Papers

Levene's test for homoscedasticity is a standard procedure used to evaluate whether multiple groups of independent observations share a common variance. Because Levene's test relies on an Analysis of Variance (ANOVA) applied to transformed absolute or squared deviations, it structurally mirrors the statistical properties of ANOVA itself. Let $\mu_j$ and $\sigma^2_j$ denote the expectation and variance of the observations in group $j$. It was previously established that for a given significance level $\alpha$, ANOVA can result in a logical contradiction: failing to reject the global null hypothesis $H_0: \mu_1 = \mu_2 = \mu_3$ while simultaneously rejecting the localized hypothesis $H_0': \mu_1 = \mu_2$ with the same or higher confidence. In this paper, we show that Levene's test directly inherits this same “paradox” regarding group variances $\sigma^2_j$. We provide theoretical reasoning and a numerical illustration of this inconsistency, demonstrating how the addition of a well-behaved third group can dilute the test statistic and mask a significant localized variance discrepancy.

Speculative sampling accelerates diffusion model generation by drafting inexpensive candidate states and correcting them under a coupling that preserves the target distribution exactly, reducing the number of expensive target evaluations. Existing diffusion samplers, notably those based on reflection maximal coupling, are topologically constrained: their lookahead drafts form a chain graph, a single linear sequence, which inherently limits the acceptance rate per target evaluation. We connect speculative sampling in diffusion models to relative entropy coding (REC). This perspective shows the lookahead need not be linear and motivates our central contribution, draft trees, which enrich the candidates considered per round and lower the target function evaluations. We further adopt greedy rejection sampling, an REC algorithm, as the draft-target coupling, improving acceptance while guaranteeing exact target samples. Experiments across diverse target and draft models demonstrate up to 8.3% acceleration over the reflection coupling baseline in practical settings.

Theoretical analysis and Monte Carlo simulation separate local-epoch effects from averaging and tuning effects in federated Lasso. How much local work should precede averaging when fitting a sparse regression? An orthogonal calculation shows that extra epochs can have no effect while averaging still enlarges the selected set. A correlated two-site construction gives an exact, nonmonotone limiting objective gap and its minimizing epoch count. We then compare coordinate-descent averaging, two thresholding modifications, and adapted FedDualAvg across twelve scenarios and 600 replicates. Methods share a penalty, independent validation samples, selection rules, and resource limits. FedDualAvg generally achieves smaller objective gaps but does not always recover variables better. Thresholding gains depend strongly on selection rules. Epoch effects vary with correlation, signal strength, site allocation, and the outcome measured. These results distinguish faster iteration from better optimization, prediction, and variable selection.

In an important recent work, Blanc (2026) gave an algorithm for robustly learning Boolean concept classes with respect to a fixed distribution that outputs a (randomized) classifier achieving the optimal error of $\eta + \varepsilon$ where $\eta$ is the noise rate. In contrast, it is well known that deterministic hypotheses cannot achieve error less than $2\eta + \varepsilon.$ Blanc's algorithm is computationally inefficient, and the main problem left open in his work is to find a polynomial-time algorithm given access to an oracle for empirical risk minimization (ERM). In this paper, we resolve this problem and give such an algorithm. Perhaps surprisingly, our techniques make crucial use of various types of no-regret learners. Additionally, we give an efficient algorithm (no ERM oracle required) for robustly learning any function class that admits sandwiching polynomials with respect to hypercontractive distributions. As one consequence, we give the first polynomial-time algorithm for robustly learning a halfspace with respect to Gaussian marginals that achieves error $\eta + \varepsilon$ for any constant $\varepsilon$.

Alzheimer's disease is clinically staged as a three-class progression: cognitively normal (CN), mild cognitive impairment (MCI) and Alzheimer's disease (AD). The intermediate MCI class is biologically heterogeneous and overlaps with both extremes and it is the dominant source of classification error. This work develops a leakage-controlled framework for subject-level CN/MCI/AD classification that fuses four sources of information: a 3D Tau PET model, a 3D structural MRI model, a tabular ROI-and-plasma model and a hierarchical PET–plasma model. All branches are derived from a custom preprocessing and feature-extraction pipeline applied to $881$ aligned subjects from the ADNI cohort. They are aligned at the subject level and evaluated under identical, no-leak cross-validation folds, so that the multimodal fusion, built entirely on synchronized out-of-fold predictions, avoids optimistic bias. The final model is a fixed, parameter-free convex combination of the four branches. A nested superlearner serves as a robustness analysis and reproduces, rather than improves on, the fixed weights. The fusion improves significantly over every individual branch, with the largest gain on the MCI class, while a logistic meta-learner provides a complementary, MCI-oriented operating point on the same accuracy-sensitivity frontier. The work is presented as an honest baseline. It shows that multimodal fusion improves robustness, but that MCI remains the central bottleneck. This reflects a limit of the cross-sectional information rather than of the fusion mechanism and motivates reformulating the task as a continuous progression problem.

We study computable error bounds and certified early stopping for regularized inverse problems, where a data-fidelity term is traded against a regularizer. The analysis relies on an exact duality-gap identity that splits the total gap of $F(\Phi\mu)+\lambda R(\mu)$ into a data-fidelity Fenchel–Young loss and a regularizer Fenchel–Young loss, $ \Delta(\mu,h)=L_F(\Phi\mu\parallel h)+\lambda L_R(\mu\parallel\eta),\qquad \eta=-\Phi^\star h/\lambda, $ valid for any primal point $\mu$ and any dual point $h$. The data-fidelity term $F$ is strictly convex, so wherever $F^\star$ is differentiable the loss $L_F(\Phi\mu\parallel h)$ is the Bregman divergence of~$F$ between the prediction $\Phi\mu$ and $\nabla F^\star(h)$, and it vanishes exactly at Mirror Alignment $h=\nabla F(\Phi\mu)$. Evaluated at a dual-feasible point $\tilde h$, the gap~$\Delta(\mu,\tilde h)$ is computable and oracle-free, meaning that it uses no knowledge of the solution, and it bounds the suboptimality of $\mu$. Under the source condition, the same Fenchel–Young losses give a priori bounds on the estimation and prediction errors. Their scale is the irreducible model and noise error $L_F(\Phi\mu^\star\parallel h^\star)$, which vanishes exactly when Mirror Alignment holds at the certificate. This gives an early-stopping rule: run the algorithm until the regularizer Fenchel–Young loss falls below a tolerance $\epsilon$. A constructive version of the Brøndsted–Rockafellar theorem then turns the current pair into an exact dual-feasible one, and this proxy lifts to an exact primal certificate. We build the proxy by a proximal step in the geometry of the fidelity, with Bregman kernel $F^\star$ and tilted by the prediction $\Phi\mu$: it recovers the Euclidean step of Carlier when $F$ is the squared error, and it reduces the duality gap by the regularizer Fenchel–Young loss, up to a second-order remainder that vanishes in the quadratic case. Our running example is the Generalized Beurling–Lasso (GBL), where $R$ is the total-variation norm on signed measures. It contains the classical Beurling–Lasso, obtained with the squared error, and also covers robust, logistic, entropic and inverse-optimal-transport losses. The same duality gap certifies deep-learning optimizers such as Lion-K and Muon, in their proximal form, as solvers of the regularized program. A companion paper by the same authors builds on these error bounds to establish exact support recovery for the GBL under a non-degenerate source condition.

Galactic foregrounds represent a major contamination to the measurement of primordial $B$-modes from observations of the Cosmic Microwave Background polarisation. Even after the application of component separation algorithms, foreground residuals may potentially still bias the estimate of the tensor-to-scalar ratio $r$, causing a false detection. In this work, we present the methodology of a robustness test for the validation of an eventual detection of primordial $B$-modes, as obtained by a future, LiteBIRD-like satellite experiment. The goal of the test is to identify the foreground residuals contamination by looking for non-Gaussian properties in the CMB $B$-modes map, recovered through blind component separation algorithms. We adopt scattering transforms (ST) as our summary statistics sensitive to the non-Gaussian features of foreground residuals and to their correlation with foregrounds tracer maps. We characterise and validate the methodology on realistic sky simulations with different levels of foregrounds complexity. The proposed test is able to identify a bias on the tensor-to-scalar ratio of $\gtrsim 10^{-3}$ in $\sim 90\%$ of our simulations, with this bias value being of the same order of the accuracy targeted by LiteBIRD. Additionally, for our particular experimental configuration, the test is passed when the bias is lower than the sensitivity on the $r$ parameter, and no warning is raised. These results provide an important step forward in the development of statistical tools for validating future measurement of cosmological parameters, against foregrounds contamination.

For any fixed $k$, we prove a lower bound on the $k$th largest modulus of an eigenvalue of the non-backtracking matrix $B$. Specifically, consider any deterministic or random family of graphs that converges locally to the unimodular Galton-Watson tree with root degree distribution $D$, and set $\kappa:=\mathbb E[D(D-1)]/\mathbb E[D]$. Given $\kappa>1$ and an exponential-moment bound on the empirical degree distributions, we show that $|\lambda_k(B)|\geq\sqrt{\kappa}-o_N(1)$, where $N$ is the number of vertices. When restricted to locally tree-like regular graphs, this recovers a well-known consequence of the Ihara-Bass formula. In the specific case where the graph is generated through the Erdős-Rényi model with expected degree $d>1$, this proves a conjecture of Bordenave, Lelarge, and Massoulié. To do this, we show that the normalized log-determinant of the Bethe-Hessian of the graph is bounded by that of the Bethe-Hessian of its local limit. This bound is violated if the eigenvalues of the non-backtracking matrix are too small. We establish this using an effective-conductance interpretation of the tree Green's function recursion.

Clinical protocols may require rescue medication when a patient's condition crosses a prespecified threshold. The outcome under continued non-rescue is then unobserved after first crossing, preventing point identification without extrapolation assumptions. In this paper, we develop sharp partial identification and inference for this hypothetical estimand while retaining the original no-rescue intervention. We show that bounding this mean reduces exactly to restricting conditional mean effects of withholding rescue at first-crossing histories. Building on this representation, restrictions on prespecified crossing-stratum averages yield three results. (i) A finite convex program gives sharp bounds without modeling post-crossing trajectories, attained by full-data laws that preserve the observed distribution. (ii) Confidence intervals incorporate sampling uncertainty and numerical error to cover the entire sharp set, including strata with no observed crossings. (iii) Under local feasibility conditions, sampling and numerical errors control endpoint accuracy even with zero-probability strata, whereas valid coverage requires no such conditions. Our experiments show that the stratum-average restrictions can yield narrower confidence intervals in larger samples, and that accounting for uncertainty in empty strata maintains valid coverage when rescue is rare.

Large language model (LLM) distillation aims to transfer the capabilities of a powerful teacher to a smaller student. Direct imitation, however, can also transfer the teacher's systematic bias and errors. This challenge is particularly pronounced under covariate shift, when the teacher's reliability on target questions is uncertain and target-domain reward feedback is unavailable. We propose Coupled Calibration and Learning (CCL), an LLM distillation algorithm that couples teacher calibration with student updates through token-level branching, using reward feedback only on source questions. Each iteration calibrates the teacher using source feedback and then uses the calibrated teacher to train the student on target questions. The updated student, in turn, informs subsequent calibration. In an autoregressive policy framework, we prove that the output student's expected average Kullback-Leibler divergence to the oracle student converges to zero at a polynomial rate in the number of iterations. The oracle maximizes the true reference-regularized target reward within the student class, which need not represent the unrestricted optimal policy. Our analysis quantifies the progress of projected student gradient updates while controlling the error in teacher calibration. We further establish a separation from regularized direct matching: its error relative to the oracle student can remain bounded away from zero even when the teacher achieves higher regularized target reward than every student policy. These results demonstrate that LLM distillation can overcome persistent teacher bias and recover the optimal student through coupled calibration and learning, without target-domain reward feedback.

Covariates missing not at random generally prevent point identification of regression coefficients without untestable restrictions. This paper studies linear regression when every missing covariate is restricted to a prespecified compact interval. The resulting population target is the set of best linear predictor coefficients compatible with the observed-data law. Under a non-atomic observed-data law and the stated regularity conditions, we can represent this identified set as the image, under the least-squares moment map, of the Aumann expectation of a random moment set. For the population set and its empirical analogue, we derive explicit bounds under bounded, sub-exponential and polynomial envelope conditions. The bounds show their dependence on sample size, dimension, confidence level and tail parameters. Under a Donsker condition and a uniform zero-set error bound, an oracle enlargement of the set-valued Z-estimator also converges at the root n rate. We separately study a Hadamard-design method for exploring the effect of admissible imputations without enumerating all vertices and build intuition to detect situations where the population target can be approximated by a zonotope. A numerical experiment illustrates how co-missingness and the width of the imputation intervals affect this diagnostic.

Variational quantum circuits (VQCs) are parameterized quantum circuits optimized classically. We propose a hybrid variational quantum circuit (HVQC) extending VQCs with a classical affine post-measurement layer, enabling vector-valued regression without the linear overhead of independent scalar circuits. Theoretically, we show that elementary one-and two-qubit circuits can approximate quadratic functions and products via data re-uploading and entanglement, providing the foundations of the full architecture. Experimentally, on two synthetic image reconstruction datasets and the Friedman1 benchmark (40,568 test samples), our HVQC matches Gaussian Process Regression and outperforms XGBoost and Random Forest. An ablation study confirms that both quantum and classical components are essential, and results highlight the central role of the feature map in hybrid quantum-classical models.

By consuming multiple copies of an unknown qubit state, one can modify its purity while preserving the direction of its Bloch vector. We determine the maximum linear rate at which qubit states of different purities can be interconverted, allowing a nonzero error, quantified, for instance, by the trace distance, provided that it vanishes in the limit of infinitely many copies. Interestingly, the optimal conversion rate is determined by the two eigenvalues of the complex right-logarithmic-derivative (RLD) Fisher information matrix associated with $\mathrm{SU}(2)$ rotations of the qubit state. When the output qubits have higher purity, corresponding to concentration, the optimal rate is given by the ratio of the maximum eigenvalues of the input and output RLD matrices. In contrast, when the output qubits have lower purity, corresponding to dilution, the optimal rate is given by the ratio of their minimum eigenvalues. Remarkably, both concentration and dilution can be implemented using SWAP tests as the only nontrivial two-qubit measurement primitive, together with ancillary qubits initially prepared in maximally mixed states, without requiring any additional two-qubit gates. Our work thus provides a novel operational interpretation of the full complex RLD Fisher information matrix. Crucially, its antisymmetric, purely imaginary part encodes geometric information beyond the statistical distance between density operators and plays an essential role in determining the optimal state-conversion rates.

Policy learning aims to determine who should be treated based on individual characteristics. In high-stakes settings such as medicine and public policy where safety is a central concern, improving the average outcomes alone may not be sufficient: decision makers may also seek to protect individuals from harm, in line with the Hippocratic principle of “do no harm.” In this paper, we propose conformal policy learning (CPL), a policy learning procedure with a new distribution-free safety guarantee that controls the probability of assigning treatment to an individual who would be harmed relative to control. CPL views each treatment decision as testing a hypothesis of counterfactual harm and assigns treatment by thresholding conformal p-values. These p-values use observable proxies and selective calibration to address the challenge that the potential outcomes under comparison are never simultaneously observed. For randomized experiments, under standard exchangeability conditions, CPL provides finite-sample safety guarantee at a user-specified level, without imposing any outcome modeling assumptions. Moreover, when the outcome model is consistently estimated, CPL achieves asymptotically optimal welfare subject to the safety constraint. In observational studies, CPL with learn-then-balance weights achieves doubly robust safety guarantees. We evaluate CPL through extensive simulations and apply it to an empirical study of AI-powered interventions designed to reduce conspiracy beliefs.

Two features intrinsic to survey sampling complicate semiparametric efficiency analysis: design-induced dependence among sampling indicators and the randomness of finite-population targets under the superpopulation law. For general semiparametric full-data models, we show that the observed-data experiment under a broad class of dependent designs is locally asymptotically normal with the tangent-space structure of a reference Poisson experiment. Under the joint superpopulation-design law, first-order efficiency depends on the design only through the limiting inclusion-probability function. Standard missing-at-random projection in the reference experiment characterizes the observed-data efficient influence function. Finite-population targets are treated through first-order asymptotic expansions, extending the analysis beyond exact random sums to nonlinear census characteristics. Superpopulation and finite-population centerings yield equivalent notions of local regularity and efficiency, with their bounds linked by a Pythagorean decomposition that gives a generalized finite-population correction. We then give general and design-specific conditions under which cross-fitted estimators with estimated nuisance functions attain both efficiency bounds. For the finite-population mean, the bound equals the large-sample limit of the Godambe-Joshi anticipated-variance lower bound. For scalar targets, we characterize optimal limiting inclusion probabilities. Simulations and California Academic Performance Index data illustrate the theory.

Statistical heterogeneity limits federated learning when a single global classifier cannot represent client-specific label distributions. In this work, we propose Personalized Federated Knowledge Distillation with Head Adaptation (pFedKDH), which aggregates only the shared backbone, keeps persistent client-specific heads, and uses a recalibrated global head as a teacher during local training. Across MNIST, Fashion-MNIST, CIFAR10, and CIFAR100 under class-wise Dirichlet partitions, pFedKDH obtains the best accuracy in most settings, with accuracy gaps up to 37.67% over the weakest baseline and consistently low standard deviation across repetitions. Component-wise diagnostics and convergence results support the role of persistent heads and distillation-guided local optimization under label-skewed data.

Comparative analyses of mortality dynamics across countries have long shaped our understanding of mortality inequalities and patterns of divergence and convergence over time. However, most existing studies focus on either mortality differences across countries at a single point in time or on country trajectories, without considering both factors simultaneously. In this paper, we address this gap by introducing a multi-stage clustering framework, applied to 94 countries by sex over the period 1960-2019 using data from the World Population Prospects. Specifically, we model life-table probabilities of death using a time-dependent beta latent variable model and identify a small set of “mortality states” that characterise the mortality pattern of each country at any given point in time. We then cluster countries based on their sequences of mortality states. Our results reveal substantial heterogeneity in the timing and pace of transitions between mortality states, including a persistent East-West divide in Europe and distinctive Latin American patterns associated with elevated young-adult male mortality. Cross-country inequality increased for both sexes until the 1990s, before plateauing and subsequently declining. Our proposed multi-stage clustering framework provides an interpretable, time-explicit description of mortality dynamics that jointly captures within-country change and between-country differences, and can be applied to other settings where trajectories of categorical or discretised profiles are of interest.

High-dimensional data create challenges for causal effect estimation because identifying the covariates needed for correct model specification becomes increasingly difficult. Double/debiased machine learning (DML) facilitates the use of machine learning (ML) for causal inference by mitigating regularization and overfitting bias, but comparatively less attention has been given to covariate selection in relation to the double robustness (DR) property possessed by some DML estimators. In particular, ML-based covariate selection may result in differential covariate selection or in misspecification of both models, thereby limiting the practical utility of the DR property. To address these issues, we propose using the union of the covariates selected by the propensity score (PS) and outcome ML models to re-estimate both models. Simulation results show that using the union consistently reduces more confounding bias than using separate selected covariate sets. The results also show that ML-based estimation does not uniformly outperform conventional DR estimation, even under conditions favorable to the Lasso, and that post-Lasso reduces more confounding bias than standard Lasso. These findings demonstrate that successful use of ML for causal inference depends not only on the ML algorithm but also on how the information obtained through covariate selection is incorporated into causal effect estimation.

We investigate Empirical Bayes (EB) methods in the context of compound adaptive experiments, where the arm distribution in each experiment follows a normal distribution with an unknown mean that we seek to estimate. There are two main EB strategies: $g$-modeling, which estimates the prior by maximizing the marginal likelihood, and $f$-modeling, which derives posterior means directly from the empirical distribution of the observations. We show that $g$-modeling continues to be a valid EB procedure even when it incorrectly assumes that data are collected exogenously; its validity does not depend on the particular sampling algorithm or on whether sample sizes are endogenous. In practice, one can apply standard $g$-modeling techniques by acting as though the data were exogenously sampled. We extend regret guarantees from exogenous sampling to adaptively generated data. By contrast, naively applying the Tweedie formula based on the marginal density of the observed data, as in standard $f$-modeling, can produce biased rules under adaptive sampling. We corroborate the robustness of $g$-modeling through simulations with widely used adaptive algorithms and demonstrate its applicability using a real-world dataset consisting of multiple sequential experiments.

Accurate electricity load forecasting is essential to ensure the real-time balance between supply and demand, especially in systems increasingly influenced by weather conditions and renewable energy integration. In this paper, we propose an adaptive probabilistic forecasting framework that leverages multiple meteorological forecast providers to mitigate the impact of weather prediction errors on load forecasts. Our approach combines an adaptive generalized additive model (GAM) with a state-space formulation estimated via the Viking algorithm, enabling dynamic parameter updates over time. To improve uncertainty quantification, we extend the framework with a probabilistic layer based on GAM for location, scale and shape (GAMLSS), allowing the residual variance to depend on explanatory variables. We further introduce a short-term correction mechanism to enhance performance across different forecast horizons. A key contribution of this work is the aggregation of multiple meteorological inputs, whose diversity is shown to improve both point and probabilistic forecasts, particularly in capturing extreme quantiles by incorporating forecast dispersion. The methodology is evaluated on French national electricity consumption data and demonstrates consistent gains over single-provider approaches, with significant improvements in probabilistic forecasting accuracy.

In this paper we study a linear drift perturbed by a superposition of $m$ independent fractional Brownian motions with known Hurst parameters and a common scale, observed at $N$ equidistant times. Inference for such models is usually asymptotic; we show that here it is exact. We derive the maximum likelihood estimators of the drift $\theta$ and of the scale $\alpha^{2}$ in closed form and obtain their exact finite-sample joint law: $\widehat\theta$ is Gaussian, $N\widehat\alpha^{\,2}/\alpha^{2}$ is chi-square with $N-1$ degrees of freedom, and the two are independent. As this law is free of every model parameter, we deduce Student and chi-square confidence intervals and tests of exact level for every $N\ge2$, whatever the Hurst vector. We also prove that the estimators are uniformly minimum variance unbiased with $\widehat\theta$ attaining the Cramér–Rao bound at every $N$, that both are strongly consistent and asymptotically normal, and that the drift estimators form, in law, a Brownian motion run along their own variance scale. A sharp non-asymptotic bound shows that the accuracy of the drift is governed by the length of the observation window and not by the mesh, and a Monte Carlo study confirms exact coverage, even at small sample sizes, and quantifies what is lost when the Hurst vector is misspecified.

In various fields, such as medicine and marketing, accurately predicting individual treatment effects holds significant promise. However, achieving reliable predictions alone is often insufficient for making informed decisions; it is equally important to understand why the treatment effect is higher for some individuals than for others. To address this two-fold challenge of prediction and interpretation, we introduce an algorithm based on decision trees and random forests for estimating individual treatment effects. Our algorithm is simple: it operates exactly like a standard random forest, but with a different splitting criterion, and requires no additional workarounds such as double machine learning or orthogonalization as used in Generalized random forests. It handles observational studies with varying treatment propensities without requiring separate estimation of the full propensity function. This is achieved by combining two splitting criteria—one targeting heterogeneity in the treatment effect, the other targeting bias correction for the average treatment effect—which together improve split point selection and automatically distinguish confounders from features responsible for heterogeneity. As a result, interpretation follows directly from the fitted tree structure itself, that is, from which features the trees split on and with which split statistics, without requiring separate post-hoc analysis. For the theoretical analysis of this algorithm, we consider a change point model with step functions for potential outcomes and treatment propensity and provide insights into the theoretical underpinnings of our approach. Simulation studies show that our simple algorithm achieves comparable, and often better, prediction accuracy than existing methods, while substantially improving interpretability.

Feature-based multi-product pricing uses customer characteristics to identify demand heterogeneity and tailor prices across products. Choice model trees segment customers through interpretable feature rules and fit a demand model within each leaf. Existing methods typically construct these trees greedily, selecting one myopic split at a time. We develop optimal choice model trees with multinomial logit leaves (OCMT-MNL), jointly optimizing the tree and leaf models within a prescribed depth. Our exact dynamic program derives closed-form Fenchel lower bounds during constrained Newton iterations and propagates them across nested and disjoint customer subsets, avoiding new fits and resuming unfinished fits without repeating completed work. In synthetic experiments, it reduces exact leaf fits by 99.98% and leaf evaluations by 86.13%, achieving up to 7.15-fold speedups over unpruned dynamic programming. One-dimensional lookup tables translate offline estimation into real-time pricing, with a revenue-loss bound quadratic in grid spacing under the fitted model. Compared with greedy trees, OCMT-MNL achieves lower revenue loss with fewer leaves on synthetic data and better predictive fit on real data. In a 23-week randomized experiment on ancillary seat pricing across 48 airline markets and 190,220 passengers, OCMT-MNL increases seat revenue per passenger by a statistically significant 11.3% over static pricing.

Langevin-based Markov chain Monte Carlo (MCMC) algorithms use gradient information to improve sampling, particularly in high dimensions. Classical optimal scaling theory for these algorithms has largely focused on the Metropolis-Hastings (MH) acceptance rule. However, there has been a recent surge in acceptance rules beyond MH for applications spanning differential privacy, stochastic MCMC, diffusion models, and molecular dynamics. We develop optimal scaling results for Langevin proposals employed with generalized acceptance rules belonging to a suitable class. For high-dimensional targets, we recover the usual $O(d^{-1/3})$ scaling, while different acceptance rules lead to different optimal acceptance probabilities.

We propose Low-Rank Quantile Surfaces (LRQS), a bivariate causal model in which, in the causal direction, an unknown monotone transformation of the conditional quantile surface admits a low-rank functional decomposition. LRQS subsumes location-scale noise models and post-nonlinear heteroscedastic noise models, while allowing multiple quantile bases to represent changes beyond location-scale effects. We prove generic identifiability of LRQS: the transformed quantile surface is low rank in the causal direction, whereas reverse representability under the corresponding constraints occurs only for exceptional, fine-tuned cause marginals. We provide a simple-yet-powerful causal score using a nonparametric fitting procedure that alternates between rank-constrained approximation of discretized quantile surfaces and isotonic estimation of the unknown monotone transformation. Experiments on synthetic mechanisms with higher-rank distributional shape variation and strong nonlinear distortions, together with standard bivariate benchmarks, show that LRQS is especially effective when conditional distributional shape or observation distortion goes beyond existing location-scale assumptions.

Hyperbolic geometry provides a natural inductive bias for genomic representation learning, but existing hyperbolic genomic models primarily use Lorentz convolutions to learn local sequence representations, while their residual pathways do not directly aggregate full Lorentz representations. We propose HyCoSeq, a contextual hyperbolic representation learning framework for genomic sequences. HyCoSeq incorporates weighted Lorentzian residual aggregation into multi-curvature Lorentz encoding, allowing full Lorentz representations to participate directly in geometry-consistent local aggregation. It further introduces a bidirectional long short-term memory network that integrates information from both sequence directions to learn contextual relationships among local representations at different positions within a genomic sequence, thereby extending local hyperbolic convolutional encoding to sequence-level contextualized representations. Extensive experiments across diverse genomic tasks show that HyCoSeq outperforms existing hyperbolic baselines and, without large-scale genomic pretraining, achieves competitive performance against substantially larger pretrained DNA language models.

For multivariate scale and location–scale models with independent components, we extend the univariate results of Zhou and Nayak (2012) and derive optimum equivariant estimators under the generalized Pitman closeness criterion. We first show, by a counterexample, that in the multivariate case the Pitman closeness comparison within the class of equivariant estimators is not transitive, so that a Pitman closest equivariant estimator does not exist in general. We then enlarge the transformation group by the coordinate permutations—equivalently, impose the formal equivariance principle of Berger (1985) across isomorphic component problems, in the spirit of the separable rules of Robbins' (1951) compound decision theory—and show that within the resulting restricted class an optimum is restored. A multivariate median lemma based on strictly convex losses then yields explicit Pitman closest equivariant estimators of the scale parameters, powers of the scale parameters, and the location parameters, given by median-adjusted versions of any given equivariant estimator. Applications to the multivariate uniform and multivariate normal distributions are worked out in detail. Monte Carlo experiments for the Rayleigh distribution and for a competing risks model with Rayleigh component lifetimes confirm that the proposed estimators dominate the maximum likelihood and Bayes estimators under the Pitman closeness criterion, and a real industrial data set on ball bearing failure times illustrates the feasibility of the method in practice.

Equivariance is increasingly used in machine learning and statistics, often without systematic justification. In a companion article, the equivariance criterion was applied to the normal linear model with a fixed design matrix (fixed-$X$), yielding the minimum risk equivariant (MRE) estimators of the coefficient vector and of the condensed diagonal covariance matrix under a multivariate invariant location–scale group. We extend these results to the random-$X$ case, with covariates sampled from a population. The extension hinges on a distinction vacuous for fixed-$X$ but fundamental for random-$X$: whether risk and unbiasedness are evaluated conditionally on the realized design or after averaging over the design distribution. Under conditional evaluation, the fixed-$X$ group applies given $X$: least squares remains the best equivariant estimator of the coefficient vector, and the MRE estimators of the population variances keep their fixed-$X$ forms with population sizes at the realized design. Under absolute evaluation with an i.i.d.\ design, the picture changes qualitatively: the natural scale group acting jointly on $(Y,X)$ fixes the coefficient vector, the induced parameter-space action is intransitive, equivariant risks are constant only along orbits indexed by the signal-to-noise ratio $\rho=\|\beta\|^2/\sigma^2$, and no uniformly minimum risk equivariant estimator exists. In the scalar case the optimal equivariant weight is the oracle shrinkage factor $w^*(\rho)=\rho/(\rho+E[T^{-1}])$, with least squares recovered as the infinite-signal limit $\rho\to\infty$—explaining and refining the known failure of the Gauss–Markov theorem with random regressors. For a centered design under location–scale transformations, least squares remains optimal within the natural invariant-contrast class, and the MRE estimator $S^2/(n-p+2)$ of the error variance is valid under both modes.

Protecting individual privacy has become a central and urgent concern in modern data analysis, given the vast quantities of data now generated and processed. In this paper, we develop a systematic theory of exact asymptotic efficiency for regular parametric estimation under central zero-concentrated differential privacy. In the privacy regime, the governing object is a diameter-constrained information region: the set of information matrices generated by statistics with diameter at most one. For weighted quadratic loss, we show that the exact local minimax risk is an inverse information variational functional over this region. More generally, a mixed information region yields a unified efficiency constant across different regimes, covering the privacy regime and classical Fisher efficiency. A matching estimator releases the empirical mean of a nearly optimal bounded statistic with Gaussian noise and locally inverts its population moment map. Our theory differs from classical efficiency theory in its set-valued information geometry and loss-dependent efficient estimator. As examples, we provide closed-form constants and optimal procedures for various concrete models, including one-dimensional regular families, Gaussian means, categorical probability vectors, and regression models among others. The theory also transfers to Gaussian differential privacy through an exact parameter rescaling.

Weighted majority votes are central to many successful ensemble methods. PAC-Bayesian theory provides tight generalization guarantees for such models by analyzing the expected risk of stochastic classifiers, while analyzing the risk of deterministic majority votes relies on surrogate bounds. To avoid these surrogates, Zantedeschi et al. ( 2021) introduced guarantees for stochastic majority votes, but the resulting models remain randomized. In this paper, we propose a derandomization framework for stochastic majority votes. To do so, we apply recent advances in disintegrated PAC-Bayesian theory directly to the space of majority vote weight vectors, transforming stochastic guarantees into certificates for a single deterministic majority vote. We derive two families of high-probability generalization bounds, covering both data-independent and data-dependent constructions of the ensemble, which naturally lead to a self-bounding learning algorithm optimizing deterministic majority vote guarantees.