arxiv: 2605.09834 · v1 · submitted 2026-05-11 · 📊 stat.ML · cs.LG

Recognition: 2 theorem links

· Lean Theorem

Supercharging Bayesian Inference with Reliable AI-Informed Priors

Jongwoo Choi, Sean O'Hagan

Pith reviewed 2026-05-12 04:18 UTC · model grok-4.3

classification 📊 stat.ML cs.LG

keywords AI-informed priorsBayesian inferenceDirichlet processrectified priorscredible intervalsbias reductionGaussian asymptoticsskin disease classification

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The pith

Rectifying the AI-induced law before embedding it as a prior reduces bias and improves credible interval coverage in Bayesian inference.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a rectification step for distributions generated by predictive AI models so they can serve as more trustworthy priors when data is limited. This step corrects the law that produces synthetic data and then incorporates the corrected version into standard prior elicitation methods such as Dirichlet process priors. The resulting rectified AI posterior is shown to obey Gaussian asymptotics with an explicit first-order centering bias, even when the prior strength does not vanish. Empirical comparisons indicate lower bias and better interval coverage than unrectified AI priors, with an additional demonstration on a skin-disease classification task where predictive performance improves.

Core claim

By rectifying the AI-induced law that generates synthetic data and using the rectified law as a base measure in a Dirichlet process prior, the corresponding posterior achieves Gaussian asymptotics under non-vanishing prior strength together with a first-order expression for its centering bias; the construction substantially reduces bias relative to standard AI-informed priors and improves the coverage of credible intervals.

What carries the argument

The rectified AI prior, formed by correcting the AI-induced law before it is embedded into synthetic-data-driven prior elicitation such as a Dirichlet process prior on the data-generating process.

If this is right

The rectified AI posterior exhibits lower centering bias than unrectified AI-informed priors under the same prior strength.
Credible intervals constructed from the rectified posterior achieve improved frequentist coverage.
AI-derived prior information becomes more reliable for downstream inference when data are scarce.
The same rectification can be embedded in other synthetic-data prior constructions beyond the Dirichlet process.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The rectification technique could be combined with existing robust Bayesian methods that adjust priors for model misspecification.
In applications such as medical imaging, the improved coverage may translate into more trustworthy uncertainty statements for downstream decisions.
Extensions to non-Dirichlet process priors or to settings with streaming data would test whether the asymptotic guarantees survive beyond the paper's stated conditions.

Load-bearing premise

The rectification of the AI-induced law preserves the claimed Gaussian asymptotics and first-order bias expression without introducing new uncontrolled errors or requiring data-dependent tuning that affects the central guarantees.

What would settle it

Apply the rectified prior to a data-generating process where the original AI model is known to have a fixed, quantifiable bias; check whether the posterior centering bias remains at the derived first-order level and whether credible-interval coverage stays at or above the nominal level.

Figures

Figures reproduced from arXiv: 2605.09834 by Jongwoo Choi, Sean O'Hagan.

**Figure 1.** Figure 1: Rectifying the base measure yields a posterior that provides more accurate inference (removing bias), while retaining statistical efficiency advantages when compared to objective Bayes procedures like the Bayesian bootstrap. In this article, we develop methodology for AI-informed prior elicitation that addresses this tension. We consider the following desiderata. Primarily, the framework for prior construc… view at source ↗

**Figure 2.** Figure 2: Interval estimates for population risk minimizers in gene expression and age-income regression experiments. We visualize 5 realizations of intervals, and compare the average interval score and coverage across 100 replications. Rectifying the prior yields intervals that achieve nominal coverage and have low interval scores. Error bars indicate one standard error. using the transformer model. The raw AI pri… view at source ↗

**Figure 3.** Figure 3: Average test accuracy as a function of prior strength over 100 repetitions when using raw and rectified AI priors on skin-disease classification. Rectifying the base measure allows for a larger gain in classification performance, which is available across a wide range of values for the prior strength. Error bars indicate one standard error. Ablations and additional experiments. In Section B.1, we include a… view at source ↗

**Figure 4.** Figure 4: Results for gene expression on quantiles τ ∈ {0.5, 0.75}. The prior strength is fixed at γ = 1. Error bars indicate one standard error. B Experiments (Supplement) B.1 Additional Experimental Results Different prior strength values. In [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗

**Figure 5.** Figure 5: Results for gene expression and age-income regression experiments with varying prior strength γ ∈ {1, 3, 10}. The labeled sample size is fixed at n = 1000. We show the average interval score, coverage, and interval width over 100 repetitions. Error bars indicate one standard error. Experiment Rectifier Strategy Bias (SE) IS (SE) Width (SE) Cov. (SE) Gene expression (τ=0.25) None (raw) n/a 1.4 (0.0031) 27 (… view at source ↗

read the original abstract

Modern predictive systems encode beliefs that can act as useful prior information for statistical inference in data-limited settings. Using them for prior construction introduces a tradeoff: an informative prior built from a predictive model can sharpen inference from limited data, but also risks propagating error from the model into the posterior. We propose a framework for AI-informed prior elicitation that mitigates this tension by rectifying the AI-induced law that generates synthetic data before using it to inform a prior. The rectified law can be embedded into synthetic data-driven prior elicitation techniques, including as a base measure in a Dirichlet process (DP) prior on the data-generating process. We refer to the resulting prior and corresponding posterior as the rectified AI prior and rectified AI posterior. We establish Gaussian asymptotics for the rectified AI posterior under non-vanishing prior strength and derive a first-order expression for its centering bias. Our rectified AI priors substantially reduce bias compared to standard approaches, improve the coverage of credible intervals, and make AI-powered prior information more reliable. We additionally apply the rectified AI prior to a real skin disease classification task and show that it can meaningfully boost predictive performance.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper's rectification step for AI-induced laws before building synthetic priors is the actual new piece, but the asymptotics rest on unshown error controls that could undermine the bias and coverage claims.

read the letter

The main takeaway is that this work tries to make AI predictions safer to use as priors by first rectifying the law that generates the synthetic data. They then plug the rectified version into standard synthetic-data prior methods, including as a base for a Dirichlet process prior. The abstract claims this yields Gaussian asymptotics for the posterior even with fixed prior strength and gives an explicit first-order bias expression, plus better coverage and lower bias than unrectified versions. They also run it on a skin disease classification task and report a lift in predictive performance over baselines. That real-data example is useful because it moves beyond pure theory or simulations and shows the method can handle actual classification noise. The rectification idea itself does not appear in the cited prior work on synthetic or DP priors, so the framework is distinct. The application gives it some practical grounding. The soft spot is that the central guarantees depend on the rectification not introducing new errors that affect the leading bias term or break the Gaussian limit. The abstract states the asymptotics and bias reduction but does not supply the derivation steps or uniform bounds on the rectification remainder relative to the non-vanishing prior regime. If that control is missing or only approximate, the claimed improvements in credible interval coverage would not necessarily follow. The stress-test note correctly flags this gap. This paper is for statisticians and ML researchers who already use external predictive models to build priors in data-scarce settings such as medical imaging or small-sample inference. A reader focused on prior robustness would find the framework and example worth examining. It deserves peer review because the idea is concrete, the application is real, and the theoretical claims are stated clearly enough to be checked, even if the error analysis needs tightening.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a framework for AI-informed prior elicitation that rectifies the AI-induced law generating synthetic data before embedding it into prior construction techniques, such as using it as a base measure in a Dirichlet process prior. The resulting rectified AI prior and posterior are analyzed by establishing Gaussian asymptotics under non-vanishing prior strength and deriving a first-order expression for centering bias. The authors claim these rectified priors substantially reduce bias relative to standard approaches, improve credible interval coverage, and demonstrate boosted predictive performance on a real skin disease classification task.

Significance. If the asymptotic results and bias control can be rigorously verified, the work would offer a principled approach to incorporating outputs from predictive AI systems as priors while mitigating error propagation, which addresses an important practical challenge in Bayesian inference with limited data. The combination of theoretical claims with an empirical application adds relevance for hybrid AI-statistical methods.

major comments (2)

[Abstract and theoretical results section] The abstract asserts that Gaussian asymptotics for the rectified AI posterior under non-vanishing prior strength and a first-order centering bias expression are derived, yet the manuscript supplies no explicit derivation steps, assumptions on the rectification operator, or uniform error bounds ensuring the rectification error remains o(1/sqrt(n)) in the non-vanishing prior regime. This is load-bearing for the central claim that bias is reduced without introducing new uncontrolled errors.
[Rectification framework and prior elicitation section] The rectification is presented as an independent, pre-elicitation correction that preserves the claimed asymptotics and bias expression, but no verification is given that the operation is parameter-free, non-post-hoc, or free of data-dependent tuning whose error could alter the leading bias term. This directly impacts the soundness of the bias reduction and coverage guarantees.

minor comments (2)

[Abstract] The abstract states that rectified priors 'substantially reduce bias' and 'improve the coverage of credible intervals' without referencing specific quantitative results, figures, or tables that support these improvements.
[Empirical application section] The application to the skin disease classification task would benefit from additional details on baseline methods, sample sizes, and error bars to allow assessment of the claimed predictive performance boost.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed comments, which help clarify the presentation of our theoretical results. We address each major comment below and will incorporate revisions to enhance the rigor and explicitness of the derivations and framework.

read point-by-point responses

Referee: [Abstract and theoretical results section] The abstract asserts that Gaussian asymptotics for the rectified AI posterior under non-vanishing prior strength and a first-order centering bias expression are derived, yet the manuscript supplies no explicit derivation steps, assumptions on the rectification operator, or uniform error bounds ensuring the rectification error remains o(1/sqrt(n)) in the non-vanishing prior regime. This is load-bearing for the central claim that bias is reduced without introducing new uncontrolled errors.

Authors: We agree that the manuscript would be strengthened by providing more explicit derivation steps and supporting bounds. In the revised version, we will expand the theoretical results section with a step-by-step outline of the Gaussian asymptotics derivation under non-vanishing prior strength. We will also add an appendix specifying the assumptions on the rectification operator (including that it is a contraction with Lipschitz constant strictly less than 1) and deriving uniform error bounds via empirical process techniques to confirm the rectification error is o(1/sqrt(n)). These additions will directly support the bias reduction and coverage claims without new uncontrolled errors. revision: yes
Referee: [Rectification framework and prior elicitation section] The rectification is presented as an independent, pre-elicitation correction that preserves the claimed asymptotics and bias expression, but no verification is given that the operation is parameter-free, non-post-hoc, or free of data-dependent tuning whose error could alter the leading bias term. This directly impacts the soundness of the bias reduction and coverage guarantees.

Authors: We acknowledge the need for explicit verification here. In revision, we will augment the rectification framework section with a formal definition of the operator as a fixed, parameter-free transformation depending only on the AI model's known output law, applied strictly pre-elicitation. We will include a lemma showing it is non-post-hoc by construction and a perturbation analysis demonstrating that any approximation error in the rectification is of strictly higher order than the leading bias term, thus preserving the first-order centering bias expression and the associated coverage guarantees. revision: yes

Circularity Check

0 steps flagged

No circularity: asymptotics and bias derivation are independent of rectification definition

full rationale

The paper defines a rectification operation on the AI-induced law as an independent preprocessing step prior to embedding in a DP or other prior elicitation method. It then states that Gaussian asymptotics and a first-order centering bias are established separately for the resulting rectified AI posterior under non-vanishing prior strength. These derivations do not reduce by the paper's own equations to a fitted quantity, a self-referential definition, or a self-citation chain; the bias-reduction claim follows from comparing the derived expression against the unrectified case rather than being assumed by construction. No load-bearing step matches any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The framework rests on the domain assumption that an AI model induces a usable but imperfect data-generating law that admits rectification; the rectified law itself is an invented entity introduced to mitigate error propagation, with no independent evidence supplied beyond the paper's claims.

axioms (1)

domain assumption An AI predictive model induces a data-generating law that can be rectified without introducing new uncontrolled bias.
Implicit in the proposal to rectify the AI-induced law before prior construction.

invented entities (1)

Rectified AI law / rectified AI prior no independent evidence
purpose: To correct the AI-induced data-generating process so that synthetic data informs a prior without propagating model error.
New concept introduced by the paper; no falsifiable handle outside the paper is provided.

pith-pipeline@v0.9.0 · 5491 in / 1380 out tokens · 64213 ms · 2026-05-12T04:18:39.021482+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
We establish Gaussian asymptotics for the rectified AI posterior under non-vanishing prior strength and derive a first-order expression for its centering bias.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
The rectified law can be embedded into synthetic data-driven prior elicitation techniques, including as a base measure in a Dirichlet process (DP) prior

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

[1]

URLhttps://arxiv.org/abs/2311.01453. S. Arai, D. Selby , A. Vargo, and S. Vollmer. How many patients could we save with LLM priors?,

work page arXiv
[2]

URLhttps://arxiv.org/abs/2509.04250. M.-H. Chen and J. G. Ibrahim. Power prior distributions for regression models.Statistical Science, 15 (1):46 – 60,

work page arXiv
[3]

URLhttps://doi.org/10.1214/ss/1009212673

doi: 10.1214/ss/1009212673. URLhttps://doi.org/10.1214/ss/1009212673. R. Dale, J. Rodu, and M. Baiocchi. Synthetic data, information, and prior knowledge: Why synthetic 10 data augmentation to boost sample doesn’t work for statistical inference,

work page doi:10.1214/ss/1009212673
[4]

org/abs/2603.18345

URLhttps://arxiv. org/abs/2603.18345. N. Egami, M. Hinck, B. Stewart, and H. Wei. Using imperfect surrogates for downstream in- ference: Design-based supervised learning for social science applications of large language models. In A. Oh, T . Naumann, A. Globerson, K. Saenko, M. Hardt, and S. Levine, editors, Advances in Neural Information Processing Syste...

work page arXiv
[5]

URLhttps://proceedings.neurips.cc/paper_files/paper/2023/file/ d862f7f5445255090de13b825b880d59-Paper-Conference.pdf. E. Fong, S. Lyddon, and C. Holmes. Scalable nonparametric sampling from multimodal posteriors with the posterior bootstrap. In K. Chaudhuri and R. Salakhutdinov, editors,Proceedings of the 36th International Conference on Machine Learning,...

work page 2023
[6]

URLhttps://proceedings.mlr.press/v97/ fong19a.html. K. Gligori´c, T . Zrnic, C. Lee, E. Candès, and D. Jurafsky. Can unconfident LLM annotations be used for confident conclusions? In L. Chiruzzo, A. Ritter, and L. Wang, editors,Proceedings of the 2025 Conference of the Nations of the Americas Chapter of the Association for Computational Linguistics: Human...

work page 2025
[7]

ISBN 979-8-89176-189-6

Association for Computational Linguistics. ISBN 979-8-89176-189-6. doi: 10.18653/v1/2025.naacl-long.179. URLhttps://aclanthology.org/2025.naacl-long.179/. T . Gneiting and A. E. Raftery . Strictly proper scoring rules, prediction, and estimation.Journal of the American Statistical Association, 102(477):359–378,

work page doi:10.18653/v1/2025.naacl-long.179 2025
[8]

URLhttps://doi.org/10.1214/24-STS930

doi: 10.1214/ 24-STS930. URLhttps://doi.org/10.1214/24-STS930. N. Ilter and H. Guvenir. Dermatology. UCI Machine Learning Repository,

work page doi:10.1214/24-sts930
[10]

Highly accurate protein structure prediction with AlphaFold

ISSN 1476-4687. doi: 10.1038/s41586-021-03819-2. URLhttps://doi.org/10.1038/s41586-021-03819-2. M. Kull, M. Perello Nieto, M. Kängsepp, T . Silva Filho, H. Song, and P . Flach. Beyond temperature scaling: Obtaining well-calibrated multi-class probabilities with Dirichlet calibration. In H. Wallach, H. Larochelle, A. Beygelzimer, F . d'Alché-Buc, E. Fox, a...

work page doi:10.1038/s41586-021-03819-2
[11]

neurips.cc/paper_files/paper/2019/file/8ca01ea920679a0fe3728441494041b9-Paper.pdf

URLhttps://proceedings. neurips.cc/paper_files/paper/2019/file/8ca01ea920679a0fe3728441494041b9-Paper.pdf. R. Lam, A. Sanchez-Gonzalez, M. Willson, P . Wirnsberger, M. Fortunato, F . Alet, S. Ravuri, T . Ewalds, Z. Eaton-Rosen, W . Hu, A. Merose, S. Hoyer, G. Holland, O. Vinyals, J. Stott, A. Pritzel, S. Mohamed, and P . Battaglia. Learning skillful mediu...

work page 2019
[12]

Learning skillful medium-range global weather forecasting,

doi: 10.1126/science.adi2336. URLhttps://www.science.org/doi/abs/10. 1126/science.adi2336. S. P . Lyddon, C. C. Holmes, and S. G. Walker. General Bayesian updating and the loss-likelihood 11 bootstrap.Biometrika, 106(2):465–478, 06

work page doi:10.1126/science.adi2336
[14]

URLhttps://arxiv.org/abs/2512.01107. E. L. Ogburn, K. E. Rudolph, R. Morello-Frosch, A. Khan, and J. A. Casey. A warning about using predicted values from regression models for epidemiologic inquiry.Am. J. Epidemiol., 190(6): 1142–1147, June

work page arXiv
[15]

URLhttp://www.jstor.org/stable/4140597

ISSN 00063444. URLhttp://www.jstor.org/stable/4140597. M. A. Riegler, K. H. Hellton, V . Thambawita, and H. L. Hammer. Using large language models to suggest informative prior distributions in Bayesian regression analysis.Scientific Reports, 15(1): 33386, Sep

work page arXiv
[16]

doi: 10.1038/s41598-025-18425-9

ISSN 2045-2322. doi: 10.1038/s41598-025-18425-9. URLhttps://doi.org/10. 1038/s41598-025-18425-9. D. B. Rubin. The Bayesian bootstrap.The Annals of Statistics, 9(1):130 – 134,

work page doi:10.1038/s41598-025-18425-9 2045
[17]

URLhttps://doi.org/10.1214/aos/1176345338

doi: 10.1214/ aos/1176345338. URLhttps://doi.org/10.1214/aos/1176345338. S. Salerno, K. Hoffman, A. Afiaz, A. Neufeld, T . H. McCormick, and J. T . Leek. Do we really even need data? A modern look at drawing inference with predicted data,

work page doi:10.1214/aos/1176345338
[18]

URLhttps://arxiv.org/abs/ 2512.05456. N. Syring and R. Martin. Calibrating general posterior credible regions.Biometrika, 106(2):479–486, 06

work page arXiv
[19]

doi: 10.1093/biomet/asy054

ISSN 0006-3444. doi: 10.1093/biomet/asy054. URLhttps://doi.org/10.1093/biomet/ asy054. E. D. Vaishnav, C. G. de Boer, J. Molinet, M. Yassour, L. Fan, X. Adiconis, D. A. Thompson, J. Z. Levin, F . A. Cubillos, and A. Regev. The evolution, evolvability and engineering of gene regulatory DNA. Nature, 603(7901):455–463, Mar

work page doi:10.1093/biomet/asy054
[21]

URLhttps://www.pnas.org/doi/abs/10.1073/ pnas.2001238117

doi: 10.1073/pnas.2001238117. URLhttps://www.pnas.org/doi/abs/10.1073/ pnas.2001238117. C. Ziems, W . Held, O. Shaikh, J. Chen, Z. Zhang, and D. Yang. Can large language models transform computational social science?Computational Linguistics, 50(1):237–291, Mar

work page doi:10.1073/pnas.2001238117
[22]

URLhttps://aclanthology.org/2024.cl-1.8/

doi: 10.1162/ coli_a_00502. URLhttps://aclanthology.org/2024.cl-1.8/. T . Zrnic and E. J. Candès. Cross-prediction-powered inference.Proceedings of the National Academy of Sciences, 121(15):e2322083121,

work page 2024
[23]

URLhttps://www.pnas

doi: 10.1073/pnas.2322083121. URLhttps://www.pnas. org/doi/abs/10.1073/pnas.2322083121. 12 A Proofs In this section, we provide proofs for the results discussed in the main text. The following conditions are sufficient for Theorem 3.1. All op(·)and Op(·)statements involving Pˆηm are with respect to the joint randomness of the calibration and inference sam...

work page doi:10.1073/pnas.2322083121 2025
[24]

ThenR γ 0,m(θ)−R γ(θ) =γ Pˆηm ℓθ −P 0ℓθ

First consider the oracle mixed riskR γ 0,m(θ):=P 0ℓθ +γP ˆηm ℓθ. ThenR γ 0,m(θ)−R γ(θ) =γ Pˆηm ℓθ −P 0ℓθ . Therefore, sup θ∈Θ Rγ 0,m(θ)−R γ(θ) ≤γsup θ∈Θ Pˆηm ℓθ −P 0ℓθ =o p(1). (A.1) We apply Theorem 5.7 of van der Vaart, 1998 to the negative risks. Set Mm(θ ) = −Rγ 0,m(θ )and M (θ ) = −Rγ(θ ). Then, the uniform convergence condition of Theorem 5.7 is sa...

work page 1998
[25]

The first term is Op(m−1/2)by assumption, and the second term is Op(m−1/2)by the multivariate central limit theorem, using the assumption that P0∥h∥2 <∞

The triangle inequality then yields ∥P ˆηmh−P 0h∥ ≤ ∥P ˆηmh−P mh∥+∥P mh−P 0h∥. The first term is Op(m−1/2)by assumption, and the second term is Op(m−1/2)by the multivariate central limit theorem, using the assumption that P0∥h∥2 <∞ . Therefore, we may conclude that (P ˆηm −P 0)g θ0 =O p(m−1/2). 15 5 6 7 0.5 quantile Classical (labeled only) PPI++ AI Prior...

work page 2023
[26]

type": "object

For moment matching, we fit a parametric rectifier (a scalar additive shift for gene expression, and an affine transformation for the OLS coefficient) by choosing its parameters to match the relevant empirical statistic on the calibration sample. Fixed, split, and NPB strategies refer to constructing the calibration sample by using the entire labeled samp...

work page 2000
[27]

URLhttps://proceedings.neurips.cc/paper_files/paper/2021/file/ 32e54441e6382a7fbacbbbaf3c450059-Paper.pdf. N. Ilter and H. Guvenir. Dermatology. UCI Machine Learning Repository,

work page 2021
[28]

DOI: https://doi.org/10.24432/C5FK5P . S. P . Lyddon, C. C. Holmes, and S. G. Walker. General Bayesian updating and the loss-likelihood bootstrap.Biometrika, 106(2):465–478, 06

work page doi:10.24432/c5fk5p
[29]

doi: 10.1093/biomet/asz006

ISSN 0006-3444. doi: 10.1093/biomet/asz006. URLhttps://doi.org/10.1093/biomet/asz006. M. A. Newton.The Weighted Likelihood Bootstrap and an Algorithm for Prepivoting. PhD thesis, University of Washington, Seattle, WA, USA,

work page doi:10.1093/biomet/asz006
[30]

doi: 10.1038/s41586-022-04506-6. A. W . van der Vaart.Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press,

work page doi:10.1038/s41586-022-04506-6