Papers

In uncertainty quantification, evaluating sensitivity measures under specific conditions (i.e., conditional Sobol' indices) is essential for systems with parameterized responses, such as spatial fields or varying operating conditions. Traditional approaches often rely on point-wise modeling, which is computationally expensive and may lack consistency across the parameter space. This paper demonstrates that for a pre-trained global Polynomial Chaos Expansion (PCE) model, the analytical conditional Sobol' indices are inherently embedded within its basis functions. By leveraging the tensor-product property of PCE bases, we reformulate the global expansion into a set of analytical coefficient fields that depend on the conditioning variables. Based on the preservation of orthogonality under conditional probability measures, we derive closed-form expressions for conditional variances and Sobol' indices. This framework bypasses the need for repetitive modeling or additional sampling, transforming conditional sensitivity analysis into a purely algebraic post-processing step. Numerical benchmarks indicate that the proposed method ensures physical coherence and offers superior numerical robustness and computational efficiency compared to conventional point-wise approaches.

This paper studies uncertainty quantification for large language models (LLMs) under black-box access, where only a small number of responses can be sampled for each query. In this setting, estimating the effective semantic alphabet size–that is, the number of distinct meanings expressed in the sampled responses–provides a useful proxy for downstream risk. However, frequency-based estimators tend to undercount rare semantic modes when the sample size is small, while graph-spectral quantities alone are not designed to estimate semantic occupancy accurately. To address this issue, we propose SHADE (Soft-Hybrid Alphabet Dynamic Estimator), a simple and interpretable estimator that combines Generalized Good-Turing coverage with a heat-kernel trace of the normalized Laplacian constructed from an entailment-weighted graph over sampled responses. The estimated coverage adaptively determines the fusion rule: under high coverage, SHADE uses a convex combination of the two signals, while under low coverage it applies a LogSumExp fusion to emphasize missing or weakly observed semantic modes. A finite-sample correction is then introduced to stabilize the resulting cardinality estimate before converting it into a coverage-adjusted semantic entropy score. Experiments on pooled semantic alphabet-size estimation against large-sample references and on QA incorrectness detection show that SHADE achieves the strongest improvements in the most sample-limited regime, while the performance gap narrows as the number of samples increases. These results suggest that hybrid semantic occupancy estimation is particularly beneficial when black-box uncertainty quantification must operate under tight sampling budgets.

The Nobel Prize in literature 1965 was awarded Mikhail Sholokhov (1905-1984), for the epic novel Tikhij Don about Cossack life and the birth of a new Soviet society (And Quiet Flows the Don, or The Quiet Don, in different translations). Sholokhov has been compared to Tolstoy and was at least one and two generations ago called `the greatest of our writers' in the Soviet Union. In Russia alone his books have been published in more than a thousand editions, selling in total more than sixty million copies. He was an elected member of the USSR Supreme Soviet, the USSR Academy of Sciences, and of the CPSU Central Committee. But in the autumn of 1974 an article was published in Paris, Stremya `Tihogo Dona' (Zagadki romana (`The Rapids of Quiet Don: the Enigmas of the Novel'), by the author and critic D$^*$. He claimed that Tikhij Don was not at all Sholokhov's work, but that it rather was written by Fiodor Kriukov, a more obscure author who fought against bolshevism and died in 1920. The article was given credibility and prestige by none other than Aleksandr Solzhenitsyn (a Nobel prize winner five years after Sholokhov), who wrote a preface giving full support to D$^*$'s conclusion. Scandals followed, also touching the upper echelons of Soviet society, and Sholokhov's reputation was faltering abroad (see e.g. Doris Lessing's (1997) comments; `vibrations of dislike instantly flowed between us'). Are we in fact faced with one of the most flagrant cases of plagiarism in the history of literature?

Statistical analysis of network data has attracted considerable attention in recent years, due to the rapid advancement of well-trained network models and the accessibility of large public network datasets. In this article, we propose a transfer learning procedure for boosting estimation accuracy of a target network structure based on the well-known Degree-Corrected Mixed-Membership (DCMM) model in the literature. By leveraging useful information from informative source datasets, we theoretically prove that the transfer learning procedure greatly improve the estimation accuracy for the target connection probability matrix. Our theoretical analysis also reveals that the benefits from knowledge transfer in this context attributes to the enlarged eigenvalue gap of the target connection probability matrix. Additionally, we propose a random projection step in conjunction with the conventional aggregation procedure to alleviate the heavy computational burden in practice. In the presence of potentially harmful sources, we further provide an iterative truncation algorithm for selecting useful datasets and avoiding negative transfer. Numerical results showcase the practical utility of our methods in real-world network dataset analysis, including journal citation network dataset and international trade network dataset.

Loss-based priors assign probability mass to parameter values according to the inferential loss incurred when they are excluded from the parameter space, and provide a general solution for discrete parameters. Extending this idea to continuous settings is challenging, as the exclusion of a single point induces no loss. We propose a neighbourhood-exclusion framework in which inferential loss is defined by removing a local region around each parameter value. Under standard regularity conditions, this yields a class of prior distributions driven by the local geometry of the Kullback–Leibler divergence. In one dimension, the resulting prior coincides with Jeffreys' prior, while in higher dimensions it leads to a family of priors indexed by the geometry of the exclusion region. The proposed formulation provides a unified extension of loss-based priors and offers a geometric interpretation of objective prior construction beyond isotropic settings.

Estimating the number of components is a fundamental challenge in unsupervised learning, particularly when dealing with high-dimensional data with many components or severely imbalanced component sizes. This paper addresses this challenge for classical Gaussian mixture models. The proposed estimator is simple: center the data, compute the singular values of the centered matrix, and count those above a threshold. No iterative fitting, no likelihood calculation, and no prior knowledge of the number of components are required. We prove that, under a mild separation condition on the component centers, the estimator consistently recovers the true number of components. The result holds in high-dimensional settings where the dimension can be much larger than the sample size. It also holds when the number of components grows to the smaller of the dimension and the sample size, even under severe imbalance among component sizes. Computationally, the method is extremely fast: for example, it processes ten million samples in one hundred dimensions within one minute. Extensive experimental studies confirm its accuracy in challenging settings such as high dimensionality, many components, and severe class imbalance.

Semi-supervised learning with manifold regularization is a classical framework for jointly learning from both labeled and unlabeled data, where the key requirement is that the support of the unknown marginal distribution has the geometric structure of a Riemannian manifold. Typically, the Laplace-Beltrami operator-based manifold regularization can be approximated empirically by the Laplacian regularization associated with the entire training data and its corresponding graph Laplacian matrix. However, the graph Laplacian matrix depends heavily on the prespecified similarity metric and may lead to inappropriate penalties when dealing with redundant or noisy input variables. To address the above issues, this paper proposes a new Semi-Supervised Meta Additive Model (S$^2$MAM) based on a bilevel optimization scheme that automatically identifies informative variables, updates the similarity matrix, and simultaneously achieves interpretable predictions. Theoretical guarantees are provided for S$^2$MAM, including the computing convergence and the statistical generalization bound. Experimental assessments across 4 synthetic and 12 real-world datasets, with varying levels and categories of corruption, validate the robustness and interpretability of the proposed approach.

Recent empirical observations suggest that network transitivity is highly correlated with community structure in many real-world networks. In this paper, we theoretically investigate this relationship by deriving the limits of the global and average clustering coefficients for the geometric block model (GBM). Both limits exhibit a phase transition; specifically, the functional forms of the limit functions differ between the weak and strong community structure strength regimes. For a GBM with balanced communities, the limits of the global and average clustering coefficients are identical, whereas these limits differ for unbalanced communities. In general, the clustering coefficients do not exhibit a monotonic relationship with community structure strength. Particularly, for a balanced GBM where the within-community edge probability is a constant multiple of the between-community edge probability, the limit decreases from $3/4$ to $3/5$ and subsequently increases toward an asymptotic upper bound of $3/4$ as the multiple grows from one. A similar pattern is observed for the global clustering coefficient in unbalanced settings, where both limits exhibit an explicit dependence on community size.

Physical activity (PA) is widely recognized as protective against mental distress, yet whether this benefit varies systematically across population subgroups remains poorly understood. Using pooled data from ten consecutive annual waves of the U.S. Behavioral Risk Factor Surveillance System (2015-2024; n = 3,242,218), we investigate heterogeneity in the association between leisure-time PA and frequent mental distress (FMD, >=14 days/month) across age groups. Survey-weighted logistic regression reveals a striking age gradient: the adjusted odds ratio for PA ranges from 0.89 among young adults (18-24) to 0.50 among adults aged 55-64, with the protective association strengthening monotonically with age. Temporal analysis across all ten years shows that the young-adult PA effect has been eroding over the past decade, with the 18-24 OR reaching 1.01 (null) in both 2018 and 2024 – paralleling the deepening youth mental health crisis. Causal Forest via Double Machine Learning independently identifies age as the dominant driver of treatment effect heterogeneity (feature importance = 0.39, 2.5x the next predictor). E-value sensitivity analysis, propensity score overlap checks, placebo tests, and imputation comparisons confirm the robustness of the findings. These results suggest that the well-documented exercise–mental health link may not generalize to the youngest adult population, whose distress appears increasingly driven by stressors that PA alone cannot mitigate.

We establish finite-time last-iterate guarantees for vanilla stochastic gradient descent in co-coercive games under noisy feedback. This is a broad class of games that is more general than strongly monotone games, allows for multiple Nash equilibria, and includes examples such as quadratic games with negative semidefinite interaction matrices and potential games with smooth concave potentials. Prior work in this setting has relied on relative noise models, where the noise vanishes as iterates approach equilibrium, an assumption that is often unrealistic in practice. We work instead under a substantially more general noise model in which the second moment of the noise is allowed to scale affinely with the squared norm of the iterates, an assumption natural in learning with unbounded action spaces. Under this model, we prove a last-iterate bound of order $O(\log(t)/t^{1/3})$, the first such bound for co-coercive games under non-vanishing noise. We additionally establish almost sure convergence of the iterates to the set of Nash equilibria and derive time-average convergence guarantees.

Inference-time LLM alignment methods, particularly activation steering, offer an alternative to fine-tuning by directly modifying activations during generation. Existing methods, however, often rely on non-anticipative interventions that ignore how perturbations propagate through transformer layers and lack online error feedback, resulting in suboptimal, open-loop control. To address this, we show empirically that, despite the nonlinear structure of transformer blocks, layer-wise dynamics across multiple LLM architectures and scales are well-approximated by locally-linear models. Exploiting this property, we model LLM inference as a linear time-varying dynamical system and adapt the classical linear quadratic regulator to compute feedback controllers using layer-wise Jacobians, steering activations toward desired semantic setpoints in closed-loop with minimal computational overhead and no offline training. We also derive theoretical bounds on setpoint tracking error, enabling formal guarantees on steering performance. Using a novel adaptive semantic feature setpoint signal, our method yields robust, fine-grained behavior control across models, scales, and tasks, including state-of-the-art modulation of toxicity, truthfulness, refusal, and arbitrary concepts, surpassing baseline steering methods. Our code is available at: https://github.com/trustworthyrobotics/lqr-activation-steering

Air pollution is a worldwide public health threat that can cause or exacerbate many illnesses, including respiratory disease, cardiovascular disease, and some cancers. However, epidemiological studies and public health decision-making are stymied by the inability to assess pollution exposure impacts in near real time. To address this, developing accurate digital twins of environmental pollutants will enable timely data-driven analytics - a crucial step in modernizing health policy and decision-making. Although other models predict and analyze fine particulate matter exposure, they often rely on modeled input data sources and data streams that are not regularly updated. Another challenge stems from current models relying on predefined grids. In contrast, our deep-learning approach interpolates surface level PM2.5 concentrations between sparsely distributed US EPA monitoring stations in a grid-free manner. By incorporating additional, readily available datasets - including topographic, meteorological, and land-use data - we improve its ability to predict pollutant concentrations with high spatial and temporal resolution. This enables model querying at any spatial location for rapid predictions without computing over the entire grid. To ensure robustness, we randomize spatial sampling during training to enable our model to perform well in both dense and sparse monitored regions. This model is well suited for near real-time deployment because its lightweight architecture allows for fast updates in response to streaming data. Moreover, model flexibility and scalability allow it to be adapted to various geographical contexts and scales, making it a practical tool for delivering accurate and timely air quality assessments. Its capacity to rapidly evaluate multiple scenarios can be especially valuable for decision-making during public health crises.

We study finite-horizon continuous-time policy evaluation from discrete closed-loop trajectories under time-inhomogeneous dynamics. The target value surface solves a backward parabolic equation, but the Bellman baseline obtained from one-step recursion is only first-order in the grid width. We estimate the time-dependent generator from multi-step transitions using moment-matching coefficients that cancel lower-order truncation terms, and combine the resulting surrogate with backward regression. The main theory gives an end-to-end decomposition into generator misspecification, projection error, pooling bias, finite-sample error, and start-up error, together with a decision-frequency regime map explaining when higher-order gains should be visible. Across calibration studies, four-scale benchmarks, feature and start-up ablations, and gain-mismatch stress tests, the second-order estimator consistently improves on the Bellman baseline and remains stable in the regime where the theory predicts visible gains. These results position high-order generator regression as an interpretable continuous-time policy-evaluation method with a clear operating region.

Discovering optimal designs through sequential data collection is essential in many real-world applications. While Bayesian Optimization (BO) has achieved remarkable success in this setting, growing attention has recently turned to context-specific optimal design, formalized as Contextual Bayesian Optimization (CBO). Unlike BO, CBO is inherently more challenging as it must approximate an entire mapping from the context space to its corresponding optimal design, requiring simultaneous exploration across contexts and exploitation within each. In many modern applications, such tasks arise across multiple potentially heterogeneous but related clients, where collaboration can significantly improve learning efficiency. We propose CCBO, Collaborative Contextual Bayesian Optimization, a unified framework enabling multiple clients to jointly perform CBO with controllable contexts, supporting both online collaboration and offline initialization from peers' historical beliefs, with an optional privacy-preserving communication mechanism. We establish sublinear regret guarantees and demonstrate, through extensive simulations and a real-world hot rolling application, that CCBO achieves substantial improvements over existing approaches even under client heterogeneity. The code to reproduce the results can be found at https://github.com/cchihyu/Collaborative-Contextual-Bayesian-Optimization

Postmarketing safety surveillance relies on data from spontaneous reporting systems (SRS) such as FAERS, EudraVigilance and VigiBase, and commonly uses SRS data mining methods to assess the associations between drugs and adverse events (AEs). Traditionally, these analyses have focused on signal detection framed as a binary decision problem, whereas more recent work has emphasized more nuanced inference involving signal strength estimation and uncertainty quantification. In this paper, we review contemporary SRS data mining approaches and their statistical underpinnings for safety assessment using data from major pharmacovigilance databases worldwide. In addition to methodological review, we provide practical guidance on data preprocessing for such analysis, including construction of SRS contingency tables using only aggregated AE-drug counts, as are publicly available from databases such as VigiBase and EudraVigilance. We illustrate the guidance via opioid-related datasets obtained from FAERS and VigiBase, complied with subsequent downstream SRS data analyses.

Generalized Additive Models (GAMs) can be used to create non-linear glass-box (i.e. explicitly interpretable) models, where the predictive function is fully observable over the complete input space. However, glass-box interpretability itself does not allow for the incorporation of expert knowledge from the modeller. In this paper, we present ParamBoost, a novel GAM whose shape functions (i.e. mappings from individual input features to the output) are learnt using a Gradient Boosting algorithm that fits cubic polynomial functions at leaf nodes. ParamBoost incorporates several constraints commonly used in parametric analysis to ensure well-refined shape functions. These constraints include: (i) continuity of the shape functions and their derivatives (up to C2); (ii) monotonicity; (iii) convexity; (iv) feature interaction constraints; and (v) model specification constraints. Empirical results show that the unconstrained ParamBoost model consistently outperforms state-of-the-art GAMs across several real-world datasets. We further demonstrate that modellers can selectively impose required constraints at a modest trade-off in predictive performance, allowing the model to be fully tailored to application-specific interpretability and parametric-analysis requirements.

Penalized generalized estimating equations (PGEE) stabilize point estimation for longitudinal binary data under near-separation, but inference still depends on how the sandwich variance is corrected. Existing corrections for PGEE can overadjust in high-leverage directions, require restrictive pooling assumptions, or add global regularization without explaining the bias. We establish first-order asymptotics for PGEE along convergent interior-root sequences and derive a matrix characterization of the parameter-specific overcorrection induced by full leverage adjustment. Finite-sample calibration is limited by both mean bias and the variability of leverage-corrected variance estimates. We propose $\hat{V}_{AR}$, which keeps the score-level leverage correction and adds a finite-sample upward translation dominated at first order by the finite-population factor, with a smaller centering term. In simulations, $\hat{V}_{AR}$ gives conservative or near-nominal type I error in low-event, small-$N$ settings, including $N = 10$, where several standard corrections remain anti-conservative and pooling estimators are unavailable for unbalanced designs.

Understanding and mapping extreme heat is critical for risk management and public health planning, particularly in regions with complex terrain and heterogeneous climate. We present a case study of extreme heat in the Four Corners region of the United States, using high-resolution surface skin temperature data from the North American Land Data Assimilation System to characterize spatially heterogeneous and seasonally varying extremes across complex terrain, and to assess their implications for heat-related public health risks. Spatial extremes exhibit complex dependencies across geographic regions, which require sophisticated statistical models to capture. While recent advances in spatial extreme value modeling provide flexible representations of joint tail dependencies, statistical inference remains computationally demanding, especially for datasets with a large number of locations. To address this, we propose a random scale mixture process that facilitates Bayesian inference of spatial extremes, and develop scalable inference strategies that leverage advances in spatial modeling and amortized learning. We evaluate the proposed inference methods through large-scale simulation studies, representing the first such extensive study in spatial extremes, and a high-resolution surface skin temperature application in the Four Corners region. Surface skin temperature is particularly useful as a predictor for air temperature, for studying heatwaves and related environmental phenomena, and to calculate heat indices reflecting downstream health risks at any location. Our findings provide insights into efficient, data-driven approaches for modeling spatial extremes, and serve as guidelines for practitioners in the fields of climate science, environmental risk assessment, and beyond.

Air quality monitoring in Italy relies on sparse, irregular, ground-based stations that provide high-quality but incomplete measurements of pollution. Chemical transport models (CTMs) offer full spatial and temporal coverage but smooth over local variability. We develop a spatial transfer-learning framework that integrates these two data sources to produce daily, fine-grid predictions of nitrogen dioxide (NO$_2$) concentrations across Italy for 2023, with uncertainty quantification. The resulting maps provide a resource for decision making in downstream applications such as epidemiology and environmental policy. Our approach builds on the geostatistical LatticeKrig framework, which uses compactly supported basis functions and coefficients governed by a sparse precision matrix. We learn a nonstationary, anisotropic correlation structure from the gridded CTM outputs using an image-to-image neural architecture that estimates millions of spatially varying parameters in a matter of seconds. The basis-function representation enables this covariance structure to be transferred to the point-level station data and projected onto a finer prediction grid, a key extension for handling the change of support between data sources. A likelihood-based refinement step then adjusts the correlation range to recover fine-scale variability smoothed out by the gridded data. The proposed methodology results in a flexible, non-stationary, and anisotropic representation of the spatial process, better accommodating the complex geography of Italy. Performance is assessed through experiments on both gridded CTM outputs and point-level station measurements, demonstrating improvements over the stationary formulation.

Recovering latent structure from count data has received considerable attention in network inference, particularly when one seeks both cross-group interactions and within-group similarity patterns in bipartite networks, which is widely used in ecology research. Such networks are often sparse and inherently imperfect in their detection. Existing models mainly focus on interaction recovery, while the induced similarity graphs are much less studied. Moreover, sparsity is often not controlled, and scale is unbalanced, leading to oversparse or poorly rescaled estimates with degrading structural recovery. To address these issues, we propose a framework for structured sparse nonnegative low-rank factorization with detection probability estimation. We impose nonconvex $\ell_{1/2}$ regularization on the latent similarity and connectivity structures to promote sparsity within-group similarity and cross-group connectivity with better relative scale. The resulting optimization problem is nonconvex and nonsmooth. To solve it, we develop an ADMM-based algorithm with adaptive penalization and scale-aware initialization and establish its asymptotic feasibility and KKT stationarity of cluster points under mild regularity conditions. Experiments on synthetic and real-world ecological datasets demonstrate improved recovery of latent factors and similarity/connectivity structure relative to existing baselines.

Popular software packages report four generalizations of the ANOVA F test when conducting a multivariate analysis of variance (MANOVA). The reported operating characteristics of these fours tests vary widely depending on which research article the reader chooses. Some studies report extremely high type I error rates for a particular test even under ideal assumptions of multivariate normality and homoskedasticity; other studies report rates near the nominal level despite violations of the model assumptions. This simulation study seeks to clarify this apparent contradiction by providing a systematic evaluation of the type I error rates of the four statistics used to test for a group effect in MANOVA.

Nonlinear machine-learning models are increasingly used to discover causal relationships in time-series data, yet the interpretation of their outputs remains poorly understood. In particular, causal scores produced by regularized neural autoregressive models are often treated as analogues of regression coefficients, leading to misleading claims of statistical significance. In this paper, we argue that causal relevance in nonlinear time-series models should be evaluated through forecast necessity rather than coefficient magnitude, and we present a practical evaluation procedure for doing so. We present an interpretable evaluation framework based on systematic edge ablation and forecast comparison, which tests whether a candidate causal relationship is required for accurate prediction. Using Neural Additive Vector Autoregression as a case study model, we apply this framework to a real-world case study of democratic development, modeled as a multivariate time series of panel data - democracy indicators across 139 countries. We show that relationships with similar causal scores can differ dramatically in their predictive necessity due to redundancy, temporal persistence, and regime-specific effects. Our results demonstrate how forecast-necessity testing supports more reliable causal reasoning in applied AI systems and provides practical guidance for interpreting nonlinear time-series models in high-stakes domains.

Spatially resolved transcriptomics is a fast-developing set of technologies that enables the measurement of localized gene expression across spatial locations in a sample. Detecting spatially varying genes is critical for analyzing such data, yet existing methods often fail to account for inter-gene correlations, leading to inflated false positive and false negative rates. Additionally, most prominent methods rely on predefined spatial covariance kernels, making them sensitive to the complexity of spatial expression patterns. Motivated by a human breast cancer dataset, we address these limitations in existing literature through JASPER (Joint Bayesian Analysis of SPatial Expression via Regression), a Bayesian framework that jointly models spatial expression patterns across multiple genes using a spatial basis function regression approach. We demonstrate the superior performance of JASPER compared to existing methods in several real-world spatial transcriptomic datasets and supporting simulation experiments. JASPER identifies genes with stronger spatial correlation and greater biological relevance, as validated by overlap comparison, enrichment analysis, and pathway analysis using independent biological databases. Our results highlight the ability of JASPER to improve the statistical and biological interpretability of spatial transcriptomics data, making it a powerful tool for uncovering spatial gene expression patterns in complex biological systems.

Masked diffusion large language models (dLLMs) are a promising alternative to autoregressive generation. While reinforcement learning (RL) methods have recently been adapted to dLLM fine-tuning, their objectives typically depend on sequence-level marginal likelihoods, which are intractable for masked diffusion models. To address this, we derive Discrete Tilt Matching (DTM), a likelihood-free method that recasts dLLM fine-tuning as state-level matching of local unmasking posteriors under reward tilting. DTM takes the form of a weighted cross-entropy objective with explicit minimizer, and admits control variates that improve training stability. On a synthetic maze-planning task, we analyze how DTM's annealing schedule and control variates affect training stability and prevent mode collapse. At scale, fine-tuning LLaDA-8B-Instruct with DTM yields strong gains on Sudoku and Countdown while remaining competitive on MATH500 and GSM8K.

Local prediction-error-based curiosity rewards focus on the current transition without considering the world model's cumulative prediction error across all visited transitions. We introduce Curiosity-Critic, which grounds its intrinsic reward in the improvement of this cumulative objective, and show that it admits a tractable per-step surrogate: the difference between the current prediction error and the asymptotic error baseline of the current state transition. We estimate this error baseline online with a learned critic co-trained alongside the world model; since the critic only has to learn how hard a transition is to predict, its estimate of the irreducible noise floor converges well before the world model saturates, redirecting exploration toward learnable transitions. The reward is higher for learnable transitions and collapses toward zero for stochastic ones, thereby separating epistemic (reducible) from aleatoric (irreducible) prediction error online. Prior prediction-error curiosity formulations, from Schmidhuber (1991) to learned-feature-space variants, emerge as special cases corresponding to specific approximations of this error baseline. Experiments on a stochastic grid world show that Curiosity-Critic outperforms prediction-error, visitation-count, and Random Network Distillation methods in training speed and final world model accuracy.

This paper develops a nonparametric density estimator with parametric overtones. Suppose $f(x,\theta)$ is some family of densities, indexed by a vector of parameters $\theta$. We define a local kernel smoothed likelihood function which for each $x$ can be used to estimate the best local parametric approximant to the true density. This leads to a new density estimator of the form $f(x,\hat\theta(x))$, thus inserting the best local parameter estimate for each new value of $x$. When the bandwidth used is large this amounts to ordinary full likelihood parametric density estimation, while for moderate and small bandwidths the method is essentially nonparametric, using only local properties of data and the model. Alternative ways more general than via the local likelihood are also described. The methods can be seen as ways of nonparametrically smoothing the parameter within a parametric class. Properties of this new semiparametric estimator are investigated. Our preferred version has approximately the same variance as the ordinary kernel method but potentially a smaller bias. The new method is seen to perform better than the traditional kernel method in a broad nonparametric vicinity of the parametric model employed, while at the same time being capable of not losing much in precision to full likelihood methods when the model is correct. Other versions of the method are equivalent to using particular higher order kernels in a semiparametric framework. The methodology we develop can be seen as the density estimation parallel to local likelihood and local weighted least squares theory in nonparametric regression.

A crucial question throughout statistics is whether an observed correlation between two variables is a direct correlation or only an indirect one mediated by a confounder. We organize the existing nonlinear measures of direct correlation into two families, each with a systematic construction: (i) removing the direct correlation from the joint distribution and quantifying the resulting distributional shift, and (ii) intervening on one variable via do-calculus and quantifying the response of the other. For every Kullback-Leibler-based measure in either family we propose a Jensen-Shannon-based regularized analogue; the regularized measures take values in $[0,1]$, satisfy the metric property, and are free of the singularities of the Kullback-Leibler divergence. We analyze the achievable upper bound of each regularized measure under the observed marginals, and derive the maximal value each measure can attain when only the alphabet sizes of the variables are fixed; the maxima admit closed forms built on a single binary-entropy function. The measures are compared on a decision-making model and on three public datasets (Titanic survival, UCI Adult income, and the 1973 Berkeley graduate admissions), with bootstrap confidence intervals for every reported value.

In this work, we revisit the problem of active sequential prediction-powered mean estimation, where at each round one must decide the query probability of the ground-truth label upon observing the covariates of a sample. Furthermore, if the label is not queried, the prediction from a machine learning model is used instead. Prior work proposed an elegant scheme that determines the query probability by combining an uncertainty-based suggestion with a constant probability that encodes a soft constraint on the query probability. We explored different values of the mixing parameter and observed an intriguing empirical pattern: the smallest confidence width tends to occur when the weight on the constant probability is close to one, thereby reducing the influence of the uncertainty-based component. Motivated by this observation, we develop a non-asymptotic analysis of the estimator and establish a data-dependent bound on its confidence interval. Our analysis further suggests that when a no-regret learning approach is used to determine the query probability and control this bound, the query probability converges to the constraint of the max value of the query probability when it is chosen obliviously to the current covariates. We also conduct simulations that corroborate these theoretical findings.

Verification of model outputs is rapidly emerging as a key primitive for both training and real-world deployment of large language models (LLMs). In practice, this often involves using imperfect LLM judges and reward models since ground truth acquisition can be time-consuming and expensive. We introduce Fully Unsupervised Score Ensembling (FUSE), a method for improving verification quality by ensembling verifiers without access to ground truth correctness labels. The key idea behind FUSE is to control conditional dependencies between verifiers in a manner that improves the unsupervised performance of a class of spectral algorithms from the ensembling literature. Despite requiring zero ground truth labels, FUSE typically matches or improves upon semi-supervised alternatives in test-time scaling experiments with diverse sets of generator models, verifiers, and benchmarks. In particular, we validate our method on both conventional academic benchmarks such as GPQA Diamond and on frontier, unsaturated benchmarks such as Humanity's Last Exam and IMO Shortlist questions.

The spiked Wigner ensemble is a prototypical model for high-dimensional inference. We study the spectral properties of an inhomogeneous rank-one spiked Wigner model in which the variance of each entry of the noise matrix is itself a random variable. In the high-dimensional limit, we derive exact equations for the spectral edges, the outlier eigenvalue, and the distribution of the components of the outlier eigenvector. These equations determine the BBP transition line that separates the gapped phase, where the signal is detectable, from the gapless phase. In the gapped regime, the distribution of the outlier eigenvector provides a natural estimator of the spike. We solve the equations for a noise matrix whose variances are generated from a truncated power-law distribution. In this case, the BBP transition line is non-monotonic, showing that an inhomogeneous noise can enhance signal detectability.