Generalized Prediction-Powered Inference

• Generalized PPI is a family of inferential techniques that uses model-based predictions combined with correction from a small labeled sample to reduce bias and variance.
• It implements a 'predict, then debias' approach by augmenting empirical losses with a rectifier, accommodating methods like cross-fitting, inverse-probability weighting, and conformal prediction.
• The framework improves estimation efficiency in practical applications and adapts to various missing-data mechanisms and distribution shifts while retaining valid inference.

Generalized prediction-powered inference (PPI) is a family of inferential procedures for partially labeled or partially observed outcome settings in which a small set of gold-standard labels is combined with a larger set of covariates and model-based predictions. Its common mechanism is to use predictions to form a low-variance plug-in quantity and then to debias that quantity with a labeled-sample correction, often called a rectifier, so that validity does not depend on the prediction model being correct. In the current literature, the framework is presented both as a general “predict, then debias” recipe and as a class of concrete estimators extending the original mean- and convex-loss constructions to broader estimands, broader missingness regimes, and broader inferential objects such as regular asymptotically linear estimators, e-values, conformal procedures, and confidence sequences (Angelopoulos et al., 2023, Song et al., 28 Jan 2026, Zou et al., 10 Feb 2026).

1. Canonical formulation and basic statistical structure

Generalized PPI is most naturally defined through the standard semi-supervised data layout. A labeled sample contains pairs $(X_i,Y_i)$, an unlabeled sample contains $X_i$ only, and a prediction rule $f$ or $\hat Y_i=f(X_i)$ is available for all units. In the simplest mean-estimation case, the PPI estimator is the familiar “prediction average + rectifier” decomposition,

$$\hat{\theta}_{\text{PPI}} = \frac{1}{N}\sum_{i=1}^{N}\hat Y_i + \frac{1}{n}\sum_{i=1}^{n}(Y_i-\hat Y_i),$$

or equivalently,

$$\hat\theta^{\rm PP} = \frac{1}{N}\sum_{i=1}^N f(X_i) - \frac{1}{n}\sum_{i=1}^n \bigl(f(X_i)-Y_i\bigr).$$

The first term uses the large prediction-rich sample, and the second term corrects the prediction bias using the labeled residuals (Angelopoulos et al., 2023, Mozer, 19 Mar 2026).

The generalized viewpoint replaces this specific estimator by a loss-based template. If

$$\theta^\star = \arg\min_{\theta} \, \mathbb{E}\{\ell(Y,X;\theta)\},$$

then PPI forms an augmented empirical loss,

$$\hat\theta_{\mathrm{PPI}} = \arg\min_{\theta} \left[ \frac{1}{n_u}\sum_{i:S_i=0}\ell(\hat Y_i,X_i;\theta) - \frac{1}{n_\ell}\sum_{i:S_i=1}\{\ell(\hat Y_i,X_i;\theta)-\ell(Y_i,X_i;\theta)\} \right].$$

This formulation makes explicit that generalized PPI is not a claim that $\hat Y$ is correct; it is an augmented, bias-corrected empirical objective in which the predictor is used as a variance-reducing surrogate rather than as a substitute for the outcome (Song et al., 28 Jan 2026).

A closely related tuning refinement is PPI++, written for the mean case as

$$\hat{\theta}_{\text{PPI++}} = \frac{1}{n}\sum_{i=1}^n Y_i + \lambda\left( \frac{1}{N}\sum_{i=1}^N \hat Y_i - \frac{1}{n}\sum_{i=1}^n \hat Y_i \right),$$

with $X_i$0 recovering original PPI and $X_i$1 recovering the labeled-only estimator. More generally, the literature treats generalized PPI as a family whose members differ mainly in how they protect validity and efficiency under different data-generating and missingness regimes (Mozer, 19 Mar 2026, Song et al., 28 Jan 2026).

The original framework isolates three baseline conditions: distribution comparability between labeled and unlabeled samples, an external and independent prediction model, and complete covariate information so that predictions are available for all units. Later variants relax these through MAR-based weighting, cross-fitting, and imputation-based generalizations (Song et al., 28 Jan 2026).

2. From convex estimation to regular asymptotically linear estimators

The broadest formal generalization currently stated in the literature extends PPI from M-estimation and Z-estimation to any regular asymptotically linear estimator. If a labeled-only estimator satisfies

$X_i$2

and an auxiliary $X_i$3-only estimator satisfies

$X_i$4

then the rectified estimator

$X_i$5

is asymptotically normal, and with $X_i$6 its asymptotic variance is

$X_i$7

The variance-minimizing weight is

$X_i$8

and the best auxiliary influence function is $X_i$9. Standard PPI arises as the special case in which the auxiliary estimator is obtained by replacing $f$0 by $f$1 (Zou et al., 10 Feb 2026).

For population means, these generalized formulas recover a much older estimator. The mean PPI estimator is algebraically equivalent to the survey-sampling difference estimator of Cassel et al. (1976), and PPI++ is equivalent to the generalized regression (GREG) estimator of Sarndal et al. (2003). The equivalence is exact at the estimator level: $f$2 What differs is the inferential framing: survey sampling treats the finite population as fixed and randomness as coming from the sampling design, whereas PPI typically treats the data as i.i.d. from a superpopulation. The same formula therefore targets different estimands under different sources of randomness (Mozer, 19 Mar 2026).

This survey-sampling identification also clarifies efficiency claims. Under simple random sampling without replacement,

$f$3

so residual variance, rather than outcome variance, drives precision. At the same time, the modern literature emphasizes that standard PPI is generally not semiparametrically efficient outside restrictive oracle conditions; it is better viewed as a computationally simple alternative to full efficient AIPW-style constructions (Mozer, 19 Mar 2026, Zou et al., 10 Feb 2026).

A further efficiency refinement appears in the linear-regression setting studied through prediction-based inference after prediction. There, the original unweighted PPI augmentation is valid but not always efficient, while a Chen–Chen style weighted augmentation is asymptotically normal and guaranteed to be at least as efficient as labeled-only OLS within the considered linear-augmentation class (Gronsbell et al., 2024).

3. Labeling mechanisms, missing-data interpretation, and distribution shift

The original PPI setup is effectively MCAR. In the notation used for generalized informative-labeling extensions, labels are observed only when $f$4, with

$f$5

Standard PPI is valid when the labeled residual mean is representative of the population residual mean, which is the simple-random-sampling or MCAR case. When labeling is informative, the unweighted residual correction is biased, and the rectifier must be replaced by an inverse-probability-weighted version (Datta et al., 13 Aug 2025).

For the finite-population mean

$f$6

the informative-labeling extension imports Horvitz–Thompson and Hájek weighting into the PPI rectifier. Writing $f$7,

$f$8

and the generalized estimators become

$f$9

When $\hat Y_i=f(X_i)$0, the usual PPI rectifier is exactly the Hájek form applied to residuals, so standard PPI is a special case of the weighted construction (Datta et al., 13 Aug 2025).

The missing-data interpretation is explicit. Standard PPI corresponds to MCAR, generalized IPW-PPI corresponds to MAR with

$\hat Y_i=f(X_i)$1

and MNAR is outside the method’s scope. When $\hat Y_i=f(X_i)$2 is unknown, the literature considers estimated propensities $\hat Y_i=f(X_i)$3 from a correctly specified propensity model such as logistic regression of $\hat Y_i=f(X_i)$4 on $\hat Y_i=f(X_i)$5, and reports that estimated-propensity performance closely matches the known-probability case in simulations (Datta et al., 13 Aug 2025).

A second line of generalization concerns covariate distribution shift. In the generalized ALE treatment, the observed data are $\hat Y_i=f(X_i)$6, with $\hat Y_i=f(X_i)$7 indicating that $\hat Y_i=f(X_i)$8 is observed and with

$\hat Y_i=f(X_i)$9

Three targets are distinguished: the full population target $$\hat{\theta}_{\text{PPI}} = \frac{1}{N}\sum_{i=1}^{N}\hat Y_i + \frac{1}{n}\sum_{i=1}^{n}(Y_i-\hat Y_i),$$0, the unlabeled-population target $$\hat{\theta}_{\text{PPI}} = \frac{1}{N}\sum_{i=1}^{N}\hat Y_i + \frac{1}{n}\sum_{i=1}^{n}(Y_i-\hat Y_i),$$1, and the labeled-population target $$\hat{\theta}_{\text{PPI}} = \frac{1}{N}\sum_{i=1}^{N}\hat Y_i + \frac{1}{n}\sum_{i=1}^{n}(Y_i-\hat Y_i),$$2. These lead, respectively, to IPW and inverse-odds-weighted generalized PPI estimators that transport the prediction-powered correction to shifted covariate regimes (Zou et al., 10 Feb 2026).

The same missing-data framing links generalized PPI to augmented inverse probability weighting. Under MCAR, one generalized PPI form can be written as

$$\hat{\theta}_{\text{PPI}} = \frac{1}{N}\sum_{i=1}^{N}\hat Y_i + \frac{1}{n}\sum_{i=1}^{n}(Y_i-\hat Y_i),$$3

which places PPI directly inside the broader AIPW family while preserving its appeal as a simple prediction-assisted construction (Zou et al., 10 Feb 2026).

4. Expansion of inferential targets and validity regimes

A central motivation for generalized PPI is that original PPI was largely tied to Z-estimation and related convex-optimization targets such as means, quantiles, and regression coefficients. One major extension replaces the Z-estimation viewpoint by e-values. If a valid e-value has product form

$$\hat{\theta}_{\text{PPI}} = \frac{1}{N}\sum_{i=1}^{N}\hat Y_i + \frac{1}{n}\sum_{i=1}^{n}(Y_i-\hat Y_i),$$4

then the prediction-powered component is

$$\hat{\theta}_{\text{PPI}} = \frac{1}{N}\sum_{i=1}^{N}\hat Y_i + \frac{1}{n}\sum_{i=1}^{n}(Y_i-\hat Y_i),$$5

and the full prediction-powered e-value is

$$\hat{\theta}_{\text{PPI}} = \frac{1}{N}\sum_{i=1}^{N}\hat Y_i + \frac{1}{n}\sum_{i=1}^{n}(Y_i-\hat Y_i),$$6

This extension yields anytime-valid inference, post-hoc valid significance levels, sequential stopping, multiple testing, change-point detection, and causal discovery, because every inference procedure expressible in terms of e-values inherits a prediction-powered counterpart (Csillag et al., 6 Feb 2025).

A second extension replaces point prediction with conformal set prediction. Instead of a single $$\hat{\theta}_{\text{PPI}} = \frac{1}{N}\sum_{i=1}^{N}\hat Y_i + \frac{1}{n}\sum_{i=1}^{n}(Y_i-\hat Y_i),$$7, a set predictor

$$\hat{\theta}_{\text{PPI}} = \frac{1}{N}\sum_{i=1}^{N}\hat Y_i + \frac{1}{n}\sum_{i=1}^{n}(Y_i-\hat Y_i),$$8

is calibrated, and inference is conducted through lower and upper envelopes $$\hat{\theta}_{\text{PPI}} = \frac{1}{N}\sum_{i=1}^{N}\hat Y_i + \frac{1}{n}\sum_{i=1}^{n}(Y_i-\hat Y_i),$$9 and $$\hat\theta^{\rm PP} = \frac{1}{N}\sum_{i=1}^N f(X_i) - \frac{1}{n}\sum_{i=1}^n \bigl(f(X_i)-Y_i\bigr).$$0. For bounded $$\hat\theta^{\rm PP} = \frac{1}{N}\sum_{i=1}^N f(X_i) - \frac{1}{n}\sum_{i=1}^n \bigl(f(X_i)-Y_i\bigr).$$1, with $$\hat\theta^{\rm PP} = \frac{1}{N}\sum_{i=1}^N f(X_i) - \frac{1}{n}\sum_{i=1}^n \bigl(f(X_i)-Y_i\bigr).$$2,

$$\hat\theta^{\rm PP} = \frac{1}{N}\sum_{i=1}^N f(X_i) - \frac{1}{n}\sum_{i=1}^n \bigl(f(X_i)-Y_i\bigr).$$3

which yields valid confidence intervals for means and then extends to Z-estimation, M-estimation, and e-values. In the e-value setting, the conformal paper states that its construction is the first general prediction-powered procedure that operates off-line (Csillag et al., 17 Oct 2025).

Sequential generalization proceeds differently. In anytime-valid, Bayes-assisted PPI, the core decomposition is

$$\hat\theta^{\rm PP} = \frac{1}{N}\sum_{i=1}^N f(X_i) - \frac{1}{n}\sum_{i=1}^n \bigl(f(X_i)-Y_i\bigr).$$4

with $$\hat\theta^{\rm PP} = \frac{1}{N}\sum_{i=1}^N f(X_i) - \frac{1}{n}\sum_{i=1}^n \bigl(f(X_i)-Y_i\bigr).$$5 estimable from abundant unlabeled data and $$\hat\theta^{\rm PP} = \frac{1}{N}\sum_{i=1}^N f(X_i) - \frac{1}{n}\sum_{i=1}^n \bigl(f(X_i)-Y_i\bigr).$$6 from the labeled sample. Confidence intervals are replaced by confidence sequences $$\hat\theta^{\rm PP} = \frac{1}{N}\sum_{i=1}^N f(X_i) - \frac{1}{n}\sum_{i=1}^n \bigl(f(X_i)-Y_i\bigr).$$7 satisfying

$$\hat\theta^{\rm PP} = \frac{1}{N}\sum_{i=1}^N f(X_i) - \frac{1}{n}\sum_{i=1}^n \bigl(f(X_i)-Y_i\bigr).$$8

The construction uses the method of mixtures and Ville’s inequality, and it allows prior information about prediction quality to enter through a prior on the rectifier $$\hat\theta^{\rm PP} = \frac{1}{N}\sum_{i=1}^N f(X_i) - \frac{1}{n}\sum_{i=1}^n \bigl(f(X_i)-Y_i\bigr).$$9 rather than on $$\theta^\star = \arg\min_{\theta} \, \mathbb{E}\{\ell(Y,X;\theta)\},$$0 directly (Kilian et al., 23 May 2025).

These extensions suggest that “generalized PPI” now denotes more than a larger class of point estimators. It denotes a transfer principle: whenever a valid inferential object can be expressed through a plug-in prediction component and a labeled-data correction, the prediction-powered logic can often be imported into that object with its native validity notion preserved (Csillag et al., 6 Feb 2025, Csillag et al., 17 Oct 2025).

5. Computational variants, tuning strategies, and shrinkage-based generalizations

The computational diversification of generalized PPI is substantial. Some variants seek wider applicability, some protect against data reuse, and some exploit prior or cross-task structure.

Extension	Central mechanism	Representative papers
PPBoot / bootstrap PPI	One bootstrap around the PPI-style correction; percentile intervals	(Zrnic, 2024, Efron, 26 Jun 2026)
Cross-PPI / Cross-PPBoot	Cross-fitting to avoid leakage when the predictor is trained on inference data	(Song et al., 28 Jan 2026)
Bayesian PPI / FAB-PPI	Posterior-composable proxy estimands; Bayes-assisted rectifier shrinkage	(Hofer et al., 2024, Cortinovis et al., 4 Feb 2025)
PAS	Within-task PPI++ debiasing plus across-task empirical Bayes shrinkage for many means	(Li et al., 20 Feb 2025)

Bootstrap-based generalization appears in two distinct forms. PPBoot defines the debiased point estimator

$$\theta^\star = \arg\min_{\theta} \, \mathbb{E}\{\ell(Y,X;\theta)\},$$1

then resamples labeled and unlabeled samples and forms percentile bootstrap intervals. Its stated advantage is applicability to arbitrary estimation problems without problem-specific CLT derivations, and its empirical behavior is often nearly identical to or sometimes better than asymptotic PPI/PPI++ when those are available (Zrnic, 2024). A later bootstrap paper takes a more model-based route, fitting a calibration model linking $$\theta^\star = \arg\min_{\theta} \, \mathbb{E}\{\ell(Y,X;\theta)\},$$2 to $$\theta^\star = \arg\min_{\theta} \, \mathbb{E}\{\ell(Y,X;\theta)\},$$3 and then using a two-stage bootstrap to obtain uncertainty for broad functionals $$\theta^\star = \arg\min_{\theta} \, \mathbb{E}\{\ell(Y,X;\theta)\},$$4, again with the aim of avoiding asymptotics (Efron, 26 Jun 2026).

The literature also generalizes PPI by tuning how much of the prediction signal is used. PPI++ introduces the scalar $$\theta^\star = \arg\min_{\theta} \, \mathbb{E}\{\ell(Y,X;\theta)\},$$5, while later variants include stratified PPI, matrix-tuned or estimating-equation variants, generalized tuning-function methods, and tuned cross-fitted versions. The practical message is that prediction-powered efficiency gains are not automatic; tuning is used to guarantee asymptotic performance no worse than complete-case analysis or to adapt to differential prediction quality across strata (Song et al., 28 Jan 2026).

Bayesian generalizations re-express PPI as a “proxy estimand + posterior sampling” framework. Bayesian PPI treats component quantities such as imputed means, residual means, conditional probabilities, or discrete judge outputs as posterior random variables and then propagates uncertainty by Monte Carlo integration. The resulting constructions handle settings with discrete, abstaining, or nonlinear autoraters more naturally than the classical difference-estimator form (Hofer et al., 2024). FAB-PPI keeps frequentist validity but replaces the raw rectifier by a Bayes-assisted version,

$$\theta^\star = \arg\min_{\theta} \, \mathbb{E}\{\ell(Y,X;\theta)\},$$6

and uses a heavy-tailed prior, particularly the horseshoe prior, so that the method shrinks strongly when the predictor is likely good but reverts to standard PPI behavior in low-prior-probability regions (Cortinovis et al., 4 Feb 2025).

A separate multi-problem generalization appears in Prediction-Powered Adaptive Shrinkage (PAS). For many related mean-estimation tasks, PAS first computes within-task PPI++-style unbiased estimators

$$\theta^\star = \arg\min_{\theta} \, \mathbb{E}\{\ell(Y,X;\theta)\},$$7

then shrinks them across tasks toward the prediction means $$\theta^\star = \arg\min_{\theta} \, \mathbb{E}\{\ell(Y,X;\theta)\},$$8 using a global parameter $$\theta^\star = \arg\min_{\theta} \, \mathbb{E}\{\ell(Y,X;\theta)\},$$9 chosen by minimizing the correlation-aware unbiased risk estimate

$$\hat\theta_{\mathrm{PPI}} = \arg\min_{\theta} \left[ \frac{1}{n_u}\sum_{i:S_i=0}\ell(\hat Y_i,X_i;\theta) - \frac{1}{n_\ell}\sum_{i:S_i=1}\{\ell(\hat Y_i,X_i;\theta)-\ell(Y_i,X_i;\theta)\} \right].$$0

The compound-estimation viewpoint is new relative to single-task PPI: within-task debiasing is preserved, but strength is borrowed across tasks through empirical Bayes shrinkage (Li et al., 20 Feb 2025).

6. Diagnostics, applications, limitations, and recurring controversies

Generalized PPI is accompanied by a diagnostic literature because several of its core assumptions are untestable or only partially testable. Recommended checks include comparing labeled and unlabeled covariate distributions through standardized mean differences, Kolmogorov–Smirnov tests, and energy distance; auditing training provenance to detect overlap between training and inference data; inspecting missingness patterns; and examining whether labeled residuals appear representative of the unlabeled pool. When no trustworthy external model exists, the recommended remedy is cross-fitting through Cross-PPI or Cross-PPBoot rather than direct reuse of the same data for training and inference (Song et al., 28 Jan 2026).

One recurrent warning concerns subgroup estimands. For average treatment effects or subgroup means, a pooled rectifier can be biased when prediction error differs across groups. If $$\hat\theta_{\mathrm{PPI}} = \arg\min_{\theta} \left[ \frac{1}{n_u}\sum_{i:S_i=0}\ell(\hat Y_i,X_i;\theta) - \frac{1}{n_\ell}\sum_{i:S_i=1}\{\ell(\hat Y_i,X_i;\theta)-\ell(Y_i,X_i;\theta)\} \right].$$1, then a pooled-arm treatment effect estimator has bias

$$\hat\theta_{\mathrm{PPI}} = \arg\min_{\theta} \left[ \frac{1}{n_u}\sum_{i:S_i=0}\ell(\hat Y_i,X_i;\theta) - \frac{1}{n_\ell}\sum_{i:S_i=1}\{\ell(\hat Y_i,X_i;\theta)-\ell(Y_i,X_i;\theta)\} \right].$$2

The recommended fix is to compute separate rectifiers within each subgroup or arm, which aligns generalized PPI with standard stratified model-assisted survey practice (Mozer, 19 Mar 2026).

Another recurrent warning concerns double-dipping and MNAR. Reusing the same data for predictor training and for inference can make intervals too narrow and coverage too low; in the Mosaiks study, under double-dipping, confidence intervals were systematically too narrow, and for income and nightlights coverage fell to about $$\hat\theta_{\mathrm{PPI}} = \arg\min_{\theta} \left[ \frac{1}{n_u}\sum_{i:S_i=0}\ell(\hat Y_i,X_i;\theta) - \frac{1}{n_\ell}\sum_{i:S_i=1}\{\ell(\hat Y_i,X_i;\theta)-\ell(Y_i,X_i;\theta)\} \right].$$3 when $$\hat\theta_{\mathrm{PPI}} = \arg\min_{\theta} \left[ \frac{1}{n_u}\sum_{i:S_i=0}\ell(\hat Y_i,X_i;\theta) - \frac{1}{n_\ell}\sum_{i:S_i=1}\{\ell(\hat Y_i,X_i;\theta)-\ell(Y_i,X_i;\theta)\} \right].$$4. Under missing not at random, all methods, including classical inference using only labeled data, yielded biased estimates in the reported simulations (Song et al., 28 Jan 2026). Informative labeling is therefore distinguishable from MNAR: the former can be handled by IPW-style generalized PPI under MAR, while the latter is not handled by the current framework (Datta et al., 13 Aug 2025).

A further controversy concerns novelty. For means, the survey-sampling reinterpretation states directly that PPI is the difference estimator and PPI++ is GREG, so generalized PPI does not derive its importance from a new mean estimator. Its newer contributions lie in packaging, extensions to broader estimands such as general M-estimators and regular asymptotically linear estimators, accessible software ecosystems, cross-fitting, e-values, conformal procedures, and modern ML or LLM workflows (Mozer, 19 Mar 2026, Zou et al., 10 Feb 2026).

Empirically, the framework has repeatedly been shown to improve efficiency when predictions are informative and the assumptions hold. The original paper reports, for example, that to reject a null odds ratio of at most $$\hat\theta_{\mathrm{PPI}} = \arg\min_{\theta} \left[ \frac{1}{n_u}\sum_{i:S_i=0}\ell(\hat Y_i,X_i;\theta) - \frac{1}{n_\ell}\sum_{i:S_i=1}\{\ell(\hat Y_i,X_i;\theta)-\ell(Y_i,X_i;\theta)\} \right].$$5 in the proteomics example, PPI needed $$\hat\theta_{\mathrm{PPI}} = \arg\min_{\theta} \left[ \frac{1}{n_u}\sum_{i:S_i=0}\ell(\hat Y_i,X_i;\theta) - \frac{1}{n_\ell}\sum_{i:S_i=1}\{\ell(\hat Y_i,X_i;\theta)-\ell(Y_i,X_i;\theta)\} \right].$$6 labels versus $$\hat\theta_{\mathrm{PPI}} = \arg\min_{\theta} \left[ \frac{1}{n_u}\sum_{i:S_i=0}\ell(\hat Y_i,X_i;\theta) - \frac{1}{n_\ell}\sum_{i:S_i=1}\{\ell(\hat Y_i,X_i;\theta)-\ell(Y_i,X_i;\theta)\} \right].$$7 for classical inference; for galaxy morphology it needed $$\hat\theta_{\mathrm{PPI}} = \arg\min_{\theta} \left[ \frac{1}{n_u}\sum_{i:S_i=0}\ell(\hat Y_i,X_i;\theta) - \frac{1}{n_\ell}\sum_{i:S_i=1}\{\ell(\hat Y_i,X_i;\theta)-\ell(Y_i,X_i;\theta)\} \right].$$8 labels versus $$\hat\theta_{\mathrm{PPI}} = \arg\min_{\theta} \left[ \frac{1}{n_u}\sum_{i:S_i=0}\ell(\hat Y_i,X_i;\theta) - \frac{1}{n_\ell}\sum_{i:S_i=1}\{\ell(\hat Y_i,X_i;\theta)-\ell(Y_i,X_i;\theta)\} \right].$$9; for Amazon deforestation it needed $\hat Y$0 labels versus $\hat Y$1; and under covariate shift in an OLS example it reduced the labeled sample requirement from $\hat Y$2 to $\hat Y$3 (Angelopoulos et al., 2023). At the same time, the literature is explicit that if the predictor is weak, the unlabeled sample is not much larger than the labeled sample, or the relevant missingness assumptions fail, classical inference may be preferable or all methods may fail (Angelopoulos et al., 2023, Song et al., 28 Jan 2026).

In this sense, generalized prediction-powered inference is best viewed neither as a single estimator nor as a universal efficiency guarantee. It is a modular family of bias-corrected, prediction-assisted inferential constructions whose statistical meaning is determined by the estimand, the labeling mechanism, the source of validity, and the way in which predictions enter the rectifier.