Papers

In this paper we consider parameter estimation for discretely observed diffusion processes. In particular, we focus on data that are observed at low frequency and methodology that can estimate parameters with uncertainty quantification. Most statistical work in this domain develops advanced Markov chain Monte Carlo (MCMC) algorithms for sampling from the posterior of the parameters, a task which is often complicated by the fact that one seldom has access to the transition density of the diffusion process; one has to combine sophisticated MCMC methods which are robust to the required time discretization of the diffusion, which can yield expensive algorithms. We focus on developing the martingale posterior method for the context of interest, when one can only numerically approximate the transition density of the diffusion. Based on using types of diffusion bridges we introduce a new martingale posterior method for parameter estimation for discretely observed diffusion processes. We prove that this algorithm approximates, in some sense, the martingale posterior which has no time-discretization bias up-to $\mathcal{O}(\Delta)$ if $\Delta$ is the time discretization step. Our approach is illustrated on several examples, showing orders of magnitude speed up versus state-of-the-art MCMC algorithms.

Reliable inference for spatial regression remains challenging because it requires the correct specification of the spatial dependence structure, the mean trend, and the error distribution. Existing parametric testing methods rely on restrictive assumptions that are difficult to verify in practice and can lead to inaccurate conclusions under misspecification. To address this, we develop a robust nonparametric Monte Carlo testing framework for spatial regression based on random shifts. We construct test statistics that measure the dependence between residuals, obtained after removing the effects of nuisance covariates, and the covariate of interest. This allows us to assess the significance of the covariate in the sense of partial correlation. The proposed framework enables robust inference across various models without requiring parametric assumptions or even a closed-form distribution of the test statistics. Furthermore, we establish the asymptotic exactness of the random shift test in the increasing-domain setting when the sample covariance is used as the test statistic. Through extensive numerical experiments, we demonstrate that our method maintains the nominal significance level while achieving competitive power, whereas parametric methods can exhibit inflated type I error rates, even when they are correctly specified.

Face recognition from a single image per person is a challenging problem because the training sample is extremely small. We consider a variation of this problem. In our problem, we recognize only one person, and there are no labeled data for any other person. This setting naturally arises in authentication on personal computers and mobile devices, and poses additional challenges because it lacks negative examples. We formalize our problem as one-class classification, and propose and analyze an algorithm that learns a non-parametric model of the face from a single labeled image and a stream of unlabeled data. In many domains, for instance when a person interacts with a computer with a camera, unlabeled data are abundant and easy to utilize. This is the first paper that investigates how these data can help in learning better models in the single-image-per-person setting. Our method is evaluated on a dataset of 43 people and we show that these people can be recognized 90% of time at nearly zero false positives. This recall is 25+% higher than the recall of our best performing baseline. Finally, we conduct a comprehensive sensitivity analysis of our algorithm and provide a guideline for setting its parameters in practice.

Policy gradient methods are reinforcement learning algorithms that adapt a parameterized policy by following a performance gradient estimate. Conventional policy gradient methods use Monte-Carlo techniques to estimate the gradient, which tend to have high variance, requiring many samples and resulting in slow convergence. We first propose a Bayesian framework for policy gradient, based on modeling the policy gradient as a Gaussian process. This reduces the number of samples needed to obtain accurate gradient estimates. Moreover, estimates of the natural gradient and a measure of the uncertainty in the gradient estimates, namely, the gradient covariance, are provided at little extra cost. Since the proposed framework considers system trajectories as its basic observable unit, it does not require the dynamics within trajectories to be of any particular form, and can be extended to partially observable problems. On the downside, it cannot exploit the Markov property when the system is Markovian. To address this, we supplement our Bayesian policy gradient framework with a new actor-critic learning model in which a Bayesian class of non-parametric critics, based on Gaussian process temporal difference learning, is used. Such critics model the action-value function as a Gaussian process, allowing Bayes rule to be used to compute the posterior distribution over action-value functions, conditioned on the observed data. Appropriate choices of the policy parameterization and of the prior covariance (kernel) between action-values yield closed-form expressions for the posterior of the gradient of the expected return with respect to the policy parameters. We perform detailed experimental comparisons of the proposed Bayesian policy gradient and actor-critic algorithms with classic Monte-Carlo based policy gradient methods, on a number of reinforcement learning problems.

This paper proposes an algorithm for real-time learning without explicit feedback. The algorithm combines the ideas of semi-supervised learning on graphs and online learning. In particular, it iteratively builds a graphical representation of its world and updates it with observed examples. Labeled examples constitute the initial bias of the algorithm and are provided offline, and a stream of unlabeled examples is collected online to update this bias. We motivate the algorithm, discuss how to implement it efficiently, prove a regret bound on the quality of its solutions, and apply it to the problem of real-time face recognition. Our recognizer runs in real time, and achieves superior precision and recall on 3 challenging video datasets.

Bayesian online learning provides a coherent framework for sequential inference. However, its theoretical understanding remains limited, particularly in the one-pass setting. Existing theoretical guarantees typically require the mini-batch sample size to diverge, a condition that fails in the one-pass regime. In this paper, we propose a new Bayesian online learning algorithm tailored to the one-pass setting, which incorporates a warm-start phase to ensure stable sequential updates. For this algorithm, we show that the sequentially updated posterior attains the optimal convergence rate. Building on this, we establish an online analogue of the Bernstein-von Mises theorem, which guarantees valid uncertainty quantification without diverging mini-batch sample sizes. Our analysis is based on a novel theoretical framework that differs fundamentally from existing approaches in the online learning literature. Numerical experiments on generalized linear models show that the proposed method matches the performance of the batch estimator while outperforming existing online procedures.

In diagnostic test accuracy meta-analysis (DTA-MA), standard inference methods using bivariate random-effects models for jointly synthesizing sensitivity and specificity can be sensitive to outlying studies and may yield misleading conclusions. In this article, we propose frequentist outlier-robust statistical inference methods for DTA-MA based on density power divergence. The proposed methods automatically downweight influential outlying studies by modifying the estimating function using the robust divergence with a tuning parameter. To achieve robust yet statistically efficient inference in the presence of outlying studies, the proposed methods incorporate practical strategies for selecting the tuning parameter, including a data-adaptive criterion based on the Hyvärinen score. We also quantify the contributions of individual studies to the robust pooled estimates, facilitating interpretation of how outlying studies affect the results. We illustrate the effectiveness of the proposed methods through an application to a DTA-MA of the Mini-Mental State Examination. Simulation studies showed that the proposed methods reduced bias and root mean squared error relative to existing methods and improved coverage probability in the presence of outliers. The proposed methods enable a sensitivity analysis to assess whether the main results obtained using standard methods are driven by outlying studies.

Conditional Average Treatment Effect (CATE) estimation in practice demands three properties simultaneously: heterogeneous effects $\tau(x)$, calibrated uncertainty over them, and robustness to the heavy tails that contaminate real outcome data. Meta-learners (Künzel et al., 2019) give (i); causal forests and BART give (i)-(ii) with Gaussian-tail assumptions; no widely used tool gives all three. We present Bayesian X-Learner, an X-Learner built on cross-fitted doubly robust pseudo-outcomes (Kennedy, 2020) with a full MCMC posterior over $\tau(x)$ via a Welsch redescending pseudo-likelihood. On Hill's IHDP benchmark the default configuration attains mean $\sqrt{\varepsilon_{\mathrm{PEHE}}} = 0.56$ on 5 replications (lowest mean; differences from S-/T-/X-learners, full-config Causal BART, and a causal forest baseline are not significant at $\alpha=0.05$, and rank ordering is unstable at 10 replications – IHDP comparisons are competitive rather than dominant). On contaminated "whale" DGPs with up to 20-25% tail density, a one-flag extension (contamination_severity) that selects a Huber-$\delta$ nuisance loss per Huber's minimax-$\delta$ relation recovers RMSE $\approx 0.13$ with tight credible intervals (single-cross-fit 30-seed coverage 83% [Wilson 66%, 93%] at 20% density; modular-Bayes pooling with Bayesian-bootstrap nuisance draws restores nominal 95% coverage).

This article introduces novel methodologies for estimating contextual exposure to HIV population viral load using GPS data. We propose a comprehensive analytical framework comprising (i) local (grid-cell level) estimation of HIV population viral load, (ii) derivation of individual activity spaces from GPS trajectories, and (iii) quantification of contextual exposure to HIV within these activity spaces. We integrate HIV surveillance and sociodemographic survey data with GPS-based mobility data collected in rural KwaZulu-Natal, South Africa, to characterize mobility patterns among young adults aged 20-30 years. Using derived measures of mobility and contextual exposure, we assess whether participants' sex and age systematically influence the magnitude, configuration, and heterogeneity of their mobility patterns. Furthermore, we describe analytical approaches to examine how contextual exposure to HIV evolves as activity spaces extend beyond static residential locations, outlining procedures to identify GPS-tracked participants at elevated risk of HIV acquisition. KEYWORDS: Population viral load exposure; GPS-based mobility analysis; Activity space

Causal inference is essential for data-driven decision-making, as it aims to uncover causal relationships from observational data. However, identifying causality remains challenging due to the potential for confounding and the distinction between correlation and causation. While recent advances in causal machine learning and matching algorithms have improved estimation accuracy, these methods often face trade-offs between interpretability and computational efficiency. This paper proposes a novel approach that combines a tree-based discretization technique, tailored for causal inference, with an integer linear programming-based matching algorithm. The discretization ensures approximately linear relationships for control datasets within strata, enabling effective matching, while the optimization framework optimizes for global balance. The resulting algorithm yields computational efficiency and less biased ATT estimates compared to state-of-the-art algorithms. Empirical evaluations demonstrate the proposed method's practical advantages over existing techniques in causal inference scenarios.

Regression models with both high-dimensional responses and covariates have attracted growing attention. Standard multivariate regression models become inadequate when the response variables depend not only on observed covariates but also on latent variables that capture key unobserved characteristics. To draw statistical inferences on covariate effects while accounting for latent variables, we consider a high-dimensional generalized latent variable model that accommodates mixed-type responses and allows for flexible dependence between covariates and latent variables, which is more suitable for many real-world applications than existing methods that either rely on a linear regression form or restricted assumptions on the dependence between covariates and latent variables. We develop an alternating algorithm that iteratively updates the regression parameters and the latent variables, transforming an intractable nonconvex problem into a sequence of tractable convex subproblems. Theoretically, we provide algorithmic guarantees by establishing statistical consistency of the resulting estimator and deriving an error bound for it. Further, building on this estimator, we construct a debiased estimator for the covariate effect and establish its asymptotic normality. The effectiveness of the proposed method is demonstrated through an application to evaluating the fairness of the Programme for International Student Assessment (PISA).

Pretrial risk assessment tools are used on over one million U.S. defendants each year, yet their use for predicting rare violent re-offense faces a basic statistical barrier. We derive a universal precision bound – the Likelihood Ratio Wall – showing that when violent re-arrest rates are low (2-5%), achieving even a 50% hit rate among people labeled "high risk" (positive predictive value, or PPV) would require tools far more discriminative than current instruments appear to be. For rare outcomes, a tool can have respectable-looking performance metrics and still be wrong most of the time it flags someone as "high risk for violence." We show that post-hoc score recalibration cannot solve this problem because it does not improve the tool's underlying ability to separate true positives from false positives. We further prove a Surveillance Ceiling: when over-policing inflates recorded "risk factors" among those who would not re-offend, the maximum achievable precision is structurally lower for over-policed groups, even at equal offense rates. We translate these results into the Number Needed to Detain (how many people must be detained to prevent one violent offense), and propose that risk reports should communicate this uncertainty explicitly. Our findings suggest that for rare violent outcomes, debates about fairness metrics alone are incomplete: under current data regimes, the available features may not support high-confidence individualized detention decisions.

Nonstationary Gaussian processes (GPs) are essential for modeling complex, locally heterogeneous spatial data. A common modeling approach is the spatial deformation method that warps the domain to recover isotropy. However, this static method does not account for changes in spatial correlation induced by covariates, limiting its ability to predict nonstationary GPs under new covariate conditions. To enable predictive modeling of the deformation method, we propose to model the spatial deformation as a function of covariates. The spaces of diffeomorphic deformations and Euclidean covariate vectors are connected by characterizing deformations as generated by velocity fields living in a Lie algebra. To overcome the estimation instability caused by high-order interactions between multiple covariates in a general Lie algebra, we prove that those interactions can be truncated with a moderate physical assumption. Based on the theoretical results, a concise functional form of deformations driven by multiple covariates can be established, and an efficient estimation-inference algorithm is developed for out-of-sample nonstationary GP prediction with limited covariate-deformation sample pairs. The effectiveness and generalizability of the method are demonstrated on a simulation study and two case studies, in the fields of manufacturing and geostatistics, respectively.

Malaria, childhood acute respiratory infection, and child undernutrition together account for over two million deaths annually in children under five, with the burden concentrated in low and middle-income countries where climate variability modulates transmission, exposure, and nutritional outcomes. Routine health surveillance in these settings remains sparse and reactive. Satellite-derived representations of the Earth's surface offer a scalable, low-cost complement to traditional covariates, yet their utility as predictors of population health outcomes is poorly characterised. We summarise findings from three studies evaluating AlphaEarth Foundations 64-dimensional satellite embeddings as predictors of population health outcomes, focusing on vulnerable populations. The studies span infectious disease (malaria, respiratory infection) and stunting. In each study, embeddings provide predictive value at sufficient spatial granularity: (i) malaria prediction across Nigeria shows consistent per-region R^2 gains; (ii) childhood acute respiratory infection prediction across 11 DHS countries increases pooled R^2 from 0.157 to 0.206 across three tree-based estimators; (iii) stunting prediction across 35 countries is neutral at country level due to collinearity with fixed effects. The stunting case is currently limited by lack of DHS cluster-level coordinates, which is the next key experiment.

Groundwater in the Densu Basin is increasingly threatened by heavy metal contamination, but conventional methods fail to capture the statistical complexity and spatial heterogeneity of pollution indicators. A key challenge is modelling the Heavy Metal Pollution Index (HPI), which is typically skewed and affected by correlated contaminants, leading to biased predictions without transformation. This study develops a predictive framework integrating response transformations with nested cross-validated ensemble machine learning. Three transformations (raw, log, and Gaussian copula) were applied to HPI and evaluated across six learners: support vector regression (SVM), $k$-nearest neighbours (k-NN), CART, Elastic Net, kernel ridge regression, and a stacked Lasso ensemble. Raw-scale models produced deceptively high fits (Elastic Net and stacked ensemble $R^2 \approx 1.0$), suggesting over-optimism. The log transformation stabilised variance (SVM: $R^2 = 0.93$, RMSE $= 0.18$; k-NN: $R^2 = 0.92$, RMSE $= 0.20$). The Gaussian copula gave the most reliable results: stacked ensemble $R^2 = 0.96$ (RMSE $= 0.19$), with other learners maintaining high accuracy. Copula-based models improved residuals and produced spatially plausible maps. DBSCAN clustering revealed Fe and Mn as primary HPI contributors, consistent with regional hydrogeochemistry. Limitations include reliance on random (not spatial) cross-validation and basin-specific scope. Future work should explore spatial validation and other geological settings. Overall, distribution-aware ensembles with clustering diagnostics offer robust, interpretable assessments of groundwater contamination.

Engineering design problems are often modeled as multi-objective optimization tasks in which a scalarized utility function selects an optimal design from the Pareto set. In practice, preferences are imperfectly known, so uncertainty in the preference model leads to uncertainty in the resulting optimal design. This paper proposes a probabilistic framework that treats preference parameters as random variables and examines how preference uncertainty propagates to decision uncertainty. A random preference vector induces a probability distribution over optimal designs, allowing us to identify which regions of the Pareto front are most likely to be selected and to assess recommendation stability under preference variability. To explain the sources of this variability, we apply variance-based global sensitivity analysis to the induced optimal solutions, using Sobol' indices and Shapley values to quantify the contributions of individual design variables and their dependencies. We further summarize the overall dispersion of the optimal-design distribution using the Fréchet variance, which provides a scalar measure of decision stability under a given preference model. Two vehicle design case studies demonstrate how problem structure can lead to discrete versus continuous decision distributions and show how the proposed quantities support preference-aware design analysis.

We study the effective estimation of the diffusivity and Hurst parameter for the homogenized limit of a class of slow/fast systems. Depending on the system parameters, this limit solves a stochastic differential equation driven by either a Wiener process or a Hermite process. In the class of models we consider, the fast variable is a fractional Ornstein–Uhlenbeck process. We show that estimators constructed from the homogenized limit remain consistent when applied to appropriately subsampled data generated by the original slow/fast system. A key tool in our analysis is the consistency of renormalized quadratic variations for a family of additive functionals of the fast process. Using Wiener chaos expansions, we obtain an \(L^2\)-orthogonal decomposition of these renormalized quadratic variations. This allows us to show that, under appropriate subsampling conditions, the consistency properties of the estimators are preserved even when the data is generated by the slow/fast system rather than the homogenized limit. We also show that, under stricter subsampling conditions, a non-central limit theorem is preserved in the case where the fluctuations of the estimator around the true value are non-Gaussian. As a direct consequence of convergence in \(L^2\), we obtain consistency of an estimator for the limiting self-similarity that does not require knowledge of the limiting diffusivity. Finally, we show that our results apply to a class of one-dimensional fluctuation models.

In Alzheimer's disease research, a clinically important question is how much longer individuals would remain dementia-free beyond a given time under different baseline amyloid statuses. We address this question using observational data from the Alzheimer's Disease Neuroimaging Initiative (ADNI) treating baseline amyloid status as the exposure. Estimation is challenging because amyloid status is confounded, time to dementia onset is heterogeneous and heavily right censored, and the target population depends on joint potential event times. At each time point, we consider the always-survivor principal stratum comprising individuals who would remain dementia-free under both amyloid status and estimate quantile contrasts in residual time to dementia onset. We model the joint distribution of event time, exposure, and baseline covariates using an enriched Dirichlet process mixture and conduct posterior inference via Bayesian g-computation. The framework accommodates partially observed covariates under a within-subcluster missing-at-random assumption, estimates contrasts across multiple time points and quantiles from one posterior fit and supports sensitivity analyses for unmeasured confounding, cross-world dependence, and informative censoring. Simulations show favorable finite-sample performance under heterogeneity and heavy censoring. In ADNI, residual time to dementia onset was shorter under elevated than non-elevated baseline amyloid status, both overall and within baseline subgroups.

Recent results have shown that for a linear tilt to a reference measure, the scores that would be produced under convolution with a normal variable can be expressed in terms of convolutions of the original density. Here, we extend that result to include constant negative diagonal tilts as well. The relationship follows from relating the denoisers of the two densities, which define the scores via Tweedie formula. A linear tilt results in a location shift to the score operator, while a quadratic tilt results in both a location shift and a time shift. Thus the scores of the tilted density can be understood as the scores of the original convolution process at a different location and noise level. These results are of interest to those in the score based diffusion community, and may lead to better score estimators which take advantage of these tilted score relationships.

Variable selection in linear regression models has been a problem since hypothesis testing began. Which variables to include or exclude from a model is not an easy task. Techniques such as Forward, Back ward, Stepwise Regression sequentially add or delete variables from a model. Penalized likelihood methods such as AIC, BIC, etc. seek to choose variables that have a significant contribution to the likelihood. Penalized sum of square methods such as LASSO and Elastic Net have been used to penalize small coefficients to only allow variables with large coefficients in the model. This work introduces an Artificial Intelligence approach to model selection where an ANN is trained to determine the significance of the variables based on OLS estimates. A simulation study shows the accuracy across various sample sizes and variances. Furthermore, a simulation study is conducted to compare the performance of the approach against Forward, Backward, AIC, BIC and LASSO. The approach is illustrated using a dataset from the World Health Organization regarding Life Expectancy. A github link is provided to the pretrained ANN that can handle up to 100 predictor variables, the original WHO dataset and the subset used in this work.

The Normalized Transformer, or nGPT (arXiv:2410.01131) achieves impressive training speedups and does not require weight decay or learning rate warmup. However, despite having hyperparameters that explicitly scale with model size, we observe that nGPT does not exhibit learning rate transfer across model dimension and token horizon. To rectify this, we combine numerical experiments with a principled use of alignment exponents (arXiv:2407.05872) to revisit and modify the $\mu$P approach to hyperparameter transfer (arXiv:2011.14522). The result is a novel nGPT parameterization we call $\nu$GPT. Through extensive empirical validation, we find $\nu$GPT exhibits learning rate transfer across width, depth, and token horizon.

Automatic feature engineering can improve predictive performance on tabular data by generating diverse feature transformations. However, the candidate space induced by combinations of input features and operators grows rapidly with dimensionality, resulting in substantial computational cost. We propose SCOPE-FE, a framework that controls the search space before candidate generation. SCOPE-FE combines FeatureClustering, a structural pair gate based on mixed-type feature association, with OperatorProbing, a dataset-specific utility control over operators. Unlike conventional expand-and-reduce approaches that generate a large candidate set and prune it afterward, SCOPE-FE focuses computation on a smaller, data-dependent candidate pool. Across ten OpenFE benchmark datasets, SCOPE-FE achieves a median candidate-space reduction of 82.9% and lowers component-summed feature-engineering time-including separately measured FeatureClustering overhead-on all ten datasets, yielding a geometric-mean speedup of 2.66x and a maximum speedup of 5.48x. Despite this reduction, SCOPE-FE is within the stated practical-equivalence margin of OpenFE on 8 of 10 datasets. An exhaustive candidate audit shows enrichment above uniform-random expectation on 8 of 10 datasets, with a median enrichment of 1.35x. Against Random-Pair, SCOPE-FE has higher enrichment on 6 of 10 datasets, with 5 of 10 significant; against Random-Operator and Random-Joint, it has higher enrichment on 8 of 10 datasets, with 8 of 10 significant for each. These results demonstrate that pre-generation search-space control can substantially reduce feature-engineering time while retaining a utility-enriched candidate pool under the OpenFE-compatible evaluation protocol.

Deep learning models for 12-lead electrocardiogram (ECG) analysis achieve high diagnostic performance but lack the intuitive interpretability required for clinical integration. Standard feature attribution methods are limited by the inherent difficulty in mapping abstract waveform fluctuations to physical anatomical pathologies. To resolve this, we propose a cross-modal method that projects feature attributions from high-performance 12-lead ECG models onto the CineECG 3D anatomical space. Our study reveals that while models trained directly on CineECG signals suffer from reduced accuracy and incoherent attributions, the proposed mapping mechanism effectively recovers clinically relevant feature rankings. Validated against a ground-truth dataset of 20 cases annotated by domain experts, the mapped explanations yield a Dice score of 0.56, significantly outperforming the 0.47 baseline of standard 12-lead attributions. These findings indicate that cross-modal averaging mapping effectively filters attribution instability and improves the localization of pathological features, combining the diagnostic expressiveness of standard ECG with the intuitive clarity of anatomical visualization.

We introduce Hyper Input Convex Neural Networks (HyCNNs), a novel neural network architecture designed for learning convex functions. HyCNNs combine the principles of Maxout networks with input convex neural networks (ICNNs) to create a neural network that is always convex in the input, theoretically capable of leveraging depth, and performs reliable when trained at scale compared to ICNNs. Concretely, we prove that HyCNNs require exponentially fewer parameters than ICNNs to approximate quadratic functions up to a given precision. Throughout a series of synthetic experiments, we demonstrate that HyCNNs outperform existing ICNNs and MLPs in terms of predictive performance for convex regression and interpolation tasks. We further apply HyCNNs to learn high-dimensional optimal transport maps for synthetic examples and for single-cell RNA sequencing data, where they oftentimes outperform ICNN-based neural optimal transport methods and other baselines across a wide range of settings.

In Orabona and Pál [2016], we introduced the shifted KT potentials, to remove the $\ln \ln T$ factor in the parameter-free learning with expert bound. In this short technical note, I show that this is equivalent to changing the prior in the Krichevsky–Trofimov algorithm. Then, I show how to use the same idea to remove the $\ln \ln T$ factor in the data-independent bound for the Squint algorithm.

Learning curves are a fundamental primitive in supervised learning, describing how an algorithm's performance improves with more data and providing a quantitative measure of its generalization ability. Formally, a learning curve plots the decay of an algorithm's error for a fixed underlying distribution as a function of the number of training samples. Prior work on revenue-maximizing learning algorithms, starting with the seminal work of Cole and Roughgarden [STOC, 2014], adopts a distribution-free perspective, which parallels the PAC learning framework in learning theory. This approach evaluates performance against the hardest possible sequence of valuation distributions, one for each sample size, effectively defining the upper envelope of learning curves over all possible distributions, thus leading to error bounds that do not capture the shape of the learning curves. In this work we initiate the study of learning curves for revenue maximization and provide a near-complete characterization of their rate of decay in the basic setting of a single item and a single buyer. In the absence of any restriction on the valuation distribution, we show that there exists a Bayes-consistent algorithm, meaning that its learning curve converges to zero for any arbitrary valuation distribution as the number of samples $n \to \infty$. However, this convergence must be arbitrarily slow, even if the optimal revenue is finite. In contrast, if the optimal revenue is achieved by a finite price, then the optimal rate of decay is roughly $1/\sqrt{n}$. Finally, for distributions supported on discrete sets of values, we show that learning curves decay almost exponentially fast, a rate unattainable under the PAC framework.

We prove pathwise convergence of the layerwise evolution of tokens in a finite-depth, finite-width transformer model with MultiLayer Perceptron (MLP) blocks to a continuous-time stochastic interacting particle system. We also identify the stochastic partial differential equation describing the evolution of the tokens' distribution in this limit and prove propagation of chaos when the number of such tokens is large. The bounds we establish are quantitative and the limits we consider commute. We further prove that the limiting stochastic model displays synchronization by noise and establish exponential dissipation of the interaction energy on average, provided that the common noise is sufficiently coercive relative to the deterministic self-attention drift. We finally characterize the activation functions satisfying the former condition.

Accurate, localised rainfall information is essential for agricultural planning, climate risk assessment, and water resources management. Gridded climate products provide rainfall information over large areas but can lack the accuracy needed at local scales, often requiring bias correction before use in local impact studies. Local intensity scaling (LOCI) and quantile mapping (QM) are two widely used bias correction methods which adjust both rainfall frequency and intensity, but do not account for the temporal structure of daily rainfall. This can lead to biases in the representation of wet and dry spells. This study proposes integrating a two-state first-order Markov chain into existing bias correction methods through state-dependent rain day thresholds and rainfall adjustments, aimed at improving temporal structure. Two implementations of this framework are presented: Markov chain local intensity scaling (MC LOCI) and Markov chain quantile mapping (MC QM). The proposed methods were applied to AgERA5 reanalysis data with rainfall data from five stations in Zimbabwe. Results showed that the Markov chain methods improved the representation of rainfall persistence, onset, and wet and dry spell characteristics compared to LOCI and QM, while maintaining improvements in rain day frequency, mean and total rainfall. Improvements in event timing and daily rainfall amounts were limited. Results from five locations in Zimbabwe demonstrate that the proposed methods could be beneficial for crop simulation, hydrological modelling and other applications requiring accurate rainfall sequencing. Evaluation across additional regions and gridded products would establish the broader applicability of the proposed methods under a range of conditions.

We introduce the ARCH-m(X) model, a semiparametric extension of the ARCH-X framework in which the effect of a multivariate exogenous covariate vector X on the conditional variance is modeled through an unknown nonparametric function m(), accommodating complex nonlinear relationships between external predictors and financial volatility. Within this model, we develop a novel hypothesis test for the significance of covariates constructed with an artificial one-way ANOVA. Under some regularity conditions, the test statistic is shown to converge in distribution to the standard Normal. Another key contribution of this paper is the construction of a variable selection procedure based on the Benjamini-Yekutieli false discovery rate correction applied to covariate-level p-values. We show that the resulting index set coincides with the true set of relevant covariates with probability tending to one as n goes to infinity. Extensive simulations confirm that the proposed methods outperform existing competitors, and an empirical application to SP500 return volatility illustrates the practical utility of the proposed variable selection framework.

The maximum likelihood estimator in nonlinear panel data models with interactive fixed effects is biased. Several bias correction methods, such as analytical and jackknife approaches, have been proposed to enable valid inference. This paper shows that the parametric bootstrap also enables valid inference in such models. In particular, we show that the parametric bootstrap replicates the asymptotic distribution of the maximum likelihood estimator. Therefore, it yields asymptotically unbiased estimates and confidence sets with asymptotically correct coverage. We also propose a transformation-based bootstrap confidence interval that delivers improved finite-sample performance. Simulation results support the theoretical findings. Finally, we apply the proposed method to examine technological and product market spillover effects on firms' innovation behavior.