Papers

Adaptive sample size re-estimation, early stopping, and trial re-design at interim analyses can reduce expected sample sizes in randomised trials. Cluster randomised trials, in which groups of participants are randomly allocated to treatment status, may particularly benefit as they can be costly and their required sample sizes depend on one or more auxiliary parameters governing correlations within and between clusters, which are often estimated with high uncertainty. We adapt a combination test approach to the cluster trial setting allowing for early stopping for futility or efficacy and accounting for correlations between trial stages and other nuisance parameters. We consider design decisions for multi-dimensional sample sizes involving clusters, participants, and time and allowing for modifications to intervention roll-out patterns. We use a Pareto optimality approach to balance objectives relating to different components of the sample size and costs. We also examine the interim estimation of auxiliary parameters and trial re-design for efficiency. We illustrate the methods including an example of a parallel cluster trial re-design and a re-analysis of the large cluster randomised trial E-MOTIVE.

Estimating heterogeneous treatment effects (HTEs) from right-censored survival data is critical in high-stakes applications such as precision medicine and individualized policy-making. Yet, the survival analysis setting poses unique challenges for HTE estimation due to censoring, unobserved counterfactuals, and complex identification assumptions. Despite recent advances, from Causal Survival Forests to survival meta-learners and outcome imputation approaches, evaluation practices remain fragmented and inconsistent. We introduce SurvHTE-Bench, the first comprehensive benchmark for HTE estimation with censored outcomes. The benchmark spans (i) a modular suite of synthetic datasets with known ground truth, systematically varying causal assumptions and survival dynamics, (ii) semi-synthetic datasets that pair real-world covariates with simulated treatments and outcomes, and (iii) real-world datasets from a twin study (with known ground truth) and from an HIV clinical trial. Across synthetic, semi-synthetic, and real-world settings, we provide the first rigorous comparison of survival HTE methods under diverse conditions and realistic assumption violations. SurvHTE-Bench establishes a foundation for fair, reproducible, and extensible evaluation of causal survival methods. The data and code of our benchmark are available at: https://github.com/Shahriarnz14/SurvHTE-Bench .

Singular statistical models-including mixtures, matrix factorization, and neural networks-violate regular asymptotics due to parameter non-identifiability and degenerate Fisher geometry. Although singular learning theory characterizes marginal likelihood behavior through invariants such as the real log canonical threshold and singular fluctuation, these quantities remain difficult to interpret operationally. At the same time, widely used criteria such as WAIC and WBIC appear disconnected from underlying singular geometry. We show that posterior tempering induces a one-parameter deformation of the posterior distribution whose associated observables generate a hierarchy of thermodynamic response functions. A universal covariance identity links derivatives of tempered expectations to posterior fluctuations, placing WAIC, WBIC, and singular fluctuation within a unified response framework. Within this framework, classical quantities from singular learning theory acquire natural thermodynamic interpretations: RLCT governs the leading free-energy slope, singular fluctuation corresponds to curvature of the tempered free energy, and WAIC measures predictive fluctuation. We formalize an observable algebra that quotients out non-identifiable directions, allowing structurally meaningful order parameters to be constructed in singular models. Across canonical singular examples-including symmetric Gaussian mixtures, reduced-rank regression, and overparameterized neural networks-we empirically demonstrate phase-transition-like behavior under tempering. Order parameters collapse, susceptibilities peak, and complexity measures align with structural reorganization in posterior geometry. Our results suggest that thermodynamic response theory provides a natural organizing framework for interpreting complexity, predictive variability, and structural reorganization in singular Bayesian learning.

The emergence of generative AI models has dramatically expanded the availability and use of synthetic data across scientific, industrial, and policy domains. While these developments open new possibilities for data analysis, they also raise fundamental statistical questions about when synthetic data can be used in a valid, reliable, and principled manner. This paper reviews the current landscape of synthetic data generation and use from a statistical perspective, with the goal of clarifying the assumptions under which synthetic data can meaningfully support downstream discovery, inference, and prediction. We survey major classes of modern generative models, their intended use cases, and the benefits they offer, while also highlighting their limitations and characteristic failure modes. We additionally examine common pitfalls that arise when synthetic data are treated as surrogates for real observations, including biases from model misspecification, attenuated uncertainty, and difficulties in generalization. Building on these insights, we discuss emerging frameworks for the principled use of synthetic data. We conclude with practical recommendations, open problems, and cautions intended to guide both method developers and applied researchers.

Causal structure learning from observational data is central to many scientific and policy domains, but the time series setting common to many disciplines poses several challenges due to temporal dependence. In this paper we focus on score-based causal discovery for multivariate time series and introduce TS-BOSS, a time series extension of the recently proposed Best Order Score Search (BOSS) (Andrews et al. 2023). TS-BOSS performs a permutation-based search over dynamic Bayesian network structures while leveraging grow-shrink trees to cache intermediate score computations, preserving the scalability and strong empirical performance of BOSS in the static setting. We provide theoretical guarantees establishing the soundness of TS-BOSS under suitable assumptions, and we present an intermediate result that extends classical subgraph minimality results for permutation-based methods to the dynamic (time series) setting. Our experiments on synthetic data show that TS-BOSS is especially effective in high auto-correlation regimes, where it consistently achieves higher adjacency recall at comparable precision than standard constraint-based methods. Overall, TS-BOSS offers a high-performing, scalable approach for time series causal discovery and our results provide a principled bridge for extending sparsity-based, permutation-driven causal learning theory to dynamic settings.

While globally optimal empirical risk minimization (ERM) decision trees have become computationally feasible and empirically successful, rigorous theoretical guarantees for their statistical performance remain limited. In this work, we develop a comprehensive statistical theory for ERM trees under random design in both high-dimensional regression and classification. We first establish sharp oracle inequalities that bound the excess risk of the ERM estimator relative to the best possible approximation achievable by any tree with at most $L$ leaves, thereby characterizing the interpretability-accuracy trade-off. We derive these results using a novel uniform concentration framework based on empirically localized Rademacher complexity. Furthermore, we derive minimax optimal rates over a novel function class: the piecewise sparse heterogeneous anisotropic Besov (PSHAB) space. This space explicitly captures three key structural features encountered in practice: sparsity, anisotropic smoothness, and spatial heterogeneity. While our main results are established under sub-Gaussianity, we also provide robust guarantees that hold under heavy-tailed noise settings. Together, these findings provide a principled foundation for the optimality of ERM trees and introduce empirical process tools broadly applicable to other highly adaptive, data-driven procedures.

Modern predictive systems combine predictors, sequential monitors, prediction sets, and online strategies, each with a different certificate of optimality. We study four criterion-relative geometries: Blackwell risk dominance, anytime-valid admissibility, fixed-level marginal coverage with expected-length efficiency within a declared rank-indexed family, and choice-based approachability (CApp) boundary-feasibility. We embed the four procedure types in a common product space and prove witness-based pairwise non-nesting: for every ordered pair of criterion classes, an explicit predictive system is active in both relevant coordinates, belongs to one class, and fails the other. The result records non-nesting across different object spaces and partial orders; it does not assert practical incompatibility. We separate three measure-relative coherence notions. In conditionally i.i.d. models, posterior predictive means under a single prior are martingales under the prior predictive law. For a point null, anytime-valid admissibility within e-processes is equivalent to the nonnegative martingale property. Self-consistency under a predictor's own predictive law does not imply Blackwell admissibility, as shown by a Bernoulli log-loss counterexample. Coverage admissibility is certified by exchangeability ranks within the declared rank-indexed family, while CApp boundary-feasibility uses Cesaro steering. A constrained-Bayes design schema organizes the four paradigms without collapsing their distinct decision spaces, partial orders, or risk functionals. Admissibility is criterion-relative.

In clinical prediction settings the importance of a high-dimensional feature like genomics is often assessed by evaluating the change in predictive performance when adding it to a set of traditional clinical variables. This approach is questionable, because it does not account for collinearity nor known directionality of dependencies between variables. We suggest to use asymmetric Shapley values as a more suitable alternative to quantify feature importance in the context of a mixed-dimensional prediction model. We focus on a setting that is particularly relevant in clinical prediction: disease state as a mediating variable for genomic effects, with additional confounders for which the direction of effects may be unknown. We derive efficient algorithms to compute local and global asymmetric Shapley values for this setting. The former are shown to be very useful for inference, whereas the latter provide interpretation by decomposing any predictive performance metric into contributions of the features. Throughout, we illustrate our framework by a leading example: the prediction of progression-free survival for colorectal cancer patients.

We investigate the extreme values of a sparse and equicorrelated Gaussian field on a triangle: the correlations on every vertical or horizontal line are all equal to a parameter $r \in [0,1/2]$ and are zero everywhere else. This problem is closely linked with various problems in high-dimensional statistics and extreme-value theory. We identify the threshold for $r$ at which the standard Gumbel law breaks down. Our result is based on a subtle application of the Chen-Stein method for Poisson approximation. As applications, we discuss the implication of our results on multiple testing and resolve several questions that were left open in \cite{heiny2024maximum}, \cite{tang2022asymptotic} and \cite{Jiang19}.

Finding patient subgroups with similar characteristics is crucial for personalized decision-making in various disciplines such as healthcare and policy evaluation. While most existing approaches rely on unsupervised clustering methods, there is a growing trend toward using supervised clustering methods that identify operationalizable subgroups in the context of a specific outcome of interest. We propose Bayesian Supervised Causal Clustering (BSCC), with treatment effect as outcome to guide the clustering process. BSCC identifies homogenous subgroups of individuals who are similar in their covariate profiles as well as their treatment effects. We evaluate BSCC on simulated datasets as well as real-world dataset from the third International Stroke Trial to assess the practical usefulness of the framework.

Recent studies have observed that intermediate layers of foundation models often yield more discriminative representations than the final layer. While initially attributed to autoregressive pretraining, this phenomenon has also been identified in models trained via supervised and discriminative self-supervised objectives. In this paper, we conduct a comprehensive study to analyze the behavior of intermediate layers in pretrained vision transformers. Through extensive linear probing experiments across a diverse set of image classification benchmarks, we find that distribution shift between pretraining and downstream data is the primary cause of performance degradation in deeper layers. Furthermore, we perform a fine-grained analysis at the module level. Our findings reveal that standard probing of transformer block outputs is suboptimal; instead, probing the activation within the feedforward network yields the best performance under significant distribution shift, whereas the normalized output of the multi-head self-attention module is optimal when the shift is weak.

Most statistical process monitoring methods for multichannel profiles focus solely on the mean and are almost ineffective when changes involve the covariance structure. Although it is known to be crucial, covariance monitoring requires estimating a much larger number of parameters, which may shift in a subtle and sparse fashion. That is, an out-of-control (OC) state may manifest with small deviations and affect only a very limited subset of these parameters. To address these difficulties, we propose a multichannel profile covariance (MPC) control chart based on functional graphical models that provide an interpretable representation of conditional dependencies between profiles. A nonparametric combination of the likelihood-ratio tests corresponding to different sparsity levels is then used to draw an overall inference and signal whether an OC state may have occurred. Between-profile relationships that are likely to have shifted are naturally identified at no additional computational cost. An extensive Monte Carlo simulation study compares the MPC control chart with state-of-the-art competitors, and a case study on monitoring multichannel temperature profiles in a roasting machine illustrates its practical applicability.

Individualized decision rules (IDRs) have become increasingly prevalent in societal applications such as personalized marketing, healthcare, and public policy design. However, a critical ethical concern arises from the potential discriminatory effects of IDRs trained on biased data. These algorithms may disproportionately harm individuals from minority subgroups defined by sensitive attributes like gender, race, or language. To address this issue, we propose a novel framework that incorporates demographic parity (DP) and conditional demographic parity (CDP) constraints into the estimation of optimal IDRs. We show that the theoretically optimal IDRs under DP and CDP constraints can be obtained by applying perturbations to the unconstrained optimal IDRs, enabling a computationally efficient solution. Theoretically, we derive convergence rates for both policy value and the fairness constraint term. The effectiveness of our methods is illustrated through comprehensive simulation studies and an empirical application to the Oregon Health Insurance Experiment.

Data normalisation, a common and often necessary preprocessing step in engineering and scientific applications, can severely distort the discovery of governing equations by magnitudebased sparse regression methods. This issue is particularly acute for the Sparse Identification of Nonlinear Dynamics (SINDy) framework, where the core assumption of sparsity is undermined by the interaction between data scaling and measurement noise. The resulting discovered models can be dense, uninterpretable, and physically incorrect. To address this critical vulnerability, we introduce the Sequential Thresholding of Coefficient of Variation (STCV), a novel, computationally efficient sparse regression algorithm that is inherently robust to data scaling. STCV replaces conventional magnitude-based thresholding with a dimensionless statistical metric, the Coefficient Presence (CP), which assesses the statistical validity and consistency of candidate terms in the model library. This shift from magnitude to statistical significance makes the discovery process invariant to arbitrary data scaling. Through comprehensive benchmarking on canonical dynamical systems and practical engineering problems, including a physical mass-spring-damper experiment, we demonstrate that STCV consistently and significantly outperforms standard Sequential Thresholding Least Squares (STLSQ) and Ensemble-SINDy (E-SINDy) on normalised, noisy datasets. The results show that STCV-based methods can successfully identify the correct, sparse physical laws even when other methods fail. By mitigating the distorting effects of normalisation, STCV makes sparse system identification a more reliable and automated tool for real-world applications, thereby enhancing model interpretability and trustworthiness.

The purpose of this paper is to estimate the limiting variance of asymptotically stationary Gaussian processes observed at high frequency, using the second moment estimator (SME). We study rates of convergence of the central limit theorem for the SME in terms of the total variation, Kolmogorov and Wasserstein distances, using some novel techniques and sharp estimates for cumulants. We apply our approach to provide Berry-Esseen bounds in Kolmogorov and Wasserstein distances for estimators of the drift parameters of Gaussian Ornstein-Uhlenbeck processes. Moreover, we prove that most of our estimates are strictly sharper than the ones obtained in the existing literature.

Due to the conceptual simplicity, the linear filtering framework, notably the autoregressive (AR) process, has a long history in simulating clutter sequences with specified probability density functions (PDFs) and autocorrelation functions (ACFs). However, linear filtering inevitably distorts the input distribution, which may lead to inaccurate PDF reproduction or restrict applicability to very simple ACFs. To address these challenges, this study proposes a series-based analytic continuation strategy that revitalizes AR process clutter simulation by accurately precomputing the input pre-distortion required to compensate for AR filtering. First, the moments and cumulants of the AR input are derived based on the input-output relationship of the AR process, facilitating the moment and cumulant expansions of the Laplace transform (LT) and the logarithmic LT around zero, respectively. Second, both series expansions are analytically continued via the Padé approximation (PA) to recover the LT over the full complex plane. Notably, the PA-based continuation of the moment expansion, a conventional choice, can be highly inaccurate when the LT exhibits strong oscillations. By contrast, given the logarithmic LT generally has a simpler structure, the continuation of the cumulant expansion provides a more stable and accurate alternative. Third, the LT recovered from the cumulant expansion facilitates fast simulation of the AR input non-Gaussian white sequence via a random variable transformation method, thereby enabling an efficient AR process. Finally, simulations demonstrate that the proposed strategy enables accurate and fast simulation of non-Gaussian correlated clutter sequences.

Causal discovery across multiple datasets is often constrained by data privacy regulations and cross-site heterogeneity, limiting the use of conventional methods that require a single, centralized dataset. To address these challenges, we introduce fedCI, a federated conditional independence test that rigorously handles heterogeneous datasets with non-identical sets of variables, site-specific effects, and mixed variable types, including continuous, ordinal, binary, and categorical variables. At its core, fedCI uses a federated Iteratively Reweighted Least Squares (IRLS) procedure to estimate the parameters of generalized linear models underlying likelihood-ratio tests for conditional independence. Building on this, we develop fedCI-IOD, a federated extension of the Integration of Overlapping Datasets (IOD) algorithm, that replaces its meta-analysis strategy and enables, for the fist time, federated causal discovery under latent confounding across distributed and heterogeneous datasets. By aggregating evidence federatively, fedCI-IOD not only preserves privacy but also substantially enhances statistical power, achieving performance comparable to fully pooled analyses and mitigating artifacts from low local sample sizes. Our tools are publicly available as the fedCI Python package, a privacy-preserving R implementation of IOD, and a web application for the fedCI-IOD pipeline, providing versatile, user-friendly solutions for federated conditional independence testing and causal discovery.

This paper develops a robust parametric framework for jump detection in discretely observed CKLS-type jump-diffusion processes with high-frequency asymptotics, based on the minimum density power divergence estimator (MDPDE). The methodology exploits the intrinsic asymptotic scale separation between diffusion increments, which decay at rate $\sqrt{\Delta_n}$, and jump increments, which remain of non-vanishing stochastic magnitude. Using robust MDPDE-based estimators of the drift and diffusion coefficients, we construct standardized residuals whose extremal behavior provides a principled basis for statistical discrimination between continuous and discontinuous components. We establish that, over diffusion intervals, the maximum of the normalized residuals converges to the Gumbel extreme-value distribution, yielding an explicit and asymptotically valid detection threshold. Building on this result, we prove classification consistency of the proposed robust detection procedure: the probability of correctly identifying all jump and diffusion increments converges to one under proper asymptotics. The MDPDE-based normalization attenuates the influence of atypical increments and stabilizes the detection boundary in the presence of discontinuities. Simulation results confirm that robustness improves finite-sample stability and reduces spurious detections without compromising asymptotic validity. The proposed methodology provides a theoretically rigorous and practically resilient robust approach to jump identification in high-frequency stochastic systems.

Modern data analysis across diverse disciplines increasingly relies on time series. Many of these datasets exhibit cyclostationarity, where patterns approximately repeat in a regular manner, often across multiple time scales, such as daily, weekly or yearly cycles. In this context, statistical inference is essential to distinguish genuine underlying effects from random variability. While tools like Analysis of Variance (ANOVA) provide such inference, they often lack interpretability and struggle with the complexities of multivariate data. To address these limitations, we propose a unified pipeline for the exploratory analysis of cyclostationary times series using ANOVA Simultaneous Component Analysis (ASCA). ASCA is an extension of ANOVA that is able to work in both univariate and multivariate cases. Combining inference with the visualization capabilities of Principal Component Analysis (PCA), ASCA provides powerful options for interpretability. ASCA's capabilities have been well-established in the analysis of experimental data, but they remain largely unexplored for observational data like time series. Our workflow introduces an algorithmic approach to modeling time-dependent data using ASCA, enabling control over multiple cyclostationary time scales while also accounting for the specific challenges of this type of data, such as autocorrelation. Furthermore, we observed that ASCA provides a better separation of variability across factors than ANOVA in unbalanced designs due to its multivariate nature. We demonstrate the efficacy of this methodology through two real-world case studies: water temperature trends in mountain lakes in Sierra Nevada, Spain, and airborne pollen trends over 30 years recorded in the city of Granada, Spain.

The Edge of Stability (EoS) is a phenomenon where the sharpness (largest eigenvalue) of the Hessian approaches and then hovers near the stability threshold $2/\eta$ during gradient descent (GD) with step size $\eta$. Despite (apparently) violating classical smoothness assumptions, EoS has been widely observed in deep learning, but its theoretical foundations remain incomplete. We provide an interpretation of EoS through the lens of Directional Smoothness [Mishkin et al., 2024]. This interpretation naturally extends to non-Euclidean norms, which we use to define generalized sharpness under an arbitrary norm. Our generalized sharpness measure includes previously studied vanilla GD and preconditioned GD as special cases, as well as methods for which EoS has not been studied, such as $\ell_{\infty}$-descent, Block CD, Spectral GD, and their normalized versions. Through experiments on neural networks, we show that non-Euclidean GD with our generalized sharpness also exhibits progressive sharpening followed by oscillations around or above the threshold $2/\eta$. Practically, our framework provides a geometry-aware spectral diagnostic that can be applied across a broad class of non-Euclidean gradient methods.

Overparameterized ML models, including neural networks, typically induce underdetermined training objectives with multiple global minima. The implicit bias refers to the limiting global minimum that is attained by a common optimization algorithm, such as gradient descent (GD). In this paper, we characterize the implicit bias of GD for training a shallow ReLU model with the squared loss on high-dimensional random features. Prior work (Vardi and Shamir, 2021) showed that the implicit bias does not exist in the worst-case, or corresponds exactly to the minimum-$\ell_2$-norm interpolating solution under exactly orthogonal data (Boursier et al., 2022). Our work interpolates between these two extremes and shows that, for sufficiently high-dimensional random data, the implicit bias approximates the minimum-$\ell_2$-norm solution with high probability with a gap on the order $\Theta(\sqrt{n/||\lambda||_1})$, where $n$ is the number of training examples and $\lambda$ denotes the spectrum of the data covariance matrix. Our results are obtained through a novel primal-dual analysis that carefully tracks the evolution of predictions, data-span coefficients, as well as their interactions, and show that the ReLU activation pattern quickly stabilizes with high probability over random data.

Gradient descent on overparameterized neural networks typically operates at the Edge of Stability (EoS), where the largest Hessian eigenvalue hovers around a step-size-dependent threshold. We study how sparse connectivity changes generalization below this threshold in two-layer ReLU networks. Prior results have shown that for fully-connected networks (FCNs), generalization guarantees in this regime degrade and become vacuous on high-dimensional spherical inputs. Our analysis reveals that sparse connectivity fundamentally alters this picture. Under sparse connectivity, the network processes a collection of low-dimensional patches rather than the full input vector, so the effective constraint imposed by the stability condition is governed by the geometry of the training patch collection. We prove that when the receptive fields are small relative to the ambient dimension, the effective constraint yields non-vacuous generalization bounds in precisely the spherical regime where FCNs provably fail. The same framework also reveals a contrasting failure mode: if the patch collection lacks geometric structure, the constraint becomes unable to prevent overfitting. We corroborate this theory by analyzing the patch geometry of natural images, showing that standard convolutional designs produce patch multiset with low-dimensional structure that facilitates generalization. This provides a principled explanation for the generalization advantage of convolutional networks. Thus, our analysis yields a unified framework that identifies how architecture, data geometry, and gradient descent jointly govern generalization performance.

Causal discovery with latent variables is a fundamental task. Yet most existing methods rely on strong structural assumptions, such as enforcing specific indicator patterns for latents or restricting how they can interact with others. We argue that a core obstacle to a general, structural-assumption-free approach is the lack of an equivalence characterization: without knowing what can be identified, one generally cannot design methods for how to identify it. In this work, we aim to close this gap for linear non-Gaussian models. We establish the graphical criterion for when two graphs with arbitrary latent structure and cycles are distributionally equivalent, that is, they induce the same observed distribution set. Key to our approach is a new tool, edge rank constraints, which fills a missing piece in the toolbox for latent-variable causal discovery in even broader settings. We further provide a procedure to traverse the whole equivalence class and develop an algorithm to recover models from data up to such equivalence. To our knowledge, this is the first equivalence characterization with latent variables in any parametric setting without structural assumptions, and hence the first structural-assumption-free discovery method. Code and an interactive demo are available at https://equiv.cc.

This paper deals with the problem of outliers in high frequency observation data from diffusion processes. Robust estimation methods are needed because the inclusion of outliers can lead to incorrect statistical inference even in the diffusion process. To construct a robust estimator, we first approximate the transition density of the diffusion process to the Gaussian density by using Kessler's approach and then employ two types of minimum robust divergence estimation methods. In this paper, we provide the asymptotic properties of the robust estimator using $\gamma$-divergence. Furthermore, we derive the conditional influence functions of the estimation using divergences and discuss its boundness.

Numerical models are widely used to simulate the Earth system, but they are computationally expensive and often depend on many uncertain input parameters. Calibration and uncertainty quantification require running them across many input configurations, so their effective use is costly. Statistical emulation provides a practical alternative for efficiently exploring model behavior. We are motivated by the Arctic sea-ice component of the Energy Exascale Earth System Model (MPAS-Seaice), which generates large spatiotemporal outputs at multiple spatial resolutions, with high-resolution (or high-fidelity, HF) simulations being more accurate but computationally more expensive than lower-resolution (low-fidelity, LF) simulations. Multi-fidelity (MF) emulation integrates information across resolutions to construct efficient and accurate surrogate models, yet existing approaches struggle to scale to large spatiotemporal data. We develop a MF emulator that combines tensor decomposition for dimensionality reduction, Gaussian process priors for flexible function approximation, and an additive discrepancy model to capture systematic differences between LF and HF data. The proposed framework scales to complex spatiotemporal fields while keeping predictions accurate and uncertainty well calibrated. In the MPAS-Seaice analysis, it attains the lowest prediction error of all emulators considered, including separable-covariance and neural-network approaches, especially throughout the summer melt season, when sea-ice concentration varies most across model settings and matters most scientifically, and best reproduces the spatiotemporal dependence of the output. By combining many inexpensive low-resolution runs with a few expensive high-resolution ones, the emulator substantially reduces the simulation cost of exploring model behavior, and suits large-scale studies involving complex physical models.

In this study, we focus on a generalized nonparametric scalar-on-function regression model for heterogeneously distributed and strongly mixing data. We provide almost complete convergence rates for the local linear estimator of the regression function. We show that, under our conditions, the pointwise and uniform convergence rates are the same on a compact set. On the other hand, when the data is dependent, it is proved that the convergence rate can be slower than those obtained for independent data. A simulation study shows the good performance and finite sample properties of the functional local linear estimator (FLL) in comparison to the local constant estimator (FLC). In addition, a one step ahead energy consumption forecasting exercise illustrates that the forecasts of the FLL estimator are significantly more accurate than those of the FLC.

Standard accounts of memory consolidation emphasise the stabilisation of stored representations, but struggle to explain representational drift, semanticisation, or the necessity of offline replay. Here we propose that high-capacity neocortical networks optimise stored representations for generalisation by reducing complexity via predictive forgetting, i.e. the selective retention of experienced information that predicts future outcomes or experience. We show that predictive forgetting formally improves information-theoretic generalisation bounds on stored representations. Under high-fidelity encoding constraints, such compression is generally unattainable in a single pass; high-capacity networks therefore benefit from temporally separated, iterative refinement of stored traces without re-accessing sensory input. We demonstrate this capacity dependence with simulations in autoencoder-based neocortical models, biologically plausible predictive coding circuits, and Transformer-based language models, and derive quantitative predictions for consolidation-dependent changes in neural representational geometry. These results identify a computational role for off-line consolidation beyond stabilisation, showing that outcome-conditioned compression optimises the retention-generalisation trade-off.

We introduce an augmented form of the van Trees inequality, that yields uniformly tighter lower bounds on the minimax squared Bayes risk of estimators compared with the classical van Trees inequality. Our augmented inequality also accommodates prior distributions whose densities need not vanish at the boundaries of their supports. We demonstrate how this refinement can be utilised for elementary proofs of a number of minimax lower bounds for nonparametric estimands, that also often attain sharper constants than those obtained by the alternative Le Cam convergence of experiments theory and the classical van Trees inequality, and in some cases obtain exact constants. As an example, our augmented van Trees inequality can be used to obtain the asymptotic minimax pointwise mean squared error when estimating the regression function in the model with normal errors: when the regression function is univariate and differentiable with Lipschitz derivative we obtain this quantity up to a constant factor of $1.37$; and in the high dimensional regime with a Hölder smooth regression function of smoothness $\beta\in(0,2]$ we obtain exact constants. Both these results do not follow from an application of the classical van Trees inequality. The flexibility of our augmented van Trees inequality accommodates lower bounds for models beyond Gaussianity, loss functions beyond the squared error loss, and we are also able to incorporate this augmentation into generalised versions of the van Trees inequality for irregular models.

We study sequential multiple testing with independent data streams, where the goal is to identify an unknown subset of signals while controlling commonly used error metrics, including generalized familywise rates and false discovery and non-discovery rates. For these problems, procedures that are first-order optimal are known, in the sense that the ratio of their expected sample size (ESS) to the minimal achievable ESS converges to one as the error tolerance levels vanish. In this work, we develop a unified theory of second-order asymptotic optimality. We establish general sufficient conditions under which second-order Bayesian optimality implies second-order frequentist optimality for broad classes of sequential testing procedures. As a consequence, several procedures previously known to be first-order optimal are shown to be second-order optimal: for every signal configuration, the difference between their ESS and the minimal achievable ESS remains uniformly bounded as the error tolerance levels tend to zero. In addition, we derive a second-order asymptotic expansion of the minimal achievable ESS, which refines the classical first-order approximation by identifying the second-order correction term arising from a boundary-crossing problem for a multidimensional random walk. We apply this result to several commonly used error metrics.

We study curvature-driven edge reweighting for community recovery in the balanced two-block stochastic block model. Given a graph G with initial weights equal to the adjacency matrix, we iteratively update edge weights using Lin-Lu-Yau (Ollivier-type) Ricci curvature, while all transportation costs are computed in the unweighted graph metric. In a moderate-density regime we prove uniform concentration of edge curvatures and show that a single Ricci reweighting step produces a two-level weighting that amplifies within-block connectivity relative to across-block connectivity. As a consequence, spectral clustering on the reweighted graph has a strictly larger population eigengap, and we obtain corresponding non-asymptotic perturbation bounds and Davis-Kahan misclustering guarantees. We further analyze a fixed finite horizon of iterated reweighting, where the random iterates track a deterministic two-weight recursion uniformly over the time horizon. This yields a principled finite-horizon curvature flow interpretation for community detection in a canonical random graph model.