exchangeability condition derivation

Existing conformal prediction theory relies on deterministic nonconformity measures, which are ill-suited for machine learning settings involving randomness. Moreover, the commonly adopted condition of exchangeability in distribution is insufficient to guarantee valid predictions in such stochastic contexts. This work addresses these limitations by introducing a stricter sufficient condition for validity—combining conditional independence with distributional exchangeability—and develops a novel conformal prediction framework grounded in probability theory and statistical learning theory. The proposed approach establishes rigorous validity conditions applicable to a broad class of randomized machine learning procedures, thereby providing a solid theoretical foundation for practical applications involving stochastic training processes.

This paper addresses the weak theoretical foundations, fragmented proof strategies, and high entry barrier of conformal prediction by systematically constructing a distribution-free finite-sample uncertainty quantification framework. Methodologically, it unifies permutation testing, the exchangeability principle, and distribution-free inference, integrating techniques from probability theory, statistical learning, and reliability analysis. Key contributions include: (1) the first systematic survey and pedagogical reconstruction of core proof strategies in conformal prediction; (2) establishment of a formal, reproducible theoretical framework with a transparent logical chain; and (3) provision of rigorous finite-sample guarantees for predictive set construction—without assuming any parametric form of the data-generating distribution—and seamless integration into complex machine learning pipelines. Collectively, these advances substantially lower both theoretical comprehension and practical implementation barriers.

This work addresses the failure of standard exchangeability—and consequent unreliability of distribution-free inference—under hierarchical data structures (e.g., grouped or repeated-measures designs). We introduce *hierarchical exchangeability*, the first formal theoretical foundation for distribution-free inference in non-i.i.d. hierarchical settings. Methodologically, we extend conformal prediction and the jackknife+ framework to hierarchical structures and propose a second-moment coverage mechanism, strengthening guarantees from marginal coverage to *conditional second-moment coverage*. Experiments demonstrate that our approach substantially reduces conditional miscoverage rates; under model misspecification, prediction interval width increases only marginally, while under correct model specification, the overhead is negligible. Our core contribution is the first provably reliable, distribution-free inference framework tailored specifically for hierarchical data.

This paper addresses the problem of testing exchangeability of random variables and invariance under compact groups. Methodologically, it introduces a novel e-value-based framework for posterior-valid p-values: (i) it derives the first exact analytic expression for posterior-valid p-values in group-invariance testing; (ii) it designs two data-dependent sampling schemes that unify and extend validity guarantees to arbitrary stopping times; and (iii) it integrates group representation theory, sequential analysis, and the likelihood ratio principle to establish new optimality characterizations under group invariance. Contributions include: (i) substantially improving the statistical power of the t-test in spherical symmetry testing; (ii) uncovering an intrinsic connection between exchangeability testing and the softmax function; and (iii) proposing a new sign-symmetry test whose power dominates existing approaches.

This work addresses the lack of theoretical guarantees on training-conditional coverage—i.e., coverage under the training data distribution—of conformal prediction under covariate shift. We first systematically investigate its upper-bound characterization and controllability. We derive a weighted Dvoretzky–Kiefer–Wolfowitz inequality to establish tight, provable training-conditional coverage bounds for split conformal prediction under nearly assumption-free conditions. Furthermore, leveraging algorithmic uniform stability, we provide the first training-conditional coverage guarantees for full conformal and jackknife+ methods. Our results demonstrate that all three mainstream conformal prediction frameworks achieve controllable training-conditional coverage under covariate shift, with split conformal yielding bounds that are both minimally assumption-dependent and tight. This work fills a critical theoretical gap in conditional coverage analysis of conformal prediction beyond the i.i.d. setting.

This work addresses the limitation of traditional conformal prediction, which guarantees only marginal coverage and often exhibits poor conditional coverage, leading to calibration bias in specific regions of the covariate space. To overcome this, the authors propose Randomized Localized Conformal Prediction (RLCP), a method that performs local calibration within neighborhoods of test points, thereby enhancing conditional coverage while preserving marginal validity. The paper establishes, for the first time, finite-sample, high-probability uniform guarantees for such localized approaches, simultaneously controlling both conditional coverage error and oracle length error. By leveraging Hölder continuity, kernel density estimation, data-splitting-based score learning, and conformal quantile regression, the authors develop a theoretical framework for local coverage, deriving finite-sample bounds on the conditional coverage gap and length error, and demonstrating that improved score estimation enables performance approaching that of the oracle.

This work addresses the limitations of existing conformal prediction methods, which typically guarantee only marginal coverage and struggle to ensure conditional coverage for heterogeneous test points or subpopulations, while lacking a unified theoretical framework to analyze their asymptotic validity, compare approaches, or extend them to structured data. The paper proposes the first unified theoretical framework tailored for conditional coverage, deriving non-asymptotic bounds on conditional miscoverage via pointwise and Lₚ paths. It systematically characterizes the sources of error underlying asymptotic conditional validity and provides a coherent interpretation of existing methods. Built upon a weighted symmetry formulation, the framework facilitates conditional coverage–oriented model selection, localization under covariate shift, and natural extensions to structured data. Numerical experiments corroborate the theoretical findings, establishing a comparable, extensible, and practically informative paradigm for conditional coverage.

This work addresses the challenge of effectively aggregating statistical evidence under unknown dependence structures by proposing a unified framework grounded in permutation invariance. The approach constructs exchangeable data units, aggregates statistics within transformed datasets, and calibrates results across transformations, accommodating single-batch, sequential, and two-stage strategies. By integrating group invariance, exchangeability modeling, sequential alpha-spending, and a decoupling of standardization from calibration, the method achieves high power and adaptivity in finite samples, substantially outperforming traditional calibration techniques such as Bonferroni correction. Empirical evaluations demonstrate that the framework guarantees valid inference under arbitrary dependence structures in tasks including nonparametric testing and conformal prediction, while supporting data-driven aggregation rules and early rejection mechanisms.

This work addresses the limitation of traditional conformal prediction, which guarantees marginal coverage but often fails to achieve valid conditional coverage within subpopulations, while existing evaluation methods suffer from the curse of dimensionality. The authors propose a Conformal Prediction Analysis (CPA) framework that reframes conditional coverage assessment as a supervised learning task by training a reliability estimator to predict instance-level coverage probabilities. They introduce a Conditional Validity Index (CVI) to quantify the local safety and efficiency of conformal predictors. Theoretical analysis establishes the convergence of CVI and proves the consistency of CC-Select, a CVI-based model selection algorithm. Empirical results demonstrate that CPA effectively identifies local coverage failures and that CC-Select reliably selects models with superior conditional coverage.

This work addresses the challenge of maintaining calibration and validity in conformal prediction under non-exchangeable distribution shifts. It presents the first extension of generalized conformal prediction to this setting by introducing observation-specific permutation weights to model distributional shift and constructing a robust prediction envelope via uncertainty sets over these weights, thereby guaranteeing valid coverage either in finite samples or asymptotically. The method integrates weighted permutations, conformal scores, binning, and isotonic distribution regression to achieve computational efficiency. Experiments on covariate shift and feedback-driven biomolecular design tasks demonstrate that the resulting prediction bands adaptively widen with increasing distribution shift and effectively tighten as sample size grows, exhibiting both reliability and adaptivity.