Conformal Prediction = Bayes?

TL;DR

This paper critically examines conformal prediction, revealing fundamental limitations and distinctions from Bayesian methods, especially regarding calibration, conditionality, and sequential inference, challenging its interpretation as a form of Bayesian conditioning.

Contribution

The paper provides a rigorous theoretical analysis distinguishing conformal prediction from Bayesian conditioning, highlighting its violations of conditional extensionality and issues with sequential updating.

Findings

Conformal prediction sets depend on marginal design even with fixed conditional distributions.

Finitely additive extensions preserving calibration are nonconglomerable, leading to vulnerabilities.

Rank-calibrated updates cannot be realized as regular conditionals of any countably additive exchangeable law.

Abstract

Conformal prediction (CP) is widely presented as distribution-free predictive inference with finite-sample marginal coverage under exchangeability. We argue that CP is best understood as a rank-calibrated descendant of the Fisher-Dempster-Hill fiducial/direct-probability tradition rather than as Bayesian conditioning in disguise. We establish four separations from coherent countably additive predictive semantics. First, canonical conformal constructions violate conditional extensionality: prediction sets can depend on the marginal design P(X) even when P(Y|X) is fixed. Second, any finitely additive sequential extension preserving rank calibration is nonconglomerable, implying countable Dutch-book vulnerabilities. Third, rank-calibrated updates cannot be realized as regular conditionals of any countably additive exchangeable law on Y^infty. Fourth, formalizing both paradigms as families of one-step predictive kernels, conformal and Bayesian kernels coincide only on a Baire-meagre subset of the space of predictive laws. We further show that rank- and proxy-based reductions are generically Blackwell-deficient relative to full-data experiments, yielding positive Le Cam deficiency for suitable losses. Extending the analysis to prediction-powered inference (PPI) yields an analogous message: bias-corrected, proxy-rectified estimators can be valid as confidence devices while failing to define transportable belief states across stages, shifts, or adaptive selection. Together, the results sharpen a general limitation of wrappers: finite-sample calibration guarantees do not by themselves supply composable semantics for sequential updating or downstream decision-making.