mathbb{P}(B\mid A) > \mathbb{P}(B\mid \neg A) \] whenever all four quantities are defined. In other words, $B$ is evidence for $A$ if and only if $A$ is evidence for $B$. This note gives seven different proofs of this fact – by cross-multiplication, covariance, coupling parameters, odds ratios, pointwise mutual information, combinatorial double counting, and mixed discrete derivatives – and develops a surrounding web of interpretations. Once the marginals $\mathbb{P}(A)$ and $\mathbb{P}(B)$ are fixed, a $2\times 2$ table has only one degree of freedom, so every scalar notion of positive association must be governed by the same signed parameter.

We consider testing zero pricing errors in high-dimensional linear factor pricing models. Existing methods are mainly based on either an $L_2$ statistic, which is effective under dense alternatives, or an $L_\infty$ statistic, which is powerful under very sparse alternatives. To bridge these two regimes, we develop a class of $L_q$-based tests for finite $q$, including the practically useful $L_4$ and $L_6$ cases. We show that larger $q$ leads to greater sensitivity to sparse alternatives. We further establish the asymptotic independence between the $L_\infty$ statistic and the $L_q$ statistic for any finite $q$, which motivates a Cauchy combination test that adapts to a broad range of sparsity levels. Simulation studies and a real-data analysis show that the proposed methods are more robust to the unknown sparsity of the alternative and can outperform existing procedures in finite samples.

We present mlr3mbo, a modular toolbox for Bayesian optimization in R. mlr3mbo supports single- and multi-objective optimization, multi-point proposals, batch and asynchronous parallelization, and robust error handling. While it can be used for many standard Bayesian optimization variants in applied settings, researchers can also construct custom Bayesian optimization algorithms from its flexible building blocks. In addition to an introduction to the software, its design principles, and its building blocks, the paper presents two extensive empirical evaluations on the surrogate-based benchmark suite YAHPO Gym. To identify robust default configurations for both numeric and mixed-hierarchical optimization regimes, and to gain further insights into the respective impacts of individual settings, we run a coordinate descent search over the mlr3mbo configuration space and analyze its results. Furthermore, we benchmark mlr3mbo against a wide range of established optimizers, including HEBO, SMAC3, Ax, and Optuna, and find that it performs on par with state-of-the-art.

This paper focuses on the problem of unbounded density ratio estimation – an understudied yet critical challenge in statistical learning – and its application to covariate shift adaptation. Much of the existing literature assumes that the density ratio is either uniformly bounded or unbounded but known exactly. These conditions are often violated in practice, creating a gap between theoretical guarantees and real-world applicability. In contrast, this work directly addresses unbounded density ratios and integrates them into importance weighting for effective covariate shift adaptation. We propose a three-step estimation method that leverages unlabeled data from both the source and target distributions: (1) estimating a relative density ratio; (2) applying a truncation operation to control its unboundedness; and (3) transforming the truncated estimate back into the standard density ratio. The estimated density ratio is then employed as importance weights for regression under covariate shift. We establish rigorous, non-asymptotic convergence guarantees for both the proposed density ratio estimator and the resulting regression function estimator, demonstrating optimal or near-optimal convergence rates. Our findings offer new theoretical insights into density ratio estimation and learning under covariate shift, extending classical learning theory to more practical and challenging scenarios.

To reduce the cost of estimating the probability of a rare event involving a very large number of random parameters, we propose a new strategy for dimension reduction coupled with a surrogate model for the expensive part of the algorithm. To this end, we extend the Ordinary Kriging Active Subspace (OK-AS) method into a sequential version. Our approach consists of iteratively re-estimating the active subspace using a Kriging surrogate trained in a rotated coordinate system until the active subspace stabilises. This method allows for a reduction in prediction error and a better approximation of the active subspace on a benchmark of test problems. Furthermore, we integrate our algorithm into an efficient pre-existing approach for estimating the probability of a rare event. This approach is based on learning the active subspace associated with the random event whose probability is to be estimated. The sequential learning of an importance sampling density is necessary and corresponds to the expensive part of this strategy. To circumvent this issue, we integrate our sequential OK-AS version into the estimation of the importance sampling density. The numerical results indicate that our method allows for reducing the cost required to obtain a precise estimate of the rare event probability.

Nonnegative matrix factorization (NMF) approximates a nonnegative matrix, X, by the product of two nonnegative factors, WH, where W has r columns and H has r rows. In this paper, we consider NMF using the component-wise L1 norm as the error measure (L1-NMF), which is suited for data corrupted by heavy-tailed noise, such as Laplace noise or salt and pepper noise, or in the presence of outliers. Our first contribution is an NP-hardness proof for L1-NMF, even when r=1, in contrast to the standard NMF that uses least squares. Our second contribution is to analyze, under simplified probabilistic assumptions, how the sparsity in the data enforces zero solution in the optimal scalar update in the factors of L1-NMF when all the other entries are kept fixed. This provides an intuition of the connection between the sparsity of the L1-NMF factors with the sparsity of the input. Even though sparsity favors interpretability, if the data is affected by false zeros, too sparse solutions might degrade the model. Our third contribution is a new, more general, L1-NMF model for sparse data, dubbed weighted L1-NMF (wL1-NMF), where the sparsity of the factorization is controlled by adding a penalization parameter to the entries of WH associated with zeros in the data. The fourth contribution is a new coordinate descent (CD) approach for wL1-NMF, denoted as sparse CD (sCD), where each subproblem is solved by a weighted median algorithm. Although it lacks convergence guarantees to a stationary point, sCD is, to the best of our knowledge, the first algorithm for L1-NMF whose complexity scales with the number of nonzero entries in the data, making it efficient in handling large-scale, sparse data. We perform extensive numerical experiments on synthetic and real-world data, including imaging mass spectrometry and topic modeling, to show the effectiveness of our new proposed model (wL1-NMF) and algorithm (sCD).

Aligning human-interpretable concepts with the internal representations learned by modern machine learning systems remains a central challenge for interpretable AI. We introduce a geometric framework for comparing supervised human concepts with unsupervised intermediate representations extracted from foundation model embeddings. Motivated by the role of conceptual leaps in scientific discovery, we formalise the notion of concept frustration: a contradiction that arises when an unobserved concept induces relationships between known concepts that cannot be made consistent within an existing ontology. We develop task-aligned similarity measures that detect concept frustration between supervised concept-based models and unsupervised representations derived from foundation models, and show that the phenomenon is detectable in task-aligned geometry while conventional Euclidean comparisons fail. Under a linear-Gaussian generative model we derive a closed-form expression for Bayes-optimal concept-based classifier accuracy, decomposing predictive signal into known-known, known-unknown and unknown-unknown contributions and identifying analytically where frustration affects performance. Experiments on synthetic data and real language and vision tasks demonstrate that frustration can be detected in foundation model representations and that incorporating a frustrating concept into an interpretable model reorganises the geometry of learned concept representations, to better align human and machine reasoning. These results suggest a principled framework for diagnosing incomplete concept ontologies and aligning human and machine conceptual reasoning, with implications for the development and validation of safe interpretable AI for high-risk applications.

Dynamic structural equation modeling (DSEM) is widely used for analyzing intensive longitudinal data (ILD). Although many ILD have categorical (Bernoulli or binomially distributed) responses, currently available Metropolis-within-Gibbs samplers for estimating DSEMs are limited to using the probit link and the Bernoulli distribution. These samplers scale poorly with increasing model complexity and/or data size. Here, we present a hybrid sampler – alternating between one step of the No-U-Turn Sampler (NUTS) and one Gibbs step – which solves both of these problems: the Gibbs step naturally handles Pólya-Gamma distributed latent variables arising from binomially distributed responses with a logit link, and the NUTS step utilizes a Kalman filter to exactly marginalize over latent states, alleviating the need to sample these variables. We demonstrate in simulation experiments that the proposed sampler is more efficient than alternative algorithms, and that it makes DSEM estimation with binomial data feasible for larger data and models than what has previously been possible. We also illustrate its use in an example application of predicting panic attacks.

Deep learning is widely deployed for time series learning tasks such as classification and forecasting. Despite the empirical successes, only little theory has been developed so far in the time series context. In this work, we prove that if the network inputs are generated from short-range dependent linear processes, the outputs of fully convolutional neural networks (FCNs) with global average pooling (GAP) are asymptotically Gaussian and the limit is attained if the length of the observed time series tends to infinity. The proof leverages existing tools from the theoretical time series literature. Based on our theory, we propose a generalization of the GAP layer by considering a global weighted pooling step with slowly varying, learnable coefficients.

This paper studies transfer learning for ridge-regularized robust linear regression in the moderate-dimensional regime, where the number of predictors is of the same order as the sample size and the regression coefficients are not assumed to be sparse. We propose Trans-RR, which combines a robust ridge estimator from a source study with a robust ridge correction based on the target study. Under mild assumptions, we characterize the asymptotic estimation error of the proposed estimator and show that leveraging source data can substantially improve estimation accuracy relative to the traditional single-study ridge-regularized robust estimator. Simulation results and a real-data analysis support the theory and illustrate both positive and negative transfer as the discrepancy between the source and target studies varies.

Randomstrasse101 is a blog dedicated to Open Problems in Mathematics, with a focus on Probability Theory, Computation, Combinatorics, Statistics, and related topics. This manuscript serves as a stable record of the Open Problems posted in 2025, with the goal of easing academic referencing. The blog can currently be accessed at randomstrasse101.math.ethz.ch

Background: In clinical research, the Bland-Altman analysis is commonly used to assess agreement of metric measurements made by two or more techniques, devices or methods. The approach can also deal with repeated measurements per subject or observational unit. However, a strong and implicit assumption is that agreement of methods is homogeneous across subjects. Objective: To extend the previously introduced multivariable modeling of conditional method agreement with single measurements per subject to the frequent case of repeated measurements. Methods: Appropriate regression trees, called conditional method agreement trees (COAT), are generalized to capture the dependence of the parameters of the Bland-Altman analysis on covariates. These parameters, the expectation and variance of the differences between the methods, are decomposed into subject-specific components to account for repeated measurements. Whilst the theoretical, asymptotic properties of tree models are known, a simulation study was carried out to assess the performance of COAT in finite samples. A comparison of devices measuring cardiac output serves as an application example. Results: COAT is applicable to the two relevant cases of paired and unpaired repeated measurements. In the simulation study, it controlled the type-I error at the nominal level and could detect covariate-dependent method agreement with increasing sample size. The Adjusted Rand Index, a measure of concordance between the estimated and true subgroups, reached very high values close to the maximum of 1. The analysis of cardiac output showed that patients' characteristics may influence the agreement between measuring devices, with implications for use in patient care. Conclusion: COAT can explicitly define subgroups of heterogeneous method agreement in dependence of covariates with appropriate statistical testing in case of repeated measurements.