Papers

We address the problem of observation noise misspecification in Bayesian filtering of dynamical systems via recent advances in generalised Bayesian inference. Mis-match in tail decay between the true data generating process and an assumed observation model, often showing via frequent outliers, can strongly impact Bayesian updates and analysis in Kalman filtering. Existing approaches often employ detect-and-delete-schemes or covariance inflation to avoid assimilation of influential instances of mis-specification. In challenging settings where the analysis updates are barely sufficient to counteract the induced forecast uncertainty, these strategies may destabilize or struggle to provide reliable uncertainty quantification. We consider a novel Kalman filter adjusting information processing in the analysis step by employing diffusion score matching for inference to obtain robustness while maintaining well-quantified uncertainties. We provide theoretical properties of the diffusion score matching Kalman filter in linear Gaussian state space systems covering conjugacy and closed form parameter update in the analysis step, robustness, covariance stability, and tuning as well as high-dimensional consistency. We derive ensemble approximations via stochastic and deterministic coupling as well as implementing localization to obtain EnKF, ESRF and LETKF varieties. We evaluate the methods in appropriate simulation studies on target-tracking, the chaotic Lorenz 63 system and the Lorenz 96 system in 40 dimensions. Our insights highlight a critical trade-off between robustness and stability in Bayesian filtering. Methods employing generalized Bayesian inference can navigate this balance and improve data assimilation in challenging environments combining non-linear dynamics and potentially non-Gaussian observation noise.

Metabolic syndrome is a complex clinical condition characterized by the simultaneous presence of multiple metabolic risk factors and represents a major public health concern. The syndrome develops silently and may remain undiagnosed for long periods, highlighting the importance of investigating early metabolic alterations before overt disease onset. Longitudinal monitoring of predominantly healthy individuals may help identify metabolic risk early. The paper proposes a Bayesian statistical model to estimate the probability of metabolic syndrome among blood donors during pre-donation screening, incorporating information collected at previous visits. Using longitudinal data from one of the main blood donor associations in Italy, AVIS Milan, we analyze repeated clinical and lifestyle measurements from a predominantly healthy population of donors. In particular, we fit a Bayesian multivariate model that jointly represents the logarithm of the five diagnostic components of metabolic syndrome. The model accounts for within-donor dependence across repeated visits and provides probabilistic estimates of individual risk. Our framework aims to provide clinicians at AVIS Milan with an interpretable traffic-light warning system (low, intermediate, high risk) during pre-donation screening to facilitate the identification of individuals at risk of metabolic syndrome at future visits and to support targeted preventive interventions during routine donor assessment, ultimately contributing to a long-term reduction in healthcare costs for the Italian national healthcare system.

Sampling is a fundamental algorithmic task in wide-ranging applications across multiple disciplines such as scientific computing, statistics and machine learning. In this paper, an efficient stochastic Runge-Kutta scheme is proposed to accelerate the Schrödinger-Föllmer sampler, designed for sampling from complex and high-dimensional multimodal distributions. The resulting stochastic Runge-Kutta Schrödinger-Föllmer sampler (SRKSFS) is proved to achieve a convergence rate of order $\mathcal{O} ( h^{3/2} |\ln h|)$ in the $L^2$-Wasserstein distance, considerably improving the order $\mathcal{O}(h)$ of the existing Euler type sampler. Obtaining the enhanced convergence rate is, however, not trivial, by noting that the drift of the diffusion process is not differentiable but only $\frac{1}{2}$-Hölder continuity with respect to the time variable. To address the difficulty, we rely on delicate error estimates to overcome the singularity due to time derivatives of the drift, at the expense of the logarithmic factor. Furthermore, the framework is extended to data-driven Schrödinger-Föllmer generation with empirical measures, enabling data-driven sampling without known density. A variety of numerical experiments are reported to validate the effectiveness of the proposed sampling algorithms.

Protesting mildly against the notion of an exactly correct parametric model the view is adopted that the logistic regression equation is merely an approximation to the underlying, true function. The behaviour of likelihood based estimators is investigated in such a general framework. The maximum likelihood estimator is shown to be consistent for a certain least false parameter value minimising a weighted average of quantities that measure the distance from the true to the parametric model. Asymptotic normality is also demonstrated. Finally a number of additional remarks are offered, some pointing to natural generalisations and some to new questions for research, like weighted and local likelihood estimation methods.

For Huber contamination on a known finite sample space, the unrestricted contaminating law is a probability vector on the support atoms, and domination over all measurable subsets reduces to atomwise inequalities. Placing a Dirichlet prior on this probability vector and a Beta prior on the contamination proportion gives an exact marginal likelihood for the structural parameter after analytic integration of both nuisance quantities. The likelihood is a finite weighted sum over allocations of the observed counts between the structural and contaminating components. For fixed support size, this sum and its score can be evaluated by a dynamic program with quadratic cost in the sample size, enabling gradient-based posterior sampling.

Prior-data fitted networks (PFNs) have recently emerged as a powerful approach for Bayesian prediction tasks, approximating the posterior predictive distribution (PPD) through in-context learning. Despite their strong empirical performance and ability to go beyond point predictions, theoretical understandings of the algorithmic capability of transformers to learn distributions in context are still lacking. Focusing on Gaussian process regression problems, we show by construction that transformers can implement a gradient descent algorithm targeting the posterior predictive mean and variance, followed by nonlinear mappings that yield binned probabilities of PPD. We study the error bounds of the approximated PPD in terms of attention depth and bin resolution. Based on these results, we further demonstrate the key role of normalization and the choice of attention depth in enabling the extrapolation abilities of transformers beyond the pretraining sample size range. We conduct simulations that corroborate our findings, providing insight into the expressivity of PFNs targeting PPDs and how architectural choices may influence generalization capabilities.

The classic concept of "calibrated forecasts" and its more recent refinement, "calibeating," are defined with respect to the standard quadratic scoring rule. We extend these notions to the class of proper scoring rules (for which the true distribution is an optimal forecast) and define proper calibration and proper calibeating by requiring the corresponding guarantees to hold uniformly over all bounded proper scoring rules. We first establish that calibration always implies proper calibration, whereas calibeating need not imply proper calibeating. Second, we show how to guarantee proper calibeating and proper multicalibeating; in particular, complete calibeating—a strong form of calibeating that calibeats the joint binning—is always proper. Finally, we consider decision-making under uncertainty, where one best replies to the forecasts. We establish that (proper) calibration is equivalent to universal no regret, and (proper) complete calibeating is equivalent to universal no regret together with universal subsuming of the reference sequence, where universal refers to all bounded utility functions.

Model merging offers a promising avenue for knowledge integration and parallel development without retraining. Yet, existing methods either ignore the geometry of the loss landscape or rely on intractable full-space Hessian approximations. We propose EpiMer, a framework that casts model merging as solving the Fréchet mean on a Riemannian manifold and restricts the computation to a low-rank subspace spanned by the task vectors. With the expected Hessian as the metric, we reveal a connection between local curvature and epistemic uncertainty of the parameters. Our theoretical analysis decomposes the merging error bound into the subspace Fréchet variance and the residual energy, and provides a closed-form characterization of when curvature-aware merging provably outperforms flat-geometry methods. In addition, our framework unifies both curvature-aware methods and recent spectral methods as special cases of the subspace Fréchet mean with different geometric metrics. Merging fine-tuned CLIP-ViT models on eight image classification tasks, Epistemic Merging strictly outperforms the baselines on all three CLIP-ViT backbones at matched rank, improving the across-task average accuracy and worst-task accuracy on every backbone.

CART random forests are among the most widely used modern predictive methods, with well-documented empirical success. Yet, at the mechanistic level, the algorithm is often treated as a black box because of its complexity. In this paper, we develop a stochastic-control perspective on feature-subsampled CART random forests, named CART random opportunity-set allocation (CART-ROSA). At each node, the random subset of features is interpreted as a random feasible action set, and the CART split rule as a masked-action allocation policy. This policy induces a controlled stochastic process over informative split-count states, whose terminal law determines both single-tree error and cross-tree interaction terms in the forest mean squared error (MSE). Such representation opens the black box of CART-forests by separating two design levers: the informative-opportunity rate induced by feature subsampling, and the contraction strength from the within-mask split policy. We establish that the CART policy is locally stabilizing: it contracts imbalances in informative split allocations and concentrates terminal tree geometry. At the system level, however, it can be globally suboptimal for the forest objective. Specializing to the linear model, we derive the MSE risk expansion explicitly. Our results show how an operations-research perspective makes tractable a theoretical gap difficult to access from the standard algorithmic description of CART forests.

Reinforcement learning (RL) often has a hierarchical structure, where an upper-level (UL) learner selects model parameters and a lower-level (LL) decision-making process responds, naturally leading to a bilevel optimization problem. Most existing bilevel RL methods assume a single-policy LL Markov decision process (MDP), and therefore fail to capture competitive structures arising in applications such as incentive design, where multiple policies interact. We study bilevel optimization problems in which the LL problem is a regularized min-max zero-sum Markov game and the UL objective is optimized through the saddle-point equilibrium induced by the LL game. In this work, we propose penalty-augmented Nikaido-Isoda descent-ascent (PANDA), a penalty-based first-order policy-gradient method based on the Nikaido-Isoda function. By exploiting the min-max game structure, PANDA avoids computing UL hypergradients and does not require second-order information. We prove that PANDA converges to stationary points without convexity assumptions on either the UL or LL objectives. Moreover, PANDA reaches an $\epsilon$-stationary point in $\tilde{\mathcal{O}}(\epsilon^{-1})$ iterations with sample complexity $\tilde{\mathcal{O}}(\epsilon^{-3})$, matching the best-known rates for bilevel RL with single-policy LL MDPs. Experiments demonstrate the superior performance of PANDA over closely related baselines.

In regression problems where covariates are naturally organized in a hierarchical tree structure, a central challenge is to select the resolution at which covariates enter the model. Determining this level of feature aggregation is of intrinsic scientific interest and can improve statistical efficiency by inducing sparsity. While a rich literature addresses this problem in the linear setting, extending feature aggregation to the nonlinear setting remains an open challenge. In this work, we propose to simultaneously perform model selection and feature aggregation through a penalized Nadaraya-Watson-type estimator. Our proposed estimator, Kernel Regression with Tree-EXploring AggregationS (KR-TEXAS), constructs adaptive penalty weights for the features based on pilot estimators of the regression function's partial derivatives. Under mild conditions, we establish model selection consistency for a well-defined target aggregation set, and our simulations show strong performance in both model selection and prediction. Finally, we demonstrate the utility of our procedure by applying it to a microbiome data set to predict short chain fatty acids. A user-friendly implementation of our procedure is available in the R package krtexas.

Feedforward network (FFN) layers account for a large fraction of parameters and nonlinear expressivity in Transformer-based large language models (LLMs). Despite the evolution from ReLU and GELU to gated variants such as SwiGLU, most FFN designs still use a single fixed activation function, applying the same nonlinear transformation to all tokens. In this work, we propose Mixture of Activations (MoA), a token-adaptive FFN design that mixes a dictionary of activation functions using lightweight input-dependent gates while sharing the same linear projections. As an input-independent counterpart, we also introduce learnable activations (LA), which form linear combinations of activation functions for both ReLU-type and SwiGLU-type FFNs. Theoretically, we establish strict finite-width expressive separations among fixed-activation FFNs, LA, and MoA: LA strictly contains fixed-activation FFNs, while MoA strictly contains LA, with the additional expressivity arising from input-dependent nonlinear hybridization. Empirically, we evaluate MoA through extensive pre-training experiments on dense and MoE language models ranging from 0.12B to 2B parameters under different token budgets, optimizers, and learning rate schedules. MoA consistently achieves lower terminal loss and exhibits more favorable scaling behavior than well-tuned baselines, with minimal parameter and computational overhead. These results suggest that token-adaptive activation mixing is a simple and effective mechanism for improving FFN expressivity in LLMs.

We study the sample complexity of policy gradient for log-growth control – the problem of learning, from observed state transitions, a feedback gain that optimally stabilizes a scalar linear system driven through a multiplicative-noise actuation channel. The objective $J(K) = \mathbb{E}[\log|1+BK|]$ is the top Lyapunov exponent of the closed loop. This problem carries a structural difficulty we call the cusp obstruction: the optimal gain $K^*$ always places the noise singularity $b_{\rm sing}(K) = -1/K$ in the interior of the support. At this singular optimum the policy gradient exists only as a Cauchy principal value, not as a Lebesgue integral, and the natural single-sample gradient estimator has infinite variance. Standard first-order stochastic-optimization analysis is thus inapplicable at the optimum, and merely smoothing the objective does not resolve the difficulty. The obstruction, however, has an exploitable symmetry: the Cauchy kernel is an odd function of the displacement from the moving pole, so pairing each observation with its reflection through the pole cancels the divergent part. This one cancellation simultaneously controls the population curvature, the gradient-estimator variance, and the bias incurred when the noise density is estimated. Combining these bounds with a closed-form single-transition gradient oracle, we prove that projected mini-batch policy gradient, initialized in any compact subset of the stabilizing region, attains total sample complexity $\tilde{O}(1/\eta)$ when the noise density is known and $\tilde{O}(\eta^{-(2s+1)/(2s)})$ when it must be estimated, for $C^s$ noise densities with $s \geq 2$.

We propose KO-PDE-IDENT, a data-driven framework for identifying parsimonious partial differential equations (PDEs) with false discovery rate (FDR) control. PDE discovery from noisy observations is often hindered by extreme multicollinearity among candidate terms, which causes typical sparse-regression methods to select spurious terms. To address this problem, KO-PDE-IDENT initially mines a support set of potential candidate terms via model-X knockoff filters with finite-sample FDR control, then refines and ranks the surviving PDE alternatives. The framework integrates three components. First, knockoff feature statistics are constructed by coupling $\ell_{0}$-constrained adaptive best-subset selection with SHapley Additive exPlanations (SHAP), yielding an effective and computationally efficient difference statistic. Second, a recursive feature elimination (RFE) procedure removes terms whose marginal contributions are dispensable and assesses statistical necessity through knockoff-perturbed hypothesis testing. Third, the final model selection is formulated as a multi-criteria decision-making (MCDM) problem, where the optimal governing equation is the alternative that best balances a wide range of criteria such as predictive accuracy, model complexity and coefficient uncertainty. We evaluate KO-PDE-IDENT on five canonical PDEs under severe noise corruption. Empirical results show that our framework can exactly recover the true PDE structure, eliminating false discoveries while retaining all true underlying terms, with low coefficient estimation error.

Monte Carlo simulation is used to assess a likelihood-based EM procedure for diagnosing higher-order temporal dependence in multivariate event streams. Multivariate Hawkes processes provide a likelihood-based framework for modelling self- and cross-excitation in continuous-time event streams, but their standard pairwise formulation can confound dense dyadic excitation with genuinely group-triggered temporal dependence. We introduce a hyperedge-triggered Hawkes model in which a candidate group contributes additional intensity only after all of its member nodes fire within a prescribed trigger window. A most-recent-anchor convention makes this higher-order component non-accumulating and leads to a corresponding piecewise compensator in the likelihood. The model is estimated by a latent-branching EM algorithm that assigns event-level responsibilities to background, pairwise, and hyperedge-triggered sources. Controlled simulations evaluate recovery, regularisation, convergence, separation from pairwise baselines, trigger-window and kernel-timescale sensitivity, computational scaling, and comparison with a parameter-matched third-order surrogate. The results show that higher-order excitation can be recovered in informative regimes, while pairwise–hyperedge confounding remains the main statistical limitation. Applications to retinal and visual-cortex spike trains are presented as exploratory predictive analyses: the hyperedge-triggered component can improve candidate-penalised held-out fit, but the fitted effects should be interpreted as diagnostics of higher-order temporal dependence rather than calibrated discoveries of individual biological hyperedges.

The use of dual system estimation (DSE) is heavily used in Census Bureau operations. With DSE methods, it is important to implement methods to infer the population size among those with missing data from one or both data sources. The use of log-linear models, calculated through EM algorithms, promises a way for estimation of counts among all groups with incomplete recorded data, as displayed by Van der Heijden et al. 2022. Unfortunately, the numerical computations involved scale very poorly the more the population is divided, to the point where simultaneous analysis of several demographic and geographic factors, such as state of residence and ethnicity, becomes computationally infeasible. Here, an alternative method to calculate the log-linear estimates will be provided, which can calculate the maximum likelihood estimator in orders of computation lower than through the EM algorithm.

Childhood anemia affects around 40% of children aged 6-59 months globally and arises from heterogeneous factors, limiting model generalizability. We evaluate a transformer-based tabular foundation model against classical supervised methods under cross-country and data-scarce settings. We used DHS data from 16 countries across Africa, Asia, Latin America, the Caucasus, and the Middle East (n=68,856). We compared Logistic Regression, XGBoost, LightGBM, and TabPFN v2.6. Performance was assessed using AUC-ROC, Brier score, and ECE. Generalization was evaluated using leave-one-country-out (LOCO), reverse-LOCO, and few-shot settings. Subgroup analyses included sex, age, residence, maternal education, and wealth. Feature importance was estimated using SHAP. TabPFN outperformed classical models in low-data regimes (<200 samples), showing higher discrimination and better calibration. Across countries, it achieved the lowest Brier score (0.042) and ECE (0.203). Under full-data settings, AUC-ROC ranged from 0.59-0.76 with small between-model differences ($\leq 0.05$). LOCO performance was stable (0.58-0.69), driven by country context. Reverse-LOCO showed asymmetric transferability. Subgroup performance was consistent with no systematic demographic bias. SHAP identified child age, altitude, and height-for-age z-score as dominant predictors, followed by wealth and maternal education. Performance in childhood anemia prediction is driven more by population variation than model choice. TabPFN provides advantages in low-resource settings through improved discrimination and calibration, highlighting foundation models as promising tools for data-scarce global health prediction.

Control charts for process monitoring are widely used in practice. Most control charts require the monitored (residuals) process to be serially independent (and to satisfy specified distributional assumptions), whereas undetected dependence (or violations of distributional assumptions) may severely affect the charts' performances. Therefore, (distribution-free) control charts for monitoring serial dependence are of utmost relevance for practice. Recently, various nonparametric control charts have been proposed for this purpose, which are based on ordinal patterns, and which showed an appealing performance in detecting different types of serial dependence. In this research, we further progress in this direction and develop novel nonparametric control charts being based on transcripts and algebraic distances (as derived from ordinal patterns). The performance of the newly proposed control charts is evaluated in a simulation study, and their application in practice is illustrated with a real-world data example from chemical industry.

Statistical procedures rarely retain all features of the observed data. A sufficient statistic removes information irrelevant to a parameter; a maximum likelihood estimate compresses an empirical objective into an optimizing point; and a hidden state in a sequential model compresses past observations into a learned representation. This article develops these practices under the unified notion of target-oriented statistical compression: a useful summary preserves what matters for an inferential, predictive, or decision-relevant target, rather than every detail of the realized data path. The central object is the conditional target process \(M_n=\E(Z\given\G_n)\), where \(Z\) is the target and \(\G_n=\sigma(T_n)\) is the information retained by the compression map \(T_n\). When \((\G_n)\) is a decreasing filtration, \((M_n)\) is a reverse martingale with limit \(M_\infty=\E(Z\given\G_\infty)\). Exact sufficiency corresponds to lossless compression, while approximate summaries such as penalized estimators, principal components, and neural-network hidden states produce reverse quasi-martingale defects measuring coherence loss across compression levels. The diagnostic \(r_n=|M_n-M_{n-1}|\) is treated as an observable stability proxy, not as an unbiased estimator of the theoretical defect. Boundary degeneracy in sequential binary problems is developed as a central application. Practical boundary claims require joint assessment of boundary closeness, uncertainty control, and trajectory stability. The companion paper \citet{chang2025rm} develops the corresponding stopping procedures, finite-sample bounds, and numerical evidence; the present paper provides the broader theoretical infrastructure and extends the framework to Gaussian, Poisson, and quasi-martingale monitoring problems.

Individualized randomized experiments are central to online platforms for optimizing personalized decisions in complex environments. In two-sided markets, however, standard treatment effect estimation is often invalid due to strong temporal and cross-unit interference, a challenge compounded when only aggregated data are available because of privacy or system constraints. To address these issues, we identify the Global Average Treatment Effect (GATE) using only group-level data from treatment and control groups. We first establish identification conditions based on aggregated observations, and then propose the Individualized Randomized Experiment Varying Coefficient Decision Process (IRE-VCDP) model, which accounts for interference through supply-demand dynamics. Building on this framework, we develop a complete procedure for estimation and statistical inference of the GATE, along with theoretical guarantees for the proposed test. Extensive simulations and real-world experiments using data from a leading ridesharing platform demonstrate the effectiveness of our approach.

This paper studies causal discovery for a directed acyclic graph under a structural equation model with additive heteroscedastic errors. We first establish new identifiability results for location-scale noise models, showing that heteroscedasticity can be leveraged to recover causal directions. Based on these insights, we propose a novel iterative procedure, Residual Simultaneous Quantile Estimation (RESQUE), where each iteration consists of a residual-construction stage and a composite quantile regression stage, enabling recursive identification of sink nodes via the invariance of conditional scale coefficients across quantiles. We then establish its theoretical guarantees for recovering topological order and graph structure, even when the number of variables diverges with the sample size. Simulation studies and application to benchmark datasets show that RESQUE performs favorably compared with existing methods, especially when causal information is partly encoded in the variance component. These results highlight exploiting structured variance signals for causal discovery and provide a principled framework for multivariate causal discovery beyond mean-based modeling.

Gaussian processes (GPs) provide a principled Bayesian framework for uncertainty estimation, but their computational complexity severely limits scalability to large datasets. We propose SIKA-GP, which accelerates GP inference using sparse inducing kernel approximations based on a dyadic ordered template basis, incurring only ${O}(\log M)$ complexity dependence on the number of inducing points. Our approach constructs compact and expressive kernel representations from sparsely activated bases, enabling efficient tensorized GPU computation and seamless integration with modern large-scale models. SIKA-GP can be naturally embedded into Bayesian neural networks (BNNs) with sparse activations, yielding significant speedups in both training and inference without sacrificing predictive performance. The method naturally extends to deep feature learning, addressing the scalability challenges introduced by deep architectures and high-dimensional feature representations. Empirical results on vision and transformer-based language benchmarks demonstrate that our approach consistently delivers fast and accurate GP models, providing a principled path toward scalable kernel learning.

Hierarchical composite endpoints are increasingly used in clinical trials to compare patients first on the most clinically important outcome and then, only when that comparison is tied, on lower priority outcomes. Under right censoring, a lower priority comparison may already be observed but still cannot contribute because the higher priority genuine tie required for descent through the hierarchy is not confirmed. Existing restricted win-statistic estimators address censoring by requiring such ties from higher priority to be observed as genuine ties. This all-or-nothing rule preserves the restricted-time estimand, but excludes pairs with censoring-induced ties even when their lower priority comparisons contain useful information. We propose conditional tie weighting, which replaces the unavailable higher priority genuine-tie indicator by its conditional probability given the observed pairwise data. The resulting estimator targets the same restricted-time win probabilities while allowing partially observed pairs to contribute fractionally when their lower priority comparison is informative. We establish identification and large-sample theory for the resulting two-sample U-statistics with estimated nuisance functions, and derive sandwich variance estimators for the win ratio, net benefit, and win odds. Simulations show substantial efficiency gains, especially under heavier censoring and longer restriction horizons. A reanalysis of the HF-ACTION trial illustrates how conditional tie weighting recovers information from censoring-induced ties in death-first hospitalization comparisons further apply our estimator to reanalyze a completed randomized clinical trial.

We show that LLMs encode syntactic distinctions not present in the Universal Dependencies (UD) tree distances that structural probes are trained to recover. On English wh-movement stimuli, we measure the probe distance between an embedded subject and its verb, whose UD tree distance is invariant across conditions. That distance is shorter than baseline when the embedded clause is finite and longer when it is infinitival – a within-clause sign asymmetry present in all 13 models across four families we test, at a majority of layers. No account based on UD distance, linear order, or monotone structural complexity can produce a sign reversal, while Minimalist phase theory can. Holding the matrix verb fixed while varying only the complement type reproduces the same finite-infinitival ordering in every model, ruling out a lexical-semantic explanation. The cross-clause pair separately reproduces the phase-count ordering of earlier work, validating the probes. A single activation patch, interchanging the clause-selecting matrix verb, moves the two pairs in opposite directions in 8 of 13 models. Because the causal intervention leaves the target string unchanged, it also rules out the clause-length and surface-cue explanations. Together these results reveal syntactic structure in LLMs beyond the UD probe target, and a Minimalist phase account is consistent with it.

This paper addresses structured out-of-distribution (OOD) testing in high-stakes machine learning applications. Traditional conformal methods rely on joint exchangeability, making it difficult to incorporate auxiliary information such as spatiotemporal or grouping structures. To overcome this limitation, we propose the structure-adaptive conformal q-value (SCQ), a significance index that integrates individual test evidence with structural patterns. We also develop pseudo-score-guided transductive automated model selection (P-TAMS), which adapts conformalized model selection to structured OOD testing across a toolbox of candidate models. Together, SCQ and P-TAMS form a unified framework under pairwise exchangeability, providing finite-sample error-rate control, improved power, and enhanced interpretability. Experiments on simulated and real data demonstrate that the proposed approach controls the false discovery rate and performs well across diverse settings.

Estimating equations arise in a wide range of statistical applications, including longitudinal and clustered data analysis, survival analysis, econometrics, and semiparametric inference. In high-dimensional settings, adding sparsity-inducing regularization often leads to computational challenges that are not fully addressed by standard penalized optimization routines. These challenges are closely tied to the structural form of the underlying estimating problem: mainly, the estimating function needs not be the gradient of a scalar objective and may involve asymmetric Jacobians, overidentification, nonsmoothness, nonconvexity, or nested optimization. This article first reviews the application areas of estimating equations, and then the computational methods for regularized estimating equations by organizing them into four broad formulations: minimization-type, Dantzig-type, regularization-type, and fixed-point-type approaches. We discuss the main numerical strategies associated with each formulation, including penalized optimization, constrained linear programming, iterative root-solving, and proximal fixed-point iteration. We also highlight the connection between regularized estimating equations and fixed-point problems, which provides a unified computational perspective for analyzing and solving regularized estimating equations.

Understanding the effects of interventions is central to scientific progress, with randomized controlled trials (RCTs) regarded as the gold standard for causal inference in many applied fields. However, RCTs are costly, time-consuming, and often constrained by ethical or practical limitations, motivating the need for causal methods able to draw conclusions from observational data. While such data is collected at ever larger scale, making its use for causal inference is often hindered by the fact that not all variables affecting treatment allocation and the outcome are observed: an issue known as unobserved confounding. In this paper, we introduce a new study design called confounder detection via treatment intent. The idea is to query a human expert who makes treatment decisions, and ask them to compare pairs of units proposed by a principled matching strategy, with the goal of eliciting unobserved variables that explain why treatment decisions differ. We provide a theoretical basis for such a procedure, ascertaining conditions under which such a study design may elicit unobserved confounders. Building on this newly established foundations, we study treatment effects of interventions in the intensive care unit (ICU). First, we show empirical evidence strongly indicating that electronic health records (EHRs) collected in ICUs are subject to unobserved confounding. By using clinical text notes as a proxy for physicians' knowledge and leveraging natural language processing, we provide a proof of concept for our methodology in a semi-synthetic environment with a known ground truth.

Causal discovery in time series is increasingly performed using nonlinear machine-learning models, yet the resulting causal relationships are almost always summarized by scalar edge scores. We argue that this practice obscures the true object learned by nonlinear autoregressive models: a state-dependent function whose effect varies across regimes, magnitudes, and contexts. We formalize function-valued causal influence for additive, contribution-decomposable architectures and show that scalar causal scores constitute a severe information bottleneck, conflating between-state variation with within-state residual noise. Using Neural Additive Vector Autoregression as a representative architecture, we introduce a practical framework based on Individual Conditional Expectation for estimating causal response functions directly from trained models. Through controlled synthetic experiments, we demonstrate that edges with indistinguishable scalar scores can exhibit qualitatively different functional behaviors, including monotonic, thresholded, saturating, and sign-changing effects. An applied case study on democratic development further shows that function-valued analysis reveals regime-specific and asymmetric causal structure systematically missed by score-centric approaches.

Small-area precipitation forecasts support real-time decisions for reservoir operation, irrigation planning, drought monitoring, and flash-flood response. Operational value depends not only on point accuracy, but also on calibrated exceedance probabilities and warning rules that remain stable when local weather regimes depart from the training climatology. We evaluate a reverse-martingale regularized recurrent neural network (\RMRNN) for probabilistic precipitation forecasting and sequential early warning. A backward-coherence penalty is added to the recurrent hidden state; the resulting residual process drives a Shiryaev–Roberts (SR) detector, so the same latent trajectory that produces the forecast also supplies a continuously updated drought or flood-regime indicator. The framework is tested on the Taiwan CWA dense rain-gauge network, CHIRPS v2 daily gridded precipitation over Taiwan and the Horn of Africa, and NOAA GHCN-Daily stations over the Texas Hill Country. Across 1{,}000 replications, \RMRNN{} matches or slightly improves the GRU baseline in RMSE, MAE, and CRPS at 1~h–72~h lead while substantially improving alarm characteristics. The SR detector reduces false-alarm ratios by a factor of three to five at matched detection power. In the 2020–2021 Taiwan drought, onset is flagged 8–12 days earlier than SPI-3 thresholding; in the 2023 Typhoon Haikui flood, flash-flood risk is signalled 4~h before the CWA operational alert.

Gaussian processes with stationary kernels on bounded domains exhibit inflated posterior variance near the boundary. Despite being a long-recognized artifact in geostatistics and a source of over-exploration in Bayesian optimization, the causes and effects of boundary-induced acquisition bias are underexplored. We trace the root cause to a simple geometric mechanism: the truncation of the kernel correlation neighborhood at the domain boundary creates an observation-independent distortion that worsens with dimensionality. We show how this distortion manifests across three acquisition classes: variance maximization concentrates selections at the corners, whereas negative integrated posterior variance and expected predictive information gain move selections inward to axis-aligned interior shells. These patterns arise without reference to any objective function, meaning that acquisition behavior can be dominated by kernel geometry rather than the desired task-specific uncertainty. To quantify this, we introduce a function-free selection-profile diagnostic for arbitrary acquisitions, kernels, and bounded-domain geometries.