Papers

Operator learning for partial differential equations (PDEs) aims to learn solution operators on infinite-dimensional function spaces from finite-resolution data. In this setting, it is important for the learned model to be discretization-invariant, or resolution-robust, and to reflect PDE-specific structure. It is therefore natural to ask how such structure should be encoded in the model architecture, hypothesis class, or learning procedure. In this paper, we study operator learning for solution operators of nonlinear parabolic PDEs based on Duhamel–Picard iteration. We formulate Picard iteration as an abstract state-transition model and present a theoretical framework for Picard-type operator learning. We derive implementation-agnostic generalization error bounds that separate the implementation error from the estimation error associated with the abstract state-transition model induced by Picard iteration. A key consequence is that increasing the Picard depth reduces the Picard truncation error without causing an unbounded growth of the entropy-based estimation error. We also extend the analysis to long-time prediction by rolling out the same learned local model over successive time blocks. Finally, we illustrate the theory for nonlinear heat equations on the torus using a Picard-type Fourier neural operator as a concrete implementation.

The simulation of physical phenomena with computer models relies on the estimation of physical and/or numerical parameters calibrated to fit experimental data. The approximations within the computer model and the errors in the measurements lead to uncertainties in the calibrated parameters. Bayesian calibration offers a well-studied framework to provide reliable uncertainty quantification on the calibrated parameters. When dealing with complex computer codes whose outputs are infinite-dimensional, Bayesian calibration may be extended by providing a relevant distance in the output space. In this paper, Bayesian calibration is performed using distances from the large deformation diffeomorphic metric matching (LDDMM) framework. LDDMM distances can provide a suitable metric for infinite-dimensional shapes such as scalar fields (i.e. images) or function graphs. This metric can be interpreted as the minimal energy deformation required to transform one shape into another. As such, it provides a readily interpretable metric for Bayesian calibration. On top of this, the representation of the diffeomorphism group as an exponential transformation of an RKHS is compatible with Bayesian inference and allows to define a predictive posterior distribution on the infinite-dimensional space shape.

We study the problem of estimating the intensity function of a covariate-driven point process based on observations of the points and covariates over a large window. We consider the nonparametric Bayesian approach, and show that a wide class of Gaussian priors, combined with flexible link functions, achieves minimax-optimal posterior contraction rates in the increasing domain asymptotics and under the assumption that the covariates be ergodic. We also employ Besov-Laplace priors, which are popular in imaging and inverse problems due to their edge-preserving and sparsity-promoting properties. We prove that these yield optimal estimation of spatially inhomogeneous intensities belonging to Besov spaces with low integrability index. These results are based on a general concentration theorem that extends recent findings from the literature. To corroborate the theory, we provide extensive numerical simulations, implementing the considered procedures via suitable posterior sampling schemes. Further, we present two real data analyses motivated by applications in forestry and the environmental sciences.

Distributional causal inference requires estimating not only average treatment effects but also interventional outcome distributions, including quantiles, tail risks, and policy-dependent uncertainty. As a method for distributional causal inference, generative adversarial network (GAN)-based counterfactual methods are flexible tools for this task. However, these methods have several limitations. First, the objectives of certain techniques do not coincide with the statistical risk of the identifiable causal target, and therefore provide limited theoretical guarantees regarding estimable counterfactual distributions or optimality. Second, they tend to rely on unstable density-based methods, such as density ratio estimation. In this paper, we propose GANICE (GAN for Interventional Conditional Estimation) with several advantages: it (i) clarifies the conditional interventional distribution for each treatment–covariate state as the causal estimation target; (ii) estimates the conditional distribution such that its averaged Wasserstein risk is minimized; (iii) establishes minimax optimality. GANICE achieves these advantages through the introduction of the extended Wasserstein distance, the incorporation of a cellwise critic in its dual, and an optimality proof based on Besov space theory. Our experiments demonstrate that GANICE consistently outperforms existing methods.

Dense Associative Memory (DenseAM) is a promising family of AI architectures that is represented by a neural network performing temporal dynamics on an energy landscape. While hyperparameter transfer methods are well-studied for feed-forward networks, these methods have not been developed for settings in which weights are shared across layers and within the layer, which is common in DenseAMs. Additionally, DenseAMs utilize rapidly peaking activation functions that are rarely used in feed-forward architectures. The confluence of these aspects makes DenseAM a challenging framework for using existing methods for hyperparameter transfer. Our work initiates the development of hyperparameter transfer methods for this class of models. We derive explicit prescriptions for how the hyperparameters tuned on small models can be transferred to models trained at scale. We demonstrate excellent agreement between these theoretical findings and empirical results.

Recent work on causal abstraction, in particular graphical approaches focusing on causal structure between clusters of variables, aims to summarize a high-dimensional causal structure in terms of a low-dimensional one. Existing methods for learning such summaries from data assume that both the high- and low-dimensional structures are acyclic, which is helpful for causal effect identification and reasoning but excludes many high-dimensional models and thus limits applicability. We show that in the linear non-Gaussian (LiNG) setting, the high-dimensional acyclicity assumption can be relaxed while still allowing recovery of a low-dimensional causal directed acyclic graph (DAG). We further connect identifiability of this low-dimensional DAG to existing results: LiNG models with cycles are observationally identifiable only up to an equivalence class whose members differ by reversals of directed cycles; our low-dimensional DAG, which is invariant across all members of a given equivalence class, thus forms a natural representative of the class. While existing approaches for learning this observational equivalence class over high-dimensional variables have exponential time complexity, our low-dimensional summary is learned in worst-case cubic time and comes with explicit bounds on the sample complexity. We provide open source code and experiments on synthetic data to corroborate our theoretical results.

Thompson sampling is a widely used strategy for contextual bandits: at each round, it samples a reward function from a Bayesian posterior and acts greedily under that sample. Prior-data fitted networks (PFNs), such as TabPFN v2+ and TabICL v2, are attractive candidates for this purpose because they approximate Bayesian posterior predictive distributions in a single forward pass. However, PFNs predict noisy future rewards, while Thompson sampling requires uncertainty over the latent mean reward function. We propose PFN-TS, a Thompson sampling algorithm that converts PFN posterior predictives into mean-reward samples using a subsampled predictive central limit theorem. The method estimates posterior variance from a geometric grid of $O(\log n)$ dataset prefixes rather than the full $O(n)$ predictive sequence used in previous predictive-sequence approaches, and reuses TabICL's cached representations across rounds. We prove consistency of the subsampled variance estimator and give a Bayesian regret bound that decomposes PFN-TS regret into exact posterior-sampling regret under the PFN prior plus approximation terms. Empirically, PFN-TS achieves the best average rank across nonlinear synthetic and OpenML classification-to-bandit benchmarks, remains competitive on linear and BART-generated rewards, and attains the highest estimated policy value in an offline mobile-health evaluation. Code is available at https://anonymous.4open.science/r/PFN_TS-36ED/.

This paper develops power and sample size formulas for causal inference with time-to-event outcomes. The target estimand is the marginal hazard ratio: the coefficient of a marginal structural Cox proportional hazard model with treatment as the only predictor. We extend the robust sandwich variance theory and derive the analytical form of the asymptotic variance for the inverse probability weighted partial likelihood estimator. Building on this, we derive a new analytical sample size formula valid at any prespecified effect size, applicable to both randomized trials and observational studies. For randomized trials, the formula requires only the canonical inputs of treatment proportion, effect size, and event rate. The new formula corrects the mischaracterization of classic log-rank-based formulas. For observational studies, one additional input suffices: an overlap coefficient summarizing covariate similarity between comparison groups. We further develop a variance inflation approach applicable to any propensity score balancing weights, anchored to the corrected baseline variance. We provide an online calculator and an R package 'PSpower' to implement the method.

In infectious disease modeling during the early phase of a pandemic, SEIR-type compartmental models are standard tools, and they require epidemiological parameters as inputs. Because these parameters are subject to uncertainty, different research groups often report different epidemic curves, and obtaining a representative curve that captures the characteristics of these trajectories is important for decision-making. However, simple pointwise summaries can attenuate epidemic peaks under temporal misalignment and generally do not preserve the dynamical structure of the underlying compartmental models. To address these limitations, we propose a method for summarizing multiple solutions to SEIR-type compartmental models on a functional space by computing a constrained power Fréchet mean with temporal shift registration. In our method, we regard the pairs of exposed and infectious compartments as objects in a Hilbert space, and the consensus curve is defined as the solution to a constrained optimization problem. Differential equation constraints and population constraints are incorporated in the optimization to preserve a partially mechanistic interpretation regarding the infectious compartment. We develop an implementable block-optimization algorithm based on basis function expansion. In simulation studies based on early COVID-19 parameter estimates, the proposed method produced consensus curves with a single epidemic peak, whereas pointwise summaries exhibited attenuated or multiple peaks. We further applied the method to six literature-derived parameter sets from early COVID-19 studies and obtained a representative trajectory with interpretable epidemiological parameters. The proposed approach provides a generalized trajectory-summarization method that includes mean- and median-type estimators and preserves mechanistic interpretability.

In many real-world settings such as online recommendation or consumer choice modeling, individuals make repeated choices from a fixed set of options. Accurately estimating their underlying preferences is essential for generating personalized future recommendations. Probabilistic models for understanding user choice behavior from past decisions can serve as a valuable addition to existing recommender systems and choice prediction methods. To this end, in this article, we introduce a novel statistical framework for predicting user preferences based on their past choices, under a natural monotonicity assumption: options that were chosen more frequently or more intensely in the past are more likely to be chosen again in the future. Our approach builds on a parametric model proposed by Le Goff and Soulier (2017), originally used to describe how ants in an ant colony select a path among many pre-existing paths. We propose a non-parametric generalization of this model, drawing inspiration from the generalized elephant random walk introduced by Maulik et al. (2024). We develop a method of maximum likelihood estimation of the user preference probabilities under the above-mentioned monotonicity constraint. We also derive theoretical guarantees for our estimator and demonstrate the effectiveness of our method through both simulated experiments and real-world datasets.

Generative models trained on finite data face a fundamental tension: their score-matching or next-token objective converges to the empirical training distribution rather than the population distribution we seek to learn. Using rule-valid synthetic tasks, we trace this tension across two training timescales: $\tau_{\mathrm{rule}}$, the step at which generations first become rule-valid, and $\tau_{\mathrm{mem}}$, the step at which models begin reproducing training samples. Focusing on parity and extending to other binary rules and combinatorial puzzles, we characterize how these two clocks, $\tau_{\mathrm{rule}}$ and $\tau_{\mathrm{mem}}$, depend on key aspects of the learning setup. Specifically, we show that $\tau_{\mathrm{rule}}$ increases with rule complexity and decreases with model capacity, while $\tau_{\mathrm{mem}}$ is approximately invariant to the rule and scales nearly linearly with dataset size $N$. We define the innovation window as the interval $[\tau_{\mathrm{rule}}, \tau_{\mathrm{mem}}]$. This window widens with increasing $N$ and narrows with rule complexity, and may vanish entirely when $\tau_{\mathrm{rule}} \geq \tau_{\mathrm{mem}}$. The same two-clock structure arises in both diffusion (DiT) and autoregressive (GPT) models, with architecture-dependent offsets. Dissecting the learned score of DiT models reveals a corresponding evolution of the optimization landscapes, where rule-valid samples' basins expand substantially around $\tau_{\mathrm{rule}}$, while training samples' basins begin to dominate around $\tau_{\mathrm{mem}}$. Together, these results yield a unified and predictive account of when and how generative models exhibit genuine innovation.

In this paper, we study the problem of sampling from a distribution under the constraint of differential privacy (DP). Prior works measure the utility of DP sampling with density ratio-based measures such as KL divergence. However, such formulations suffer from two key limitations: 1) they fail to capture the geometric structure of the support, and 2) they are not applicable when the supports of the distributions differ. To deal with these issues, we develop a novel framework for DP sampling with Wasserstein distance as the utility measure. In this formulation, we propose Wasserstein Projection Mechanism (WPM), a minimax optimal mechanism based on Wasserstein projection. Furthermore, we develop efficient algorithms for computing the proposed mechanisms approximately and provide convergence guarantees.

Hospitals, companies, and research consortia want language models that learn from, and answer questions over, data that must stay where it is held, over links with limited bandwidth, such as the satellite links of remote clinics and ships. Averaging the sites' model parameters makes each message grow with the model and requires one shared architecture; methods that exchange predictions avoid both limits, but, to our knowledge, none bounds what compressing those predictions costs. We introduce two methods that compress every message at a small and predictable cost. Federated Probe-Logit Distillation (FPLD) trains a shared language model: each site sends compressed logits on shared unlabeled probes, and a hub distills one model from their average. Federated Conformal Retrieval-Augmented Generation (FC-RAG) answers questions: each site scores candidate answers using its own documents and sends compressed scores, and the hub returns a conformal prediction set that contains the correct answer at a guaranteed rate. Both use one codec, which centers each message and rounds it after adding a random offset known to the receiver. For logits within a fixed clipping range, we bound the expected log-loss added by compression independently of vocabulary size; the bound depends only on the bits, the range, and the number of sites, falls as more sites take part, and lets the bit budget be set before deployment. We also prove that recalibrating after compression keeps the coverage guarantee, and that FPLD's training error bounds how far coverage can slip when a threshold is reused after a model update. In experiments with GPT-2 and Qwen2.5, 8 bits per number cause no detectable loss in the trained model or in coverage. FPLD also trains a shared model across different architectures, where parameter averaging does not apply; with one shared architecture, parameter averaging remains more accurate.

Every adaptive learning system must alternate between two operations: consolidating what it already knows and expanding into new evidence. We propose Consolidation-Expansion Operator Mechanics (OpMech), a framework that makes this structure precise. The central object is the order-gap $\Ogap(\theta; e)$, the degree to which a consolidation operator~$Q$ and an expansion operator~$P_e$ fail to commute at a given knowledge state. Because the order-gap is computable from the system's own trajectory, it serves as a real-time control signal: large values indicate that the system is still sensitive to the ordering of consolidation and expansion; once the order-gap falls and stays small, further processing is unlikely to change the outcome. Three results give the signal precise meaning: the order-gap decays along convergent trajectories; a persistently large order-gap implies the system is far from its settled state; and an order-gap-based stopping rule terminates with provable guarantees in both noiseless and bounded-noise settings. The framework applies across five domains: bandits, reinforcement learning, stochastic optimization, continual learning, and recursive language models. We give conditions under which the order-gap reliably tracks convergence in three representative cases. We develop the recursive language model application in detail, showing how OpMech replaces heuristic stopping rules and fixed recursion budgets with principled, evidence-driven alternatives.

False discovery rate (FDR) is a cornerstone of modern multiple testing. However, it often fails to guarantee the reliability of "marginal" discoveries that lie at the boundary of the rejection set, which are often crucial in high-precision applications. While recent works (Soloff et al., 2024; Xiang et al., 2025) introduced the boundary false discovery rate (bFDR) to control the error probability at the marginal discovery, their method relies on restrictive assumptions such as independence or specific prior distributions. In this paper, we first propose $k$-bFDR, a novel generalization that controls the error probability of the $k$ least significant discoveries. We then provide a systematic investigation into the theoretical relationship between $k$-bFDR and existing error metrics. Furthermore, building upon the closure principle, we develop Domino, a unified framework that guarantees $k$-bFDR control under arbitrary dependence, applicable for both p-values and e-values. We prove the theoretical validity of the proposed Domino algorithm and demonstrate through extensive numerical experiments that it consistently achieves rigorous $k$-bFDR control while identifying trustworthy marginal discoveries. Analyses of real data reveal that $k$-bFDR control yields higher-quality rejection sets with greater practical significance.

In this paper we consider the parameter estimation problem associated to partially-observed time changed SDEs, with observations that are given at discrete times. In particular we consider both likelihood and Bayesian estimation. We develop new Markov chain Monte Carlo (MCMC) algorithms which allow an unbiased score-based stochastic approximation method to provide likelihood-type parameter estimators. We also use a variant of this MCMC algorithm to perform multilevel-based Bayesian parameter estimation. We prove that this latter method achieves a mean square error of $\mathcal{O}(\epsilon^2)$ ($\epsilon>0$) with a cost of $\mathcal{O}(\epsilon^{-2}\log(\epsilon)^2)$. Our methodologies are tested numerically on both simulated and real data.

Multicalibration requires predicted scores to agree with label probabilities across rich families of subgroups and score-dependent tests, but existing methods require clean input-label pairs for evaluation and post-processing. This assumption fails in weakly supervised learning (WSL) regimes – including positive-unlabeled, unlabeled-unlabeled, and positive-confidence learning – where clean labels are costly or unavailable even though reliable uncertainty estimates may be crucial. We address this gap by developing estimators of multicalibration error and post-hoc correction methods for WSL settings in which clean input-label pairs are unavailable. We propose a unified framework for estimating and correcting multicalibration under weak supervision by combining contamination-matrix risk rewrites with witness-based calibration constraints, yielding corrected multicalibration moments with finite-sample guarantees. We further propose weak-label multicalibration boost (WLMC), a generic post-hoc recalibration algorithm under weak supervision. Finally, we conduct experiments across multiple weak-supervision settings to evaluate multicalibration behavior and offer empirical insight into uncertainty estimation under weak supervision.

Methods that rely on proxies, without imposing strong parametric structure, are increasingly used to deal with unobserved variables in causal inference. One influential line of this work reconstructs latent distributions used to identify the target functional by exploiting eigenvalue eigenvector structure. Within this framework, we first establish identification of the full data law in the presence of hidden outcomes, and then develop influence function based estimators for causal effects. To the best of our knowledge, this is the first work to develop influence function based estimators in this setting without relying on unbiased proxy measurements or partial observation, while achieving multiple robustness and desirable efficiency properties. We demonstrate the performance of our approach through simulation studies.

Modern predictive systems encode beliefs that can act as useful prior information for statistical inference in data-limited settings. Using them for prior construction introduces a tradeoff: an informative prior built from a predictive model can sharpen inference from limited data, but also risks propagating error from the model into the posterior. We propose a framework for AI-informed prior elicitation that mitigates this tension by rectifying the AI-induced law that generates synthetic data before using it to inform a prior. The rectified law can be embedded into synthetic data-driven prior elicitation techniques, including as a base measure in a Dirichlet process (DP) prior on the data-generating process. We refer to the resulting prior and corresponding posterior as the rectified AI prior and rectified AI posterior. We establish Gaussian asymptotics for the rectified AI posterior under non-vanishing prior strength and derive a first-order expression for its centering bias. Our rectified AI priors substantially reduce bias compared to standard approaches, improve the coverage of credible intervals, and make AI-powered prior information more reliable. We additionally apply the rectified AI prior to a real skin disease classification task and show that it can meaningfully boost predictive performance.

We study nearest-neighbor-based estimators of Tsallis entropy associated with Poisson and binomial point processes on general metric measure spaces. Using stabilization techniques based on flexible add-one cost operators together with second-order Poincaré inequalities, we establish asymptotic normality and derive explicit convergence rates for the Kolmogorov distance. Our analysis avoids explicit score-function decompositions and instead relies on flexible localizations of add-one costs, which simplify the treatment of higher-order terms. Under natural stabilization and moment conditions, the resulting bounds recover the classical normal approximation rates \(s^{-1/2}\) and \(n^{-1/2}\) and extend corresponding results for Shannon and Rényi entropy estimators. We further illustrate the scope of the framework through examples involving Tsallis entropy functionals, weighted \(k\)-nearest-neighbor Shannon entropy estimators. The examples provided highlight the benefits of stabilization-based normal approximations for non-parametric statistical inference in complex spatial and high-dimensional settings.

RNA sequencing (RNA-seq) is the conventional genome-scale approach used to capture the expression levels of all detectable genes in a biological sample. This is now regularly used for population-based studies designed to identify genetic determinants of various diseases. Naturally, the accuracy of these tests should be verified and improved if possible. In this study, we aimed to detect and correct for expression level-dependent errors which vary from sample to sample, and are not corrected by conventional normalization techniques . We examined several RNA-seq datasets from the Cancer Genome Atlas (TCGA), Stand Up 2 Cancer (SU2C), and GTEx databases with various types of preprocessing. By applying local averaging, we found sample by sample expression-level dependent biases in all datasets studied. Using simulations, we show that these biases corrupt gene-gene correlation estimations and $t$ tests between subpopulations. To mitigate these biases, we introduce two different nonlinear transforms based on statistical considerations that correct these observed biases. We demonstrate that that these transforms effectively remove the observed per-sample biases, reduce sample-to-sample variance, and improve the characteristics of gene-gene correlation distributions. Using a novel simulation methodology that creates controlled differences between subpopulations, we show that these transforms reduce variability and increase sensitivity of two population tests. The improvements in sensitivity and specificity were of the order of 3-5% in most instances after the data was corrected for bias. Altogether, these results improve our capacity to understand gene-gene relationships, and may lead to novel ways to utilize the information derived from clinical tests.

We present GenusSink, a new class of approximate generalized Sinkhorn algorithms with shortest-path-distance costs for bounded genus (e.g. planar) graphs, providing near-linear time: (1) pre-processing, (2) iteration step, (3) final transport plan matrix querying and near-linear memory. Graphs handled by GenusSink include in particular planar graphs and bounded-genus meshes approximating 3D objects. GenusSink addresses total quadratic time complexity of its brute-force counterpart by leveraging separator-based decomposition of graphs, computational geometry techniques, and new results on fast matrix-vector multiplications with generalized distance matrices, using, in particular, Fourier analysis and low displacement rank theory. It is inspired by recent breakthroughs in graph theory on approximating bounded genus metrics with small treewidth metrics \citep{minor-free-paper}. The graph-centric approach enables us to target optimal transport problem with the corresponding distributions defined on the manifolds approximated by weighted graphs and with cost functions given by geodesic distances. We conduct rigorous theoretical analysis of GenusSink, provide practical implementations, leveraging newly introduced in this paper separation graph field integrators (S-GFIs) data structures and present empirical verification. GenusSink provides orders of magnitude more accurate computations than other efficient Sinkhorn algorithms, while still guaranteeing significant computational improvements, as compared to the baseline. As a by-product of the developed methods, we show that GenusSink is numerically equivalent to the brute-force geodesic Sinkhorn algorithm on $n$-vertex graphs with treewidth $O(\log \log (n))$ (e.g. on trees).

Providing non-conservative uncertainty quantification for function estimates derived from noisy observations remains a fundamental challenge in statistical machine learning, particularly for applications in safety-critical domains. In this work, we propose novel non-asymptotic probabilistic uniform error bounds for kernel-based regression. Compared to related bounds in the literature that are restricted to (conditionally) independent sub-Gaussian noise, our bounds allow to consider a broad class of non-Gaussian distributions, such as sub-Gaussian, bounded, sub-exponential, and variance/moment-bounded noise. Moreover, our results apply to correlated and uncorrelated noise. We compare our proposed error bounds with existing results in terms of the induced uncertainty region and their performance in safe control, demonstrating the tightness of the proposed bounds.

The power method is one of the most fundamental tools for extracting top principal components from data through low-rank matrix approximation. Yet, when the target rank is large, the cost of matrix multiplication associated with this procedure becomes a major bottleneck. We develop an algorithmic and theoretical framework for accelerating the power method using fast sketching, which is a popular paradigm in randomized linear algebra. Our framework leads to simple and provably efficient methods for singular value decomposition, low-rank factorization, and Nyström approximation, which attain strong numerical performance on benchmark problems. The key novelty in our analysis is the use of regularized spectral approximation, a property of fast sketching methods which proves more flexible in generalizing power method guarantees than traditional arguments.

Understanding effect modification – how treatment effects vary across subpopulations – is practically important in observational studies, as it helps identify which subgroups are likely to benefit from a given treatment. In this paper, we study the discovery of effect modification in matched observational studies, where each treated unit may be matched to multiple controls. We develop a finite-sample valid procedure for identifying and selecting covariate-interpretable subgroups, with exact control of the subgroup-level false discovery rate (FDR). Our method explicitly accounts for unmeasured confounding via sensitivity models, and leverages multiple matched controls to improve statistical power. We demonstrate the favorable performance of our method relative to baseline methods through extensive simulation studies and a real-world application to the economic returns to college education.

Needless to say, linear dynamics are pervasive in economic time series, particularly autoregressive ones. While gradient boosting with trees excels at capturing nonlinearities, it is inefficient in small samples when much of the predictive content is linear, expending splits to approximate relationships better captured by simple linear terms. This paper proposes LGB+, a boosting procedure operating on a more inclusive set of basis functions. The idea comes in two flavors. LGB+ evaluates a tree and a linear candidate at each step against out-of-bag data; only the winner advances. The simpler variant, LGB^A+, alternates on a fixed schedule: a block of tree updates, then a greedy linear correction, repeat. Both designs avoid ex ante commitments to any particular functional form or predictor selection. Because the prediction is the sum of a linear and a tree component, forecasts decompose natively into linear and nonlinear contributions, and so does permutation-based variable importance and historical proximity weights. In a quarterly U.S. macroeconomic forecasting exercise, LGB+ delivers strong gains for targets with pronounced autoregressive dynamics or mixed linear-nonlinear signals. Variables dominating the linear channel are those operating through autoregressive persistence or near-accounting relationships to the target (e.g., initial claims for unemployment and building permits for housing starts).

Researchers use interference models based on exposure mappings to facilitate estimation of causal effects in randomized experiments with interference. To test the veracity of such models, researchers can use specification tests that aim to detect departures from the stipulated model. However, existing tests suffer from poor power and are often unable to detect important model violations. The main result in this paper is to show that the specification testing problem for exposure mapping models is inherently difficult, and the poor power of existing tests is inescapable. In particular, the worst-case Type I and Type II error rates must sum to one for any specification test of such models, ruling out the existence of a uniformly consistent test. This is the worst-case overall error rate achieved by a naive test that discards all data and arbitrarily rejects the null at random; the testing problem is in this sense impossible. This negative result holds true for all exposure mappings, all sample sizes, for uniformly bounded outcomes, and for alternatives that are maximally separated from the null. While some tests can detect some type of departures from the null model, there will always be relevant departures from the null that are undetectable. Informative specification tests must therefore restrict the alternative model against which they seek to attain power for, beyond the restrictions imposed by the exposure mappings alone. We illustrate this by providing a uniformly consistent test for differentiating no-interference from a network-linear-in-means model.

Many systems in physics, engineering, and biology exhibit multiscale stochastic dynamics, where low-dimensional slow variables evolve under the influence of high-dimensional fast processes. In practice, observations are often limited to a single trajectory of the slow component, while the fast dynamics remain unobserved, making statistical learning challenging. Approaches based on partial differential equations (PDE), such as Fokker-Planck formulations, aim to characterize the evolution of probability densities, typically requiring dense space-time data or grid-based solvers. In contrast, we adopt a trajectory-based perspective and develop a data-driven framework for learning effective stochastic dynamics from a single observed path. We model the dynamics by coupled multiscale stochastic differential equations (SDEs) and first obtain a principled model reduction through stochastic averaging. Unlike generic model reduction techniques such as PCA, this respects the dynamical structure of the original system and explicitly incorporates the interaction between slow and fast scales. A central challenge, however, is that the reduced model depends on the invariant distribution of the fast process, which is a solution to an intractable and often unknown PDE. We introduce a novel learning framework that parameterizes the invariant distribution using normalizing flows, enabling expressive density modeling in the latent fast-variable space. The flow is trained end-to-end by optimizing a penalized likelihood objective induced by the reduced stochastic dynamics. Furthermore, we develop a Bayesian variational inference procedure for uncertainty quantification, employing a second normalizing flow to approximate the posterior distribution over model parameters. This yields a scalable approach to capturing epistemic uncertainty in multiscale systems.

The general regularisation scheme, a versatile approach for nonparametric estimation, has been successfully applied to regression, density ratio, and score estimation. In this paper, we introduce a unified framework encompassing these settings and extend it to conditional density estimation, deriving a new estimator with rigorously established convergence rates. We implement the Landweber regularisation, which is computationally more tractable than Tikhonov regularisation in this context. Numerical experiments demonstrate that our estimator matches or outperforms the Nadaraya-Watson estimator in various scenarios, including time series models.

Average forecast accuracy is not the same as forecast reliability. I treat forecast loss differentials relative to a benchmark as a return series. I then evaluate these returns using risk-adjusted performance measures from finance, including the Sharpe ratio, Sortino ratio, Omega ratio, and drawdown-based metrics. I also introduce the Edge Ratio capturing a model's propensity to deliver uniquely informative predictions relative to the forecasting frontier. I apply this framework to U.S. macroeconomic forecasting, comparing econometric benchmarks, machine learning models, a foundation model (TabPFN), and the Survey of Professional Forecasters. While it is often feasible to beat professional forecasters in terms of average accuracy, it is much harder to beat them on a risk-adjusted basis. They rarely exhibit catastrophic failures and often achieve high Edge Ratios, plausibly reflecting the value of contextual judgment. Nonetheless, selected machine learning methods deliver attractive risk profiles for specific targets. The framework naturally extends to meta-analyses across targets, horizons, and samples, illustrated with a density forecast evaluation and the M4 competition.