Papers

Hybrid switching Lévy-driven stochastic differential equations with pure-jump noise and state-dependent switching rates are studied under high-frequency observation. A three-stage inference procedure is proposed for the drift, scale, and switching-rate parameters, combining a staged Gaussian quasi-likelihood with an intensity-type contrast. Checkable sufficient conditions for weighted exponential ergodicity are established for the hybrid process; the proof does not rely on Brownian smoothing, but uses a fixed skeleton-chain argument combining small-jump accessibility and regime connectivity. Under ergodicity and the high-frequency sampling scheme, consistency, joint asymptotic normality, and a polynomial-type large deviation inequality are proved for the full estimator. The joint limit exhibits a transparent covariance structure: the drift and scale blocks are coupled through the third moment of the driving Lévy noise, whereas the switching-rate block is asymptotically uncorrelated with the continuous-coefficient blocks. Numerical experiments for models driven by normal inverse Gaussian noise illustrate the finite-sample behavior of the proposed estimators.

We establish approximation and learning guarantees for Fourier neural operators (FNOs) applied to time-$T$ solution operators of dissipative evolution equations. The analysis builds on the premise that FNOs can efficiently approximate and learn solution operators whenever these operators admit stable and accurate spectral discretizations. To formalize this idea, we introduce classes of evolution operators defined through spectral methods and derive FNO approximation bounds and polynomial sample complexity guarantees for these classes. For equations with polynomial nonlinearities, the learning rates depend primarily on the smoothness of the input space and the dimension of the physical domain. Our results hold uniformly over broad families of dissipative equations, rather than for a single fixed PDE, and apply in particular to the Navier–Stokes, Allen–Cahn, and Cahn–Hilliard equations. For equations with non-polynomial smooth nonlinearities, we prove that polynomial sample complexity still holds with rates that now additionally depend on the smoothness of the nonlinear terms and the dissipation strength. Overall, we connect classical spectral approximation theory with modern operator learning and explain when FNOs can learn nonlinear evolution operators efficiently.

In this paper, we look at the relationships that economic variables have with adverse health outcomes in the western counties of Washington, Idaho, Oregon, California, and Nevada, with specific emphasis on how suicide rate relates to such economic variables. Data was first gathered from Census and County Health Rankings for the entire United States (for website use and usefulness for future research), cleaned and regression-imputed, and then various exploratory data analysis methods were used, such as PCA, clustering, correlation gathering, linear fittings, and LASSO. PCA and clustering suggested that counties may group according to broader state-level economic patterns, although political interpretations would require additional electoral data. Correlation Analysis, along with LASSO and linear fittings, showed us the destructive variables that connected the most with economic variables (in terms of $R^2$ and correlation values seen), the economic variables that are most and least important in predicting suicide rate, and the possible relationships that suicide rate has with these economic variables.

Researchers often use the density of connections between groups of agents, such as communities, blocs, or markets, to characterize the structure of a social or economic network. In many cases, these groups are selected using the network data, making conventional fixed-group inference procedures potentially invalid. To address this issue, we develop two new confidence intervals that are universally valid post-selection in the sense that they guarantee simultaneous coverage asymptotically over all pairs of groups whose relative sizes do not vanish. Our first interval builds on a strategy of Berk et al. (2013). Our second interval is based on a Talagrand-type concentration inequality for empirical processes. Both intervals are simple to compute and scalable to large networks, but a key technical contribution of our paper is to show that the second interval is rate-optimal over a broader class of intervals. Three empirical illustrations show that accounting for selection can matter in practice. Some evidence for homophily in a social network and a hub-and-spoke structure in a trade network survives our correction, while evidence for a segmented market structure in a worker transition network does not.

Airbnb is a community based on connection and belonging – many hosts on Airbnb are everyday people who share their worlds to provide guests with the feeling of connection and being at home; Airbnb strives to connect people and places. Among our efforts to connect guests and hosts, we provide tools to enable hosts to set competitive prices, which helps improve affordability for guests while helping hosts get more bookings. We also personalize the guest experience to show them the listings that match their needs. To help inform these efforts, we combine economic modeling and causal inference techniques to understand how guests book stays based on the prices hosts set, among other factors, and how that preference varies across different guests and listings. Such understanding helps us identify opportunities for Airbnb to support the marketplace and better connect guests and hosts. For example, understanding how much guests respond to different prices helps optimize the tools that we provide to hosts, in order to enable hosts to choose and set competitive prices that further balance demand and supply. As another example, understanding heterogeneity in guest preferences helps us personalize the guest experience and better match them with the listings that meet their needs, based on how much they respond to different prices and other factors.

Parameter-Efficient Fine-Tuning (PEFT) commonly adapts pretrained weights through low-rank updates, and recent methods further exploit the singular value decomposition (SVD) of the base weight for initialization or subspace selection. However, these methods do not explicitly preserve the coupled geometry between the pretrained left and right singular bases. Motivated by recent minimum-perturbation theory, which shows that stable finetuning follows a coherent SVD rotation in which a single orthogonal $Q$ acts on both the left singular basis $U_0$ and the right singular basis $V_0$, we prove a per-slice analogue: each row slice of $W_0$ can be adapted by a shared orthogonal rotation $Q_i$ on its left basis $U_i$ and right basis $V_i$ together with a diagonal spectrum shift. We implement this form as CORA (Coherent Orthogonal Rotation Adaptation), which applies per-slice orthogonal rotations and a per-layer diagonal scale to the rank-$r$ SVD truncation of $W_0$. CORA uses $\tfrac{1}{2}m(r{-}1)$ trainable parameters per linear layer, about $4{\times}$ fewer than LoRA at the same rank. CORA outperforms LoRA, DoRA, PiSSA, and MiLoRA on commonsense reasoning and code generation while using about $8{\times}$ fewer parameters.

Variational Autoencoders (VAEs) commonly assume a standard isotropic Gaussian prior over the latent space, an assumption that often fails to capture the true distribution of latent representations for complex datasets. This mismatch can limit reconstruction accuracy, reduce sample quality, and constrain the expressive power of the learned latent space. We propose the eXact-Prior Variational Autoencoder (X-VAE), a framework that replaces the conventional standard normal prior with a Gaussian prior derived from the latent representations of a pretrained autoencoder (AE). Specifically, the empirical mean and standard deviation of the AE latent codes are used to parameterize a data-adaptive prior that more closely reflects the underlying structure of the training data. During generation, X-VAE introduces a latent scaling factor that enables explicit control over the variance of the sampled latent vectors, providing a simple mechanism for balancing sample diversity and fidelity. This flexibility makes the proposed approach particularly well suited for applications such as industrial and engineering design, where generated solutions must satisfy strict structural or functional constraints while still permitting meaningful design exploration. We present the mathematical formulation of well-suited X-VAE, derive the corresponding KL divergence objective for the proposed prior, and evaluate the method on standard benchmark datasets. Experimental results demonstrate that X-VAE preserves reconstruction quality while producing latent representations that better align with the empirical data distribution, leading to improved controllability and more realistic generated samples.

Federated Learning (FL) is a distributed machine learning (ML) paradigm with collaboration among multiple clients without sharing data. FL is challenging under data heterogeneity and partial client participation. Learning sparse models is useful for communication and computational efficiency in FL, but it is especially difficult in the small-sample high-dimensional regime (d >> N) where optimization can yield parameter configurations that fail to generalize to unseen test data. While magnitude-based pruning doesn't account for uncertainty exploration in the parameter space, a formulation with probabilistic gates and an L0 constraint allows sampling from competing sparse configurations during training. In this work, we study entropy regularization of gate distributions as a mechanism to maintain uncertainty in sparse federated optimization by preventing early commitment to sparse support. We examine its impact under data heterogeneity, client participation heterogeneity, and sparsity. Experiments on synthetic and real-world benchmarks show consistent improvements over federated iterative hard thresholding (Fed-IHT) and pruning after dense federated averaging (FedAvg) training, both in statistical performance on test data and in sparsity recovery accuracy.

This paper establishes finite-sample worst-case maximal inequalities for averages of independent centered heavy-tailed random vectors. The object of interest is the expected top-$k$ Euclidean norm of the sample average, which includes the expected coordinate-wise maximum as the special case $k=1$. Under coordinatewise variance constraints and tail-envelope constraints, the worst-case value is characterized up to universal constants over the class of distributions satisfying a finite $q$:th envelope moment condition. Analogous bounds are obtained for the sub-Weibull envelope class and the marginal sub-Weibull class.

We present an algorithm for the group distributionally robust (GDR) least squares problem. Given $m$ groups, a parameter vector in $\mathbb{R}^d$, and stacked design matrices and responses $\mathbf{A}$ and $\mathbf{b}$, our algorithm obtains a $(1+\varepsilon)$-multiplicative optimal solution using $\widetilde{O}(\min\{\mathsf{rank}(\mathbf{A}),m\}^{1/3}\varepsilon^{-2/3})$ linear-system-solves of matrices of the form $\mathbf{A}^{\top}\mathbf{B}\mathbf{A}$ for block-diagonal $\mathbf{B}$. Our technical methods follow from a recent geometric construction, block Lewis weights, that relates the empirical GDR problem to a carefully chosen least squares problem and an application of accelerated proximal methods. Our algorithm improves over known interior point methods for moderate accuracy regimes and matches the state-of-the-art guarantees for the special case of $\ell_{\infty}$ regression. We also give algorithms that smoothly interpolate between minimizing the average least squares loss and the distributionally robust loss.

In many service systems, an estimation of customers' waiting times for the service can assist in decision making focused on enhancing the operational efficiency, improving the customers' experience, and ensuring efficient resource allocation. In this paper, we study the customers' waiting times in a finite-capacity service system with a finite number of parallel servers and a shared waiting area. We consider two customer types, Type 1 and Type 2, with dynamic admission priorities, dynamically evolving customer type, and abandonment. We model the system under such assumptions using a continuous-time Markov chain (CTMC) and develop a methodology based on Krylov subspace approximation methods to evaluate the conditional waiting time distributions of Type 1 and Type 2 customers in the system. This methodology (CTMC-Krylov) offers a scalable computational approach that is well suited for analysing large complex systems. Next, we model this system using a quasi-birth-and-death (QBD) process and derive analytical expressions building on matrix-analytic methods to evaluate the conditional and long-run waiting time distributions using recursion. We illustrate the practical applicability of our models in a hospital system through a suite of numerical examples based on a large dataset obtained from a tertiary referral hospital in Australia, considering two types of patients, complex (Type 1) and other (Type 2).

Watermarking promises statistical traceability of large language model (LLM) uses, but real documents rarely arrive as purely human-written or purely LLM-generated. This motivates a quantitative question beyond detection: what proportion of a document is generated from a pre-specified watermarked LLM? We study this watermark proportion estimation problem under the Gumbel–max watermarking mechanism, treating the next-token prediction distributions as unknown and arbitrary nuisance parameters subject to a non-degeneracy condition. We compare two observation regimes: in the full observation regime, the estimator observes the pseudorandom vector and the selected token at each position; in the more prevalent setting of pivotal reduction, it observes only a scalar pivot, which follows a one-dimensional Uniform–Beta mixture distribution. Under pivotal reduction, we develop a Laguerre-polynomial estimator and establish a matching information-theoretic lower bound for the sample complexity. For full observation, we introduce an event-counting estimator and show a matching lower bound, yielding a substantially smaller sample complexity. As our results imply, although reducing to pivotal statistics is an elegant and prevalent choice, it is not always sample-efficient for estimating the proportion of watermarks.

In longitudinal studies, outcomes of interest are often truncated by death, meaning that they are only observed or well-defined conditional on intercurrent events such as survival. Existing strategies face a trade-off: causally interpretable estimands, such as survivor average causal effects, target a latent subgroup, whereas while-alive and composite summaries apply to the full population but are difficult to interpret as causal effects on the non-mortality outcome. We address these challenges by introducing methodology for a new set of estimands that (i) concern the entire population, (ii) remain causally interpretable, and (iii) leverage the longitudinal data commonly available in studies with outcomes truncated by death. The set of estimands includes single-world marginal separable effects that generalize conditional separable effects to full-population summaries. We develop identification and estimation results for these estimands and apply the methodology in a reanalysis of a prostate cancer trial, highlighting how different estimands can yield different treatment conclusions.

This short review examines the primary approaches for estimating the predictive distribution of Laplace-approximated Bayesian neural networks, with particular focus on the Generalized Linear Model (GLM) formulation. We survey the landscape of estimation strategies, from exact GLM computations requiring full Jacobian evaluations to Monte Carlo approximations that trade computational cost for statistical efficiency. The review covers the theoretical foundations of the Laplace approximation, the Kronecker-factored approximate curvature (KFAC) method for scalable posterior inference, and the various predictive estimation techniques developed in the literature. We provide a unified presentation that clarifies the relationships between methods and highlights their respective computational and statistical trade-offs.

We develop a framework for analyzing the learning dynamics of $\ell_2$-adversarial training of single-index models on Gaussian mixtures in the high-dimensional limit under streaming stochastic gradient descent (SGD). We derive deterministic equivalents for a broad class of statistics of the SGD iterates, including the adversarial risk and distance to adversarial optimality, in terms of the solution to a system of ODEs. We use them to study two idealized learning rate schedules: the Polyak stepsize and exact line search. In the case of $\ell_2$-adversarial least squares with a single class, we show that, unlike noiseless standard least squares, no constant learning rate guarantees monotone descent of SGD towards a minimizer of the adversarial risk. We identify anisotropic covariance and a mismatch in ridge parameters as the main sources of suboptimality of exact line search relative to the Polyak stepsize. We also introduce a stochastic differential equation (SDE), called adversarial homogenized SGD, that captures the evolution of statistics of the iterates of SGD. For $\ell_2$-adversarial least squares, using this SDE, we show the evolution of the risk is equivalent, up to dimension-free constants, to that of SGD on standard least squares with an adaptive learning rate and adaptive $\ell_2$-regularization. When the dynamics converge, the limiting adversarial risk and SGD iterate are determined by a fixed-point equation, with the limiting iterate being equivalent to the solution of a ridge regression problem whose regularization parameter is the limiting effective regularization of SGD.

We devise a novel estimator for a general quantile regression model with normal measurement errors in the covariates. The method is applicable to both linear and nonlinear quantile regressions and does not impose the quantile requirement on multiple quantile levels simultaneously. We circumvent the difficulties caused by discontinuity in quantile regression through kernel smoothing, and overcome the nonlinearity inherent in quantile regression via considering extension to the complex domain and moment generating functions. We show that the resulting estimator achieves the standard root-$n$ consistency and asymptotic normality under mild conditions. The performance of the proposed method is illustrated via numerical simulations and a real data example related to Cherry Blossom times in Japan in 2024. This is the first consistent estimator in a general quantile regression problem with normal measurement errors.

Three of the most popular methods for training language models to reason look like three different tricks. They are not. All three adjust a single number: standard deviation, reflecting how much a prompt's sampled answers disagree. When such a model is trained, it answers each problem many times, and an automatic checker marks every answer right or wrong. The standard deviation of those marks measures the disagreement: largest when the answers split evenly between right and wrong, and zero when they all agree. Group Relative Policy Optimization (GRPO) divides by this number, GRPO Done Right (Dr. GRPO) drops the division, and Decoupled Clip and Dynamic Sampling Policy Optimization (DAPO) discards the groups where it is zero. Each is presented as its own fix, yet this paper proves they are three settings of one dial. That dial is not cosmetic: for right-or-wrong rewards, the disagreement is exactly the size of the training update, the group-standard-deviation identity. A split group teaches the most, while a unanimous group teaches nothing and falls silent. The same result says which problems deserve the most weight and how many tries each one needs. This paper confirms the intuition on a large real difficulty dataset (Big-Math) and in a controlled training run. What looks like a harmless normalization step is the dial that decides where learning happens and how strongly.

We study uniform-in-time propagation-of-chaos for continuous-time Stein Variational Gradient Descent (SVGD). Classical finite-time propagation-of-chaos estimates for mean-field systems typically deteriorate rapidly with time and therefore do not directly explain the long-time relation between the finite-particle system and its mean-field limit. We obtain two complementary classes of uniform-in-time propagation-of-chaos results. For broad distributional metrics, we introduce a cutoff strategy which combines finite-time propagation-of-chaos estimates up to an $N$-dependent horizon with independent quantitative long-time convergence estimates for the finite-particle and mean-field SVGD flows. This yields uniform-in-averaging-time propagation-of-chaos bounds in Langevin kernel Stein discrepancy, Wasserstein-1 distance, and Wasserstein-2 distance, with logarithmic or iterated-logarithmic rates depending on the metric, target and kernel class. We also develop a finite-dimensional theory for matrix-valued finite-rank kernels. For Gaussian targets with bilinear kernels, the SVGD dynamics close exactly on first and second moments, yielding genuine uniform-in-physical-time parametric propagation-of-chaos rates in finite-dimensional Stein-feature metrics. We then prove a conjugacy principle showing that these feature-level estimates transfer to conjugate target-kernel pairs under orientation-preserving diffeomorphisms, thereby extending the theory to broad classes of nonlinear, including multimodal, targets. Together, these results highlight the contrast between generic distributional metrics, for which our general approach yields logarithmic rates, and closed finite-dimensional Stein observables, for which parametric $N^{-1/2}$ propagation-of-chaos rates persist uniformly in time.

Prediction from sequential panel data is central to digital-twin modeling, where new panels arrive over time and the predictive system is updated sequentially. Existing methods often rely on temporal proximity, which can fail when similar input-output patterns recur at nonadjacent times or when recent panels differ from the target panel. We propose State-Local Prediction (StaLoP), a nonparametric dynamic panel prediction framework that utilizes information through target-local predictive compatibility. StaLoP represents panels using target-local state vectors, compares historical and target panels via empirical discrepancy scores to determine relevance weights for the target point, and combines these weights with covariate localization. Theoretical results, including bias-variance characterization, asymptotic normality, simultaneous prediction bands, and a target-local-GDF-corrected MSPE criterion for panel and model selection, are developed. Extensive simulations validate the performance of StaLoP and support its theoretical properties. Applications to sequence prediction, simulator calibration, variable selection, and county-to-county migration-flow forecasting demonstrate improved out-of-sample prediction and provide scientific insights into the underlying applications.

Accurate modeling of wind turbine power curves is crucial for optimal wind farm operation. Nearly all existing power curve models focus on temporal variables such as wind speed and temperature while overlooking the influence of terrain covariates, which governs inflow wind conditions and thus also affects wind power production. This paper proposes a nonparametric spatio-temporal Gaussian process model that integrates temporal environmental covariates with spatial terrain features. The model falls in the category of spatial-temporal Gaussian process models with data on a grid. The challenge to be addressed is that the spatio-temporal modeling require certain temporal alignment among the data, a property that the wind farm data does not have. Our solution strategy is to construct a shared representative temporal covariate set which not only aligns the temporal inputs but also has a size an order of magnitude smaller than the original data size. With this transformation, our resulting model is able to employ a separable kernel structure that captures both spatial and temporal dependencies. Empirical analysis on a real wind farm dataset shows that our method improves predictive accuracy over existing baselines and can be used to quantify the various impact of the terrain characteristics on turbine performance.

We study dynamic random graphs in which the set of nodes is fixed, but edges evolve over time according to an underlying stochastic mechanism. Using a maximum-entropy approach, we define a probability distribution on graph trajectories that is consistent with observed constraints, capturing the inherent uncertainty in partially observed networks. We introduce a moment-based estimator for the parameters of this distribution and establish its statistical properties, such as consistency and asymptotic normality, with explicit formulas for the covariance structure. Numerical experiments demonstrate the estimator's accuracy and robustness across various dynamic network scenarios. Our framework bridges probabilistic modeling and statistical inference in time-varying networks, providing practical tools for understanding and predicting complex edge dynamics.

Stochastic Gradient Descent ($\textsf{SGD}$) is one of the most classical optimization algorithms with favorable theoretical guarantees, yet the practical implementation of $\textsf{SGD}$ differs subtly from its well-known form and is often referred to as Shuffling Stochastic Gradient Descent ($\textsf{Shuffling SGD}$). A particularly popular strategy in $\textsf{Shuffling SGD}$ is Random Reshuffling ($\textsf{RR}$), which has achieved great empirical success across numerous experiments. Despite its strong performance, $\textsf{RR}$ has long been considered a heuristic due to a lack of theoretical support. Over the last decade, people have finally established provable convergence rates for $\textsf{RR}$, thus justifying its observed superiority. However, for smooth convex optimization, two clouds over the convergence theory of $\textsf{RR}$ remain to this day. More precisely, according to the current theory, $\textsf{Shuffling SGD}$ under $\textsf{RR}$ converges only when the stepsize is smaller than a threshold proportional to $1/n$, where $n$ is the number of summands in the objective (or the number of data points). Consequently, the optimally tuned theoretical rate of $\textsf{Shuffling SGD}$ under $\textsf{RR}$ is strictly worse than that of $\textsf{SGD}$ when the number of epochs is smaller than another threshold proportional to $n$. These two restrictions heavily limit the applicability of existing theories and leave a critical mismatch with practice. In this work, for the first time, we prove that $\textsf{RR}$ dominates $\textsf{SGD}$ in smooth convex optimization under any reasonable stepsize after any finite number of epochs, thereby addressing a longstanding open question.

Insurance payments may depend on latent micro states although only macro states and realized payments are observed. We study a sojourn-payment model for such aggregated multi-state systems under left-truncation and right-censoring. Starting from a micro-to-macro projection, we establish strong consistency and weak convergence for inverse-probability-weighted estimators of state-specific cumulative payment processes.

We develop a Bayesian area-level small area estimation framework that jointly models binomial and Gaussian survey responses through shared spatial random effects. This work is motivated by the American Community Survey (ACS), which provides useful information that contributes to federal funding and policy making decisions, and often yields direct estimates with large standard errors in small domains. The proposed Multi-type model borrows strength across outcomes and spatial neighbors to improve the precision of the associated estimates. For the binomial component, Polya-Gamma data augmentation yields a conditionally Gaussian representation, while spatial basis functions provide dimension reduction for high-dimensional spatial data. Together, these features lead to closed-form conditional posteriors and, thus, an efficient Gibbs sampler. Through empirical simulations, we show that the proposed joint model improves estimation precision relative to independent Univariate models. Applying the method to ACS median income and poverty rate data, we find that the proposed Multi-type model yields similar point estimates but smaller posterior variances than the corresponding Univariate models.

Modern LLM workflows move coordinate-indexed objects across checkpoints: steering vectors, sparse autoencoders, top-$k$ neuron sets, attribution lists, and merge alignments. This is only well posed after fixing the model's residual-stream gauge, which we show is architecture-dependent: LayerNorm residual charts have permutation gauge $S_d$ (up to a global sign flip), while RMSNorm charts with generic per-channel gain have signed-permutation gauge $B_d = S_d \ltimes \{\pm 1\}^d$. Permutation-only alignment is therefore symmetry-incomplete for RMSNorm models. We introduce sign-marginalized Hungarian matching and prove a sharp failure mode: with decorrelated coordinates, raw signed-correlation matching has a structural permutation-accuracy ceiling at the positive-sign fraction of the true gauge, which sign-marginalization removes. We then make coordinate-preserving transport, not function-level merging, the primary object: composing saved-checkpoint local $B_d$ gauges along same-base fine-tuning trajectories recovers 91.1% of cross-run coordinates at 1500 steps versus 60.3% for endpoint matching, and the gain is not explained by merely routing through the base. The recovered gauge transfers tools that permutation-only alignment breaks: TinyLlama SAE reconstruction has NMSE 0.004 under $B_d$ versus 1.08 under $S_d$; Qwen sentiment steering preserves 95.8% of its effect versus 17.2%; refusal steering reverses sign under $S_d$; coordinate-preserving merges behave the same way. The same covariance governs stateful training: signed transport of AdamW state preserves the resumed trajectory, while permutation-only state follows a different one from a functionally identical checkpoint. Finally, gauge-sweep audits show index-level interpretability claims are reproducible only relative to an explicit gauge.

Argo profiling floats measure seawater temperature and salinity in the upper 2000 meters of the ocean. These floats are uniquely capable of measuring the global Ocean Heat Content (OHC), a quantity that is of central importance for understanding Earth Energy Imbalance. Yet, producing Argo-based OHC estimates with reliable uncertainties is statistically challenging due to the complex structure and large size of the Argo dataset. Here we present an end-to-end mapping and uncertainty quantification framework for Argo-based OHC estimation using state-of-the-art methods from spatio-temporal statistics. The framework is based on modeling vertically integrated Argo temperature profiles as a locally stationary Gaussian process defined over space and time. This enables us to produce computationally tractable OHC anomaly maps based on data-driven decorrelation scales estimated from the Argo observations. Our modeling choices are validated using statistical cross-validation, which demonstrates the importance of including a climatological time trend in the mean field and accounting for time in the covariance function. We quantify the uncertainty of these maps using local conditional simulation ensembles, a novel approach that leads to principled spatially and temporally correlated uncertainty quantification. A new paired cross-validation technique is presented to validate these uncertainties. The mapping framework is implemented in an open-source codebase that is designed to be modular, reproducible and extensible. To demonstrate the mapping and uncertainty quantification capabilities of this approach, we present new Argo OHC maps with uncertainties for 2004-2022 and report on various downstream climatological estimates and their uncertainties.

Identifying patients who are likely to benefit from a treatment is central to precision medicine and can guide follow-up trials, enrichment designs, and individualized decisions. Although randomized controlled trials (RCTs) provide evidence on efficacy, they are usually powered to estimate average treatment effects rather than patient-level benefit. Meanwhile, artificial intelligence and machine learning methods offer flexible tools for estimating heterogeneous treatment effects, especially when augmented by real-world data (RWD). However, in practice, these estimated effects are often translated into decisions through simple ranking or thresholding rules, which can ignore uncertainty and multiplicity when many patients are evaluated simultaneously. Motivated by this, we propose a model-agnostic conformal inference framework for uncertainty-aware beneficiary selection. The framework reformulates CATE-based treatment-benefit selection as a multiple-testing problem. For each candidate, we test whether the conditional treatment benefit exceeds a clinically meaningful threshold and construct a conformal p-value using RCT-based calibration. These p-values are then adjusted by the Benjamini-Hochberg procedure to control the false discovery rate (FDR) among selected beneficiaries. To improve efficiency when RCT sample sizes are limited, external data, such as RWD, can be used to train flexible treatment effect models, while conformal calibration remains anchored in the RCT data. It can be paired with conventional machine learning algorithms and emerging tabular foundation models. Simulations show that the framework maintains FDR control, with power depending on the base learner and external-data comparability. A case study in early-stage non-small-cell lung cancer illustrates how the method identifies candidate profiles with evidence of benefit from limited resection to reduce overtreatment.

We develop maximal inequalities for empirical processes indexed by graph-dependent observations. Our bounds separate the complexity of the indexing class from two features specific to graph dependence: the geometry of the underlying graph and the cost of coupling graph-separated blocks to independent copies. The coupling construction combines a novel graph-adapted dependence coefficient with a coloring of a block partition. As an application, we derive Glivenko–Cantelli results and characterize the associated effective sample size. A central implication is that graph-dependent empirical processes need not exhibit a generic root-$n$ rate: convergence is jointly determined by function-class complexity, graph geometry, and the decay of dependence with graph distance. We specialize the results to graphs with polynomial and exponential growth and to directed dyadic graphs. Finally, we apply the results to network autoregressive models, nonlinear local-propagation models, and treatment-interference settings.

While conformal prediction provides a general framework for uncertainty quantification in predictive inference, its application is often limited by computational cost. Recent methods, including Jackknife+ and Jackknife-minmax, achieve faster computation by trading a slight loss of efficiency relative to full conformal prediction, but still requires computing leave-one-out refits for all observations. In this paper, we further accelerate conformal prediction by incorporating approximate leave-one-out (ALO) estimators, and establish asymptotic coverage and efficiency. While our proof draws on methods developed for analyzing the consistency of ALO cross-validation risk estimators in high-dimensional statistics, it requires adaptations to handle conformal prediction, where leave-$i$-out residuals are needed for predictions at $x_{n+1}$ rather than just at the training covariate $x_i$. Simulation results validate our theoretical findings, showing that the ALO-based methods achieve coverage and efficiency comparable to the exact methods, while significantly reducing the runtime.

Protein-protein interaction (PPI) networks, estimated from high-throughput omics data, foster biomarker discovery and precision medicine. Gaussian graphical models (GGMs) offer a principled reconstruction framework. Yet, existing applications face two limitations: they overlook the rich existing knowledge encoded in curated biological databases, and they assume a homogeneous network structure across all individuals, neglecting the influence of covariates or confounding factors on these interactions and preventing personalised representations. Even though these limitations have been addressed separately in previous work, no current approach resolves them simultaneously. We introduce a prior-informed conditional Gaussian graphical model that integrates database-derived interaction priors with covariate-dependent network modeling in a unified, scalable framework. The key methodological innovation is a structured, weighted penalty that selectively incorporates priors into population-level network estimation, while leaving context-specific perturbations entirely data-driven, as curated databases capture canonical interactions rather than disease-specific signals. Simulation studies demonstrate consistent and robust improvements in population-level network reconstruction across diverse settings, even when prior knowledge is imperfect. Applied to UK Biobank cardiometabolic proteomics (n = 49,129, p = 366 proteins), the method recovers T2D-associated network perturbations, identifying 34 network-central candidate biomarkers, several detectable only through their connectivity, not differential expression, and revealing six biologically coherent protein communities with distinct pathway enrichments spanning metabolic, cardiovascular, and cancer-related processes. Code is available at https://github.com/AlessiaMapelli/Prior-informed-conditional-GGMs.