Papers

The use of synthetic data has become increasingly popular as a privacy-preserving alternative to sharing real datasets, especially in sensitive domains such as healthcare, finance, and demography. However, the privacy assurances of synthetic data are not absolute, and remain susceptible to membership inference attacks (MIAs), where adversaries aim to determine whether a specific individual was present in the dataset used to train the generator. In this work, we propose a practical and effective method to quantify membership disclosure risk in tabular synthetic datasets using kernel density estimators (KDEs). Our KDE-based approach models the distribution of nearest-neighbour distances between synthetic data and the training records, allowing probabilistic inference of membership and enabling robust evaluation via ROC curves. We propose two attack models: a 'True Distribution Attack', which assumes privileged access to training data, and a more realistic, implementable 'Realistic Attack' that uses auxiliary data without true membership labels. Empirical evaluations across four real-world datasets and six synthetic data generators demonstrate that our method consistently achieves higher F1 scores and sharper risk characterization than a prior baseline approach, without requiring computationally expensive shadow models. The proposed method provides a practical framework and metric for quantifying membership disclosure risk in synthetic data, which enables data custodians to conduct a post-generation risk assessment prior to releasing their synthetic datasets for downstream use. The datasets and codes for this study are available at https://github.com/PyCoder913/MIA-KDE.

Tolerance intervals provide bounds that contain a specified proportion of a population with a given confidence level, yet their construction remains challenging when parametric assumptions fail or sample sizes are small. Traditional nonparametric methods, such as Wilks' intervals, lack flexibility and often require large samples to be valid. We propose a fully nonparametric approach for constructing one-sided and two-sided tolerance intervals using a calibrated Gibbs posterior. Leveraging the connection between tolerance limits and population quantiles, we employ a Gibbs posterior based on the asymmetric Laplace (check) loss function. A key feature of our method is the calibration of the learning rate, which ensures nominal frequentist coverage across diverse distributional shapes. Simulation studies show that the proposed approach often yields shorter intervals than classical nonparametric benchmarks while maintaining reliable coverage. The framework's practical utility is illustrated through applications in ecology, biopharmaceutical manufacturing, and environmental monitoring, demonstrating its flexibility and robustness across diverse applications.

Recent shrinkage estimators for linear panel models have focused on increasing flexibility beyond empirical Bayes estimators to deliver mean squared error optimality, but these estimators' performance rarely meets their theoretical guarantees in practice. To balance theoretical and practical performance, we develop a heterogeneity-adaptive shrinkage estimator for linear panel models that adapts to the amount of information shared between groups. The primary mechanism for the information-sharing is a shrinkage of group-specific distributions towards a new "centroid" distribution through a Kullback-Leibler divergence penalty. The Kullback-Leibler divergence is uniquely geometrically suited for measuring relative information between datasets, and leads to relatively simple closed form estimators with intuitive interpretations. Among other results, we show that our estimator has a provably smaller mean squared error than the group-specific maximum likelihood estimators, and establish asymptotically valid inference procedures. We also propose a new estimator of mean squared error for implementation that leads to improved finite sample performance over minimizing Stein's unbiased risk estimate (SURE). A comprehensive set of simulations highlights our estimator's versatility, and an analysis of length of stay in ICUs illustrates its performance in a real-world setting.

Since the release of the ICH E9(R1) addendum on estimands, its application in non-inferiority trials has received far less attention than in superiority settings. A key conclusion from Lynggaard et al. was that the "choice of non-inferiority margin must reflect the chosen estimand." However, current regulatory guidance predates ICH E9(R1) and therefore does not reflect how the estimand influences the historical evidence and constancy assumption (assay sensitivity) used to derive the non-inferiority margin. This paper investigates the degree to which the non-inferiority margin depends on the estimand. Using simulated patient journeys in a weight-management setting, we illustrate how different intercurrent event strategies and variations in the intercurrent event frequency affect the estimand, and consequently the estimated treatment effect. These results emphasize that the historical treatment effect of the reference treatment versus placebo, and thus the margin $M_{1}$, is specific to an estimand and may differ even when trials formally target similar questions. We further illustrate the process of determining the non-inferiority margin using two examples in non-inferiority trials for a new theoretical weight management treatment. In the first example, we focus on the setting where the historical clinical trials use the estimand framework highlighting that even when they include the estimand framework, determining the non-inferiority margin can be challenging in case the historical trials target an estimand different from the one in the planned study. A second example highlights challenges when historical trials did not employ the estimand framework and the targeted estimand cannot be fully reconstructed.

We propose novel kernel-based tests for assessing the equivalence between distributions. Traditional goodness-of-fit testing is inappropriate for concluding the absence of distributional differences, because failure to reject the null hypothesis may simply be a result of lack of test power, also known as the Type-II error. This motivates equivalence testing, which aims to assess the absence of a statistically meaningful effect under controlled error rates. However, existing equivalence tests are either limited to parametric distributions or focus only on specific moments rather than the full distribution. We address these limitations using two kernel-based statistical discrepancies: the kernel Stein discrepancy and the Maximum Mean Discrepancy. The null hypothesis of our proposed tests assumes the candidate distribution differs from the nominal distribution by at least a pre-defined margin, which is measured by these discrepancies. We propose two approaches for computing the critical values of the tests, one using an asymptotic normality approximation, and another based on bootstrapping. Numerical experiments are conducted to assess the performance of these tests.

There is a great need for evaluating screening programs, but analysing data from population screening is often complicated by a delayed screening effect. In cancer screening, only new, not yet clinically diagnosed cases, might benefit from screening through earlier treatment. Hence, mortality data following screening should be analysed based on refined mortality, separating cases based on diagnosis before and after first screening invitation. Historically, refined mortality has been implemented by selecting comparison groups from the available data to disentangle the causal effect. While giving valid estimates, the ignorance of large parts of the available data has limited study precision. In BMJ 2014, Weedon-Fekjær et al. used a new estimation approach applying all the available Norwegian mammography screening data. The estimation uses historic pre-screening data on time from clinical diagnosis to death estimating the proportion of post-screening mortality which is expected to be based on cases incident before first screening invitation, in the absence of a screening effect. Utilizing this expected proportion of post-screening incident cases, Poisson regression offsets are added to align the expected number of cases. The screening effect is then estimated adjusting for relevant covariables. While the method increases study precision, it has not been easily available and widely adopted. We here explain the method in detail, add maximum likelihood estimation, and lay the foundation for widespread use. Applying the method on Norwegian and Danish data, bootstrap confidence intervals are considerably narrower than intervals seen using other refined mortality methods, especially for the gradually introduced Norwegian program.

Data analyses are often constructed in an imperative manner, where commands representing actions taken on the data are issued sequentially. The publication of these commands, along with the data, is essential to the reproducibility of the analysis by others. However, simply presenting the code and the results of running the code can hide important details about the data analyst's premises, expectations, and assumptions about the data. Understanding this analysis reasoning can be critical to evaluating the quality of an analysis and for suggesting possible improvements. We argue that a formal representation of a data analysis that externalizes its logical construction offers more useful information for statically illustrating an analyst's reasoning. Such a formal representation would allow for the evaluation of some aspects of a data analysis without the need for the data, the visualization of the logical connections leading to a conclusion, and the ability to assess the sensitivity of an analyst's assumptions to unexpected features in the data. In this paper we describe an implementation of this formal representation and how it might be applied to some common data analysis tasks.

With the advancement of precision medicine there is an increasing need for design and analysis methods in clinical trials with the objective of investigating effect heterogeneity and estimating subgroup effects. As this requires precise estimation of interaction effects, borrowing information from external data sources including retrospective studies and early phase clinical trials to enrich the trial in sparse subgroups is pertinent. Motivated by a trial in gastric cancer we consider a practical design and analysis framework for borrowing from external data sources that only partially inform the inference. As the analysis model we propose an individually weighted model where the external data are weighted based on their fit with the target population based on the distribution of a set of covariates. In a simulation study we assess the performance of the model under various scenarios and make comparisons to dynamic borrowing. In addition, we provide a Bayesian design framework where design priors are extracted from the external data to determine decision boundaries and sample sizes. The design procedure is demonstrated within the context of our motivating example.

Deep generative models can help with data scarcity and privacy by producing synthetic training data, but they struggle in low-data, imbalanced tabular settings to fully learn the complex data distribution. We argue that striving for the full joint distribution could be overkill; for greater data efficiency, models should prioritize learning the conditional distribution $P(y\mid \bm{X})$, as suggested by recent theoretical analysis. Therefore, we overcome this limitation with ReTabSyn, a Reinforced Tabular Synthesis pipeline that provides direct feedback on feature correlation preservation during synthesizer training. This objective encourages the generator to prioritize the most useful predictive signals when training data is limited, thereby strengthening downstream model utility. We empirically fine-tune a language model-based generator using this approach, and across benchmarks with small sample sizes, class imbalance, and distribution shift, ReTabSyn consistently outperforms state-of-the-art baselines. Moreover, our approach can be readily extended to control various aspects of synthetic tabular data, such as applying expert-specified constraints on generated observations.

In this paper we propose a new method for multiple change-point detection for piecewise-constant circular signals, a setting that, despite its importance in many scientific domains, remains comparatively under-explored. The proposed method, Permutation-based Circular Isolate-Detect, denoted PCID, uses an appropriately chosen contrast function and permutation testing to detect change-points in an offline manner, for the data sequence under consideration. Prior to detection, PCID isolates the change-points. The contrast function used is derived under the assumption of von Mises distribution for the noise, but we show that the method is robust and performs well for other distributions as well. Simulations are used to showcase the usability of the method in different signal and noise structures, including serially correlated noise. In order to exhibit the practical relevance of the method in real-world applications, PCID is applied to three real-world datasets, namely flare, acrophase and wave data.

Deep neural networks (DNNs) have become integral to a wide range of scientific and practical applications due to their flexibility and strong predictive performance. Despite their accuracy, however, DNNs frequently exhibit poor calibration, often assigning overly confident probabilities to incorrect predictions. This limitation underscores the growing need for integrated mechanisms that provide reliable uncertainty estimation. In this article, we compare two prominent approaches for uncertainty quantification: a Bayesian approximation via Monte Carlo Dropout and the nonparametric Conformal Prediction framework. Both methods are assessed using two convolutional neural network architectures; H-CNN VGG16 and GoogLeNet, trained on the Fashion-MNIST dataset. The empirical results show that although H-CNN VGG16 attains higher predictive accuracy, it tends to exhibit pronounced overconfidence, whereas GoogLeNet yields better-calibrated uncertainty estimates. Conformal Prediction additionally demonstrates consistent validity by producing statistically guaranteed prediction sets, highlighting its practical value in high-stakes decision-making contexts. Overall, the findings emphasize the importance of evaluating model performance beyond accuracy alone and contribute to the development of more reliable and trustworthy deep learning systems.

This work addresses the challenge of analyzing geometric structures using Kendall's 3D Shape Space. While Riemannian geometry provides a robust framework for shape analysis (independent of scale, position, and orientation) the transition from theoretical manifolds to practical computational workflows remains difficult. Although Geomstats is currently the leading Python library for manifold-based statistics, it lacks specific utilities required for advanced 3D shape analysis. This article introduces tools designed to bridge this gap, translating complex mathematical abstractions into efficient, accessible software solutions for researchers.

Background: Subgroup analyses and meta-regression are commonly used to investigate heterogeneity in diagnostic test accuracy (DTA) meta-analyses (MA), but adherence to methodological guidance is unclear. This methodological review summarizes investigations of heterogeneity (IoH) in DTA-MAs, examining their frequency, characteristics, and alignment with recommendations. Methods: We included DTA-MAs published in 2024 reporting at least one pair of summary sensitivity and specificity. Non-DTA reviews, narrative syntheses, studies reporting only alternative measures, and overviews of systematic reviews were excluded. MEDLINE (via Ovid) was searched for English-language publications, with the final search in January 2025. Results: From 403 records, the most recent 100 DTA-MAs were included, each contributing one index test. IoH were reported in 61 analyses. The number of primary studies was positively associated with conducting an investigation (OR 1.66; p = 0.008). Subgroup analyses were used in 35/61 (57%), while 26/61 (43%) applied meta-regression alone or with subgroup analyses. Subgroup analyses examined fewer variables than meta-regression (p < 0.001). Among 44/61 (72%) analyses with sufficient detail to identify a statistical model, the bivariate model was used in 28/44 (64%), univariate random-effects models in 14/44 (32%), and the HSROC model in 5/44 (11%). Formal tests for subgroup differences were reported in 37/61 (61%). Protocols were available for 43/61 (70%) analyses, of which 19/43 (44%) fully prespecified IoH. Discussion: IoH were common and more likely when more primary studies were available, although individual subgroups were often supported by limited data. Reporting of statistical models and model choice was frequently unclear. Greater prespecification of IoH in protocols may reduce spurious findings and improve transparency in diagnostic research.

In statistics, forecast uncertainty is often quantified using a specified statistical model, though such approaches may be vulnerable to model misspecification, selection bias, and limited finite-sample validity. While bootstrapping can potentially mitigate some of these concerns, it is often computationally demanding. Instead, we take a model-agnostic and distribution-free approach, namely conformal prediction, to construct prediction intervals in high-dimensional functional time series. Among a rich family of conformal prediction methods, we study split and sequential conformal prediction. In split conformal prediction, the data are divided into training, validation, and test sets, where the validation set is used to select optimal tuning parameters by calibrating empirical coverage probabilities to match nominal levels; after this, prediction intervals are constructed for the test set, and their accuracy is evaluated. In contrast, sequential conformal prediction removes the need for a validation set by updating predictive quantiles sequentially via an autoregressive process. Using subnational age-specific log-mortality data from Japan and Canada, we compare the finite-sample forecast performance of these two conformal methods using empirical coverage probability and the mean interval score.

We establish a strong Gaussian approximation for high-dimensional non-degenerate U-statistics with diverging dimension. Under mild assumptions, we construct, on a sufficiently rich probability space, a Gaussian process that uniformly approximates the entire sequential U-statistic process. The approximation error is explicitly characterized and vanishes under polynomial growth of the dimension. The key technical contribution is a sharp martingale maximal inequality for completely degenerate U-statistics, combined with a high-dimensional strong approximation for independent sums. This coupling yields functional Gaussian limits without relying on $\mathcal{L}^\infty$-type bounds or bootstrap arguments. The theory is illustrated through three representative examples of U-statistics: the spatial Kendall's tau matrix, the multivariate Gini's mean difference, and the characteristic dispersion parameter. As applications, we derive Brownian bridge approximations for U-statistic-based change-point statistics and develop a self-normalized relevant testing procedure whose limiting distribution is fully pivotal. The framework naturally accommodates bounded kernels and therefore remains valid under heavy-tailed distributions. Overall, our results provide a unified probability-theoretic foundation for high-dimensional inference based on U-statistics.

Competing processes shape Europa's surface: geological activity replenishes material through resurfacing, while bombardment by charged particles alters surface chemical composition. Each process leaves distinct spectral signatures. We present a novel data-driven analysis of JWST NIRSpec-IFU observations of Europa's leading hemisphere across three observing geometries, targeting nine spectral bands sensitive to water ice, radiolytic products, and volatiles. Through spectral factorization, we isolate the dominant components of spectral variability and reconstruct their spatial distributions. We find that CO2 enrichment extends beyond Tara Regio, and covers multiple chaos units in a lens-like pattern. These CO2-enriched areas co-occur with anomalous ice-texture signatures. Together, these findings suggest that enrichment in volatiles on Europa may reflect retention-favorable near-surface microphysics as well as emplacement, refining how they are interpreted in the context of surface–interior exchange. This has implications for interpreting the sources and supply rates of extant carbon-bearing species and, ultimately, for assessing Europa's habitability.

Firms increasingly use randomized experiments to decide whether to scale up an intervention and, if so, how to re-optimize related operational choices such as inventory, capacity, or pricing. In many settings, experiments are performed on small samples, so the estimated effect of the intervention is uncertain. A common practice is to plug a 'significant' estimate of the effect into both (i) the rollout rule and (ii) the downstream optimization. However, this can lead to avoidable losses because the costs of over- versus under-estimating the effect are often asymmetric. The technically ideal approach is to obtain a data-dependent decision rule that minimizes the Bayes risk, but this lacks transparency and requires more computations. We propose Predict-Adjust-Then-Rollout-Optimize (PATRO), a plug-in approach that keeps the standard estimate, but makes data-independent adjustments, respectively, for the two types of decision. We show that the two adjustments can be substitutes or complements and provide an alternating-iteration method to compute the pair. PATRO performs both in theory and numerically close or equivalent to the Bayes-optimal benchmark, making it a simple, effective way to convert noisy experimental results into better rollout and operational decisions.

In this work, we study the convergence properties of the Dual Space Preconditioned Gradient Descent, encompassing optimizers such as Normalized Gradient Descent and Gradient Clipping. We consider preconditioners of the form $\nabla K$, where $K: \mathbb{R}^{d \times k} \to \mathbb{R}$ is convex and apply $\nabla K(\cdot)$ to train an over-parameterized linear model with a convex loss of the form $\ell(X W - Y)$, for weights $W \in \mathbb{R}^{d \times k}$, labels $Y \in \mathbb{R}^{n \times k}$ and data $X \in \mathbb{R}^{n \times d}$. Under the aforementioned assumptions, we prove that the iterates of the full-batch preconditioned gradient descent converge at an exponential rate to a point $W_{\infty} \in \mathbb{R}^{d \times k}$ satisfying $XW_{\infty} = Y$. We also study the implicit bias of Dual Space Preconditioned Gradient Descent. First, we demonstrate analytically and empirically that, for general $K(\cdot)$, $W_\infty$ depends on the chosen constant step size, hindering a precise characterization of the implicit bias. We also provide an approximate implicit bias property for general preconditioners, namely, $\|W_0 - W_{\infty}\|_F \le c \|W_0 - W_{\text{GD}, \infty}\|_F$ for a constant $c>0$ and $W_{\text{GD}, \infty}$ denoting the convergence point of GD initialized at $W_0$. Furthermore, for preconditioners of the form $K(G) = h(\|G\|_F)$, known as isotropic preconditioners, and for the stochastic variation of the algorithm with arbitrary batch-size, we prove linear convergence to $W_{\text{GD}, \infty}$. Finally, in the experiments, we demonstrate faster convergence on a nonlinear model obtained using the smoothed matrix elastic-net as a preconditioner.

Isotonic regression (IR) is shape-constrained regression to maintain a univariate fitting curve non-decreasing, which has numerous applications including single-index models and probability calibration. When it comes to multi-output regression, the classical IR is no longer applicable because the monotonicity is not readily extendable. We consider a novel multi-output regression problem where a regression function is cyclically monotone. Roughly speaking, a cyclically monotone function is the gradient of some convex potential. Whereas enforcing cyclic monotonicity is apparently challenging, we leverage the fact that Kantorovich's optimal transport (OT) always yields a cyclically monotone coupling as an optimal solution. This perspective naturally allows us to interpret a regression function and the convex potential as a link function in generalized linear models and Brenier's potential in OT, respectively, and hence we call this IR extension Brenier isotonic regression. We demonstrate experiments with probability calibration and generalized linear models. In particular, IR outperforms many famous baselines in probability calibration robustly.

Conditional density estimation is complicated by multimodality, heteroscedasticity, and strong non-Gaussianity. Gaussian processes (GPs) provide a principled nonparametric framework with calibrated uncertainty, but standard GP regression is limited by its unimodal Gaussian predictive form. We introduce the Generalized Gaussian Mixture Process (GGMP), a GP-based method for multimodal conditional density estimation in settings where each input may be associated with a complex output distribution rather than a single scalar response. GGMP combines local Gaussian mixture fitting, cross-input component alignment and per-component heteroscedastic GP training to produce a closed-form Gaussian mixture predictive density. The method is tractable, compatible with standard GP solvers and scalable methods, and avoids the exponentially large latent-assignment structure of naive multimodal GP formulations. Empirically, GGMPs improve distributional approximation on synthetic and real-world datasets with pronounced non-Gaussianity and multimodality.

Active learning for regression reduces labeling costs by selecting the most informative samples. Improved Greedy Sampling is a prominent method that balances feature-space diversity and output-space uncertainty using a static, multiplicative rule. We propose Weighted improved Greedy Sampling (WiGS), which replaces this framework with a dynamic, additive criterion. We formulate weight selection as a reinforcement learning problem, enabling an agent to adapt the exploration-investigation balance throughout learning. Experiments on 18 benchmark datasets and a synthetic environment show WiGS outperforms iGS and other baseline methods in both accuracy and labeling efficiency, particularly in domains with irregular data density where the baseline's multiplicative rule ignores high-error samples in dense regions.

In stepped-wedge trials, effects often change with the time since a cluster crossed over, and a delayed primary outcome is administratively censored for clusters that cross over late. The design couples them: long exposure times occur only in the last periods, where censoring is heaviest. We study exposure-time-specific effects when a short-term surrogate is always recorded and censoring is ignorable given the surrogate, exposure time, covariates and calendar time. The usual immediate-treatment estimandgives more weight to short exposures. Each exposure-time effect is identified through a surrogate bridge, and because the rollout fixes the exposure law, neither the surrogate distribution nor the bridge regression enters theremainder of the efficient influence function. The efficiency bound is a full-data part plus a censoring burden whose share grows with exposure time; as a function of the allocation of clusters to waves, it becomes a design criterion that can favor the heavily censored last wave. We propose a cross-fitted surrogate-assisted augmented estimator, prove joint asymptotic normality, show that with a consistent bridge its variance is the same for independent and fixed-wave crossover times, and justify a fixed-partition cluster jackknife and a heterogeneity test. Treating each crossover as a sequential randomization gives a sharper bound that uses a cluster's pre-crossover history; under a random-intercept working structure the optimal correction is a one-dimensional index of the cluster's own surrogate and outcome history, and the resulting estimator attains the bound with 30 clusters in simulations.

Designing service systems requires selecting among alternative configurations – choosing the best chatbot variant, the optimal routing policy, or the most effective quality control procedure. In many service systems, the primary evidence of performance quality is textual – customer support transcripts, complaint narratives, compliance review reports – rather than the scalar measurements assumed by classical optimization methods. Large language models (LLMs) can read such textual evidence and produce standardized quality scores, but these automated judges exhibit systematic biases that vary across alternatives and evaluation instances. Human expert review remains accurate but costly. We study how to identify the best service configuration with high confidence while minimizing expensive human audits, given that automated evaluation is cheap but biased. We formalize this as a sequential decision problem where a biased proxy score is observed for every evaluation, and a verified outcome can be acquired selectively at additional cost. We prove that LLM-only selection fails under arm-dependent bias, and that naive selective-audit estimators can be asymptotically biased. We develop an estimator combining proxy scores with inverse-propensity-weighted residuals and construct anytime-valid confidence sequences. Our algorithm, PP-LUCB, jointly decides which alternatives to evaluate and whether to request human audits, concentrating reviews where the LLM judge is least reliable. We prove correctness and establish instance-dependent cost bounds showing near-optimal efficiency. On a customer support ticket classification task, our algorithm correctly identifies the best model in 40/40 trials while achieving 90% audit cost reduction.

Utilizing established risk factors and prognostic models can often improve the construction of a newer risk model that uses novel biomarkers in a smaller, internal study. However, directly borrowing information from an established prognostic model is often unsuitable due to differences in study populations, patient outcomes measured, and other specific features of the internal study design. To better enable the use of established prognostic information when constructing a novel risk model, we propose an estimation approach centered around the idea that the risk rankings rather than the risk scores from an established prognostic model are often more transportable to the internal study context. To leverage external ranking information, our approach introduces the ranking parameters associated with the regression coefficients of an internal risk model and estimates the internal risk model parameters by penalizing a ranking-based discrepancy measure between the ranking parameters and the rankings implied by the established prognostic model. Our method does not require the external prognostic model to have a specific form, but only requires one to compute risk score rankings from an external model. Simulation studies demonstrate that our method leads to competitive predictive performance and performs particularly well when the true internal and external prognostic models have high rank correlation but large discrepancies between their underlying risk scores. We demonstrate the use of our approach through the development of a prognostic model for advanced prostate cancer patients who were treated with an immune checkpoint inhibitor

Local regression is widely used to explore spatial heterogeneity, but anisotropic or effectively low-dimensional neighborhoods can produce ill-conditioned local solves, causing coefficient variation driven by numerical artifacts rather than substantive structure. Such instability is often hidden when estimation relies on implicit tuning or optimization without exposing local diagnostics. This paper proposes Gimbal Regression (GR), a deterministic, geometry-aware local regression framework for stable and auditable estimation. GR constructs directional weights from neighborhood geometry using explicit orientation objects and deterministic safeguards, and computes local coefficients by a closed-form solve. Theoretical results are stated conditional on the realized neighborhood configuration, under which the estimator is a deterministic linear operator with finite-perturbation stability bounds. Simulations and empirical examples demonstrate predictable computation, transparent diagnostics, and improved numerical stability relative to common local regression baselines.

Sparse autoencoders can localize where concepts live in language models, but not how they interact during multi-step reasoning. We propose Causal Concept Graphs (CCG): a directed acyclic graph over sparse, interpretable latent features, where edges capture learned causal dependencies between concepts. We combine task-conditioned sparse autoencoders for concept discovery with DAGMA-style differentiable structure learning for graph recovery and introduce the Causal Fidelity Score (CFS) to evaluate whether graph-guided interventions induce larger downstream effects than random ones. On ARC-Challenge, StrategyQA, and LogiQA with GPT-2 Medium, across five seeds ($n{=}15$ paired runs), CCG achieves $\CFS=5.654\pm0.625$, outperforming ROME-style tracing ($3.382\pm0.233$), SAE-only ranking ($2.479\pm0.196$), and a random baseline ($1.032\pm0.034$), with $p<0.0001$ after Bonferroni correction. Learned graphs are sparse (5-6% edge density), domain-specific, and stable across seeds.

We study the fixed-budget best-arm identification (BAI) problem in non-stationary linear bandits. Concretely, given a fixed time budget $T\in \mathbb{N}$, finite arm set $\mathcal{X} \subset \mathbb{R}^d$, and a potentially adversarial sequence of unknown parameters $\lbrace \theta_t\rbrace_{t=1}^{T}$ (hence non-stationary), a learner aims to identify the arm with the largest cumulative reward $x_* = \arg\max_{x \in \mathcal{X}} x^\top\sum_{t=1}^T \theta_t$ with high probability. In this setting, it is well-known that i.i.d. sampling arms from the G-optimal design yields a minimax-optimal error probability of $\exp\left(-\Theta\left(T / H_{G}\right)\right)$, where $H_{G}$ scales proportionally with the dimension $d$. However, this notion of complexity is overly pessimistic, as it is derived from a lower bound in which the arm set consists only of the standard basis vectors, thus masking any potential advantages arising from arm sets with richer geometric structure. To address this, we establish an arm-set-dependent lower bound that, in contrast, holds for any arm set. Motivated by the ideas underlying our lower bound, we propose the Adjacent-optimal design, a specialization of the well-known $\mathcal{X}\mathcal{Y}$-optimal design, and develop the $\textsf{Adjacent-BAI}$ algorithm. We prove that the error probability of $\textsf{Adjacent-BAI}$ matches our lower bound up to constants, verifying the tightness of our lower bound, and establishing the arm-set-dependent complexity of this setting.

We present a new method for combining e-variables through demi-supermartingales, which settles an old conjecture in the literature on nonparametric mean testing. It also provides an explicit concentration bound for a certain Kullback–Leibler-type statistic arising in the stochastic multi-armed bandit literature. All of these combination results hold for independent e-variables as well as for the class of co-valid e-variables, whose dependence structure lies somewhere between independence and sequential validity. The results are further generalized to compound e-variables. The proofs proceed by analyzing elementary symmetric polynomials and their behavior as nonnegative demi-supermartingales.

We study the problem of selecting optimal two-block partitions to accelerate the mixing of finite Markov chains under group-averaging transformations. The main objectives considered are the Kullback-Leibler (KL) divergence and the Frobenius distance to stationarity. We establish explicit connections between these objectives and the induced projection chain. In the case of the KL divergence, this reduction yields explicit decay rates in terms of the log-Sobolev constant. For the Frobenius distance, we identify a Cheeger-type functional that characterises optimal cuts. This formulation recasts two-block selection as a structured combinatorial optimisation problem admitting difference-of-submodular decompositions. We further propose several algorithmic approximations, including majorisation-minimisation and coordinate descent schemes, as computationally feasible alternatives to exhaustive combinatorial search. Our numerical experiments reveal that optimal cuts under the two objectives can substantially reduce total variation distance to stationarity and demonstrate the practical effectiveness of the proposed approximation algorithms.

Let $f_\theta$ be the joint density of a random sample $X$. A frequently used criterion asserts that a statistic $T(X)$ is minimal sufficient if, for any sample points $x$ and $y$, $T(x) = T(y)$ exactly when there exists a finite constant $h_{xy} > 0$, independent of $\theta$, such that $f_\theta(y) = f_\theta(x)h_{xy}$ for all $\theta$. We show that this criterion is false in general via a counterexample exploiting the non-uniqueness of versions of Radon–Nikodym derivatives. Although Sato (1996) established sufficient regularity conditions for the validity of this criterion, these conditions are frequently intractable to verify in practice. We resolve this limitation by introducing a version-robust method applicable whenever sufficiency is known. Moreover, our method allows us to generalize Sato's approach from Euclidean settings to arbitrary analytic Borel sample spaces and separable measurable statistic spaces. We also obtain a method for exponential-family densities under verifiable hypotheses. Taken together, these results clarify when pointwise likelihood-ratio arguments for minimal sufficiency are mathematically sound in irregular settings. Finally, we construct a counterexample demonstrating that a distinct criterion for minimal sufficiency due to Pfanzagl (1994, 2017) similarly fails in the absence of supplementary hypotheses. Identifying minimal sufficient statistics is important not only for parsimonious data reduction but also because, in models admitting complete sufficiency, such statistics provide a practical route to the complete sufficient structure underlying optimal estimation and prediction.