Papers

We provide a comprehensive analysis of spot volatility inference in pure-jump semimartingales under two asymptotic settings: fixed-$k$, where each local window uses a fixed number of observations, and large-$k$, where this number grows with sampling frequency. For both active- and possibly inactive-jump settings, we derive generally nonstandard, typically non-Gaussian limit distributions and establish valid inference, including when the jump-activity index is consistently estimated. Simulations show that fixed-$k$ asymptotics offer markedly better finite-sample accuracy, underscoring their practical advantage for nonparametric spot volatility inference.

We establish a functional large deviation principle for fully connected multi-layer perceptrons with i.i.d. Gaussian weights (LeCun initialization) and general Lipschitz activation functions, including therefore the popular case of ReLU. The large deviation principle holds for the entire network output process on any compact input set. The proof combines exponential tightness for recursively defined processes, finite-dimensional large deviations, and the Dawson-Gärtner theorem, extending existing results beyond finite input sets and less general activations.

Wireframe parsing aims to recover line segments and their junctions to form a structured geometric representation useful for downstream tasks such as Simultaneous Localization and Mapping (SLAM). Existing methods predict lines and junctions separately and reconcile them post-hoc, causing mismatches and reduced robustness. We present Co-PLNet, a point-line collaborative framework that exchanges spatial cues between the two tasks, where early detections are converted into spatial prompts via a Point-Line Prompt Encoder (PLP-Encoder), which encodes geometric attributes into compact and spatially aligned maps. A Cross-Guidance Line Decoder (CGL-Decoder) then refines predictions with sparse attention conditioned on complementary prompts, enforcing point-line consistency and efficiency. Experiments on Wireframe and YorkUrban show consistent improvements in accuracy and robustness, together with favorable real-time efficiency, demonstrating our effectiveness for structured geometry perception. Our code is available at https://github.com/GalacticHogrider/Co-PLNet.

The asymptotic properties of multivariate Szász-Mirakyan estimators for cumulative distribution functions (cdf) supported on the nonnegative orthant are investigated. Explicit bias and variance expansions are derived on compact subsets of the interior, yielding sharp mean squared error characterizations and optimal smoothing rates. The analysis shows that the proposed Poisson smoothing yields a non-negligible variance reduction relative to the empirical cdf, leading to asymptotic efficiency gains that can be quantified through local and global deficiency measures. The behavior of the estimator near the boundary of its support is examined separately. Under a boundary-layer scaling that preserves nondegenerate Poisson smoothing as the evaluation point approaches the boundary of $[0,\infty)^d$, bias and variance expansions are obtained that differ fundamentally from those in the interior region. In particular, the variance reduction mechanism disappears at leading order, implying that no asymptotically optimal smoothing parameter exists in the boundary regime. Central limit theorems and almost sure uniform consistency are also established. Together, these results provide a unified asymptotic theory for multivariate Szász-Mirakyan cdf estimation and clarify the distinct roles of smoothing in the interior and boundary regions.

A widely used technique for improving policies is success conditioning, in which one collects trajectories, identifies those that achieve a desired outcome, and updates the policy to imitate the actions taken along successful trajectories. This principle appears under many names – rejection sampling with SFT, goal-conditioned RL, Decision Transformers – yet what optimization problem it solves, if any, has remained unclear. We prove that success conditioning exactly solves a trust-region optimization problem, maximizing policy improvement subject to a $\chi^2$ divergence constraint whose radius is determined automatically by the data. This yields an identity: relative policy improvement, the magnitude of policy change, and a quantity we call action-influence – measuring how random variation in action choices affects success rates – are exactly equal at every state. Success conditioning thus emerges as a conservative improvement operator. Exact success conditioning cannot degrade performance or induce dangerous distribution shift, but when it fails, it does so observably, by hardly changing the policy at all. We apply our theory to the common practice of return thresholding, showing this can amplify improvement, but at the cost of potential misalignment with the true objective.

Computation of confidence sets is central to data science and machine learning, serving as the workhorse of A/B testing and underpinning the operation and analysis of reinforcement learning algorithms. Among all valid confidence sets for the multinomial parameter, minimum-volume confidence sets (MVCs) are optimal in that they minimize average volume, but they are defined as level sets of an exact p-value that is discontinuous and difficult to compute. Rather than attempting to characterize the geometry of MVCs directly, this paper studies a practically motivated decision problem: given two observed multinomial outcomes, can one certify whether their MVCs intersect? We present a certified, tolerance-aware algorithm for this intersection problem. The method exploits the fact that likelihood ordering induces halfspace constraints in log-odds coordinates, enabling adaptive geometric partitioning of parameter space and computable lower and upper bounds on p-values over each cell. For three categories, this yields an efficient and provably sound algorithm that either certifies intersection, certifies disjointness, or returns an indeterminate result when the decision lies within a prescribed margin. We further show how the approach extends to higher dimensions. The results demonstrate that, despite their irregular geometry, MVCs admit reliable certified decision procedures for core tasks in A/B testing.

High-dimensional data often exhibit variation that can be captured by lower-dimensional factors. For high-dimensional data from multiple studies, one goal is to understand which underlying factors are common to all studies, and which factors are study-specific. As a particular example, we consider platelet gene expression data from patients in different disease groups. In this data, factors correspond to clusters of genes which are co-expressed; we may expect some clusters (or biological pathways) to be active for all diseases, while some clusters are only active for a specific disease. To learn these factors, we consider a nonlinear multi-study sparse factor model, which allows for both shared and study-specific factors. To fit this model, we propose a multi-study sparse variational autoencoder. The underlying model is sparse in that each observed feature (i.e. each dimension of the data) depends on a small subset of the latent factors. In the genomics example, this means each gene is active in only a few biological processes. We prove that the latent factor distributions are identifiable (up to element-wise transformations), and that the canonical shared and study-specific support structure is identifiable. Empirically, we demonstrate our method recovers meaningful factors in the platelet gene expression data.

Background: In medical and health professions education (HPE), AI is increasingly used to assess clinical competencies, including via virtual standardized patients. However, most evaluations rely on AI-human interrater reliability and lack a measurement framework for how cases, learners, and raters jointly shape scores. This leaves robustness uncertain and can expose learners to misguidance from unvalidated systems. We address this by using AI "simulated learners" to stress-test and psychometrically characterize assessment pipelines before human use. Objective: Develop an open-source AI virtual patient platform and measurement model for robust competency evaluation across cases and rating conditions. Methods: We built a platform with virtual patients, virtual learners with tunable ACGME-aligned competency profiles, and multiple independent AI raters scoring encounters with structured Key-Features items. Transcripts were analyzed with a Bayesian HRM-SDT model that treats ratings as decisions under uncertainty and separates learner ability, case performance, and rater behavior; parameters were estimated with MCMC. Results: The model recovered simulated learners' competencies, with significant correlations to the generating competencies across all ACGME domains despite a non-deterministic pipeline. It estimated case difficulty by competency and showed stable rater detection (sensitivity) and criteria (severity/leniency thresholds) across AI raters using identical models/prompts but different seeds. We also propose a staged "safety blueprint" for deploying AI tools with learners, tied to entrustment-based validation milestones. Conclusions: Combining a purpose-built virtual patient platform with a principled psychometric model enables robust, interpretable, generalizable competency estimates and supports validation of AI-assisted assessment prior to use with human learners.

Subsampling is a widely used and effective approach for addressing the computational challenges posed by massive datasets. Substantial progress has been made in developing non-uniform, probability-based subsampling schemes that prioritize more informative observations. We propose a novel stratification mechanism that can be combined with existing subsampling designs to further improve estimation efficiency. We establish the estimator's asymptotic normality and quantify the resulting efficiency gains, which enables a principled procedure for selecting stratification variables and interval boundaries that target reductions in asymptotic variance. The resulting algorithm, Maximum-Variance-Reduction Stratification (MVRS), achieves significant improvements in estimation efficiency while incurring only linear additional computational cost. MVRS is applicable to both non-uniform and uniform subsampling methods. Experiments on simulated and real datasets confirm that MVRS markedly reduces estimator variance and improves accuracy compared with existing subsampling methods.

Background: Multicollinearity inflates the variance of OLS coefficients, widening confidence intervals and reducing inferential reliability. Yet fixed variance inflation factor (VIF) cut-offs are often applied uniformly across studies with very different sample sizes, even though collinearity is a finite-sample problem. We quantify how collinearity and sample size jointly affect linear regression performance and provide practical guidance for interpreting VIFs. Methods: We simulated data across sample sizes N=100-100,000 and collinearity levels VIF=1-50. For each scenario we generated 1,000 datasets, fitted OLS models, and assessed coverage, mean absolute error (MAE), bias, traditional power (CI excludes 0), and precision assurance (probability the 95% CI lies within a prespecified margin around the true effect). We also evaluated a biased, misspecified setting by omitting a relevant predictor to study bias amplification. Results: Under correct specification, collinearity did not materially affect nominal coverage and did not introduce systematic bias, but it reduced precision in small samples: at N=100, even mild collinearity (VIF<2) inflated MAE and markedly reduced both power metrics, whereas at N>=50,000 estimates were robust even at VIF=50. Under misspecification, collinearity strongly amplified bias, increasing errors, reducing coverage, and sharply degrading both precision assurance and traditional power even at low VIF. Conclusion: VIF thresholds should not be applied mechanically. Collinearity must be interpreted in relation to sample size and potential sources of bias; removing predictors solely to reduce VIF can worsen inference via omitted-variable bias. The accompanying heatmaps provide a practical reference across study sizes and modelling assumptions.

We introduce BASTION (Bayesian Adaptive Seasonality and Trend DecompositION), a flexible Bayesian framework for decomposing time series into trend and multiple seasonality components. We cast the decomposition as a penalized nonparametric regression and establish formal conditions under which the trend and seasonal components are uniquely identifiable, an issue only treated informally in the existing literature. BASTION offers three key advantages over existing decomposition methods: (1) accurate estimation of trend and seasonality amidst abrupt changes, (2) enhanced robustness against outliers and time-varying volatility, and (3) robust uncertainty quantification. We evaluate BASTION against established methods, including TBATS, STR, and MSTL, using both simulated and real-world datasets. By effectively capturing complex dynamics while accounting for irregular components such as outliers and heteroskedasticity, BASTION delivers a more nuanced and interpretable decomposition. To support further research and practical applications, BASTION is available as an R package at https://github.com/Jasoncho0914/BASTION

Recent Transformer- and MLP-based models have demonstrated strong performance in long-term time series forecasting, yet Transformers remain limited by their quadratic complexity and permutation-equivariant attention, while MLPs exhibit spectral bias. We propose HaKAN, a versatile model based on Kolmogorov-Arnold Networks (KANs), leveraging Hahn polynomial-based learnable activation functions and providing a lightweight and interpretable alternative for multivariate time series forecasting. Our model integrates channel independence, patching, a stack of Hahn-KAN blocks with residual connections, and a bottleneck structure comprised of two fully connected layers. The Hahn-KAN block consists of inter- and intra-patch KAN layers to effectively capture both global and local temporal patterns. Extensive experiments on various forecasting benchmarks demonstrate that our model consistently outperforms recent state-of-the-art methods, with ablation studies validating the effectiveness of its core components.

Coarsened exact matching (CEM) is often promoted as a superior alternative to propensity score matching (PSM) for addressing imbalance, model dependence, bias, and efficiency. However, this recommendation remains uncertain. First, CEM is commonly mischaracterized as exact matching, despite relying on coarsened rather than original variables. This inexactness in matching introduces residual confounding, which necessitates accurate modeling of the outcome-confounder relationship post-matching to mitigate bias, thereby increasing vulnerability to model misspecification. Second, prior studies overlook that any imbalance between treated and untreated subjects matched on the same propensity score is attributable to random variation. Thus, claims that CEM outperforms PSM in reducing imbalance are unfounded, particularly when using metrics like Mahalanobis distance, which do not account for chance imbalance in PSM. Our simulations show that PSM reduces imbalance more effectively than CEM when evaluated with multivariate standardized mean differences (SMD), and unadjusted analyses indicate greater bias with CEM. While adjusted analyses in both CEM with autocoarsening and PSM may perform similarly when matching on few variables, CEM suffers from the curse of dimensionality as the number of factors increases, resulting in substantial data loss and unstable estimates. Increasing the level of coarsening may mitigate data loss but exacerbates residual confounding and model dependence. In contrast, both analytical results and simulations demonstrate that PSM is more robust to model misspecification and thus less model-dependent. Therefore, CEM is not a viable alternative to PSM when matching on a large number of covariates.

Shaping multi-megawatt loads, such as data centers, impacts generator dispatch on the electric grid, which in turn affects system CO2 emissions and energy cost. Substantiating the effectiveness of prevalent load shaping strategies, such as those based on grid-level average carbon intensity, locational marginal price, or marginal emissions, is challenging due to the lack of detailed counterfactual data required for accurate attribution. This study uses a series of calibrated granular ERCOT day-ahead direct current optimal power flow (DC-OPF) simulations for counterfactual analysis of a broad set of load shaping strategies on grid CO2 emissions and cost of electricity. In terms of annual grid level CO2 emissions reductions, LMP-based shaping outperforms other common strategies, but can be significantly improved upon. Examining the performance of practicable strategies under different grid conditions motivates a more effective load shaping approach: one that "cherry-picks" a daily strategy based on observable grid signals and historical data. The cherry-picking approach to power load shaping is applicable to any large flexible consumer on the electricity grid, such as data centers, distributed energy resources and Virtual Power Plants (VPPs).

Suicide is the tenth leading cause of death in the United States, yet evidence on medication-related risk or protection remains limited. Most post-marketing studies examine one drug class at a time or rely on empirical-Bayes shrinkage with conservative multiplicity corrections, sacrificing power to detect clinically meaningful signals. We introduce a unified Bayesian spike-and-slab framework that advances both applied suicide research and statistical methodology. Substantively, we screen 922 prescription drugs across 150 million patients in U.S. commercial claims (2003 to 2014), leveraging real-world co-prescription patterns to inform a covariance prior that adaptively borrows strength across pharmacologically related agents. Statistically, the model couples this structured prior with Bayesian false-discovery-rate control, illustrating how network-guided variable selection can improve rare-event surveillance in high dimensions. Relative to the seminal empirical-Bayes analysis of Gibbons et al. (2019), our approach reconfirms the key harmful (e.g., alprazolam, hydrocodone) and protective (e.g., mirtazapine, folic acid) signals while revealing additional associations, such as a high-risk opioid combination and several folate-linked agents with potential preventive benefit that had been overlooked. A focused re-analysis of 18 antidepressants shows how alternative co-prescription metrics modulate effect estimates, shedding light on competitive versus complementary prescribing. These findings generate actionable hypotheses for clinicians and regulators and showcase the value of structured Bayesian modeling in pharmacovigilance.

Boosting has garnered significant interest across both machine learning and statistical communities. Traditional boosting algorithms, designed for fully observed random samples, often struggle with real-world problems, particularly with interval-censored data. This type of data is common in survival analysis and time-to-event studies where exact event times are unobserved but fall within known intervals. Effective handling of such data is crucial in fields like medical research, reliability engineering, and social sciences. In this work, we introduce novel nonparametric boosting methods for regression and classification tasks with interval-censored data. Our approaches leverage censoring unbiased transformations to adjust loss functions and impute transformed responses while maintaining model accuracy. Implemented via functional gradient descent, these methods ensure scalability and adaptability. We rigorously establish their theoretical properties, including optimality and mean squared error trade-offs. Our proposed methods not only offer a robust framework for enhancing predictive accuracy in domains where interval-censored data are common but also complement existing work, expanding the applicability of existing boosting techniques. Empirical studies demonstrate robust performance across various finite-sample scenarios, highlighting the practical utility of our approaches.

In nutritional and environmental epidemiology, exposures are impractical to measure accurately, while practical measures for these exposures are often subject to substantial measurement error. Regression calibration is among the most used measurement error correction methods with external validation studies. The use of external studies to assess the measurement error process always carries the risk of introducing estimation bias into the main study analysis. Although the transportability of regression calibration is usually assumed for practical epidemiology studies, it has not been well studied. In this work, under the measurement error process with a mixture of Berkson-like and classical-like errors, we investigate conditions under which the effect estimate from regression calibration with an external validation study is unbiased for the association between exposure and health outcome. We further examine departures from the transportability assumption, under which the regression calibration estimator is itself biased. However, we theoretically prove that, in most cases, it yields lower bias than the naive method. The derived conditions are confirmed through simulation studies and further verified in an example investigating the association between the risk of cardiovascular disease and moderate physical activity in the health professional follow-up study.

Few-shot class-incremental learning (FSCIL) is a paradigm where a model, initially trained on a dataset of base classes, must adapt to an expanding problem space by recognizing novel classes with limited data. We focus on the challenging FSCIL setup where a model receives only a single sample (1-shot) for each novel class and no further training or model alterations are allowed after the base training phase. This makes generalization to novel classes particularly difficult. We propose a novel approach predicated on the hypothesis that base and novel class embeddings have structural similarity. We map the original embedding space into a residual space by subtracting the class prototype (i.e., the average class embedding) of input samples. Then, we leverage generative modeling with VAE or diffusion models to learn the multi-modal distribution of residuals over the base classes, and we use this as a valuable structural prior to improve recognition of novel classes. Our approach, Gen1S, consistently improves novel class recognition over the state of the art across multiple benchmarks and backbone architectures.

The Kullback-Leibler divergence, the Kullback-Leibler variation, and the Bernstein "norm" are used to quantify discrepancies among probability distributions in likelihood models such as nonparametric maximum likelihood and nonparametric Bayes. They are closely related to the Hellinger distance, which is often easier to work with. Consequently, it is of interest to characterize conditions under which the Hellinger distance serves as an upper bound for these measures. This article characterizes a necessary and sufficient condition for each of the discrepancy measures to be bounded by the Hellinger distance. It accommodates unbounded likelihood ratios and generalizes all previously known results. We then apply it to relax the regularity condition for the sieve maximum likelihood estimator.

Serving large language models requires assigning requests whose workloads grow during processing and whose completion times are unknown. With a synchronization barrier, lightly loaded workers wait for the slowest worker at every decode iteration. We model this loss through aggregate workload imbalance and formulate a joint admission and assignment problem with sticky requests and capacity limits. Short lookahead incorporates predicted completions of ongoing requests. For exact optimization without lookahead, we prove a comparison under independent geometric decode lengths, bounded independent nondegenerate prompt sizes, and batched releases satisfying a cumulative supply condition. The long-run expected imbalance improvement over first come, first served admission with request-count join-shortest-queue grows at least as the square root of the product of worker capacity and the logarithm of worker count, in the stated scaling regime. The comparison extends to capacity-aware round robin, random routing, and power of two choices. BalanceRoute approximations connect the formulation to practical routing. Experiments using up to 144 NPUs with proprietary and public traces show lower workload spread and higher throughput than the evaluated baselines. These empirical results complement the guarantee for exact optimization.

For subvector inference in the linear instrumental variables model under homoskedasticity but allowing for weak instruments, Guggenberger, Kleibergen and Mavroeidis (2019) (GKM) propose a conditional subvector Anderson and Rubin (1949) (AR) test that uses data-dependent critical values that adapt to the strength of the parameters not under test. This test has correct size and strictly higher power than the test that uses standard asymptotic chi-square critical values. The subvector AR test is the minimum eigenvalue of a data-dependent matrix. The GKM critical value function conditions on the largest eigenvalue of this matrix. Instead, this paper considers a data-dependent critical value function that conditions on the second-smallest eigenvalue, as this eigenvalue is the appropriate indicator of weak identification. We find that the data-dependent critical value function of GKM also applies to this conditioning and show that this test has correct size and power strictly higher than the GKM test when the number of parameters not under test is larger than one. This is the best feasible test, where we consider feasible to be within the class of conditioning on a single eigenvalue. Our proposed procedure further applies to the subvector AR test statistic that is robust to an approximate Kronecker product structure of conditional heteroskedasticity as proposed by Guggenberger, Kleibergen and Mavroeidis (2024), carrying over its power advantage to this setting as well.

In this work, we investigate the estimation of a parameter in PDEs using Bayesian procedures, and focus on posterior distributions constructed using Gaussian process priors, and its variational approximation. We establish contraction rates for the posterior distribution and the variational approximation in the regime of low-regularity parameters. Specifically, the ground truth is only assumed to belong to the support space of the prior rather than to its associated reproducing kernel Hilbert space. Also we derive contraction rates for the posterior distribution constructed using the randomly truncated Gaussian process prior. The analysis relies on a delicate approximation argument that approximates low-regularity ground truths by suitable elements in the reproducing kernel Hilbert space and balances various error sources. We illustrate the general theory on three nonlinear inverse problems for PDEs.

Sensitivity analysis for unmeasured confounding under incremental propensity score interventions remains relatively underdeveloped. Incremental interventions define stochastic treatment regimes by multiplying the odds of treatment, offering a flexible framework for causal effect estimation. To study incremental effects when there are unobserved confounders, we adopt Rosenbaum's sensitivity model in single time point settings, and propose a doubly robust estimator for the resulting effect bounds. The bound estimators are asymptotically normal under mild conditions on nuisance function estimation. We show that incremental effect bounds can be narrower or wider than those for mean potential outcomes, and that the bounds must lie between the expected minimum and maximum of the conditional bounds on E(Y^0|X) and E(Y^1|X). For time-varying treatments, we consider the marginal sensitivity model. Although sharp bounds for incremental effects are identifiable from longitudinal data under this model, practical estimators have not yet been established; we discuss this challenge and provide partial results toward implementation. Finally, we apply our methods to study the effect of victimization on subsequent offending using data from the National Longitudinal Study of Adolescent to Adult Health (Add Health), illustrating the robustness of our findings in an empirical setting.

We consider finite-sample inference for a single regression coefficient in the fixed-design linear model $Y = Z\beta + bX + \varepsilon$, where $\varepsilon\in\mathbb{R}^n$ may exhibit complex dependence or heterogeneity. We develop a group permutation framework, yielding a unified and analyzable randomization structure for linear-model testing. Under exchangeable errors, we place permutation-augmented regression tests within this group-theoretic setting and show that a grouped version of PALMRT controls Type I error at level at most $2\alpha$ for any permutation group; moreover, we provide an worst-case construction demonstrating that the factor $2$ is sharp and cannot be improved without additional assumptions. Second, we relate the Type II error to a design-dependent geometric separation. We formulate it as a combinatorial optimization problem over permutation groups and bound it under additional mild sub-Gaussian assumptions. For the Type II error upper bound control, we propose a constructive algorithm for the permutation strategy that is better (at least no worse) than the i.i.d. permutation, with simulations empirically indicating substantial power gains, especially under heavy-tailed designs. Finally, we extend group-based CPT and PALMRT beyond exchangeability by connecting rank-based randomization arguments to conformal inference. The resulting weighted group tests satisfy finite-sample Type I error bounds that degrade gracefully with a weighted average of total variation distances between $\varepsilon$ and its group-permuted versions, recovering exact validity when these discrepancies vanish and yielding quantitative robustness otherwise. Taken together, the group-permutation viewpoint provides a principled bridge from exact randomization validity to design-adaptive power and quantitative robustness under approximate symmetries.

We study the identification and estimation of long-term treatment effects under unobserved confounding by combining an experimental sample, where the long-term outcome is missing, with an observational sample, where the treatment assignment is unobserved. While standard surrogate index methods fail when unobserved confounders exist, we establish novel identification results by leveraging proxy variables for the unobserved confounders. We further develop multiply robust estimation and inference procedures based on these results. Applying our method to the Job Corps program, we demonstrate its ability to recover experimental benchmarks even when unobserved confounders bias standard surrogate index estimates.

Bidirectional causal relationships arising from mutual interactions between variables are commonly observed within biomedical, econometrical, and social science contexts. When such relationships are further complicated by unobserved factors, identifying causal effects in both directions becomes especially challenging. For continuous variables, methods that utilize two instrumental variables from both directions have been proposed to explore bidirectional causal effects within linear models. However, the existing techniques are not immediately applicable when the key variables of interest are binary. To address these issues, we propose a structural equation modeling approach that links observed binary variables to latent continuous variables through a constrained mapping. We firstly establish the identification results for bidirectional causal effects using a pair of instrumental variables. Then, we develop an estimation method for the corresponding causal parameters. We also conduct sensitivity analysis under scenarios where certain identification conditions are violated. Finally, we apply our approach to investigate the bidirectional causal relationship between heart disease and diabetes, demonstrating its practical utility in biomedical researches.

Empirical risk minimization (ERM) stability is usually studied via single-valued outputs, while convex non-strict losses yield set-valued minimizers. We identify Painlevé-Kuratowski upper semicontinuity (PK-u.s.c.) as the intrinsic stability notion for the ERM solution correspondence (set-level Hadamard well-posedness) and a prerequisite to interpret stability of selections. We then characterize a minimal non-degenerate qualitative regime: Mosco-consistent perturbations and locally bounded minimizers imply PK-u.s.c., minimal-value continuity, and consistency of vanishing-gap near-minimizers. Quadratic growth yields explicit quantitative deviation bounds.

Normative modeling estimates reference distributions of biological measures conditional on covariates, enabling centiles and clinically interpretable deviation scores to be derived. Most neuroimaging pipelines fit one model per imaging-derived phenotype (IDP), which scales well but discards multivariate dependence that may encode coordinated patterns. We propose denoising diffusion probabilistic models (DDPMs) as a unified conditional density estimator for tabular IDPs, from which univariate centiles and deviation scores are derived by sampling. We utilise two denoiser backbones: (i) a feature-wise linear modulation (FiLM) conditioned multilayer perceptron (MLP) and (ii) a tabular transformer with feature self-attention and intersample attention (SAINT), conditioning covariates through learned embeddings. We evaluate on a synthetic benchmark with heteroscedastic and multimodal age effects and on UK Biobank FreeSurfer phenotypes, scaling from dimension of 2 to 200. Our evaluation suite includes centile calibration (absolute centile error, empirical coverage, and the probability integral transform), distributional fidelity (Kolmogorov-Smirnov tests), multivariate dependence diagnostics, and nearest-neighbour memorisation analysis. For low dimensions, diffusion models deliver well-calibrated per-IDP outputs comparable to traditional baselines while jointly modeling realistic dependence structure. At higher dimensions, the transformer backbone remains substantially better calibrated than the MLP and better preserves higher-order dependence, enabling scalable joint normative models that remain compatible with standard per-IDP pipelines. These results support diffusion-based normative modeling as a practical route to calibrated multivariate deviation profiles in neuroimaging.

We present a method of constructing statistical intervals that obtain a natural middle ground between Bayesian and frequentist statistical intervals, previously unexplored in literature: To a p% Bayesian credible interval we should assign a p% belief after observing both the dataset and the interval, to p% frequentist intervals we can generally only assign a p% belief before observing either the data or the interval, while to the intervals proposed here we can assign a p% belief after observing the interval, but not necessarily after inspecting the full dataset ourselves. Even in fully non-parametric problems this only requires a prior over the parameter(s) of interest, not a high-dimensional prior over the full distribution, while maintaining many of the practical and philosophical advantages of Bayesian methods. We belief these methods may therefore provide significant advances in statistical methodology to a number of fields. This work is meant as a proof of principle: We concretely implement such intervals for two different problems and study the properties of resulting intervals. We discuss promising directions where the proposed type of interval may provide significant advantages.

Latent variable (LV) models are widely used in psychological research to investigate relationships among unobservable constructs. When one-stage estimation of the overall LV model is challenging, two-stage factor score regression (FSR) serves as a convenient alternative: the measurement model is fitted to obtain factor scores in the first stage, which are then used to fit the structural model in the subsequent stage. However, naive application of FSR is known to yield biased estimates of structural parameters. In this paper, we develop a generic bias-correction framework for two-stage estimation of parametric statistical models and tailor it specifically to FSR. Unlike existing bias-corrected FSR solutions, the proposed method applies to a broader class of LV models and does not require computing specific types of factor scores. We establish the root-n consistency of the proposed bias-corrected two-stage estimator under mild regularity conditions. To ensure broad applicability and minimize reliance on complex analytical derivations, we introduce a stochastic approximation algorithm for point estimation and a Monte Carlo-based procedure for variance estimation. In a sequence of Monte Carlo experiments, we demonstrate that the bias-corrected FSR estimator performs comparably to the “gold standard” one-stage maximum likelihood estimator. These results suggest that our approach offers a straightforward yet effective alternative for estimating LV models.