FAB-PPI: Frequentist, Assisted by Bayes, Prediction-Powered Inference

Fran\c{c}ois Caron; Stefano Cortinovis

REVIEW 3 major objections 4 minor 14 references

FAB-PPI: Frequentist, Assisted by Bayes, Prediction-Powered Inference

T0 review · 3 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read FAB-PPI: a prior on prediction quality shortens valid prediction-powered confidence intervals.

desk verdict FAB-PPI is a genuinely new and mostly sound extension of prediction-powered inference; the main fix needed is to align the abstract's coverage claim with the asymptotic, plug-in-scale reality. read the letter →

arxiv 2502.02363 v2 pith:5ZAHC3TL submitted 2025-02-04 stat.ML cs.LG

Stefano Cortinovis , François Caron This is my paper · ORCID

classification stat.MLcs.LG MSC 62F2562F1562G20

keywords prediction-poweredinferencefrequentist-assistedbyBayesFABconfidenceregionshorseshoepriorsemi-supervisedintervalsmachinelearningpredictions

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

FAB-PPI is a method for prediction-powered inference (PPI) that treats the quality of machine-learning predictions as something we have prior beliefs about, and then uses those beliefs to build confidence intervals that are still valid in the frequentist sense. The paper's central claim is that by placing a horseshoe prior on the rectifier — the expected gap between the predictions and the truth used to correct PPI — the resulting intervals are shorter than standard PPI intervals when predictions are good, and revert to standard PPI intervals when predictions are bad. The authors prove asymptotic coverage at level $1-\alpha$, show the reversion-to-PPI property formally, and prove that the mean estimator is consistent under the horseshoe prior but not under a Gaussian prior. A sympathetic reader would care because this gives a parameter-free way to make PPI adaptive to prediction quality without sacrificing its guarantees.

What carries the argument

The load-bearing object is the FAB confidence region, which for a scalar normal observation $W\sim N(\beta,\sigma^2)$ solves for the region with exact $(1-\delta)$ coverage that minimises expected volume under a prior $\pi_0(\beta)$; by the Neyman-Pearson lemma it is the acceptance region of a most powerful test, computable from the marginal likelihood. FAB-PPI applies this to the rectifier estimator $\widehat{\Delta}_\theta$. The horseshoe prior, a scale mixture of normals with a half-Cauchy on the scale, contributes an infinitely tall spike at zero (strong shrinkage when predictions are good) and power-law tails (so the region reverts to the classical interval when $|\widehat{\Delta}_\theta|$ is large). Tweedie's formula turns the derivative of the log-marginal likelihood into a Bayes-assisted point estimator $\widehat{\Delta}_\theta+\widehat\sigma_\theta^2\ell'(\widehat{\Delta}_\theta;\widehat\sigma_\theta,\tau_n)$.

What would settle it

Run the biased-predictor simulation from the paper with $n=N=10^4$ and a rectifier several standard errors from zero; if the horseshoe FAB-PPI interval's endpoints do not match the PPI interval's endpoints to within Monte Carlo error, the reversion property fails. Conversely, if coverage is below $1-\alpha$ in a setting where Assumption 3.1 is satisfied, the coverage theorem fails.

Watch

Extended reading notes

Core claim

Under Assumption 3.1, FAB-PPI constructs confidence regions for $\theta^\star$ by combining a standard confidence interval for the fit term $m_\theta$ with a FAB confidence region for the rectifier $\Delta_\theta$, where the FAB region is chosen to minimise expected volume under a prior whose scale is the estimated standard error. Theorem 4.1 states that these regions have correct asymptotic coverage for both the Gaussian and horseshoe priors. Proposition 4.2 states that under the horseshoe prior, when the estimated rectifier is large, the FAB-PPI region converges to the standard CLT-based PPI region; the Gaussian version instead grows without bound. Proposition 4.3 states that the FAB-PPI mean estimator is consistent under the horseshoe prior and inconsistent under the Gaussian prior. The paper presents these results as the theoretical basis for using the horseshoe as the default prior in FAB-PPI.

Load-bearing premise

The load-bearing premise is Assumption 3.1: the estimators of the fit term and the rectifier are asymptotically normal with consistently estimated variances; if n and N are too small or the variance estimators are poor, the coverage and efficiency guarantees have no support.

Editorial extensions

If this is right

FAB-PPI intervals are shorter than standard PPI intervals when the predictor is accurate, with the same nominal coverage, as shown on the AlphaFold, Galaxies, Forest, and Healthcare datasets.
If the predictor is heavily biased, the horseshoe version stops shrinking and matches standard PPI, while the Gaussian version's interval grows without bound; this makes horseshoe FAB-PPI robust to prior misspecification.
The mean estimator under the horseshoe prior is consistent as $n,N\to\infty$; under the Gaussian prior it is not, so the horseshoe is the theoretically supported default.
FAB-PPI can be combined with power tuning (PPI++) and extends to multivariate parameters through independent priors and a union bound.
The method inherits PPI's computational requirement of a grid search over $\theta$ in general convex problems, except in special cases like mean estimation and linear regression.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper leaves implicit that the horseshoe's reversion property makes FAB-PPI a shrinkage estimator whose effective prior influence automatically vanishes when data disagree; this suggests it could serve as a general adaptive bias-correction layer for semi-supervised estimators beyond PPI.
A testable extension is to replace the horseshoe with other power-law-tailed scale mixtures (e.g. beta-prime priors) and check whether the same coverage, shrinkage, and reversion properties hold with cheaper computation.
The prior scale $\tau_n=\widehat\sigma_\theta$ is justified asymptotically; in small samples, users may need to calibrate the scale by simulation, since the coverage theorem does not quantify the finite-sample error.
Although the paper focuses on mean and M-estimation, the same construction applies to any parameter defined by a convex loss whose subgradient admits the $m_\theta+\Delta_\theta$ decomposition, so the result should transfer to quantile and logistic objectives.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The core idea here is new and worth engaging with: wrap the PPI rectifier in a FAB confidence region with a horseshoe prior, so good predictions give shorter intervals and bad predictions revert to standard PPI. The theory is the main event and it largely holds up. Theorem 4.1's asymptotic coverage follows by Boole's inequality from CLT assumptions, Lemma S3.2 cleanly shows the horseshoe's reversion to the classical z-interval, and Proposition 4.3 nicely exposes why the Gaussian prior is inconsistent for mean estimation. The appendix does the work; I checked the steps and they are coherent. The parameter-free default (prior scale equal to the estimated standard error) is pragmatic, and the code being public is a plus.

The soft spot is exactly what the stress-test note flags. The abstract and introduction say FAB-PPI has 'correct coverage for any choice of prior.' That is true in the abstract FAB setup with fixed prior and known scale, but here the prior scale is the estimated standard error. Coverage is asymptotic only, and the small-n regime that motivates PPI has no finite-sample control. The paper should replace 'correct coverage' with 'asymptotically correct coverage' and state explicitly that the data-dependent prior scale is justified by the CLT and variance consistency. This is a presentation flaw, not a mathematical one, but it matters because readers will take the abstract at face value.

The experiments are adequate but not polished: averages over 1000 repetitions with no error bars, and some coverage plots (e.g., forest at small n) dip below nominal without comment. Since the method's selling point is shorter intervals at a given coverage, seeing the spread of coverage across repetitions would help. The real-data results do show the expected pattern, and the biased-predictions simulation clearly demonstrates the horseshoe's robustness advantage over the Gaussian.

Overall, this is a serious contribution from people who know the FAB and PPI literatures. The theory is solid, the prior contrast is instructive, and the method is likely to be used. It deserves a careful referee round, with the main requests being an honest coverage statement and a bit more transparency in the experiments.

My recommendation: send it to review, and expect it to be accepted after those revisions.

Referee Report

3 major / 4 minor

Summary. The paper proposes FAB-PPI, a modification of prediction-powered inference (PPI) that constructs confidence regions for the rectifier Δθ using the frequentist-assisted-by-Bayes (FAB) framework with a prior on Δθ. The method replaces the standard CLT-based confidence interval for Δθ in the PPI procedure with an FAB confidence region, and the PPI estimator with a Tweedie-style posterior-mean shrinkage estimator. The authors consider a Gaussian prior and a horseshoe prior, with the prior scale set to the estimated standard error of the rectifier estimator. They prove asymptotic coverage of the resulting PPI confidence region under a CLT assumption (Theorem 4.1), a reversion-to-PPI property under the horseshoe prior for large estimated rectifiers (Proposition 4.2), and consistency of the horseshoe-based mean estimator as opposed to the Gaussian-based one (Proposition 4.3). Experiments on synthetic and real data show that FAB-PPI with the horseshoe prior can produce shorter confidence intervals than PPI when predictions are good, while matching PPI when predictions are poor.

Significance. If the theoretical and empirical claims hold, FAB-PPI is a practically useful extension of PPI: it provides a principled way to shrink the rectifier when predictions are accurate, with an automatic fallback to standard PPI under heavy-tailed priors. The paper's main strengths are its clear derivation, the closed-form marginal likelihood for the horseshoe prior, the modularity with PPI and PPI++, the formal asymptotic proofs in the appendix, and the released code. The conceptual contribution—using the FAB framework to inject prior knowledge about prediction quality into PPI—is novel and well-motivated. However, the practical significance is tempered by the fact that the central coverage guarantee is asymptotic, while the method is explicitly motivated by settings with scarce labelled data. The experiments do not directly address the small-sample regime where the CLT and variance-estimator consistency might fail, and they omit error bars on the reported performance metrics.

major comments (3)

[§4.3, Theorem 4.1, and Abstract] The prior scale is set to the estimated standard error, τ_n = bσ_θ (Section 4.3), making the prior data-dependent. The exact-coverage property of the FAB construction described in Section 3.3 (Definition 3.2) applies only to a fixed prior and known scale σ. Theorem 4.1 therefore establishes only asymptotic coverage, under Assumption 3.1's CLT and almost-sure variance-estimator consistency. The abstract and the bullet list in Section 1, however, state that FAB-PPI 'maintains its frequentist guarantees' and has 'correct coverage for any choice of prior' without the asymptotic qualifier. Since the method is motivated by scarce labelled data (small n), the gap between the asymptotic guarantee and the advertised regime is load-bearing. The authors should either provide finite-sample coverage bounds or a Berry–Esseen-type refinement, or clearly restrict all coverage claims to the asymptotic regime and add diagnostics for checking Assumption 3.1 in practice.
[§5, Figures 1–3 and S6–S13] The experimental evaluation reports only point estimates of MSE, interval volume, and coverage, averaged over 1000 (or, in Figure S10, 100) repetitions, with no standard errors or confidence intervals. Coverage is a random quantity over simulation repetitions; without error bars or a table of worst-case coverages, the plots cannot distinguish nominal coverage from meaningful undercoverage, especially at the smaller sample sizes (n = 50, 100, 200) that are central to the method's motivation. Furthermore, no experiment probes a setting where Assumption 3.1 is questionable—e.g., where n is too small for the sample-variance estimators to be accurate or where the rectifier distribution is skewed or heavy-tailed—so the finite-sample validity claim is not empirically supported. The authors should report standard errors or confidence bands for all performance metrics and include at least one small-n or non-CLT scenario.
[§4.5 and Proposition 4.3] Proposition 4.3 states that the FAB-PPI mean estimator under the horseshoe prior is consistent while the Gaussian-prior estimator is not. The proof in Appendix S3.3 relies on the assertion that ℓ'_HS(y; 1, 1) is bounded and on the almost-sure convergence of (bσ^PP)^2 ℓ'(bΔ^PP; bσ^PP, bσ^PP) to 0. While the boundedness is plausible from the asymptotic expansion in S1.1, it is not demonstrated in the main text or in the proof itself, and the Gaussian-prior inconsistency is shown only under the specific choice τ_n = bσ_θ. The paper would be strengthened by stating the precise regularity conditions on the prior and the estimator needed for the limit (S41)–(S42) to hold uniformly, rather than as a one-line assertion.

minor comments (4)

[Equation (22)] The notation N(Δθ; 0, bσθ) in Equation (22) is inconsistent with the variance-parameter convention used elsewhere in the paper (e.g., Section 3.3 writes N(β, σ²)). Please clarify whether bσθ denotes a standard deviation or a variance; the downstream formulas in S3.3 imply the prior variance is bσθ².
[Figure S10 caption] Figure S10 reports results 'over 100 repetitions' while all other experimental figures use 1000 repetitions; the authors should state why this figure uses fewer repetitions or increase the number to match the rest of the paper.
[Section 5.1, biased predictions] The biased-predictions experiment assumes N is infinite; while this simplifies the analysis of the rectifier, it is an idealization that is not achievable in practice. A brief remark on how the results would change for finite N would improve the interpretation.
[Section 4.4, Proposition 4.2] The proposition is labeled 'informal' in the main text and the formal statement appears only in Appendix S3.2. Since the formal result is available, the authors should state the proposition with its precise assumptions (e.g., uniform convergence of the spending function) directly in the main text, or at least give a precise reference to Corollary S3.5 and its conditions.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: coverage and robustness are proved from the CLT assumption and independent FAB theorems; the self-citations are to a theorem paper whose assumptions do not include PPI.

full rationale

FAB-PPI composes PPI's CLT-based inference with the FAB confidence-region construction. The central claims are derived, not assumed: Theorem S3.3 derives asymptotic coverage from Assumption 3.1 and Lemmas S3.1/S3.2, which show that plugging the consistent variance estimator bσθ into the FAB region preserves coverage for Gaussian and horseshoe priors; Proposition 4.3 is a direct Tweedie-formula calculation; Proposition 4.2 is imported from Cortinovis & Caron (2024), a theorem paper about general FAB regions whose stated assumptions do not involve PPI, so it counts as independent support rather than circular self-citation. The data-dependent prior scale tau_n = bσθ is an acknowledged plug-in, and the paper claims only asymptotic coverage (Theorem 4.1), explicitly noting the Gaussian-prior estimator is inconsistent; the finite-sample/small-n gap identified by the skeptic is a scope limitation, not a circular derivation. No equation defines a predicted quantity in terms of the quantity it claims to predict, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The method introduces no new entities. Its main free parameter is the prior scale, set to the estimated rectifier standard error. The horseshoe prior, FAB construction, and CLT assumptions all come from the literature or from standard statistical modeling. The central derivations rely on asymptotic normality of the estimators and on previously established properties of FAB confidence regions.

free parameters (2)

Prior scale tau_n = bsigma_theta (estimated standard error of the rectifier)
The prior scale is set equal to the estimated standard error bsigma_theta, as recommended after experimental comparison in Section 4.3. It controls how much the horseshoe shrinks the rectifier estimate and is data-dependent.
Error split delta in experiments = delta = alpha
In the experiments, uncertainty in m_theta is ignored by setting delta = alpha after checking that N is large enough. This is an experimental simplification, not a general method parameter.

assumptions (5)

domain assumption Assumption 3.1: CLT for sample-mean estimators bm_theta and bDelta_theta with variance estimators converging almost surely
Invoked in Section 4 and in Lemmas S3.1 and S3.2 to establish asymptotic coverage, reversion to PPI, and consistency. It requires n and N large enough and N >> n.
domain assumption iid sampling of labelled and unlabelled data, with independence between the two samples
Standard PPI setup stated in Section 1. Needed for the sample mean estimators and the CLT.
domain assumption Convex loss and subgradient representation g_theta* = E[L'_theta*(X,Y)] = 0 (Equation 2)
Borrowed from PPI; defines the parameter of interest and the estimating equation used throughout.
standard math Properties of FAB spending function for power-law-tail priors from Cortinovis and Caron (2024)
Used in Lemma S3.2 and Proposition 4.2 to show reversion to the classical interval. The result is cited from prior work and not re-derived in this paper.
standard math Existence, uniqueness, and interval shape of FAB-CR (Cortinovis and Caron 2024, Theorem 3.3)
Needed to define the FAB confidence region RFABPP and to use interval bounds in Algorithm 2.

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Cite this review

Pith. "Pith review of FAB-PPI: Frequentist, Assisted by Bayes, Prediction-Powered Inference." pith.science (2026). https://pith.science/paper/5ZAHC3TL

@misc{pith2026250202363,
  author       = {Pith},
  title        = {Pith review of: FAB-PPI: Frequentist, Assisted by Bayes, Prediction-Powered Inference},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5ZAHC3TL}},
  note         = {Machine review of arXiv:2502.02363}
}

read the original abstract

Prediction-powered inference (PPI) enables valid statistical inference by combining experimental data with machine learning predictions. When a sufficient number of high-quality predictions is available, PPI results in more accurate estimates and tighter confidence intervals than traditional methods. In this paper, we propose to inform the PPI framework with prior knowledge on the quality of the predictions. The resulting method, which we call frequentist, assisted by Bayes, PPI (FAB-PPI), improves over PPI when the observed prediction quality is likely under the prior, while maintaining its frequentist guarantees. Furthermore, when using heavy-tailed priors, FAB-PPI adaptively reverts to standard PPI in low prior probability regions. We demonstrate the benefits of FAB-PPI in real and synthetic examples.

Figures

Figures reproduced from arXiv: 2502.02363 by the authors.

**Figure 1.** Biased predictions study. The panel shows the average CI volume as the bias level γ varies. ods, as well as MSE and coverage plots, are reported in Figure S6. Except for the version with the Gaussian prior, all the PPI procedures outperform classical inference for every bias level γ, but the behaviour exhibited by PPI++ deserves attention, as its CI volume is approximately constant across values of γ. This is due t… view at source ↗

**Figure 2.** Noisy predictions study. The left, middle and right panels show the average CI volume for noise levels σY = 0.1, 1, 2. case, the effect of power tuning matches the observations of Angelopoulos et al. (2023b): as the noise level increases, λˆ decreases and less weight is given to the predicted labels. When the noise is small, all PPI procedures perform similarly, and much better than classical inference. When the no… view at source ↗

**Figure 3.** Real data mean estimation study. The left, middle, and right panels correspond to the ALPHAFOLD, GALAXIES, and FOREST datasets. The top, middle, and bottom rows show average MSE, CI volume, and CI coverage over 1000 repetitions for α = 0.1. sidered (panel (a)), FAB-PPI underperforms the alternatives under both priors for small n. However, as n increases, the performance under the horseshoe prior improves and eventu… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

14 extracted references · 11 canonical work pages

[1]

The marginal likelihood is given by π(y) = Z ∞ −∞ N (y | β, σ2)πHS(β)dβ = 1√ 2πσ 2 Z ∞ 0 1√ 1 + ν2 e − y2 2σ2 (1+ν2 ) p(ν)dν = 2 π √ 2πσ 2 Z ∞ 0 e − y2 2σ2 (1+ν2 ) 1 (1 + ν2)3/2 dν

The rationale for this choice, along with a discussion of the general case η ̸= 1, is provided at the end of this section. The marginal likelihood is given by π(y) = Z ∞ −∞ N (y | β, σ2)πHS(β)dβ = 1√ 2πσ 2 Z ∞ 0 1√ 1 + ν2 e − y2 2σ2 (1+ν2 ) p(ν)dν = 2 π √ 2πσ 2 Z ∞ 0 e − y2 2σ2 (1+ν2 ) 1 (1 + ν2)3/2 dν. Using the change of variable u = 1 1+ν2 , we obtain ...

work page 2011
[3]

Forest. The FOREST dataset contains the following features for N = 1 596 parcels of land in the Amazon rainforest examined during field visits (Bullock et al., 2020): whether the parcel has been subject to deforestation (Yi ∈ {0, 1}) and the prediction of a gradient-boosted tree model for the probability of Yi being equal to one (f (Xi) ∈ [0, 1]). The goa...

work page 2020
[6]

Using the asymptotic expansion (Slater, 1960, Chapter 4, Eq

is given by κ(y) = 2 3 1F1 2, 5 2 , − y2 2σ2 1F1 1, 3 2 , − y2 2σ2 . Using the asymptotic expansion (Slater, 1960, Chapter 4, Eq. (4.I.3)) 1F1(a, b,−z) ∼ z−a Γ(b) Γ(b − a) as z → ∞, we find κ(y) ∼ 2σ2 y2 |E[β | y] − y| = σ2|ℓ′(y)| ∼2σ2 |y| as |y| → ∞. The horseshoe prior πHS has two key properties: an infinite spike at zero, inducing strong shrinkage near...

work page 1960
[7]

While Kummer’s 1F1 function is implemented in many standard scientific libraries, such as SciPy, Φ1 is not

For a general η, similar expressions can be derived for the marginal likelihood and posterior mean, replacing Kummer’s 1F1 function with the more general degenerate hypergeometric function of two variables, Φ1 (see (Carvalho et al., 2010, Equations (4) in the main text and (A1) in the appendix)). While Kummer’s 1F1 function is implemented in many standard...

work page 2024
[8]

Additionally, if the prior π0(β) := π0(β; σ) admits σ as a scale parameter, writing wα(β; σ) for the corresponding tail function, we have wα(β; σ) = wα β σ ; 1

priors when σ2 = 1and α = 0.1. Additionally, if the prior π0(β) := π0(β; σ) admits σ as a scale parameter, writing wα(β; σ) for the corresponding tail function, we have wα(β; σ) = wα β σ ; 1 . (S31) We now describe other properties of the spending function in the case of a Gaussian prior and of a prior with power-law tails, such as the horseshoe. Proposit...

work page 1963
[9]

Let π0(β; σ) be a symmetric prior on β such that the marginal density π(y) has power-law tails, i.e

Proposition S1.3 (FAB with a prior with power-law tails (Cortinovis & Caron, 2024, Lemma S1.1)). Let π0(β; σ) be a symmetric prior on β such that the marginal density π(y) has power-law tails, i.e. π(y) ∼ Cσ|y|−δ as |y| → ∞ for some constant Cσ and some exponent δ >1. Then, wα(β; σ) is bounded away from 0 and 1, and lim β→∞ wα(β) = lim β→−∞ wα(β) = 1 2 . ...

work page 2024
[10]

With the above two lemmas, we can now prove Theorem 4.1, which we restate here in extended form

Instead, by showing continuity of the mapping b∆θ/bσθ 7→ Cn,N ( b∆θ/bσθ) and again exploiting the CLT, one may prove that Cn,N converges in distribution to the random FAB confidence regionC(Y ) − Y , where C(y) is defined in Equation (S29), for a unit scale and a horseshoe prior, and Y ∼ N(0, 1). With the above two lemmas, we can now prove Theorem 4.1, wh...

work page 2024
[11]

Therefore, (S41) and (S42) hold almost surely by sandwiching

is bounded. Therefore, (S41) and (S42) hold almost surely by sandwiching. Under the Gaussian prior (22), ℓ′ N(y; σ, σ) = − y 2σ2 . Hence, since b∆PP → ∆ and b∆PP+ → ∆ almost surely, where we recall that ∆ = E[f (X) − Y ], we obtain (bσPP)2ℓ′ N b∆PP; bσPP, bσPP → −∆ 2 (bσPP+)2ℓ′ N b∆PP+; bσPP+, bσPP → −∆ 2 almost surely as min(n, N) → ∞, which implies that...

work page 2023

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The goal is to estimate the fraction of galaxies with spiral arms, i.e

for the probability of Yi being equal to one (f (Xi) ∈ [0, 1]). The goal is to estimate the fraction of galaxies with spiral arms, i.e. θ⋆ = E[Y ]. Genes. The GENES dataset contains the following features for N = 61 150 gene promoter sequences: the expression level of the gene...

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Mean estimation experiment on the FOREST dataset with varying horseshoe prior scale

classical PPI++ F AB-PPI++ (ˆσ) F AB-PPI++ (1/√ n) F AB-PPI++ (1) Figure S10. Mean estimation experiment on the FOREST dataset with varying horseshoe prior scale. The left, middle, and right panels show average MSE, CI volume, and CI coverage over 100 repetitions for α = 0.1. ...
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N., Bates, S., Fannjiang, C., Jordan, M

Angelopoulos, A. N., Bates, S., Fannjiang, C., Jordan, M. I., and Zrnic, T. Prediction-powered inference. Science, 382 (6671):669–674, 2023a. Angelopoulos, A. N., Duchi, J. C., and Zrnic, T. PPI++: Efficient prediction-powered inference. arXiv preprint arXiv:2311.01453, 2023b....

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Fisch, A., Maynez, J., Hofer, R. A., Dhingra, B., Globerson, A., and Cohen, W. W. Stratified prediction-powered infer- ence for hybrid language model evaluation. In Advances in Neural Information Processing Systems 37 (NeurIPS 2024),

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Cortinovis, S. and Caron, F. Bayes-assisted confidence regions: Focal point estimator and bounded-influence priors. arXiv preprint arXiv:2410.20169,

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [1]

The marginal likelihood is given by π(y) = Z ∞ −∞ N (y | β, σ2)πHS(β)dβ = 1√ 2πσ 2 Z ∞ 0 1√ 1 + ν2 e − y2 2σ2 (1+ν2 ) p(ν)dν = 2 π √ 2πσ 2 Z ∞ 0 e − y2 2σ2 (1+ν2 ) 1 (1 + ν2)3/2 dν

The rationale for this choice, along with a discussion of the general case η ̸= 1, is provided at the end of this section. The marginal likelihood is given by π(y) = Z ∞ −∞ N (y | β, σ2)πHS(β)dβ = 1√ 2πσ 2 Z ∞ 0 1√ 1 + ν2 e − y2 2σ2 (1+ν2 ) p(ν)dν = 2 π √ 2πσ 2 Z ∞ 0 e − y2 2σ2 (1+ν2 ) 1 (1 + ν2)3/2 dν. Using the change of variable u = 1 1+ν2 , we obtain ...

work page 2011

[2] [3]

Forest. The FOREST dataset contains the following features for N = 1 596 parcels of land in the Amazon rainforest examined during field visits (Bullock et al., 2020): whether the parcel has been subject to deforestation (Yi ∈ {0, 1}) and the prediction of a gradient-boosted tree model for the probability of Yi being equal to one (f (Xi) ∈ [0, 1]). The goa...

work page 2020

[3] [6]

Using the asymptotic expansion (Slater, 1960, Chapter 4, Eq

is given by κ(y) = 2 3 1F1 2, 5 2 , − y2 2σ2 1F1 1, 3 2 , − y2 2σ2 . Using the asymptotic expansion (Slater, 1960, Chapter 4, Eq. (4.I.3)) 1F1(a, b,−z) ∼ z−a Γ(b) Γ(b − a) as z → ∞, we find κ(y) ∼ 2σ2 y2 |E[β | y] − y| = σ2|ℓ′(y)| ∼2σ2 |y| as |y| → ∞. The horseshoe prior πHS has two key properties: an infinite spike at zero, inducing strong shrinkage near...

work page 1960

[4] [7]

While Kummer’s 1F1 function is implemented in many standard scientific libraries, such as SciPy, Φ1 is not

For a general η, similar expressions can be derived for the marginal likelihood and posterior mean, replacing Kummer’s 1F1 function with the more general degenerate hypergeometric function of two variables, Φ1 (see (Carvalho et al., 2010, Equations (4) in the main text and (A1) in the appendix)). While Kummer’s 1F1 function is implemented in many standard...

work page 2024

[5] [8]

Additionally, if the prior π0(β) := π0(β; σ) admits σ as a scale parameter, writing wα(β; σ) for the corresponding tail function, we have wα(β; σ) = wα β σ ; 1

priors when σ2 = 1and α = 0.1. Additionally, if the prior π0(β) := π0(β; σ) admits σ as a scale parameter, writing wα(β; σ) for the corresponding tail function, we have wα(β; σ) = wα β σ ; 1 . (S31) We now describe other properties of the spending function in the case of a Gaussian prior and of a prior with power-law tails, such as the horseshoe. Proposit...

work page 1963

[6] [9]

Let π0(β; σ) be a symmetric prior on β such that the marginal density π(y) has power-law tails, i.e

Proposition S1.3 (FAB with a prior with power-law tails (Cortinovis & Caron, 2024, Lemma S1.1)). Let π0(β; σ) be a symmetric prior on β such that the marginal density π(y) has power-law tails, i.e. π(y) ∼ Cσ|y|−δ as |y| → ∞ for some constant Cσ and some exponent δ >1. Then, wα(β; σ) is bounded away from 0 and 1, and lim β→∞ wα(β) = lim β→−∞ wα(β) = 1 2 . ...

work page 2024

[7] [10]

With the above two lemmas, we can now prove Theorem 4.1, which we restate here in extended form

Instead, by showing continuity of the mapping b∆θ/bσθ 7→ Cn,N ( b∆θ/bσθ) and again exploiting the CLT, one may prove that Cn,N converges in distribution to the random FAB confidence regionC(Y ) − Y , where C(y) is defined in Equation (S29), for a unit scale and a horseshoe prior, and Y ∼ N(0, 1). With the above two lemmas, we can now prove Theorem 4.1, wh...

work page 2024

[8] [11]

Therefore, (S41) and (S42) hold almost surely by sandwiching

is bounded. Therefore, (S41) and (S42) hold almost surely by sandwiching. Under the Gaussian prior (22), ℓ′ N(y; σ, σ) = − y 2σ2 . Hence, since b∆PP → ∆ and b∆PP+ → ∆ almost surely, where we recall that ∆ = E[f (X) − Y ], we obtain (bσPP)2ℓ′ N b∆PP; bσPP, bσPP → −∆ 2 (bσPP+)2ℓ′ N b∆PP+; bσPP+, bσPP → −∆ 2 almost surely as min(n, N) → ∞, which implies that...

work page 2023