Papers

Yun, Sra, and Jadbabaie (COLT 2021, open question) conjectured the SS–RS–GD inequalities: for well-conditioned symmetric matrices $A_1,\dots,A_n$, the operators $W_{ss}$, $W_{rs}$, and $W_{gd}$ that encode the expected iterate of single-shuffle SGD, random-reshuffle SGD, and gradient descent on a quadratic finite sum should satisfy \[ \|W_{ss}\|\le \| W_{rs}\|\le \|W_{gd}\|. \] The conjecture is resolved, $\bullet$ SS-RS inequality fails. Already for $n=3$, $K=2$, and $d=4$, we exhibit explicit PSD matrices whose condition number is arbitrarily close to $1$, yet $\|W_{ss}\|>\|W_{rs}\|$. $\bullet$ RS-GD inequality holds. For every symmetric $A_i$ with $\bigl(1-\frac1{4n^2+1}\bigr)I\preceq A_i\preceq I$, one has $\|W_{rs}\|\le\|W_{gd}\|$. The proof was found via GPT-5.5 Pro extended prompted by the author.

We introduce the Generalized Neural Distributional Regression (GNDR) framework, which seamlessly embeds deep neural networks into the parameter space of classical probability distributions. To reconcile the inherent non-identifiability of deep architectures with maximum likelihood theory, we propose a two-step semi-parametric estimation procedure. By isolating the terminal prediction heads and treating the upstream network as a fixed, non-linear basis expansion, GNDR enables the extraction of analytical Fisher Information matrices. This facilitates rigorous uncertainty quantification, generating observation-specific confidence bands and tolerance intervals via the multivariate Delta method. We demonstrate the framework's versatility and superior distributional calibration across diverse data modalities, including overdispersed clinical counts, right-censored transcriptomic survival profiles under a mixture cure framework, and zero-truncated age distributions derived directly from unstructured facial images. The methodology is natively implemented in the open-source Python package thetaflow.

Small area estimation borrows strength across domains to repair the poor precision of direct survey estimators. Two philosophies dominate the area-level literature. The first, descending from Ghosh and Rao (1994), borrows strength through structured Gaussian smoothing: an intrinsic conditional autoregression or its BYM2 reparameterization pools each area towards its neighbours. The second borrows strength globally but acts locally through a heavy-tailed global-local prior on area effects, of which the horseshoe of Carvalho et al. (2010) is the canonical instance; Tang et al. (2018) first brought this idea to small area estimation. We study the horseshoe Fay-Herriot model with known unequal sampling variances and make four contributions. First, a tail-robustness theorem: through a heteroscedastic Tweedie identity the posterior mean leaves strongly signalled areas essentially unshrunk, bounding the influence of an outlying direct estimate, unlike Gaussian random-effect models. Second, standardizing by the known design variances transfers the minimax contraction and credible-set theory of the homoscedastic sequence model to the heteroscedastic Fay-Herriot problem; the posterior contracts at the nearly-black minimax rate, with a matching lower bound. Third, we give an account of when structured smoothing and global-local shrinkage each win. Fourth, an O(m) Gibbs sampler, simulations, and an analysis of the Scottish lip cancer data confirm the account: on strongly spatial data the smoother predicts held-out districts best, yet the horseshoe flags exceptional districts that smoothing suppresses. Throughout we argue, following the regular-variation theory of Bhadra et al. (2016), that these properties make the horseshoe a sound default prior for area effects: it borrows strength aggressively yet lets genuinely exceptional areas speak for themselves, with no tuning and no neighbourhood graph.

Oscillatory biomedical signals such as electrocardiograms (ECG) and electroencephalograms (EEG) call for decompositions that are both computationally efficient and interpretable. This paper establishes a formal connection between two finite-order frameworks that have largely evolved independently: Adaptive Fourier Decomposition (AFD), based on orthonormal Takenaka-Malmquist expansions, and the Frequency-Modulated Mobius (FMM) model, a parametric decomposition built on Mobius transforms with morphologically meaningful parameters. We prove that finite-order AFD and FMM decompositions are mathematically equivalent. Under mild regularity assumptions, we further show that their associated estimation procedures solve the same underlying optimization problem when FMM is formulated with independent Gaussian noise. The results are extended to multi-channel signals, which are central in multilead bioelectric recordings. Practically, the equivalence clarifies how fast AFD approximations, including FFT-based implementations, relate to FMM-style parametrization and component interpretability. We illustrate these implications with an EEG example evaluating approximation behavior as the number of components increases, and with an ECG use case comparing five-component decompositions on representative beats, contrasting unlabeled AFD components with physiologically identified FMM components. Overall, the proposed equivalence provides a principled basis to leverage the computational advantages of AFD alongside the interpretability of FMM in biomedical signal analysis.

The probabilistic reading of the cumulative accuracy profile (CAP) has a long industry lineage. Falkenstein, Boral and Carty (2000) state, in discrete form, that the default rate at a score percentile equals the portfolio average rate times the local slope of the power curve; van der Burgt (2008, 2019) formalizes this as the continuous identity $p(D\mid x) = p_D\, dy/dx$ and imports the continuous form as a working fact; Tasche (2009) analyzes the resulting calibration method; Voloshyn and Voloshyn (2023) substitute Bayes' theorem, $f(x\mid D)=p(D\mid x) f(x)/p_D$, into the area integral and write the Gini as a functional of the calibration curve. The slope itself is already in the lineage (van der Burgt's $dy/dx$ is the ratio of the two cumulative differentials), but it enters as a cited working fact, never as Bayes' theorem. We make that identification explicit and draw out its consequences. First, the CAP slope is Bayes' theorem in cumulative coordinates: the standardized PD it recovers is the posterior probability rescaled by the prior. The weight of the paper then falls on two results this reading unlocks. The odds form places the weight of evidence (the log of the likelihood ratio, i.e. the Bayes factor) and the information value inside one geometry (the weight of evidence at a point is the log of the ratio of the "bad" and "good" CAP slopes). The accuracy ratio, Somers' $D_{xy}$, and the Gini $(2A-1)/(1-p_D)$ are revealed as one number computed three ways. Run in comparison mode (realized outcomes against model claims), the same identity recovers the reliability diagram in cumulative coordinates, with the sign of the gap between the empirical and model-implied Gini coefficients as a calibration diagnostic. A worked five-band example carries every identity in discrete form, and a kernel-density example extends them to the continuous case.

A game of chess is a stream: a time-ordered sequence of moves, each carrying an engine evaluation, a measure of accuracy, a measure of position complexity, and a clock reading. We model a game as a multivariate path and apply the signature transform of rough-path theory to obtain a reparametrization-invariant, graded feature set that records the order and interaction of in-game events without a parametric likelihood. We show that a player's law of play is identifiable from the expected signature up to tree-like equivalence, construct a signature-kernel two-sample test on path space, and recast cheating detection as an anytime-valid sequential test: a signature conformance score becomes an e-process whose error is controlled for every sample size at once by Ville's inequality, with fluctuations calibrated on the moderate-deviation scale. The discriminating information lives in the signature's Levy areas, which measure whether accuracy rises precisely when positions become hard–the fingerprint of engine assistance that aggregate match-rate statistics discard. In a controlled study the test holds exact type-I control and detection power rises from negligible for subtle assistance to 0.98 for blatant assistance, with a median detection time matching the growth-rate prediction. Calibrated to Magnus Carlsen's documented elite accuracy, the monitor does not flag world-champion-level play; and we exhibit cheating strategies that leave every aggregate statistic, including the best-move-frequency z-score of the Regan system, unchanged yet are caught cleanly by the signature–making precise how an order-aware, anytime-valid test strengthens the prevailing approach to chess anti-cheating.

Time series forecasting (TSF) underpins consequential decisions in energy, transportation, finance, and healthcare, yet TSF models are almost universally ranked by a single number (e.g., average error) on clean held-out data, under the implicit assumption that it predicts deployed reliability. However, real faults are not i.i.d noise but structured events with temporal shape, broken cross-variable dependencies, regime change coupled with missingness, and causal propagation across a sensing pipeline. Treating TSF robustness as a data-quality problem, we present TS-Fault, a benchmark that evaluates forecasting models under explicit, parameterized fault scenarios with controllable semantic difficulty. TS-Fault organizes recurring failures into four modes along two orthogonal axes (observation- vs mechanism-level; univariate vs multivariate) and injects each fault into the most prediction-critical window via a unified importance score. This design enables robustness to be tested against the structures models actually rely on, rather than reduced to generic noise sensitivity. We evaluate 21 models across 6 datasets, 4 modes, and 5 difficulty levels under a paired clean/corrupt protocol. The results reveal three findings that contradict common leaderboard intuition: (i) clean-data accuracy anti-correlates with robustness; (ii) clean rankings are preserved under observation-level faults but reshuffled under mechanism-level faults; and (iii) all catastrophic failures occur under mechanism-level faults, with foundation models achieving the highest clean-data accuracy yet exhibiting the greatest fragility. The code is publicly available at https://github.com/Ray-zyy/TS-Fault.

One of the major difficulties in the mechanistic interpretability of neural networks is the occurrence of polysemanticity, which suggests that each neuron is typically responsible for multiple different tasks, impeding a clean interpretation of their function. The seminal paper of Elhage et al. (2022) argues that this occurs due to superposition, a phenomenon where the neural network represents distinct features as non-orthogonal directions in a lower-dimensional space, a strategy that allows much greater compression of the data without sacrificing fidelity due to the feature sparsity of input vectors. Elhage et al. (2022) empirically validates these hypotheses in a rather natural and simple autoencoder with sparse inputs. The contribution of the present work is to analyze the mathematical basis for the occurrence and optimality of superposition, while rigorously corroborating some of their findings. In particular, we provide upper and lower bounds for the L2 reconstruction loss, tight in the very sparse regime, for power activation functions. A short list of interesting open problems are also included at the end.

The large-scale digitization of educational assessment has made the continuous oversight of item banks both essential and complex. This paper presents Analytics for Quality Assurance for Item Pools (AQuAP), a dashboard environment for monitoring item quality and item bank health. AQuAP supports the operational implementation of the large scale item generation procedures for high-stakes tests as included in the Item Factory, a framework for automated and human-supported test development. The paper describes AQuAP in relationship with the process of item development, outlines the broader metric framework for item-pool quality assurance, and highlights the Effective Bank Size (EBS) as one central indicator of pool vitality. EBS quantifies how many independent test sessions can be constructed before content repetition occurs and, when coupled with exposure and usage metrics, provides insight into item bank security, diversity, and efficiency. We further introduce bank-health metrics, such as maximum exposure, maximum conditional exposure, adjusted effective bank size, and the rarely-administered fraction, all of which extend this picture of item utilization. AQuAP illustrates how operational analytics can translate psychometric concepts into quality assurance tools for high-volume, AI-enabled testing programs. This work is illustrated with the Duolingo English Test (DET) processes.

Multi-cause observational studies contain information about unmeasured confounding through the dependence structure among causes. However, literal imputation of the unobserved confounder is often more complex than learning a lower-dimensional substitute score that preserves the shared assignment variation needed for stable causal adjustment. The deconfounder (Wang and Blei, 2019) and related substitute confounder methods exploit this idea, but flexible assignment models can fit the joint distribution of the causes while producing scores that over-encode the treatment vector, collapse overlap, or capture single-cause variation. We develop a Bayesian factor assignment framework for learning sparse substitute confounders that retain coarse multi-cause dependence with shrinkage priors. The theory is stated at the level of posterior concentration, factor score contraction, and overlap-preserving assignment geometry and therefore does not rely on a particular shrinkage prior. Under these conditions, the proposed regression-adjusted estimators are consistent for mean potential outcomes when the corresponding latent variable identification assumptions hold. Shrinkage priors provide a natural tool for latent structural learning: they favour low-dimensional factors supported by multiple causes, discourage effectively single-cause factors, and induce an ordering of the latent factors through progressive shrinkage. Synthetic experiments illustrate the roles of signal strength, outcome validity, and geometry-aware regularization. In an Alzheimer's Disease Neuroimaging Initiative (ADNI) baseline analysis, sparse substitute scores recover much of the adjustment obtained by directly conditioning on invasive cerebrospinal-fluid biomarkers, while collapse diagnostics identify when fitted factors reduce to individual observed measurements.

Offline reinforcement learning is typically analyzed under process-level reward supervision, yet many sequential decision datasets record only trajectory-level outcomes. We develop a statistical theory for offline policy optimization from such outcome-level supervision. We first study the canonical setting where the target remains the expected cumulative reward, but each offline trajectory provides only a scalar label whose conditional mean is the cumulative return. We propose OPAC, a pessimistic actor-critic algorithm that learns a latent reward model and optimizes a policy from trajectory-level labels. We prove a high-probability guarantee of order $\widetilde O(H^2\sqrt{C_{sa}(\pi^\star)/n})$ and a matching lower bound, characterizing the sharp statistical cost of replacing process-level rewards with one trajectory-level label. We then extend the principle to preference-based feedback, preserving the leading horizon and concentrability dependence up to preference-model constants. Finally, we study generalized outcome-based offline RL, where both the supervision and the objective are trajectory-level quantities induced by a nonlinear aggregation of latent per-step rewards. This problem is not learnable in general: for all-success objectives, any offline learner may require $\Omega(2^H)$ trajectories even with deterministic transitions and constant concentrability. We then identify a tractable regime through two structural coefficients, $\kappa_\mu(\sigma)$ and $\chi_\mu(\sigma)$, capturing information loss in outcome aggregation and generalized Bellman updates, under which generalized OPAC achieves polynomial sample complexity. Together, our results delineate when outcome-level supervision enables sample-efficient offline control and when missing process-level rewards create fundamental statistical barriers.

U-calibration studies online forecasting algorithms whose predictions can be consumed by any unknown downstream agent, guaranteeing sublinear regret simultaneously for all proper loss functions. Existing U-calibration algorithms achieve worst-case optimal $O(\sqrt{T})$ regret for every bounded proper loss, but they fail to adapt to easier losses: as we show, even for smooth losses such as squared loss, they incur $\Omega(\sqrt{T})$ regret instead of the optimal $O(\log T)$ regret. In this work, we show that this limitation is not inherent. Specifically, we design a single forecast algorithm that simultaneously achieves $\tilde O(\sqrt{T})$ regret for every bounded proper loss and $O(\log T)$ regret for every bounded smooth proper loss. More generally, our algorithm also attains logarithmic regret for losses that are smooth relative to the log-barrier, which include several non-Lipschitz examples. Our approach is based on a novel variant of Follow-the-Perturbed-Leader (FTPL) in which perturbations are applied directly in the prediction space using self-concordant noise. The resulting analysis also departs substantially from prior FTPL analyses due to the complex nature of this noise and may be of independent interest.

Computing geometric representations of data is a cornerstone of modern machine learning, typically achieved by training dual encoders which map queries and documents into a shared embedding space. Recent work of You et al. [NeurIPS '25] has extended this approach to hierarchical retrieval, where relevance is determined by the ancestor-descendant relationships in a Directed Acyclic Graph (DAG). While previous work has shown that valid embeddings exist when the number of descendants is small, these bounds degrade significantly for deep hierarchies, requiring dimensions as large as the total number of nodes. In this paper, we investigate compact reachability embeddings for more general graph classes and provide theoretical guarantees for representing hierarchies using embeddings whose dimension depends on structural graph parameters. We prove that for any directed tree, there exists a reachability embedding in constant dimension 3, independent of the tree's size or depth. We generalize this result to graphs characterized by treewidth $t$, constructing embeddings of dimension $O(t \log n)$, where $n$ is the number of nodes. Complementing these upper bounds, we provide matching or near-matching lower bounds, showing that dimension $\Omega(n)$ is necessary for general DAGs and $\Omega(t/\log(n/t))$ is required for graphs of treewidth $t$. We also obtain upper and lower bounds parameterized by the number of cross-edges in the DAG. We additionally show that our embeddings can be constructed on real world datasets, and that they give much smaller dimensions in high recall regimes compared to prior embeddings with theoretical guarantees.

Barren plateaus are stated as an average-case phenomenon: pick an ansatz, initialize it naively, and concentration follows. This has led to the common view that a potential cure for barren plateaus is simply to initialize the parameters more carefully. Here we show that the situation is subtler. We introduce a first-moment framework that gives a simple operator-level diagnostic for when an initialization may escape the fully concentrated barren-plateau fixed point, and for comparing the biases induced by different initialization strategies. Our framework recovers several known initialization schemes such as identity and Gaussian initialization, but also shows that barren-plateau avoidance is highly non-unique. Indeed, many shifted, biased, and non-symmetric parameter distributions can avoid concentration, and these choices need not be equivalent. In fact, our results show that one can generate exponentially many families of inequivalent initialization strategies. Then, our numerics indicate that different first-moment-distinct initializations can lead to different attained minima, suggesting that avoiding barren plateaus via smart initializations can trade the exponential concentration problem for the challenge of selecting the right trainable pocket amongst many options.

Conventional panel data methods recover retrospective counterfactuals within the observed horizon. Many policy decisions, however, are prospective: how would an untreated unit evolve beyond the observed panel under an intervention it has not previously experienced? We develop a causal forecasting framework that combines the counterfactual logic of synthetic controls with multivariate time-series extrapolation. Under a latent factor model, we impose low-rank structure on the treated-state time factors. This yields the Two-Way Synthetic Forecasting estimator, which learns cross-unit weights from pre-treatment outcomes and temporal dynamics from treated donors' post-treatment trajectories. We establish pointwise identification, finite-sample error bounds, consistency, asymptotic normality, and inference for prespecified linear summaries over fixed forecast horizons. Simulations support the theory, and an application to the opening of NFL stadiums during the 2020 season illustrates how the method can inform a prospective policy change using only information available at the decision date.

Reliable generalization in conditional latent variable models requires understanding both identifiability and extrapolation: how observed variation across attributes determines latent structure, and how that structure determines distributions at unseen attributes. However, existing identifiability and extrapolation guarantees are largely model-specific, with separate analyses in nonlinear ICA, causal representation learning, perturbation modeling, and related conditional latent variable models. We introduce concept modulation models (CMMs), an attribute-indexed class of conditional generative models with structure $A\to \Lambda \to C\to X$, where attributes select modulators, modulators induce latent concept laws, and concepts generate observed features. CMMs lift transition-based identifiability to conditional settings by showing that feature agreement on observed attributes induces a latent concept transition constrained by the CMM class. We express these constraints through attribute potentials, log-density ratios between attribute-conditioned concept laws, separating the generic lifting step from model-specific rigidity arguments. The same potentials control extrapolation: agreement at unseen attributes holds exactly when the transported attribute-potential identities extend to those attributes. This yields algebraic extrapolation criteria, identifies the common potential-based proof objects behind several existing identifiability and extrapolation results, and, when combined with the model-specific rigidity arguments in those works, recovers their stated conclusions.

Objective sleep assessment relies on polysomnography (PSG), yet clinical impact is often better reflected in patient-reported outcomes (PROs) such as sleepiness and fatigue. Existing summary indices, including the Apnea-Hypopnea Index (AHI), provide limited insight into the multidomain physiology underlying functional recovery. We propose an interpretable, causal-discovery-guided framework for deriving a hierarchical Sleep Recovery Score (SRS) from multimodal PSG. Using two large population cohorts (MESA: \(n=1{,}540\); MrOS: \(n=825\)), we apply directed acyclic graph (DAG) learning to identify candidate physiological drivers spanning respiratory burden, hypoxic burden, sleep fragmentation, sleep architecture, and autonomic regulation. Although derived from clinical PSG, these domains map naturally to sensing streams increasingly available in connected health technologies, including wearable ECG, oximetry, and sleep-stage estimation devices. To preserve mechanistic plausibility, we introduce a two-stage screening process that combines physiology-based constraints with constrained LLM-assisted auditing to identify and remove structural confounders and construct-overlapping variables. Across cohorts, these five domains emerge as recurrent physiological domains associated with recovery, and the resulting SRS shows up to \(3.4\times\) stronger alignment with perceived recovery than AHI. By linking multimodal sleep physiology to patient-centered outcomes through an interpretable, bias-aware, and domain-structured framework, this work provides a practical foundation for recovery modeling across both clinical sleep studies and emerging smart and connected health settings.

Remaining useful life (RUL) estimation is central to predictive maintenance, where an unplanned failure can cost far more than the asset itself. Statistical degradation models miss the strong nonlinearity of real systems, and data-driven models often converge to suboptimal solutions in high-dimensional, non-convex search spaces. We propose a Quantum Annealing enhanced Q-Learning (QAQL) framework that couples the sampling behaviour of quantum annealing with the sequential decision making of Q-learning. Each Q-value update is encoded as a small quadratic unconstrained binary optimization (QUBO) whose ground state is the greedy action; rather than acting as a deterministic optimizer, the annealer returns a distribution over near-optimal actions across many reads, and this stochastic action selection supplies the exploration that curbs premature convergence on nonlinear degradation trajectories. The QUBO is solved on the D-Wave Advantage system using minor embedding, with the annealer woven into the reinforcement-learning loop rather than bolted on after training. We validate QAQL on two public benchmarks: the NASA C-MAPSS turbofan engine datasets and a device-fleet predictive maintenance dataset. Averaged over many independent runs and across six error metrics, QAQL outperforms the classical and quantum baselines considered in this study, with statistically significant improvements. The results indicate that quantum annealing is a usable, not merely theoretical, optimizer inside a reinforcement-learning loop for industrial predictive-maintenance applications.

Bayesian model selection in cosmology and particle physics is often performed where posterior odds inherit a strong, often unacknowledged dependence on the prior assigned to competing models. Standard responses – reference priors, hierarchical priors, or appeals to naturalness – ignore relevant theoretical knowledge or rely on criteria hard to define operationally. We propose the Coherence Principle: a reproducible prescription for assigning model priors according to compatibility with the validated structure of an existing theory. This structure, or grammar, includes symmetries, conservation laws, locality, Lorentz invariance, and universality patterns. Unmotivated violations of these rules incur a coherence cost, converted into a prior weight through a maximum-entropy exponential form controlled by one calibratable parameter $\alpha$. The resulting prior is distinct from both the Bayesian Occam factor and naturalness: it penalizes not parameter volume or fine tuning, but departures from validated theoretical grammar. We illustrate the principle with examples from cosmology and fundamental physics: neutrino mass mechanisms, dark energy and modified gravity, inflation, beyond-Standard-Model sectors, and hierarchical astrophysical inference. We test it also on four historical cases – general relativity, Pauli's neutrino, parity violation, and special relativity – where evidential and theoretical contexts can be reconstructed. These examples show that it favors the historically successful choice when the proper grammar is defined in the correct domain and time. The Coherence Principle makes explicit a common but usually tacit part of physical reasoning: trust in validated structural rules. It turns this judgment into a transparent, testable, and overrulable component of Bayesian inference, leaving empirical likelihoods free to dominate when data are sufficiently constraining.

Modern AI agents retrieve documents, call tools, check intermediate information, and then produce a final answer or action. This creates a risk-control problem that is not visible from the final answer alone. A final response may look acceptable even when the retrieval was weak, a tool output was wrong, or an earlier step was unsupported. We propose ToolChain-CRC, a conformal risk-control method for retrieval-augmented and tool-using agents under drift. The method treats each agent run as a full trajectory of actions, observations, and final output. It builds step-level risk scores, combines them into a trajectory risk score, calibrates an accept-or-intervene rule, and adds an anytime alarm that can stop risky runs before the final answer. We prove trajectory-level risk control under exchangeable calibration runs, give a drift-aware extension with auditable constants, and prove an anytime escalation rule through a supermartingale construction. Experiments cover synthetic tool-chain drift, RAG/tool-use stress tests, public SQuAD-derived retrieval tasks, an API-free agentic QA case study, ablations, target-risk sensitivity checks, 20-seed robustness checks, a drift-margin audit, and a live RAG/tool-use agent benchmark. Across these settings, final-answer-only calibration can miss retrieval and tool failures, while trajectory-level calibration keeps accepted-trajectory risk below the target.

Distributed stochastic gradient descent (SGD) is limited by communication rather than computation, since each iteration requires an AllReduce across processes. Communication-avoiding SGD (CA-SGD) amortizes communication over $s$ iterations by replacing $s$ consecutive AllReduces with a single AllReduce of an $sb\times sb$ Gram matrix, trading more computation and bandwidth for fewer synchronization points. Modern GPUs with matrix hardware and reduced-precision formats offset this by accelerating the Gram GEMM and shrinking BF16 traffic. We study mixed-precision CA-SGD for generalized linear models on NVIDIA GPUs. Our finite-precision analysis decomposes the local rounding error of one CA-SGD outer iteration into nine independent precision choices, depending on the hardware only through its low-precision unit roundoffs, so the resulting recipes transfer in principle across GPU generations. The recipe stores the input matrix and margin vector in low precision, computes the Gram matrix from low-precision inputs with high-precision accumulation, communicates it in high precision, and performs the inner recurrence and weight updates in high precision. On NERSC Perlmutter A100 GPUs, mixed-precision CA-SGD matches FP32 SGD loss within $0.5\%$ on logistic, linear, and Poisson problems and reaches $5.1$–$6.8\times$ speedup over FP32 SGD on epsilon, SUSY, HIGGS, synth, and Poisson-synth. Our software is available at https://doi.org/10.5281/zenodo.20448273

Unlike traditional causal inference, which prospectively evaluates the effects of causes, apportioning causal responsibility requires a retrospective assessment to deduce the causes of an outcome that has already occurred. This paper proposes a quantitative framework for apportioning causal responsibility between two binary risk factors that jointly contribute to a realized adverse outcome. Ideally, knowing the individual's latent causal type, defined by the potential outcomes under all possible exposure combinations, would allow precise apportionment; however, these potential outcomes cannot be simultaneously observed. We therefore define the average causal responsibility of each risk factor as its expected responsibility over the distribution of latent causal types. Under the assumptions of no confounding and monotonicity, we establish nonparametric identification of this metric when the type-specific responsibilities satisfy a structural balance condition, and derive sharp bounds otherwise. We illustrate the proposed framework using the classic example of lung cancer attributable to smoking and asbestos exposures.

Covariance matrices used in astronomical and cosmological parameter inference are often estimated from a finite number of simulations, so covariance uncertainty can affect posterior calibration and parameter constraints. We study covariance regularisation from the perspective of likelihood-based inference with simulation-estimated covariance matrices. First, we analyse scalar covariance scaling under the Gaussian plug-in likelihood and the covariance-marginalised Sellentin–Heavens likelihood. Using an expected negative log-likelihood loss, we show that Hartlap covariance-side scaling is recovered as the optimum under the Gaussian plug-in likelihood, whereas the unscaled sample covariance is optimal under the Sellentin–Heavens likelihood. This shows that scalar covariance corrections are likelihood-dependent and that an additional Hartlap-type global scaling is not favoured once covariance uncertainty is marginalised through the Sellentin–Heavens likelihood. We then introduce a shrinkage formulation in which the sample covariance is regularised towards a spherical target and the shrinkage intensity is treated as an auxiliary inferential quantity. A prior is assigned to the shrinkage intensity, the likelihood induces its posterior distribution, and the final parameter posterior is obtained by marginalising over it. Monte Carlo experiments show that shrinkage substantially improves covariance conditioning, while marginalisation over the shrinkage intensity propagates uncertainty about the amount of regularisation into posterior inference. The proposed approach provides a simple way to combine covariance-marginalised likelihood inference with structural regularisation of noisy simulation-estimated covariance matrices.

We develop a spectral based coefficient of determination to measure how well the spectral density of a stationary linear process is represented by the class of MA($q$) models. Using periodogram-based estimators, we establish asymptotic normality, derive tests for the MA($q$) hypothesis, and construct procedures for determining the smallest order $q$ achieving a prescribed approximation quality.

Sparse point observations are increasingly available for precipitation nowcasting, but it is unclear how much they improve dense radar-field forecasts. We partially address this question with a multimodal graph neural network nowcasting system over the Nordic radar domain. The model predicts rain rate every five minutes up to two hours ahead and is trained with different combinations of radar history, MEPS numerical weather prediction, Netatmo surface observations, MSG satellite channels, stochastic noise, and CRPS-based ensemble losses. The study is designed as an ablation of operationally relevant information sources and training objectives. We compare radar-only, NWP-informed, station-informed, satellite-informed, noise-augmented, and CRPS-based configurations using complementary diagnostics on the radar grid, at station locations, for rain onset, and through oracle, displacement, and amplitude scores. The results show that each source improves a different part of the forecast problem. MEPS stabilises radar-only extrapolation, Netatmo observations improve local station and onset diagnostics, and satellite predictors reduce some station-level biases but may activate rain too early when used deterministically. CRPS-based configurations provide the most consistent radar-grid gains, while the combined satellite and CRPS setup gives the best overall oracle/DAS score. These results do not support the conclusion that point observations are uninformative for nowcasting, but they show that local observational skill and spatially coherent radar-field skill are distinct targets. The practical implication is that sparse observations can provide useful local constraints, but their benefit for radar-like fields depends on the training loss, uncertainty representation, and how observation support is encoded in the model.

This paper develops a variational framework for regulated language generation. Starting from autoregressive token sampling, we derive the induced distribution over complete messages and relate it to an entropy-regularized Gibbs law. Regulation is modeled as an optimal discriminator whose convex-dual value is an f-divergence, and the generator-regulator interaction is formulated as a saddle-point problem. The framework applies to moderation, censorship, AI deception detection, compliance auditing, phishing defense, and manipulation control, where regulation concerns a distribution over possible messages rather than a single output. The equilibrium clarifies the tradeoff among utility, entropy, regulatory alignment, and finite-length detectability. Two finite-vocabulary case studies, censorship filtering and phishing defense, illustrate how the theory can be evaluated through utility, entropy, divergence, receiver-side scores, and detection probability.

On biomedical tabular data, flexible models such as deep networks, gradient-boosted trees, and kernel methods are repeatedly matched or beaten by linear and logistic regression given the same features. The usual reaction is to treat this as a model-side shortfall, to be fixed with more data, a better architecture, or tuning, on the assumption that the nonlinear structure is there and the model has failed to capture it. We argue that these fixes cannot help when the binding limit is the measurement rather than the model, as it frequently is in biomedicine. Additive noise blurs the population-optimal predictor, and because blurring removes a function's fine, rapidly varying detail before its broad shape, it erases nonlinear structure faster than linear structure. A degree-$k$ interaction is attenuated by the $k$-th power of feature reliability, while the linear part is attenuated only once. At the reliabilities typical of biomedical measurement, the nonlinear advantage can vanish even when the underlying biology is strongly nonlinear, and what the noise removes cannot be recovered by a larger cohort or a more flexible model, only by better measurement. The nonlinearity is hidden, not absent, and a tie between linear and flexible models is not by itself a verdict on the biology. These pieces are classical, drawn from measurement-error statistics, psychometrics, and Gaussian analysis, and we assemble them into an exact excess-risk identity. Measurement reliability is one of three conditions, alongside sample size and feature representation, that must align for a flexible model to help, and together they leave only a narrow window that most biomedical tasks fall outside. Across 140 UK Biobank tasks, the gap between flexible and linear models, where it exists, carries the predicted noise signature, and the three conditions can be separated by intervention but not by a benchmark alone.

Anomalies in functional data arise from rare or distinct processes that deviate from the dominant data-generating mechanism. Detecting such departures is essential in applications where they may correspond to errors, structural changes, or other behavior of interest. This work introduces a Bayesian nonparametric approach for anomaly detection in multivariate functional data. We model functional data as an infinite mixture of multi-output Gaussian processes, with a finite and automatically determined number of mixture components obtained through slice sampling. Mean functions are represented using a wavelet basis and regularized through Besov priors to obtain a smooth and sparse representation of the data. Cross-functional dependence is captured using the intrinsic coregionalization model and we solve covariance kernel selection by introducing a Carlin-Chib product space step in the Markov Chain Monte Carlo algorithm. Within this model, anomalous observations are assigned to small mixture components without requiring prior specification of the number or nature of anomalies. We consider a semi-supervised setting, in which labels are available for 15% of the normal observations and a large class imbalance is present. The utility of our model is demonstrated on both univariate and multivariate functional data.

Continuous-time Markov Chains are widely used to model stochastic dynamical systems, but key summary quantities such as means and covariances are often intractable. While Monte Carlo sampling provides asymptotically exact estimates, it becomes computationally prohibitive when moments must be evaluated across many parameter values. We develop a simulation-based surrogate modeling framework that learns parameter-to-moment mappings from Monte Carlo-derived, noise-corrupted training targets, enabling efficient and accurate approximation across the parameter space. We show that Monte Carlo noise affects mean estimation primarily through additive variance, whereas covariance estimation is additionally impacted by bias arising from nonlinear transformations of empirical estimates. Using a stochastic Susceptible-Infected-Recovered model, we demonstrate that neural networks accurately learn both mean and covariance under fixed simulation budgets allocated to constructing the noisy training labels. We further characterize how to allocate computational resources between parameter-space coverage and Monte Carlo replication, showing that covariance estimation requires a balanced allocation to control both variance and bias, while mean estimation benefits more from increased parameter space coverage. Finally, we show that the learned moment mappings produce valid population-level quantities and perform well in downstream tasks such as whitening. These results highlight the importance of accounting for Monte Carlo noise in surrogate modeling and provide practical guidance for simulation-based learning in stochastic systems.

A central challenge in scientific inference is to produce uncertainty assessments that are both situation-specific and frequency-calibrated. This article examines inferential models (IMs) as a framework for prior-free probabilistic reasoning with scientific uncertainty. The central IM idea is to view the auxiliary variables in a sampling model as the source of model-based uncertainty. R. A. Fisher's fiducial inference transfers auxiliary randomness to the parameter space before applying probability calculus; IMs instead predict the unobserved auxiliary value with calibrated predictive random sets (PRSs) and transfer the resulting plausibility statements only afterward. This change in order yields valid uncertainty assessments and clarifies the relations among Fisherian fiducial reasoning, Neymanian confidence theory, Dempster-Shafer belief functions, generalized fiducial inference, and IMs. By comparing IMs with objective-prior Bayesian inference, the article argues that E. T. Jaynes' logic-of-science ambition can be continued without forcing all scientific uncertainty into a precise prior distribution because calibrated imprecision is often essential. Finally, the article suggests that a differential-geometric theory of IMs may be within reach, offering a possible route to foundational questions traditionally framed in terms of the likelihood principle.