Pivotal Statistics

Who the judge is can affect an LLM-as-judge result, but measuring that effect without confusing it with candidate quality is difficult. We study four open-weight families (Llama 3.1, Qwen 2.5, Gemma 2, and Yi 1.5) in a fully crossed pairwise design with 9,312 judgments. A common per-family statistic is strongly confounded with candidate quality and correlates with Bradley-Terry ability at r = 0.95. We derive a corrected estimator that holds the candidate family fixed and compares judges. All four families then show a positive same-family lift (3.4-8.4 percentage points), with global FPS 0.067 (95% CI [0.053, 0.084], permutation p = 0.0002). The effect remains under panel-based quality controls, an independent human-consensus anchor, and a float16 judging replication. Judge-side likelihood is closely related to the effect: adding likelihood advantage reduces the controlled coefficient by 61%, which we treat as descriptive attenuation rather than causal mediation. Position is a separate failure mode. Across the panel, 55.4% of AB/BA pairs reverse, and reversal above 50% is incompatible with a simple independent content-noise model. Relative to a family-balanced reference, panel composition changes 18.5% of pairwise outcomes. A complete reproducibility archive has been prepared for public release.

We present a simple decomposition of a neural-network-based normalizing flow that naturally uncovers a pivotal statistic (or something close) in the presence of nuisance parameters, based only on a sample generator from the distribution of interest. We show that the statistic is near-pivotal in the sense of minimum average KL-divergence of its $p$-values versus uniform and we argue that it can be expected to have good power when the dimension of the statistic equals the dimension of the parameter. It is able to incorporate prior knowledge about group invariances such as translation and scale. It can discover the one-sample $t$-test almost exactly, outperforms the Welch test in terms of worst-case size over a constrained variance-ratio range and achieves good calibration on partial biserial correlations, while showing higher power (and being much faster) on small-to-moderate samples than profile likelihood-ratio techniques.

Tabular data is used extensively in many real-world use cases. Deep learning models have been developed to deal with tabular data, but generally perform poorly when the test data distribution differs from that of the training data. Researchers have proposed test-time adaptation approaches to deal with this problem. The fully test-time adaptation (FTTA) setting involves adapting deployed classifiers to shifted target distributions using only unlabeled test data. Leading FTTA methods inherit a batch-dependent approach from computer vision literature. This paper demonstrates for the first time that such approaches degrade sharply in strict streaming regimes where examples arrive and must be classified one at a time. This occurs because at a batch size of one, batch-level statistics become unavailable or poorly estimated. We argue that singleton tabular FTTA is not merely a small-batch variant of ordinary FTTA, but a distinct identifiability problem where only the location of the model's score stream remains directly observable. To address this, we propose Prequential Logit-Origin Centering (PLOC), a lightweight approach that keeps the source model frozen and shifts the logit space at each step. PLOC stores only a single running number (the mean of past logits), requires no labels, estimates no priors, and bypasses weight updates entirely. A deferred variant applies a static shift that preserves the source ranking, and thus the AUROC, exactly. Evaluated across five tabular benchmarks, three architectures (MLP, FT-Transformer, and TabTransformer), and five independent source checkpoints, PLOC significantly outperforms strong tabular and entropy-based baselines.

Watermarking promises statistical traceability of large language model (LLM) uses, but real documents rarely arrive as purely human-written or purely LLM-generated. This motivates a quantitative question beyond detection: what proportion of a document is generated from a pre-specified watermarked LLM? We study this watermark proportion estimation problem under the Gumbel--max watermarking mechanism, treating the next-token prediction distributions as unknown and arbitrary nuisance parameters subject to a non-degeneracy condition. We compare two observation regimes: in the full observation regime, the estimator observes the pseudorandom vector and the selected token at each position; in the more prevalent setting of pivotal reduction, it observes only a scalar pivot, which follows a one-dimensional Uniform--Beta mixture distribution. Under pivotal reduction, we develop a Laguerre-polynomial estimator and establish a matching information-theoretic lower bound for the sample complexity. For full observation, we introduce an event-counting estimator and show a matching lower bound, yielding a substantially smaller sample complexity. As our results imply, although reducing to pivotal statistics is an elegant and prevalent choice, it is not always sample-efficient for estimating the proportion of watermarks.

Although pitch sequencing is a central topic in baseball analytics, previous studies have primarily focused on optimizing the final pitch within a single plate appearance, leaving the role of preceding setup pitches and their impact on long-term season-level performance insufficiently examined. To address these issues, this study conducted counterfactual analyses using MLB Statcast data. A Transformer-based machine-learning model was trained to predict whether a target pitch would result in an in-play outcome or swing-out. Counterfactual pitch sequences were then generated by replacing either the final pitch or the preceding setup pitch with alternative pitch types and locations while keeping the surrounding contextual information fixed. Optimal counterfactual selections were defined as those that minimized the predicted in-play probability, and their expected effects on pitchers' seasonal statistics were estimated using regression models linking model outputs to season statistics. The results suggest that the optimization of both final and setup pitches may substantially influence season-level performance, including improvements of more than 1.0 in K/9. The analyses also provided several practical insights, including velocity-band-specific effective locations, the importance of pitch commands, and the expansion of pitch-selection options through middle-velocity pitches. These findings quantitatively support the strategic importance of pitch sequencing in baseball.

Comparing text strings is crucial when evaluating and understanding the performance of various text processing tasks such as document recognition and audio transcription. With an increasingly complex landscape of AI-based handwritten text recognition (HTR), optical character recognition (OCR) and automatic speech recognition (ASR) models, there is a need for tools that facilitate evaluation in a flexible and reproducible way. This paper presents Stringalign, a Python library designed to simplify the evaluation process for automatic transcription projects and facilitate transparent evaluation. Stringalign's tools to examine and visualise both the rate of errors and the types of errors a model makes, give insights into possible improvements and help inform model selection for a particular task. Widely used string comparison metrics, such as the character and word error rates (CER and WER), although useful, can be ambiguous due to varying definitions of what constitutes a character and a word. Stringalign addresses this challenge by ensuring all preprocessing (i.e. normalisation and tokenisation) is transparent and easily replicable, and by providing tools to move beyond summary statistics and analyse common model errors. Moreover, Stringalign adheres to FAIR (Findable, Accessible, Interoperable, and Reusable) principles for research software while staying lightweight and easy to adapt into researchers existing workflows. In this paper, we discuss challenges with character and word level string comparisons and show through examples that where existing tools can yield opaque and sometimes confusing results, Stringalign provides an easy-to-use and unambiguous alternative.

Search agent benchmarks exemplified by BrowseComp have rapidly saturated over the past year, with the strongest models surpassing 90% accuracy. Since these benchmarks are predominantly human-authored, annotators lack a global perspective on entity statistics and cannot systematically maximize search space size and structural complexity. This creates a difficulty ceiling that is hard to break. To address this, we introduce LoHoSearch (Long-Horizon Search Agents), a challenging benchmark comprising 544 human-verified questions across 11 domains. LoHoSearch is constructed via an automated pipeline built upon a knowledge graph covering over 7 million Wikipedia entities, which selects relations with large search spaces and assembles them into structurally complex questions with KG-verified unique answers. Our evaluation demonstrates that even the strongest model achieves only 34.74% accuracy, and existing context management strategies (best +6.8%) yield far smaller gains than on prior benchmarks. LoHoSearch provides a more demanding standard for evaluating long-horizon reasoning and context management in search agents.

This paper presents a nonlinear parameter estimator for Wiener-type state-space models obtained as a fixed-point architecture that couples two affine minimum mean-squared error (MMSE) estimators: one for the unknown parameters and one for latent variables. The architecture retains the functional structure of the optimal affine MMSE parameter estimator while incorporating Dynamic Basis Statistics (DBS) estimates that summarize nonlinear basis-function evaluations. Two DBS construction strategies are developed, leading to two nonlinear estimator frameworks. The dual basis-parameter estimator combines an affine basis estimator with the affine parameter estimator, whereas the dual state-parameter estimator first computes affine state estimates and their covariances, then maps these state-estimate statistics through a Gaussian DBS operator to obtain DBS estimates. Both dual estimators admit fixed-point characterizations that alternate between estimating each component using the updated prior of the other, obtained from that component's plug-in estimate statistics from the previous iteration. The efficacy of the proposed methods is examined via extensive Monte Carlo experiments, showing that the dual basis-parameter estimator attains parameter mean-squared errors comparable to those of the purely affine parameter estimator, while the dual state-parameter estimator achieves the lowest parameter mean-squared error, outperforming both the dual basis-parameter and purely affine parameter estimators, as well as sequential Monte Carlo variants of classical Particle Gibbs and Expectation-Maximization schemes.

The ranking of recommendation algorithms is a challenging problem since model performance is sensitive to dataset characteristics such as sparsity, sequential structure, and scale. This drives a demand for a proper methodology for fair comparison between algorithms. Naive aggregation of performance metrics (e.g., averaging NDCG over benchmarks) can yield misleading rankings, undermining practical selection. To address this problem, we introduce a novel, data-driven ranking methodology based on Bradley-Terry (BT) model. We demonstrate that the obtained ranking depends on key dataset statistics. Additionally, we propose a novel metric for evaluating ranking consistency and demonstrate robustness of our ranking to incomplete data. Finally, we introduce a dataset-specific methodology for ranking algorithms on unseen datasets without running the models, relying on extensions of the Bradley-Terry framework, including BT trees and BT models with covariates.

Classical deep learning typically operates on individual cases. Despite its success, real-world usage often requires repeated inference to estimate statistical quantities for complex decision-making tasks involving uncertainty or extreme-value analysis, resulting in substantial latency. We introduce neural statistical functions, a new family of models learned from pre-trained single-sample predictors and scattered data samples, which can directly infer statistics over continuous operating condition ranges without explicit sampling. By introducing the notion of prefix statistics, we transform and unify diverse statistical functions (e.g., integrals, quantiles, and maxima) into an interval-conditional framework, in which a principled identity between the prefix statistics and the individual-case regression serves as the learning objective. Neural statistical functions achieve strong performance in estimating essential statistics of complex physical processes, including accumulated energy in dynamical systems, quantiles of aerodynamic responses, and maximum stress in crash processes, while achieving up to a 100$\times$ reduction in model evaluations.

The pursuit of autonomous driving has produced one of the richest sensor data collections in all of robotics. However, its scale and diversity remain largely untapped. Each dataset adopts different 2D and 3D modalities, such as cameras, lidar, ego states, annotations, traffic lights, and HD maps, with different rates and synchronization schemes. They come in fragmented formats requiring complex dependencies that cannot natively coexist in the same development environment. Further, major inconsistencies in annotation conventions prevent training or measuring generalization across multiple datasets. We present 123D, an open-source framework that unifies such multi-modal driving data through a single API. To handle synchronization, we store each modality as an independent timestamped event stream with no prescribed rate, enabling synchronous or asynchronous access across arbitrary datasets. Using 123D, we consolidate eight real-world driving datasets spanning 3,300 hours and 90,000 kilometers, together with a synthetic dataset with configurable collection scripts, and provide tools for data analysis and visualization. We conduct a systematic study comparing annotation statistics and assessing each dataset's pose and calibration accuracy. Further, we showcase two applications 123D enables: cross-dataset 3D object detection transfer and reinforcement learning for planning, and offer recommendations for future directions. Code and documentation are available at https://github.com/kesai-labs/py123d.

Medical experts often manually segment images to obtain diagnostic statistics and discard the resulting annotations. We aim to train segmentation models to alleviate this burden, but constrained to the retained summary statistics (e.g., the area of the annotated region). Empirical results suggest that statistics alone are insufficient for this task, but adding weak information in the form of a few pixels within the area of interest significantly improves performance. We use a novel loss function that combines terms for image reconstruction quality, matching to summary statistics, and overlap between the predicted foreground and the weak supervisory signal. Experiments on standard image, ultrasound (breast cancer), and Computed Tomography (CT) scan (kidney tumors) data demonstrate the utility and potential of the approach.

Drifting models are capable one-step generative models trained to follow a drifting field. The field combines attractive and repulsive softmax-weighted centroids over the data and current-generator distributions. In practice, only a minibatch of $n$ samples from each distribution is available, and each centroid is approximated by an empirical estimate. In this paper, we begin by showing that the minibatch centroid is in general a biased estimator of the target centroid, with a pointwise $O(1/n)$ bias arising from softmax self-normalization. Correcting this bias requires the expectation over the full distribution, which is intractable. We instead approximate the leading bias term from in-batch statistics and propose Analytical Bias Correction (ABC), a closed-form plug-in adjustment. We prove that ABC reduces the bias from $O(1/n)$ to $O(1/n^2)$, introduces no first-order increase in total variance, and preserves convex-hull containment of the corrected centroid. In practice, ABC requires only two additional lines of code and has negligible wall-time overhead under compiled execution. Toy experiments confirm the theoretical $O(1/n)$ and $O(1/n^2)$ scaling. On CIFAR-10, ABC reduces FID and trains faster, with the largest gains at small $n$, where the bias is most significant.

Wav2vec 2.0 (W2V2) has shown strong performance in pathological speech analysis by effectively capturing the characteristics of atypical speech. Despite its success, it remains unclear which components of its learned representations are most informative for specific downstream tasks. In this study, we address this question by investigating the regression of dysarthric speech descriptors using annotations from the Speech Accessibility Project dataset. We focus on five descriptors, each addressing a different aspect of speech or voice production: intelligibility, imprecise consonants, inappropriate silences, harsh voice and monoloudness. Speech representations are derived from a W2V2-based feature extractor, and we systematically compare layer-wise and time-wise aggregation strategies using attentive statistics pooling. Our results show that intelligibility is best captured through layer-wise representations, whereas imprecise consonants, harsh voice and monoloudness benefit from time-wise modeling. For inappropriate silences, no clear advantage could be observed for either approach.

Federated Learning (FL) enables multiple parties to collaboratively train machine learning models without sharing raw data. However, before training, data must be preprocessed to address missing values, inconsistent formats, and heterogeneous feature scales. This preprocessing stage is critical for model performance but is largely overlooked in FL research. In practical FL systems, privacy constraints prohibit centralizing raw data, while communication efficiency introduces further challenges for distributed preprocessing. We introduce FedPS, a framework for federated data preprocessing based on aggregated statistics. FedPS leverages data-sketching techniques to efficiently summarize local datasets while preserving essential statistical information. Building on these summaries, we design federated algorithms for feature scaling, encoding, discretization, and missing-value imputation, and extend preprocessing-related models such as Bayesian Linear Regression to both horizontal and vertical FL settings. FedPS provides flexible, communication-efficient, and consistent preprocessing pipelines for practical FL deployments.

Despite the fact that experimental neural scaling laws have substantially guided empirical progress in large-scale machine learning, no existing theory can quantitatively predict the exponents of these important laws for any modern LLM trained on any natural language dataset. We provide the first such theory in the case of data-limited scaling laws. We isolate two key statistical properties of language that alone can predict neural scaling exponents: (i) the decay of pairwise token correlations with time separation between token pairs, and (ii) the decay of the next-token conditional entropy with the length of the conditioning context. We further derive a simple formula in terms of these statistics that predicts data-limited neural scaling exponents from first principles without any free parameters or synthetic data models. Our theory exhibits a remarkable match with experimentally measured neural scaling laws obtained from training GPT-2 and LLaMA style models from scratch on two qualitatively different benchmarks, TinyStories and WikiText.

Understanding how the structure of language can be learned from sentences alone is a central question in both cognitive science and machine learning. Studies of the internal representations of Large Language Models (LLMs) support their ability to parse text when predicting the next word, while representing semantic notions independently of surface form. Yet, which data statistics make these feats possible, and how much data is required, remain largely unknown. Probabilistic context-free grammars (PCFGs) provide a tractable testbed for studying these questions. However, prior work has focused either on the post-hoc characterization of the parsing-like algorithms used by trained networks; or on the learnability of PCFGs with fixed syntax, where parsing is unnecessary. Here, we (i) introduce a tunable class of PCFGs in which both the degree of ambiguity and the correlation structure across scales can be controlled; (ii) provide a learning mechanism -- an inference algorithm inspired by the structure of deep convolutional networks -- that links learnability and sample complexity to specific language statistics; and (iii) validate our predictions empirically across deep convolutional and transformer-based architectures. Overall, we propose a unifying framework where correlations at different scales lift local ambiguities, enabling the emergence of hierarchical representations of the data.