Sequential Hypothesis Testing

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9 papers in the last four weeks, against 2 the four weeks before. 0.1% of all new papers.

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Latest papers 39

Oct 6, 2026stat.ML

Anytime-valid simulation-based hypothesis testing

For a given data distribution

(X_t)_{t \in \mathbb{N}} \sim Q

i.i.d., we investigate the hypothesis testing problem:

H_0: Q = P_0

vs.

H_1: Q = P_1

, for two different model probability distributions

P_0

and

P_1

. In contrast to the standard setting, where analytic densities

p_0

and

p_1

are given, here, we consider the density-free setting, where we only have access to i.i.d. simulations

(Z^0_t)_{t \in \mathbb{N}} \sim P_0

and

(Z^1_t)_{t \in \mathbb{N}} \sim P_1

. For this simulation-based hypothesis testing setting, we construct an e-test martingale, resulting in a sequential test with anytime-valid type-I error guarantees, approximate growth optimality, geometrically decaying type-II error bounds, and asymptotic power one. Most ingredients used in our constructions are variants of well known concepts. The value of this paper lies in the compact presentation of an effective, anytime-valid solution for the density-free simulation-based sequential hypothesis testing case.

Patrick Forré, Lydia Brenner

University of Amsterdam · Nikhef

Simulation-Based Inference Sequential Hypothesis Testing

Oct 5, 2026stat.ML

Valid Stopping in Adaptive Generator-Verifier Loops

Numerous agentic workflows are based on a generator-verifier loop: a generator proposes candidates, a cheap verifier scores them, and the workflow terminates when a proposal is verified as good enough. The verifier typically proxies a more costly ground-truth oracle, and as the generator searches adaptively against it, false acceptances may accumulate. Proposals can pass the proxy but fail under the costlier ground-truth check. We study when to stop these loops while controlling the false discovery rate of the accepted proposals. Our construction introduces tools of independent interest in distribution-free statistical testing and conformal risk control, including analysis of

e

-values constructed through index betting and a novel conformal risk control procedure for non-monotone losses. We validate the approach in synthetic settings and on a protein-design benchmark.

Mahmoud Hegazy, Michael I. Jordan, Aymeric Dieuleveut

CMAP, École polytechnique, IP Paris, France · Inria Paris, France · EECS, UC Berkeley, USA

FDR Control E-Values Optimal Stopping Conformal Risk Control Sequential Hypothesis Testing

Oct 1, 2026quant-ph

Sequential Capacity of Quantum Processes with Finite Memory

How complex can the responses of a quantum device become as it runs longer with a fixed internal memory? We quantify this complexity through sequential response capacity: how many adaptive testing stages, each using a fresh run, can continue to separate possible processes by a prescribed gap in response probabilities. For fixed system and memory sizes, we establish a tight law relating this capacity to run length and probability resolution. At fixed resolution, the capacity grows on the order of

K\log K

, where

K

is the number of time steps in each run. Our construction attains this growth using time-dependent phase rotations on a single visible qubit with no additional internal memory; its tests give response probabilities exactly zero or one. Under the same tests, classical stochastic processes that measure in a fixed basis at every step have only linear capacity at fixed sizes and resolution. For phase sequences selected by a stored classical label, we then quantify how known independent Pauli noise changes this logarithmic enhancement. With ideal controls and weak residual phase noise after correction, we prove matching capacity bounds at a fixed small probability gap. These bounds identify the inverse residual phase-flip probability as the coherence timescale that limits the extra logarithmic growth.

Yibin Wang

Graduate School of Mathematics, Nagoya University, Nagoya, 464-8601, Aichi, Japan.

Sequential Hypothesis Testing

Sep 30, 2026cs.LG

Prequential E-Values for Selected-GP Near-Optimality Certificates

When optimizing an expensive black-box function sequentially, as in hyperparameter optimization, we may want to stop once the best evaluated value is certified within

\varepsilon

of the global optimum. Such a certificate needs two ingredients: a lower confidence bound for the selected value and an upper confidence envelope over the domain, typically supplied by a Gaussian process (GP). GP-UCB-style stopping rules are valid when the kernel and constants defining this envelope are fixed before the run, but the practical temptation is to tune the envelope from the same adaptive evaluations and then certify as if it had been fixed. We use prequential e-values to make this selection auditable: starting from a predeclared set of fully specified GP/RKHS envelopes, each candidate is tested by its own one-step-ahead e-process, contradicted candidates are deleted, and certification uses the largest upper bound among the survivors. With a valid selected-point lower bound and one declared candidate having valid latent coverage and noise calibration, the rule is anytime-valid. On a 512-seed noisy RBF stress sweep, it roughly halves false-certification risk at comparable power versus fit-then-certify. Relative to random fixed GP precommitment on smooth

d=3,4

objectives, each additional false certificate is accompanied by 3.0 and 13.5 additional correct certificates, respectively.

Ami Tavory, Noa Cohen

Meta Platforms

Black-Box Optimization Anytime-Valid Inference E-Values Sequential Hypothesis Testing

Sep 30, 2026stat.ME

Always-On Experimentation

Generative AI has dramatically accelerated the rate at which new treatments---from novel pharmaceuticals to online marketing campaigns---can be conceived and deployed. As a result, modern experimentation platforms often run continuously, with treatments added as they are ready and removed when they underperform. We formalize this "Always-On" experimental setting, in which treatments can be dynamically generated, added to, and removed from a running experiment, and study the statistical problem of deciding whether to accept or reject each treatment while controlling for the false discovery rate. We develop sequential tests that achieve time-uniform Type-I error control under arbitrary stopping times and "predictable" treatment schedules. Our approach builds on the testing-by-betting framework: we construct test supermartingales for testing the average treatment effect of each treatment, and show that the construction of these test supermartingales is growth-rate optimal in an almost-sure sense.

Ricardo J. Sandoval, David Arbour, Avi Feller +1

University of California, Berkeley · Adobe Research · Inria, Paris

FDR Control A/b Testing Sequential Hypothesis Testing

Sep 28, 2026cs.IR

PEAR: Progressive Evidence-Based AutoResearch for Industrial Search Systems

AutoResearch improves systems through iterative experimentation: agents propose candidate modifications, evaluate them, and use the results to guide subsequent exploration. Applying this paradigm to industrial search presents two challenges. (1) Common AutoResearch approaches follow a keep-if-better rule, retaining the highest-scoring candidate for subsequent experiments. Under non-stationary traffic, transient gains may be mistaken for persistent improvements, impairing reliable accumulation of search knowledge. (2) Candidate modifications can be evaluated at multiple fidelity levels, from low-cost proxies to online validation, differing in cost, objective alignment, and statistical reliability. Existing methods rely on individual signals or task-specific procedures, lacking a unified basis for using evidence across levels to guide search. We introduce Progressive Evidence-Based AutoResearch (PEAR) with two complementary components. Evidence-driven AutoResearch maintains an independent, hypothesis-guided research state for each strategy task within a predefined objective and intervention scope. Each state evolves through a Plan-Execute-Evaluate-Update transition that links experimentation to context-aware evidence interpretation and hypothesis revision. Confidence-Gated Verifier Ladder organizes evaluation into four levels of increasing fidelity: Offline Replay, Shadow-Traffic Evaluation, Rapid Online Evaluation, and Decision-Grade Online Evaluation. A unified confidence-based gate promotes candidates only when evidence supports a statistically significant positive effect, enabling broad low-cost exploration while reserving costly online experiments for promoted candidates. In a real-world industrial search system, strategies optimized with PEAR significantly increased Main Order/DAU by 2.7336% and 3.2957% relative to their respective baselines in two A/B experiments.

Yifan Wang, Shipeng Zhu, Fei Xiong +6

Global E-Commerce Agentic Search Team

Information Retrieval Sequential Hypothesis Testing Autonomous Experimentation Autonomous Research Agents

Sep 27, 2026cs.LG

ICMAPE: In-Context Multiagent Pure Exploration

In some multi-agent systems, the quantity to be optimized is not an externally specified reward but the information acquired about unknown properties of the environment as done in active sequential hypothesis testing (ASHT) problems. However, the ASHT literature tends to focus on finite single-agent problems with well-specified models, while there is currently a gap for practical multi-agent methods that can perform active sequential testing. We fill this gap with ICMAPE, a Bayesian learning-based framework for decentralized multi-agent pure-exploration driven by inference objectives. ICMAPE converts the fixed-confidence identification objective into a reward derived from inference confidence, so that standard reinforcement learning machinery can be applied to decentralized pure exploration. It jointly learns a centralized neural inference network that estimates a posterior distribution over hypotheses from global trajectory data, and decentralized policies that select actions from local observation histories and learn when to stop collecting data once the target confidence is reached. On two synthetic benchmarks and a Maryland nitrate concentration monitoring task based on real-world data, ICMAPE-TD3 achieves target accuracy with fewer exploration steps.

Xinyi Hu, Alessio Russo, Aldo Pacchiano

Boston University · Broad Institute of MIT and Harvard

RL Exploration Decentralized MARL Policy Learning Sequential Hypothesis Testing

Sep 27, 2026cs.CR

The Price of Peeking: Anytime-Valid Leakage Detection on ML-KEM EM Traces

Side-channel evaluators routinely inspect leakage tests while acquisition is still running, and extend or stop the campaign based on what they see. Fixed-horizon screening such as the Welch

t

-test with threshold

|t|>4.5

gives no error guarantee for this monitored decision rule. We study anytime-valid leakage detection based on testing by betting: SKIT-type swap-pair e-processes whose false-alarm probability is controlled uniformly over time under an explicit conditional symmetry null. In matched comparisons that share the frozen witness, rows and payoff, first-crossing detection needed 1.68-2.00

\times

the traces of a fixed-horizon randomization test at 80% detection on synthetic streams, and 1.68-2.38

\times

on degraded recordings from an open ML-KEM electromagnetic dataset with the primary Ridge witness at

α=0.05

. With the same primary witness and level, on undegraded reference and pqm4 recordings the monitored procedure stopped early: its median stopping point was 62-72 and 146-316 evaluation traces, i.e. 2-8% of a conservative 4096-trace budget. Under exact designed nulls on the recorded backgrounds, repeated-look

|t|>4.5

screening over all 13000-20000 samples raised a false alarm in 2.7-12.9% of replicates, against 0.0-4.7% for terminal-only screening and no rejection by a sample-wise e-Bonferroni process, which in a prespecified follow-up detected natural-label associations in 4 of 4 backgrounds after 840-3288 traces. All recordings come from one device, and natural-label results are descriptive; we state the assumptions each claim requires.

Georgios Feretzakis, Alexandros Papaspyridis

Anytime-Valid Inference Sequential Hypothesis Testing

Sep 24, 2026cs.AI

Human-AI-Powered Hypothesis Testing: Cost-Aware Selective AI Scoring and Sequential Human Escalation

Large language models are increasingly used as inexpensive judges to evaluate outputs, label data, and assess whether a system meets a desired quality standard. Yet using AI judgments for formal statistical inference is fundamentally different from simply treating them as ground-truth labels: AI evaluations can be biased or noisy, and rigorous hypothesis testing requires explicit control of type-I and type-II errors. We study how to use AI judgments, together with selective human verification, to conduct a valid hypothesis test at minimum cost. We consider a population of items with hidden binary labels. After choosing a fixed pool of items, the decision maker can selectively query AI, send an item directly to a human, escalate an AI-scored item to a human after observing the AI report, or stop once sufficient evidence has accumulated. We derive an information-theoretic lower bound that captures the minimum cost of achieving prescribed testing errors and characterizes the value of AI information and human verification through a report-dependent information frontier. Motivated by this characterization, we develop SCALE, a sequential cost-aware policy that combines selective AI scoring with adaptive human escalation. SCALE is valid at finite sample sizes and matches the lower bound to first order as the target error probabilities vanish. We further extend the framework to an unknown AI-output model using paired AI-human pilot data. Numerically, SCALE approaches Human-only or AI-only testing when one source clearly dominates, while achieving its largest savings when inexpensive AI judgments and selective human verification are both valuable.

Dae Woong, Ham, Xuejun Zhao +2

David · University of Michigan, Ann Arbor, MI 48109, USA · University of North Carolina at Charlotte, Charlotte, NC 28223, USA

Cost-Aware Inference Human-in-the-Loop Evaluation Sequential Hypothesis Testing Statistical Hypothesis Testing

Sep 18, 2026stat.ML

How Many Posterior Samples? Calibrated Stopping for Adaptive Sensing

In classification-oriented adaptive sensing, posterior samples characterize uncertainty at the current measurement state and can serve two roles: they may guide the next sensing direction, while their class labels provide votes for the candidate classes and determine whether sensing should continue. We focus on the stopping layer that turns these votes into a declaration, without modifying the posterior sampler or sensing directions. A natural plug-in rule declares when the observed vote share exceeds a threshold. We show that this threshold is not itself a confidence guarantee: when the underlying vote mass equals the threshold, the plug-in rule declares about half the time. As alternatives, we calibrate a fixed-sample rule and a finite-horizon sequential rule to a prescribed false-declaration probability, and study exact curtailment, which stops a fixed-pool rule once its final verdict is forced. We then derive how one-round declaration probabilities determine posterior-sample cost and classification accuracy along a sensing path. On MNIST with DDRM and a fixed PCA-guided probe sequence, curtailment saves up to 62% of posterior samples. Among the evaluated rules at matched operating points, sequential stopping reduces the cost the most. At a high accuracy, that same sequential rule can trade more posterior samples for fewer measurements.

Vincent Corlay, Andriy Enttsel

Mitsubishi Electric R&D Centre Europe

Active Perception Optimal Stopping Posterior Sampling Sequential Hypothesis Testing

Sep 14, 2026stat.ML

Predictive Likelihood Ratios for Language Model Watermark Detection

Keyed watermark detection tests dependence between observed tokens and pseudorandom variables reconstructed from a secret key. Building on the pivotal framework of Li et al. (2025), we construct predictive likelihood ratios that average over uncertain probability deficits and residual-tail distributions. The aim is robust detection power across alternative specifications without requiring a single signal-strength tuning. A mixture prior combines tail shape and effective width; hierarchical extensions allow within-document variation in deficit or width. The test maximizes prior-averaged power at a fixed size, but is not generally uniformly most powerful or minimax. Under the exact conditional pivot null, normalized predictive alternatives selected before each observation yield a Bayes factor that is also a test martingale: Type I error control is unaffected by alternative misspecification and remains valid under optional stopping. This guarantee does not cover violations of the conditional null, and the interpolated implementation has no certified anytime guarantee. Gumbel marginal likelihoods are evaluated by fixed quadrature. Across the evaluated tail-shape and tail-width alternatives and three horizons, the union-tail mixture has maximum observed Type II error regret .0080, compared with .0962 for the equal-tail mixture, relative to the best tested rule. On temperature-matched outputs from two open models, it improves AUC over the equal-tail baseline in all eight non-saturated model-temperature cells, although the leading reference score generally has higher AUC. Supplementary experiments show retained power under independent null-like replacement and smaller changes from hierarchical dependence modeling. The evidence supports robustness across the evaluated alternatives, not uniform power guarantees or resistance to arbitrary text edits.

Li Ma

Department of Statistics and Data Science Institute, University of Chicago

Bayesian Inference LLM Watermarking Robust Watermarking Sequential Hypothesis Testing

Aug 10, 2026stat.ML

Deciding When to Switch: E-Processes for Adaptive Minimax Training for Generative Adversarial Nets

Modern data science increasingly gives rise to hypothesis-testing problems that are not naturally formulated in terms of parameters within prespecified statistical models. One important example is the dynamic evaluation of optimization algorithms, where decisions must be made during training about whether further updates remain beneficial or the algorithm should switch to a different phase. This issue is particularly relevant in stochastic min-max optimization. Generative adversarial networks (GANs) provide a canonical example, as their training requires repeated decisions about when to switch between discriminator and generator updates, yet existing methods typically rely on fixed update ratios or heuristic criteria. We formulate this switching problem as sequential hypothesis testing and develop an e-process-based adaptive training procedure. During discriminator updates, one e-process tests the null that the discriminator-induced separation between the empirical data distribution and the generator law remains below a target level. During generator updates, with the discriminator fixed, a second e-process tests the reverse null that this separation remains above a refresh level. Conditional on the observed training sample, we prove that fresh empirical indices and latent draws yield conditional e-values that can be accumulated into e-processes, providing anytime-valid Type I error control under adaptive model updates and data-dependent switching. Across multimodal synthetic distributions and image benchmark datasets, the proposed method matches or outperforms the best fixed-ratio baselines under several widely used GAN objectives.

Hyunjoo Kim, Sicheng Wu, Agastya Venkatraman +2

Department of Statistics, Ewha Womans University · Department of Mathematics, Purdue University · Department of Computer Science, Purdue University

Min-Max Optimization Sequential Hypothesis Testing Generative Adversarial Networks

Aug 10, 2026cs.LG

From Approachability Residuals to Anytime-Valid Evidence: The Online Convex Geometry of Testing by Betting

Betting-based sequential tests and Blackwell approachability are linked by a rate-explicit reduction through support-function residuals. For a compact convex target

S

and vector observations

r_t

, an OCO learner selects a predictable normal

w_t

and produces

q_t=\langle w_t,r_t\rangle-h_S(w_t)

. We prove the exact pathwise identity

\dist(\bar r_T,S) =\frac1T\sum_{t=1}^Tq_t+\frac{\Reg_T}{T}.

When

|q_t|\leq B

, composing this identity with one-sided betting yields a finite-time transfer: if the OCO and log-wealth regrets are at most

a_T

and

\ell_T

, respectively, then a target gap exceeding

\frac{a_T}{T} +2B\sqrt{\frac{\log(1/α)+\ell_T}{T}}

forces rejection by time

T

, while non-rejection certifies the converse radius. We then formulate a controlled stochastic experiment in which an action selected after

w_t

satisfies Blackwell's supporting-halfspace condition for every null mean payoff. The resulting wealth is an e-process under adaptive nulls; sublinear OCO regret gives stochastic approachability, whereas persistent mean separation under an alternative gives exponential wealth at rate at least

δ^2/(4B^2)

. Deterministic Blackwell games and passive tests are, respectively, the noise-free and singleton-action cases of this protocol. Bounded two-sample means, kernel MMD, and active heterogeneous data sources instantiate the reduction. The resulting connection is exact algebraically, quantitative at finite time, and operational when experiments are controlled.

Jinze Zhao

University of California, San Diego

Sequential Hypothesis Testing

Jul 24, 2026stat.ML

Efficient Online LLM Watermark Detection via Rao-Blackwellized E-Processes

As large language models (LLMs) are increasingly deployed, reliable and efficient mechanisms for distinguishing AI-generated text from human-written content have become essential. Statistical watermarking has emerged as a promising solution, yet most existing methods are typically fixed-horizon procedures, precluding valid early stopping in streaming generation. In this paper, we develop an efficient online watermark detection framework with anytime-valid inference based on Rao-Blackwellized e-processes, enabling recursive token-level evidence updates without storing the full history. In particular, we instantiate the framework for the Gumbel-max watermark and reduce the original token-level dependence testing problem to a pivot-induced sequential testing problem with an explicit null distribution. Theoretically, we prove anytime-valid Type I error control under arbitrary optional stopping and establish positive asymptotic log-growth under watermarking, implying consistency of the proposed stopping rules. Simulations and experiments on real LLM-generated text demonstrate efficient online detection with rigorous anytime-valid guarantees.

Lu Luo, Dandan Mo, Chengdong Xu +4

Yunnan Key Laboratory of Statistical Modeling and Data Analysis, Yunnan University, Kunming, China · School of Statistics and Management, Shanghai University of Finance and Economics, Shanghai, China

LLM Watermarking Sequential Hypothesis Testing

Jul 21, 2026cs.CV

Detect Early, Escalate Rarely: Anytime Detection of AI-Generated Video from the Compressed Bitstream

Detectors for AI-generated video are evaluated offline. A clip is decoded to pixels and scored once, increasingly by a large vision-language model. Detection, however, is deployed online. We recast the task as streaming perception and score the motion field the codec already wrote into the bitstream. Reading that field is a parse, not a pixel-domain forward pass. Because the running aggregate is monotone, one end-calibrated threshold is anytime-valid at the data-dependent decision time. Recalibrating at each prefix is not. Escalation is priced in closed form. A compute budget maps to a deferral window, on a frontier monotone exactly where the deferral condition holds. On matched GenVidBench the codec stage reaches full-length AUC 0.64 at five orders of magnitude less compute than a pixel CNN, on CPU. Its gate holds the stopping-time false-positive rate at target while the real data match its calibration, and drifts above it under distribution shift. Deferring 15% of clips lifts accuracy from 0.75 to 0.78 at

7\times

less compute (paired: McNemar

p<10^{-6}

). The stage-1 ordering replicates on AIGVDBench. We introduce no new detector. The contribution is the reframing, two guarantees, and the measured frontiers. Code, configurations, and evaluation splits: https://github.com/KurbanIntelligenceLab/streamdet.

Mert Onur Cakiroglu, Mehmet Dalkilic, Hasan Kurban

Luddy School of Informatics, Computing, and Engineering, Indiana University Bloomington, Bloomington, IN, USA · College of Science and Engineering, Hamad Bin Khalifa University, Doha, Qatar

Deepfake Detection Sequential Hypothesis Testing

Jul 17, 2026stat.ME

Aggregation of Statistical Evidence under Exchangeability

We study aggregation of statistical evidence under unknown and potentially complex dependence using group-invariance. Building on permutation-based constructions that treat transformed datasets as exchangeable units, we aggregate evidence across statistics for each transformed dataset and calibrate the resulting aggregates across transformations. We develop a finite-sample power and adaptivity theory for this framework, together with extensions to sequential and data-dependent aggregation that preserve validity. For single-batch aggregation, which uses one collection of transformed datasets for both standardization and calibration, we show that the critical values uniformly improve on deterministic calibrations valid under arbitrary dependence, including Bonferroni correction, while adapting to the unknown dependence structure. We also introduce a sequential alpha-spending version that permits early rejection when evidence is strong, and a two-batch extension that separates standardization from calibration to accommodate learned aggregation rules and reduce computation. Applications to adaptive nonparametric testing and conformal prediction illustrate how these results sharpen existing aggregation methods.

Antonin Schrab, Rajen Shah, Arthur Gretton +1

Department of Computer Science and Technology, University of Cambridge, UK · Statistical Laboratory, University of Cambridge, UK · Gatsby Computational Neuroscience Unit, University College London, UK +1

Sequential Hypothesis Testing Multiple Hypothesis Testing Statistical Hypothesis Testing

Jul 13, 2026cs.LG

Bet on Features: Anytime-Valid and Feature-Aware Auditing of Conditional Quantile Forecasters

Black-box conditional quantile forecasts are widely used for sequential decisions under asymmetric costs, such as inventory planning in supply chain management. Once deployed, such forecasters must be monitored continuously as data streams drift and regimes change; this invalidates standard, fixed-horizon backtests for calibration. Further, existing backtests do not take into account that the notion of calibration is, in fact, information-dependent: forecasts can look calibrated to an auditor with coarse information while being miscalibrated to an auditor with richer information. We develop a distribution-free and game-theoretic testing framework for continuously auditing black-box conditional quantile forecasters with non-i.i.d. losses, such that the resulting evidence process is powerful against predictably chosen alternatives specified by the features available to the auditor. We first formalize notions of conditional quantile calibration when different sets of features are available to the auditor, establishing that the coarseness of the auditor's information set determines the hardness of the testing problem. We then identify the sets of alternatives for which the auditor can achieve power, and focusing on contextual bets linear in the features, we derive finite-time detection guarantees for such alternatives, all without an i.i.d. assumption. The resulting evidence processes are interpretable at the feature level, as they quantify fine-grained, "feature-aware" evidence for miscalibration. We empirically validate these methods on simulated and real data, finding that a popular time series forecaster (Chronos-2) is highly miscalibrated w.r.t. multiple relevant features.

Ivane Antonov, Sohom Mukherjee, Richard Pibernik +1

Julius-Maximilians-Universität Würzburg · INSEAD

Quantile Regression Sequential Hypothesis Testing Uncertainty Calibration Black-Box Model Auditing

Jul 9, 2026cs.LG

Stop Guessing When to Stop Testing: Efficient Model Evaluation with Just Enough Data

The inherent rigidity of fixed-size benchmarks makes them an inefficient tool for model evaluation. Diverse evaluation objectives, including model ranking, model selection and testing throughout development, demand varying levels of statistical power. The mismatch between fixed sample sizes and these diverse needs results in either excessive computational cost or compromised reliability - a critical concern for model evaluation. To overcome these limitations, we call for adoption of sequential testing in our field. We provide an adaptive evaluation framework, that provides a principled way to navigate the trade-off between efficiency and reliability in model evaluation. Our framework combines the established statistical paradigm of sequential testing with stopping criteria tailored to common evaluation needs such as diminishing returns detection, and minimum detectable effect size. We demonstrate its ability to adaptively manage the efficiency-reliability trade-off on the Open VLM Leaderboard, including, for example, a 80% reduction in computational cost compared to fixed-size evaluation (with a 2.5-point CI width allowance) while maintaining statistical significance.

Ofir Arviv, Kristjan Greenewald, Yotam Perlitz +3

IBM Research

Early Stopping Optimal Stopping Automated Evaluation Sequential Hypothesis Testing Statistical Hypothesis Testing

Jun 29, 2026cs.AI

Sequential Fairness Auditing with Limited Output Access

External evaluations are becoming increasingly central to the governance of AI systems. In practice, however, independent auditors often have limited access to deployed models and must rely on query-based interactions. Most existing fairness evaluation methods assume static datasets and fixed-sample statistical tests, making them poorly suited to real-world auditing scenarios in which evidence must be collected sequentially under query constraints. In this work, we formulate fairness auditing as a tolerance-aware sequential hypothesis-testing problem under limited model output access. We develop a sequential generalized likelihood-ratio framework that allows auditors to accumulate evidence from a finite audit pool and stop once sufficient support for compliance or violation has been obtained. The framework is instantiated for decision-based Statistical Parity and Equal Opportunity audits, and extended to score- and logit-based proxy audits when richer observables are available. Our results show that both the fairness metric and the level of model access significantly affect audit efficiency, and that the benefits of richer output information are not uniform across auditing settings. In particular, richer outputs can substantially reduce the number of queries required for some fairness metrics and operating regimes, while offering limited gains in near-threshold cases. This work provides a practical statistical framework for sequential fairness auditing under realistic deployment constraints.

Ioannis Pitsiorlas, Martha V. Sourla, Marios Kountouris

EURECOM, France · DaSCI, University of Granada, Spain

Algorithmic Fairness Group Fairness Sequential Hypothesis Testing Black-Box Model Auditing

Jun 23, 2026cs.AI

Bayesian control for coding agents

Modern coding agents pair LLM generators with various tools, including cheap diagnostics and expensive verifiers. The tool-use decisions are typically governed by orchestrators that often use fixed rules and ignore uncertainty. We formulate orchestration as cost-sensitive sequential hypothesis testing: a Bayesian controller maintains a belief over candidate correctness and dynamically decides whether to gather more evidence, refine the candidate, verify it, or stop. Across six generators and nine coding benchmarks, Bayesian control proves to be most valuable when verification is costly and critics are informative but imperfect. Beyond control, the belief state yields an interpretable correctness score that outperforms token-probability and raw tool-success baselines for uncertainty quantification.

Theodore Papamarkou, Vladislav Smirnov, Viktor Mazanov +4

1PolyShape · 3MBZUAI

Language Model-Based Control Coding Agents Bayesian Decision Theory LLM Uncertainty Estimation Sequential Hypothesis Testing

Jun 20, 2026cs.LG

Ranking-and-Selection with Multiple Correct Answers and Non-Answerable Estimates

We study fixed-precision ranking-and-selection in structured settings where the answer may be non-unique and where noisy estimates may temporarily admit no valid answer at all. This phenomenon arises naturally in problems such as multi-fidelity ranking-and-selection and identifying a Condorcet winner from pairwise comparisons. To address this, we propose a unified framework based on answer-wise acceptance sets, restricted generalized likelihood ratio stopping, and an answer-pitfall decomposition that yields a max-max-min characteristic value and a common sampling principle. We introduce ENDS, a general procedure that combines estimation, nomination, pitfall detection, and cost-aware information-directed selection. We instantiate ENDS for various problems by deriving explicit formulas. Extensive numerical experiments show that this unified recipe performs well across a broad range of pure-exploration problems and offers a practical framework and proof-of-concept algorithmic recipe.

Qiaoqiao Wang, Wei You

Dept. of Industrial Engineering and Decision Analytics, The Hong Kong University of Science and Technology, Hong Kong, China

Best-Arm Identification Adaptive Sampling Sequential Hypothesis Testing

Jun 18, 2026stat.ML

Betting on Moments: Legendre Jumper Martingales for Online Exchangeability Testing

A fundamental assumption in statistics and machine learning is that ``the future looks like the past,'' formalized as exchangeability: the joint data distribution is order-invariant. In practice, this assumption is often violated due to distribution shifts over time. Early detection of exchangeability violations is crucial to prevent performance degradation and enable timely interventions like model retraining. Conformal test martingales offer a flexible, distribution-free framework for sequential exchangeability testing with guaranteed false-alarm rate control by betting against the uniformity of conformal p-values. While alternatives such as plug-in martingales and mixture-based strategies exist, computationally efficient baselines like the Simple Jumper are limited to detecting mean location shifts. We propose a family of conformal test martingales based on shifted Legendre polynomials that extend the Simple Jumper to higher-order moments. The Simple Legendre Jumper replaces linear betting functions with polynomials of arbitrary degree, enabling rapid detection of variance, skewness, and other higher-order deviations. The Product Legendre Jumper combines multiple polynomial degrees into a single betting function but suffers from exponential state-space growth, termed the jumping tax. To resolve this, we introduce the Variational Legendre Jumper, which employs a mean-field approximation to reduce complexity to constant time per step with minimal power loss, providing an expressive, scalable framework for real-time distribution shift monitoring.

Johan Hallberg Szabadváry

Department of Mathematics, Stockholm University

Conformal Prediction Distribution Shift Detection Change-Point Detection Sequential Hypothesis Testing Temporal Distribution Shift Statistical Hypothesis Testing

Jun 17, 2026stat.ML

Sequential Kernel-based Conditional Independence Testing via Adaptive Betting

Testing conditional independence is fundamental yet intrinsically difficult: without additional assumptions, Type I error control is impossible in general. The "Model-X'' paradigm addresses this difficulty by assuming exact knowledge of a relevant conditional distribution. While small deviations from this assumption can sometimes be tolerated in classical one-shot testing, existing sequential conditional independence tests typically require the Model-X conditional to be known exactly, making them fragile when it must instead be estimated. We propose a new approach that is substantially more robust to such estimation error. Our method applies testing-by-betting to an adaptively optimized Kernel Conditional Independence statistic, together with a normalization scheme and a truncate-and-shift calibration strategy. These modifications greatly reduce Type I error inflation while preserving high power across high-dimensional synthetic benchmarks and real-world fairness tasks, outperforming existing sequential Model-X approaches. Code is available at https://github.com/he-zh/SKCI.

Zheng He, Danica J. Sutherland

Department of Computer Science, University of British Columbia, Vancouver, Canada · Alberta Machine Intelligence Institute, Edmonton, Canada

Kernel Methods Conditional Independence Testing Sequential Hypothesis Testing Statistical Hypothesis Testing

Jun 6, 2026cs.AI

PACE: Anytime-Valid Acceptance Tests for Self-Evolving Agents

Self-evolving agents improve by repeatedly proposing changes to their own prompts, skills, or workflows and keeping those that score higher on a small held-out set. Almost all effort has gone into the proposer that generates candidates; we argue the weak point is the acceptor, the rule that decides whether to commit a change. Applied hundreds of times against the same noisy dev estimate, the ubiquitous "keep it if the score went up" rule is uncontrolled adaptive multiple testing: the agent effectively p-hacks itself, accumulating false commits that make it churn and drift rather than improve. We recast committing as a sequential hypothesis test and propose PACE (Paired Anytime-valid Commit Evaluation), a training-free, anytime-valid commit gate. Each candidate is compared to the incumbent on identical instances and committed only when a testing-by-betting e-process accumulates decisive evidence, stopping early to save evaluations and controlling each candidate's false-commit probability at a user-set level even under optional stopping (a per-decision guarantee). On Qwen2.5 agents (0.5B-3B) self-evolving at the prompt level on GSM8K, SVAMP, and ARC-Challenge, greedy acceptance commits 30-42% false and 10-33% harmful edits when a genuine improvement is hidden among noisy proposals, while PACE commits the real one and essentially nothing else, matching greedy's held-out accuracy at sharply lower variance and about 18% lower evaluation cost. With no real gain available, greedy commits 13-21 spurious self-modifications per run (72-100% false) and degrades the most fragile agent by 4.9 points, while PACE holds at baseline. Reliability of self-evolution depends on the acceptor, not only on the proposer.

Zayx Shawn

Independent Researcher

Sequential Hypothesis Testing LLM Agent Reliability Multiple Hypothesis Testing Self-Evolving Agents

May 29, 2026stat.ML

Correcting Split Selection in Online Decision Trees via Anytime-Valid Inference

Bagging-based ensembles, most notably Adaptive Random Forests, are among the strongest performers for learning from data streams. A common denominator across these methods is their reliance on Hoeffding Trees as base learners, which grow decision trees incrementally by testing whether a candidate split is significantly better than its alternatives using concentration inequalities. Despite their empirical success, existing variants lack valid statistical guarantees. Current analyses rely on fixed-sample concentration bounds, while split decisions are made using data-dependent stopping rules, which invalidates their guarantees and can drive the probabilty of incorrect splits to one. We introduce a principled alternative based on anytime-valid inference. Our method provides: (i) anytime-valid control of false splits under arbitrary data streams, including non-stationary settings; (ii) finite commitment time under a predictive advantage; and (iii) under stationary i.i.d. data, risk is monotone decreasing and strictly improves at every split. Empirically, we evaluate both standalone trees and their use within Adaptive Random Forests on non-stationary streams. Our method improves performance while producing substantially smaller trees.

Salim I. Amoukou, Saumitra Mishra, Manuela Veloso

J.P. Morgan AI Research.

Anytime-Valid Inference Decision Tree Learning Random Forest Sequential Hypothesis Testing

May 27, 2026cs.CR

Optimal Rates for Differentially Private Hypothesis Testing with E-values

E-values have attracted considerable interest in recent years as flexible tools for enabling anytime-valid and adaptive data analysis. Hypothesis testing is at the core of many of these applications, which can often involve private or sensitive data. In this work, we answer a simple but important question: given two distributions

\mathbb{P}

and

\mathbb{Q}

, what is the maximum achievable e-power when testing

X\sim \mathbb{P}^n

against

X\sim\mathbb{Q}^n

with e-values that satisfy

\varepsilon

-differential privacy? We characterize the optimal rate for this problem and provide an algorithm which matches it exactly. In the sequential setting, when observations arrive one-by-one and the analyst chooses when to halt, we give matching upper and lower bounds on the stopping times of any private e-process. Numerical experiments confirm the practicality of our algorithms, which require less data than the recently proposed DP-SPRT across a range of sequential testing problems and privacy levels.

Ben Jacobsen, Tomas Gonzalez, Gavin Brown +2

University of Wisconsin-Madison · Carnegie Mellon University · University of Wisconsin–Madison

E-Values Differential Privacy Sequential Hypothesis Testing Statistical Hypothesis Testing

May 27, 2026cs.LG

Semi-Supervised Hypothesis Testing by Betting on Predictions

We introduce a testing-by-betting framework that leverages predictions on unlabeled data to enhance the power of sequential hypothesis testing. Given limited samples from the joint distribution of

(X,Y)

, and additional unlabeled samples from the marginal of

X

, we ask how unlabeled data can be used to hypothesize about the distribution of

Y

, and the conditional distribution of

Y\mid X

. We introduce an e-statistic and use it to construct a sequential test. Under standard distributional assumptions -- label shift or concept shift -- we establish that the test is anytime valid. Furthermore, we show that for binary data, the e-statistic has non-trivial power. Crucially, our approach retains these properties even when the underlying predictions are inaccurate. Through simulations and applications to large language models evaluation, we demonstrate power gains over baseline approaches, including prediction-powered inference. These gains persist even with relatively limited unlabeled data and when predictions have low accuracy due to weak correlation between

X

and

Y

.

Yaniv Tenzer, Elad Tolochinsky, Yaniv Romano

Department of Computer Science, Technion – Israel Institute of Technology · Department of Electrical and Computer Engineering, Technion – Israel Institute of Technology

E-Values Semi-Supervised Learning Sequential Hypothesis Testing Statistical Hypothesis Testing

May 19, 2026cs.LG

EviTrack: Selection over Sampling for Delayed Disambiguation

Sequential prediction is challenging in regimes of delayed disambiguation, where early observations are ambiguous and multiple latent explanations remain plausible until sufficient evidence accumulates. Standard approaches based on marginal inference struggle in this setting, either collapsing uncertainty prematurely or failing to recover once informative evidence arrives. We introduce EviTrack, a test-time inference framework that operates over latent trajectories rather than marginal states. EviTrack maintains a set of competing trajectory hypotheses and applies evidence- and likelihood-ratio-based selection to delay commitment until supported by data, drawing inspiration from hypothesis management in multiple hypothesis tracking and track-before-detect. To evaluate this setting, we construct a controlled synthetic benchmark with known latent ground truth that explicitly exhibits delayed disambiguation. At matched inference budget, EviTrack substantially outperforms sampling-based baselines, achieving faster post-disambiguation recovery. These results show that, in delayed disambiguation regimes, moderate trajectory-level selection is more effective than increasing sampling coverage, highlighting selection over sampling as a key principle for reliable sequential inference.

Omer Haq

Independent Researcher · Baltimore, MD, USA

Sequential Hypothesis Testing Sequence Modeling

May 18, 2026cs.LG

Sequential Consensus for Multi-Agent LLM Debates: A Wald-SPRT compute governor with calibration-based failure detection

Multi-agent LLM debate improves factuality and reasoning, but most recipes pick a fixed round count, over-spending on easy items and under-spending on hard ones. We adapt Wald's Sequential Probability Ratio Test (SPRT) as a plug-in compute governor for LLM debates. After each round, an LLM judge emits a [0,1] consensus score on the latest agent positions; a Wald monitor accumulates the log-likelihood ratio of "useful convergence" vs "not yet useful" under a Beta likelihood family, and stops when either boundary is crossed or returns a capped best-effort outcome at R_max. Under i.i.d. assumptions the rule inherits SPRT type-I/type-II error guarantees; in deployment the calibration itself is the more important object, since it estimates whether the judge score actually separates useful from unhelpful convergence in a given domain. We evaluate two tracks: (i) a Monte-Carlo study under calibrated Beta models characterising working curves, error rates, capping behaviour, and sensitivity; and (ii) a real-LLM evaluation on 200 attempted MMLU and 200 attempted GSM8K items with three heterogeneous agents (gpt-5, claude-opus-4-6, gemini-2.5-pro) and a claude-opus-4-6 judge, using disjoint 40-item calibration subsets. On GSM8K the rule stops in 1.01 average rounds (4.06 LLM calls) at 97.0% accuracy vs 99.0% for fixed-5 debate at 15 calls: a 3.7x call reduction at -2pp accuracy. On MMLU the calibrated KL collapses to about 0 and the rule caps on 99.5% of items at 2.1x cost. The takeaway is not that SPRT makes debate more accurate, but that a classical sequential test serves as a cheap compute-control and failure-detection layer for multi-agent LLM systems.

Andrea Morandi

Cisco Systems, Inc.

LLM Inference Efficiency Multi-Agent Debate Multi-Agent Consensus Sequential Hypothesis Testing

May 13, 2026stat.ML

A Regret Perspective on Online Multiple Testing

Online Multiple Testing (OMT), a fundamental pillar of sequential statistical inference, traditionally evaluates the False Discovery Rate (FDR) and statistical power in isolation, obscuring the highly asymmetric costs of false positives and false negatives in modern automated pipelines. To unify this evaluation, we introduce

\textit{Weighted Regret}

. Under this metric, we prove the

\textit{Duality of Regret Conservation}

: purely deterministic procedures ensuring strict FDR control inevitably incur an

Ω(T)

linear regret penalty, as threshold depletion during signal-sparse cold starts forces massive false negatives. Tailored for exogenous testing streams, we propose Decoupled-OMT (DOMT) as a baseline-agnostic meta-wrapper. By incorporating a history-decoupled, strictly non-negative random perturbation, DOMT rescues purely deterministic baselines from severe threshold depletion. Crucially, it preserves exact asymptotic safety in stationary environments and rigorously bounds finite-sample error inflation during cold-starts. Guaranteeing zero additional false negatives, it yields an order-optimal

Ω(\sqrt{T})

regret reduction in bursty environments, with a derived ``Cold-Start Tax'' characterizing the exact phase transition of algorithmic superiority. Experiments validate that DOMT consistently curtails empirical weighted regret, achieving an order-optimal sublinear mitigation of threshold depletion to navigate the non-stationary Pareto frontier.

Qingyang Hao, Kongchang Zhou, Fang Kong +1

Southern University of Science and Technology

FDR Control Regret Minimization Sequential Hypothesis Testing Multiple Hypothesis Testing Statistical Hypothesis Testing