Confidence Region Estimation

Probing the hypothesis space after seeing the data remains valid under possibilistic inferential models (IMs), provided the significance level stays fixed. The price is computation, as each plausibility is a supremum of the possibility contour over the hypothesis, and the contour itself is approximated at each queried parameter value. We propose a possibilistic radial transport, which hides the contour value of a parameter in the radius of its source point. When a transport that maximizes within-shell entropy is picked, sampling parameters covering a confidence cut becomes a matter of truncating the radius. We provide a deep learning algorithm that enforces the contour depth condition while maximizing the entropy within each shell. Our amortization makes coverage and power assessments of the learned approximation practical as well as predictive check of new datasets. We also use the sampler to construct a Bel-Pl spectrum for comparing and selecting interpretable hypotheses that satisfy a prescribed Bel-Pl decision criterion. In simulations the learned contours match or improve on ellipsoidal approximations to the cuts, while the coverage and power track the exact reference. Finally, we probe hypotheses about ovarian aging using synthetic AMH records, asking for each woman how many more years her median AMH level will remain above a specified reference value.

Many two-sample tests for populations of persistence diagrams assess global differences without identifying the regions of the birth-death plane that contribute to them. We study simultaneous inference for local mean contrasts when the number of available diagrams is fixed. They are differences in expected weighted feature mass within $\ell_\infty$ neighborhoods at several centers and radii. We estimate these contrasts using additive landmark responses. A Gaussian multiplier bootstrap calibrates simultaneous confidence intervals while allowing unequal group covariances. The neighborhoods whose intervals exclude zero form a map with approximate family-wise error control, and selecting a subset of original intervals for display preserves their joint coverage guarantee. On the simultaneous coverage event, every reported neighborhood lies within twice its radius of the support of the mean-measure difference. A geometric result gives sufficient radius conditions for a displaced feature to produce a nonzero contrast. A comparison of sufficient detection thresholds quantifies the tradeoff between reducing the number of tested coordinates and reserving observations for an independent pilot. In simulations with 40 to 120 diagrams per class, the bands achieved 94%-98% simultaneous coverage under both the strict null and equal means with unequal covariances. In the latter setting, a permutation maximum and the pooled-t implementation of the two-stage persistence-image test of Moon and Lazar rejected in up to 32% and 26% of runs, respectively. In the fixed-budget simulations, spending a third of the observations on a pilot to choose landmarks or radii located changes less often than a prespecified grid at a single radius. On the MUTAG benchmark, the localized region concentrates on rings of fused-ring systems, an exploratory reading.

Repeated evaluation can estimate a benchmark score accurately while still requiring replication to certify narrow uncertainty. We characterize that requirement on a fixed grid of $M$ tasks with $L$ binary paths per task under the hard budget $(M+t)K$, where each path costs at most $K$ responses or episodes. For fixed $L \ge 3$ and $0 < α\le 1/12$, the optimal expected width on the worst pure cohort is $Θ_{α,L}([M(t+1)]^{-1/2})$ when every task is observed and $Θ_{α,L}([M(t+\sqrt{M})]^{-1/2})$ when omission is allowed. The lower bounds cover adaptive hard-budget policies, and fixed random-subset designs attain both rates through disagreement certificates. A joint mean/disagreement interval turns the task-covering law into practical finite-budget inference. In an equal-budget LiveCodeBench replay with 16 models, 880 tasks, and five outputs per task, the task-covering design reduces median point-estimation MSE by 87.0% relative to pooled uniform sampling, while the Joint certificate produces narrower confidence intervals in 15/16 panels and reduces median interval width by 30.6%. Finite-regime analyses identify task coverage as the effective choice at the evaluated scale and characterize how cohort size and within-task agreement determine the useful operating region. Together, the sharp laws and fixed-budget evidence make replication and task coverage explicit design variables for information-efficient repeated evaluation.

This paper studies multivariate conformal prediction (CP), a distribution-free uncertainty quantification framework with finite-sample coverage guarantees. The efficiency of multivariate prediction sets hinges critically on the residual geometry encoded by the nonconformity score, while existing minimum-volume methods rely on quantile thresholds that ignore tail residual severity and implicitly bind geometry learning to coverage level. We propose a tail-aware geometry learning framework for conformal ellipsoids that decouples tail sensitivity in geometry learning from the final coverage guarantee. Using a two-split design, we learn the metric matrix via volume minimization under a CVaR constraint on an estimation split, then apply standard conformal calibration on a held-out calibration split. The resulting problem is convex and admits a bounded-reweighting interpretation that prioritizes high-residual samples. Moreover, we theoretically characterize the trade-off between ellipsoidal volume and tail severity. Experimental results demonstrate the effectiveness of the proposed method.

We develop inference for manifold-valued population minimizers when each observation belongs to a different participant and only locally private messages reach the analyst. The method releases randomized tangent gradients and combines them through Riemannian stochastic approximation and Polyak-Ruppert averaging. Directly inserting a private data surrogate into a nonlinear loss can shift its population target, whereas conditional centring of the released gradient preserves the first-order equation. We introduce symmetric-pair regression (SPR) to estimate the asymptotic variance from the same private messages used for point estimation, without holding out participants or requesting a second release. We prove the central limit theorem and consistency of the fully transcript-based sandwich covariance and intrinsic Wald region under local differential privacy. Simulations across various statistical problems and manifolds support the predicted decrease in estimation error and near-nominal coverage under moderate privacy. An application to NHANES anthropometric data illustrates private estimation of a leading body-size direction and its uncertainty.

Importance weights are essential in domain adaptation under label shift, yet their utility is often undermined by the finite sample uncertainty associated with their estimation. Existing methods typically analyze this uncertainty through Gaussian elimination on interval-valued linear systems, which leads to overly conservative confidence regions and inefficient downstream applications. We propose a paradigm shift from inversion-based inference to a direct matrix constraint framework. We use this framework to define a joint confidence region and extract marginal intervals via linear programming, deriving provably tighter bounds for importance weights while maintaining exact finite-sample validity. Furthermore, we analyze the confidence region's geometry and provide the theoretical results for its diameter bounds. Evaluated across text, image, multimodal benchmarks, including AGNews, MNIST, CIFAR-10, N24News, and a real-world autonomous driving dataset, nuImages, our approach consistently yields shorter confidence intervals and smaller prediction sets than inversion-based methods.

Repeated-sampling evaluations increasingly extrapolate pass@k far beyond the number n of samples collected per problem. We show that, in the pooled/random-task conditional-Binomial model, fixed-n success counts identify only the n free moments of the latent per-task success distribution. Consequently, direct pass@k is identified for k <= n, but generic extrapolated pass@k, tail exponents, and tail constants are not identified for k > n, even with arbitrarily many exchangeable tasks at the same rollout budget. This is stronger than the observation that the usual estimator is undefined beyond n: it characterizes the information missing from the fixed-depth count-law experiment. We give exact count-law-preserving constructions with incompatible extrapolations, state the exceptional unique-extension case, and compute sharp population identified intervals through Hausdorff principal representations. On the public 10,000-rollout-per-problem release of Brown et al., counterfactual n = 16 evaluations leave failure at k = 1000 ambiguous by factors from 1.5 to over 2,600 across four MATH/GSM8K/CodeContests configurations. The calibration shows that intermediate-scale failure share alone does not determine width. Our result does not reject parametric inference-time scaling laws; it supplies the nonparametric baseline against which their assumptions can be evaluated. We give an exact, conservative one-coordinate finite-task confidence certificate and a reporting standard separating direct estimates, identified sets, and model-conditioned forecasts.

Minimum-norm confidence envelope strategies offer a nonparametric approach to constructing nonasymptotic, simultaneous confidence regions for band-limited functions, exploiting the theory of Reproducing Kernel Hilbert Spaces (RKHS). While the finite-sample coverage guarantees of these envelopes have been established, their consistency has not been analyzed so far. In this paper, we study this construction, here termed the Minimum-Norm Confidence Envelope (MiNCE) framework, and establish the strong uniform consistency of the resulting bands, both for noise-free and noisy observation models, under mild assumptions on the measurement noises. We further extend this formulation to the frequency domain, deriving nonasymptotic, simultaneous, strongly uniformly consistent confidence bands for the smoothed spectra. Numerical experiments in nonparametric regression and spectral estimation empirically confirm our theoretical results, illustrating the contraction of the confidence envelopes toward the target function as the sample size increases.

Natural-language tasks can elicit different verdicts from protocol-following evaluators that receive the same declared information. We study aggregate disambiguation systems (ADSs). Given a task and a candidate solution, each evaluator casts a binary vote on whether the solution should be accepted, and the system aggregates the votes of a finite panel. The target is protocol reproducibility relative to an explicitly declared evaluator reference, not semantic truth. We separate fixed finite censuses, probabilistic evaluator populations, and growing-census limits, since their endpoint laws and guarantees are not interchangeable. In the population setting, we use finite samples to estimate how often a finite panel reaches the same decision as the declared evaluator population. We provide a lower confidence bound on the fraction of candidate solutions for which the disagreement probability is at most a chosen tolerance. The calculation accounts separately for sampling candidate solutions and sampling evaluators. The construction permits arbitrary dependence among columns induced by shared evaluator rows and uses exact binomial intervals at the evaluator layer and an exact one-sided binomial inversion at the generator layer. Simulations check the implementation against known population coverages and expose power limitations.

Constant-stepsize temporal-difference (TD) learning is attractive for policy evaluation, but inference from a single Markov trajectory must account for serial dependence and a stepsize-dependent stationary target. For fixed-stepsize linear TD, we establish a functional central limit theorem whose covariance retains the multiplicative component induced by the random TD matrix and the stationary iterate error. We then derive a joint functional limit for parallel Richardson--Romberg (RR) recursions driven by the same trajectory. A Brownian-bridge self-normalizer yields asymptotically pivotal confidence regions for prespecified state-value contrasts without estimating the long-run covariance or selecting a bandwidth or batch length. For such a contrast, the procedure admits a one-pass implementation whose memory does not grow with the trajectory length. At a fixed stepsize, the inferential center is the RR stationary target. We also study horizon-indexed designs in which the stepsize remains constant within each run and decreases across longer horizons. Under an explicit RR-dependent rate window, the residual RR target shift, multiplicative remainder, and initialization effect are negligible at the root-$n$ scale, yielding inference for the projected Bellman solution. Experiments on FrozenLake and Garnet illustrate stationary-target coverage, RR target correction, and the finite-sample behavior of the horizon-indexed design.

Conformal changepoint localization turns any score into a confidence set for the changepoint with finite-sample coverage. Coverage is universal; efficiency is not. The oracle score is a likelihood ratio, so practical scores estimate density ratios, and set length deteriorates under heavy tails, skewness, and distribution shift, where no length guarantee applies. We propose ARC (Augmented-Rank Conformalization), a family of scores depending on the data only through within-segment ranks: rank-CUSUM location and scale channels, their fixed combinations, and a lightweight neural score frozen after synthetic training. Every ARC score inherits finite-sample coverage for every frozen weight configuration, including random initialization and mistraining. The main result is an efficiency transfer theorem: the entire ARC confidence set is almost surely invariant under strictly increasing marginal transforms, so the set length distribution depends on the data pair only through its rank structure, and lengths certified once hold verbatim across its monotone orbit, whereas a plug-in score's length changes with every re-expression. Across different rank structures lengths do change, and are reported as such. Classical rank-test theory positions ARC as targeting the optimal invariant score at bounded cost. Simulations confirm nominal coverage for all scores, including sabotaged networks, identical sets under monotone transforms where plug-in scores inflate, and smooth degradation where plug-in sets become vacuous; on the well-log benchmark ARC localizes annotated shifts to three to five candidates and flags misfit by an empty set. Two boundaries are stated rather than hidden: serial dependence destroys exactness, and trend-type alternatives lie outside the piecewise-exchangeable model.

Double Machine Learning (DML) is a popular approach for treatment effect estimation in various settings, which allows a wide range of flexible machine learning methods to be used for nuisance parameter estimation while preserving valid inference. In practice, however, applied researchers must choose among many machine learning algorithms for nuisance models, and the impact of this choice on the variance estimation of DML is not well characterized. We conduct a comprehensive simulation study to compare the coverage probability of DML confidence intervals across different machine learning algorithms. In this study, we compare (1) analytical confidence intervals derived by DML theory versus (2) bootstrap confidence interval. We use a set of learners including ordinary least squares, LASSO, Random Forest, LightGBM, and Neural Networks under different data generation settings. We evaluate the performance across difference settings by bias, confidence interval width, and most importantly, coverage probability. Our results show substantial variability in coverage performance across analytical and bootstrap confidence intervals, highlighting that learner choice plays a critical role in reliable DML inference. Surprisingly, we find that in many settings, when sample size increases, the coverage probability of both DML analytical and bootstrap confidence interval decreases. We further investigate coverage probabilities using a real dataset on rural urban differences among U.S. counties. The real data analysis discovers that (1) the model performance still varies by the learner choices and (2) greater rurality has a statistically significant increasing effect on county level obesity prevalence.

Active testing provides a label--efficient approach to risk estimation by adaptively selecting which test points should be labelled. However, existing estimators fail to exploit the informative predictions of powerful black--box models, even though such predictions are increasingly available in settings where labels remain expensive. To address this, we propose \textbf{Prediction--Powered Active Testing (PPAT)}, a novel label--efficient risk estimation framework that combines the unbiased LURE estimator \citep{farquhar2021statistical} with a prediction--powered control variate. Rather than using proxy predictions as biased pseudo--labels, PPAT uses them to residualise the loss, preserving unbiasedness while reducing variance. Beyond the estimator itself, PPAT also changes which points should be acquired: we derive oracle and practical surrogate--based acquisition rules tailored to reducing the variance of our estimator. Moreover, we establish asymptotic normality for PPAT, yielding asymptotically valid confidence intervals and thus a principled estimate of the uncertainty around our estimates. Across tabular regression and image--classification tasks, PPAT outperforms existing methods in risk estimation, while its confidence intervals attain the target coverage with substantially fewer labels and smaller widths.

Estimating heterogeneous treatment effects (CATE) requires simultaneously detecting effect modification and quantifying estimation uncertainty. Existing tree-based methods make an uneasy trade-off: significance-based approaches (Radcliffe and Surry 2011) identify subgroup interactions directly but lack valid inference; honest causal trees (Athey and Imbens 2016) deliver nominal confidence interval coverage but use outcome-agnostic splitting criteria that sacrifice interaction sensitivity. We introduce a hybrid algorithm that fuses significance-based splitting with honest sample-splitting and cross-validation. Our splitting criterion uses the squared $t$-statistic for the treatment $\times$ side interaction ($t^2$), which is shown to be directly aligned with the honest $\text{EMSE}_τ$ criterion when the interaction is strong. Post-hoc honest cross-validation selects the cost-complexity penalty, giving a single principled estimator with nominal CI coverage at the leaf level. For forests, we retain bootstrap count vectors to enable an infinitesimal jackknife (IJ) variance estimate of Monte-Carlo convergence rather than formal pointwise inference. On the three synthetic designs from (Athey and Imbens 2016) the single tree achieves approximately 90% leaf-average CI coverage at the 90% nominal level across all three designs (200 replications each); on the Criteo, Hillstrom and Starbucks uplift datasets we match Qini coefficient performance of S-, T-learner and GRF baselines. An open-source Python package with reproducible seeds, sklearn-compatible API, and full test coverage accompanies this work (https://codeberg.org/hadjipantelis/rattus).

Researchers increasingly use text classification--supervised models or large language models--to measure constructs from natural language, providing metrics such as recall and precision as evidence of their validity. Yet, though these metrics are point estimates subject to sampling variation, measures of uncertainty are inconsistently reported alongside them. Further, when they are reported, they are often estimated with methods that are not appropriate when relevant labelled datasets are small or performance is high. To increase and improve confidence interval reporting in the field, this paper evaluates confidence interval methods for performance metrics under conditions typical of social science text classification: small to moderate sample sizes, infrequent constructs, and texts nested within individuals. Across simulations, default methods such as the Wald interval and the basic percentile bootstrap are the least accurate, with coverage sometimes far below the nominal 95% level. Accuracy is improved with the use of Agresti-Coull, Wilson, Clopper-Pearson, and a novel pseudo-count regularized bootstrap (which is particularly relevant to the calculation of F1). When texts are nested within individuals, we demonstrate that adjustment for both effective N and the appropriate degrees of freedom is necessary for producing accurate analytic intervals. Among bootstrap intervals, the hierarchical bootstrap is more accurate than the cluster bootstrap when individuals produce a moderate number of texts but overly conservative when individuals produce only a few. By providing guidance to the field on appropriate interval estimation, we aim to improve the transparency of machine learning applications, and to encourage greater attention to the validation sample size at the design stage.

Joint prediction sets for multivariate time series should control a single event while adapting to cross-coordinate dependence. We study filtered conformal ellipsoids: a frozen state-space filter emits a one-step predictive mean and covariance, and split-conformal calibration is applied to the resulting Mahalanobis scores. The filter is used to choose the ellipsoid shape; conformal calibration chooses the scalar radius, so the construction benefits from a learned predictive covariance without relying on Gaussian tail probabilities for coverage. The main difficulty is that filtered scores are dependent and learned recurrent filters need not contract in their raw hidden state; we therefore analyse contraction in an observable predictive-law quotient that identifies hidden states producing the same future sequence of emitted Gaussian laws. Under a stable Bayes Gaussian-projection filter, covariance bounds, and a finite-horizon observability Fisher condition, small excess Gaussian negative log-likelihood implies contraction of the learned emitted laws. Combined with a threshold-autocovariance envelope this yields a Chebyshev-type approximate coverage bound for filtered split-conformal prediction under dependence; a sharper Bernstein-type bound requires an additional geometric-mixing concentration assumption. Under Gaussian oracle realisability we also obtain a near-oracle log-volume comparison within the class of conditionally valid Gaussian ellipsoid rules. We instantiate the framework with a GCN-GRU filter with diagonal-plus-low-rank covariance. On moderate-size graph-native traffic benchmarks (METRLA-$20$ and PEMSBAY-$50$), the learned filter gives sharper at-target ellipsoids than static-covariance and non-filter baselines; at full-graph scale and on non-graph-native datasets, factor and copula baselines can be stronger.

We propose the data augmented bootstrap (DAB), a framework for constructing confidence intervals from approximately invariant transformations of the data. As special cases, DAB recovers popular methods that rely on exact group symmetries, such as conformal prediction, wild bootstrap for Maximum Mean Discrepancy U-statistics and the recently proposed SymmPI. Meanwhile, DAB also recovers the classical bootstrap method, which exploits the dataset's approximate invariance under uniform sampling of data indices as the dataset size grows. For all DAB methods, we establish theoretical coverage results that interpolate between finite-sample and asymptotic guarantees according to the strength of the invariance, and without assuming a group structure. The approximate invariance is measured in the Kolmogorov distance and, for statistics that satisfy Gaussian universality, reduces to conditional mean and variance matching. This allows us to incorporate data augmentation (DA), a widely used machine learning heuristic based on approximate invariances, into known statistical methods. We empirically test the performance of incorporating DA into bootstrap, wild bootstrap and conformal prediction for simulated settings as well as for image, language and scientific data.

This paper introduces a distribution-free framework for constructing post-detection confidence sets for changepoints after stopping a sequential change detection procedure. It is well known that conformal test martingales can be used to sequentially detect changes in distribution, but by themselves provide no inference for the time at which a proclaimed change occurred. Past work on post-detection inference requires pre- and post-change classes of distributions to be known, but this paper accomplishes localization of the changepoint without any distributional assumptions. We establish finite-sample coverage guarantees (conditional on correct detection). We provide non-asymptotic bounds on the conditional expected size of the confidence sets. Under suitable asymptotic regimes, we prove that the conditional expected size of the confidence set remains uniformly bounded and demonstrate strong empirical performance on simulated and real data. To the best of our knowledge, this is the first general distribution-free framework for sequential changepoint localization with valid post-detection coverage.

Many applications require statistically valid inference across many related tasks, while using only a handful of high-quality labels per hypothesis. In AI evaluation, these tasks may correspond to model behaviors across prompts, subgroups, or hypotheses; in social science surveys, they may correspond to related questions, populations, or measurement conditions. Prediction-powered inference (PPI) uses abundant but inexpensive proxy measurements to improve inference from limited, ground-truth labels, but commonly used methods treat tasks independently and therefore fail to exploit shared structure across related tasks. This limitation is especially important in settings where only a small number of labels are available per task. To address this issue, we introduce a multi-task prediction-powered inference framework that uses labeled data from related tasks to improve power while preserving task-specific inference. Our methods exploit the shared structure in the proxy-ground-truth relationship through cross-task recalibration, while retaining within-task rectification and power tuning to construct accurate point estimates and confidence intervals. We prove that efficiency gains beyond power-tuned PPI are only possible when the proxy-ground-truth relationship contains nonlinear structure; affine cross-task recalibrations are asymptotically equivalent to using the original proxy. We complement our theoretical findings with experiments on synthetic and semi-synthetic datasets, as well as a case study auditing language models on election-related information during the 2024 U.S. presidential election. Using a large human-annotation study, we show that cross-task recalibration can substantially reduce confidence interval widths when labels are scarce.

Stochastic gradient descent (SGD) is a foundational algorithm for large-scale statistical learning and stochastic optimization. However, statistical inference based on SGD iterates remains challenging when stochastic gradients have infinite variance, as the relevant limiting distributions depend on unknown nuisance parameters. In this paper, we develop an efficient, model-agnostic methodology for constructing confidence regions from SGD trajectories that applies in both finite- and infinite-variance regimes. The procedure is based on a joint weak convergence result for the Polyak-Ruppert averaged estimator and an empirical second-moment normalizer constructed from stochastic gradients along the SGD trajectory. This joint limit yields a self-normalized statistic in which the leading tail-dependent scaling terms cancel. We then use a subsampling calibration scheme to estimate the relevant critical values, avoiding explicit estimation of tail indices, slowly varying functions, or stable-law parameters. The resulting confidence regions are straightforward to implement and are asymptotically valid under both the finite- and infinite-second-moment regimes. Simulation studies show reliable coverage in various settings, supporting the proposed method as a practical tool for uncertainty quantification in stochastic optimization.

Full conformal prediction is a framework that implicitly formulates distribution-free confidence prediction regions for a wide range of estimators. However, a classical limitation of the full conformal framework is the computation of the confidence prediction regions, which is usually impossible since it requires training infinitely many estimators (for real-valued prediction for instance). The main purpose of the present work is to describe a generic strategy for designing a tight approximation to the full conformal prediction region that can be efficiently computed. Along with this approximate confidence region, a theoretical quantification of the tightness of this approximation is developed, depending on the smoothness assumptions on the loss and score functions. The new notion of thickness is introduced for quantifying the discrepancy between the approximate confidence region and the full conformal one.

We investigate the statistical properties of Temporal Difference (TD) learning with Polyak-Ruppert averaging, arguably one of the most widely used algorithms in reinforcement learning, for the task of estimating the parameters of the optimal linear approximation to the value function. Assuming independent samples, we make three theoretical contributions that improve upon the current state-of-the-art results: (i) we establish refined high-dimensional Berry-Esseen bounds over the class of convex sets, achieving faster rates than the best known results, and (ii) we propose and analyze a novel, computationally efficient online plug-in estimator of the asymptotic covariance matrix; (iii) we derive sharper high probability convergence guarantees that depend explicitly on the asymptotic variance and hold under weaker conditions than those adopted in the literature. These results enable the construction of confidence regions and simultaneous confidence intervals for the linear parameters of the value function approximation, with guaranteed finite-sample coverage. We demonstrate the applicability of our theoretical findings through numerical experiments.

We propose a data-driven approach to quantify the uncertainty of models constructed by kernel methods. Our approach minimizes the needed distributional assumptions, hence, instead of working with, for example, Gaussian processes or exponential families, it only requires knowledge about some mild regularity of the measurement noise, such as it is being symmetric or exchangeable. We show, by building on recent results from finite-sample system identification, that by perturbing the residuals in the gradient of the objective function, information can be extracted about the amount of uncertainty our model has. Particularly, we provide an algorithm to build exact, non-asymptotically guaranteed, distribution-free confidence regions for ideal, noise-free representations of the function we try to estimate. For the typical convex quadratic problems and symmetric noises, the regions are star convex centered around a given nominal estimate, and have efficient ellipsoidal outer approximations. Finally, we illustrate the ideas on typical kernel methods, such as LS-SVC, KRR, $\varepsilon$-SVR and kernelized LASSO.