Prediction-Powered Conditional Inference

• The paper introduces prediction-powered conditional inference as a semi-supervised framework that combines a small labeled sample with a large unlabeled set using predictive models to reduce variance.
• It reformulates conditional targets into weighted or localized unconditional moments and employs debiasing, calibration, and empirical likelihood techniques for efficient estimation.
• The approach accommodates methods like local polynomial regression, stratified aggregation, and RKHS localization, ensuring valid asymptotic inference and robust variance reduction.

Prediction-powered conditional inference denotes a family of semi-supervised inference procedures for covariate-specific targets that combine a small labeled sample, a large unlabeled sample, and predictions from an external or cross-fitted model. The common objective is to infer quantities such as the pointwise conditional mean $\mu(x)=E[Y\mid X=x]$, conditional linear functionals, conditional treatment effects, conditional quantiles, or more general conditional moment parameters, while using predictions to reduce variance without changing the estimand. In the literature, this is achieved by recasting conditional targets as weighted, localized, stratified, or sieve-based unconditional moments and then applying prediction-powered debiasing, calibration, or empirical likelihood methods (Angelopoulos et al., 2023, Wang et al., 18 Dec 2025, Sui et al., 5 Mar 2026).

1. Statistical formulation and target parameters

The standard data structure consists of a labeled sample $\{(X_i,Y_i)\}$, an unlabeled sample $\{X_j\}$, and a prediction rule or external model that provides $\widetilde{Y}(x)=\widehat{Y}(x)$ or predictive features such as $\phi(\widetilde{Y}(x),x)$. In some formulations the labeled subset is sampled by simple random sampling without replacement from a finite population with all covariates observed, while in others the labeled and unlabeled samples are taken as independent and identically distributed draws from the same population. Across these settings, the central requirement is that labeled residual information be representative for the prediction errors encountered in the unlabeled covariates (Lee et al., 7 Jun 2026, Song et al., 28 Jan 2026).

Conditional targets appear in several equivalent forms. One form is the pointwise conditional functional, such as $\mu(x)=E[Y\mid X=x]$ or a conditional moment restriction $\eta(x_0;\theta)=E[\ell(Y;\theta)\mid X=x_0]$ with $\theta_0(x_0)$ defined by $\eta(x_0;\theta_0(x_0))=0$. Another form is a subgroup-weighted parameter,

$\theta_w=\frac{E[g(Y)w(Z)]}{E[w(Z)]},$

with $\{(X_i,Y_i)\}$0 an indicator of subgroup membership, a poststratification weight, or a kernel weight. A third form represents conditional objects through series or basis expansions, for example $\{(X_i,Y_i)\}$1, so that conditional inference reduces to inference on the coefficient vector $\{(X_i,Y_i)\}$2 (Wang et al., 18 Dec 2025, Song et al., 28 Jan 2026, Sui et al., 5 Mar 2026).

This representation is especially important because the prediction-powered machinery is fundamentally built for moments, estimating equations, or loss minimization. Conditional inference therefore proceeds by choosing a weighting or approximation device that translates the local target into a supervised estimating equation and then augmenting that equation with prediction-derived information. This suggests that “conditional” in this literature is less a separate inferential principle than a structural modification of general prediction-powered inference.

2. Weighted, localized, and sieve-based reductions

A generic conditional estimator uses the same predict-then-debias template as global PPI, but with weights that localize the target. A representative form is

$\{(X_i,Y_i)\}$3

With $\{(X_i,Y_i)\}$4, this gives subgroup-specific inference; with $\{(X_i,Y_i)\}$5, it gives local conditional inference at $\{(X_i,Y_i)\}$6. Closely related formulations define weighted conditional scores

$\{(X_i,Y_i)\}$7

so that the localized prediction-powered score is $\{(X_i,Y_i)\}$8 (Angelopoulos et al., 2023, Lee et al., 7 Jun 2026, Song et al., 28 Jan 2026).

Local polynomial regression provides the most explicit conditional construction. Let $\{(X_i,Y_i)\}$9 be the local polynomial basis and $\{X_j\}$0 the kernel weight matrix around $\{X_j\}$1. The conventional local estimator is

$\{X_j\}$2

Prediction-powered local regression replaces the response fit on unlabeled covariates by predicted outcomes and then subtracts a labeled residual fit in the same basis: $\{X_j\}$3 with

$\{X_j\}$4

For local linear regression, the intercept estimates $\{X_j\}$5 and the slope estimates $\{X_j\}$6 (Gu et al., 2024).

Series or sieve representations play an analogous role for more general conditional functionals. For conditional means or CATEs, one specifies a basis $\{X_j\}$7 and a supervised score such as

$\{X_j\}$8

so that $\{X_j\}$9 satisfies $\widetilde{Y}(x)=\widehat{Y}(x)$0 and $\widetilde{Y}(x)=\widehat{Y}(x)$1. Prediction-powered conditional inference then augments this score with auxiliary functions built from $\widetilde{Y}(x)=\widehat{Y}(x)$2, $\widetilde{Y}(x)=\widehat{Y}(x)$3, or treatment-specific predictors $\widetilde{Y}(x)=\widehat{Y}(x)$4 (Wang et al., 18 Dec 2025).

3. Principal methodological families

Several distinct frameworks instantiate this reduction.

Framework	Main mechanism	Representative conditional targets
Local Prediction-Powered Inference	local polynomial weighted least squares with prediction fit minus labeled residual fit	$\widetilde{Y}(x)=\widehat{Y}(x)$5, $\widetilde{Y}(x)=\widehat{Y}(x)$6
StratPPI	stratum-specific PPI aggregated with stratum weights $\widetilde{Y}(x)=\widehat{Y}(x)$7	$\widetilde{Y}(x)=\widehat{Y}(x)$8
PPI++ and CC-style augmentation	tuned or projected control-variate correction	conditional contrasts encoded in $\widetilde{Y}(x)=\widehat{Y}(x)$9
PPCI and EPI	RKHS localization or empirical likelihood with auxiliary moments	fixed-point conditional moments, sieve means, CATE, quantiles

Local Prediction-Powered Inference studies the model $\phi(\widetilde{Y}(x),x)$0 with $\phi(\widetilde{Y}(x),x)$1, combines a local smoothing basis with external predictions $\phi(\widetilde{Y}(x),x)$2, and proves that the resulting estimator keeps the same leading bias order as conventional local regression while reducing covariance (Gu et al., 2024).

Stratified Prediction-Powered Inference partitions the covariate space into fixed strata $\phi(\widetilde{Y}(x),x)$3, computes a local PPI estimator inside each stratum, and aggregates these using the unlabeled stratum proportions $\phi(\widetilde{Y}(x),x)$4. In the scalar mean case,

$\phi(\widetilde{Y}(x),x)$5

This directly supports conditional inference when the conditioning variable is itself used to define the strata (Fisch et al., 2024).

PPI++ introduces a scalar tuning parameter $\phi(\widetilde{Y}(x),x)$6 in the rectified estimator,

$\phi(\widetilde{Y}(x),x)$7

or, for M-estimation, the rectified objective

$\phi(\widetilde{Y}(x),x)$8

For conditional inference, the same construction is applied to weighted numerators and denominators. In a related line, the Chen–Chen weighted augmentation uses a matrix weight $\phi(\widetilde{Y}(x),x)$9 to project the score for $\mu(x)=E[Y\mid X=x]$0 onto the score for $\mu(x)=E[Y\mid X=x]$1, yielding an estimator that is asymptotically no less efficient than labels-only least squares. In that literature, conditional contrasts are represented by including interactions, subgroup indicators, or basis expansions in $\mu(x)=E[Y\mid X=x]$2 (Angelopoulos et al., 2023, Gronsbell et al., 2024).

Prediction-Powered Conditional Inference in the narrow sense of Sui, Zhou, Zhou, and Dai constructs a reproducing-kernel weight

$\mu(x)=E[Y\mid X=x]$3

and defines the localized moment

$\mu(x)=E[Y\mid X=x]$4

Predictions are incorporated through the decomposition

$\mu(x)=E[Y\mid X=x]$5

followed by two-fold cross-fitting and root-finding for $\mu(x)=E[Y\mid X=x]$6 (Sui et al., 5 Mar 2026).

Empirical Likelihood-based Prediction-Powered Inference stacks supervised estimating equations with centered auxiliary prediction moments and then optimizes empirical likelihood over labeled weights. This integrates conditional estimation by using sieve scores or localized estimating equations while bringing unlabeled predictions into the constraints through pooled centering (Wang et al., 18 Dec 2025).

4. Asymptotic theory, efficiency, and confidence procedures

The theoretical literature distinguishes validity, efficiency gain, and robustness to prediction misspecification. The original PPI framework establishes consistent and asymptotically valid inference for means, quantiles, and regression coefficients without correctness assumptions on the machine-learning predictor, and its conditional specializations inherit the same labeled-residual plus unlabeled-prediction variance decomposition under appropriate weighting (Angelopoulos et al., 2023).

The statistical optimality theory of PPI identifies an efficient influence function for the semi-supervised observed-data model and shows that PPI attains the semiparametric efficiency lower bound when the predictor is score-calibrated, meaning

$\mu(x)=E[Y\mid X=x]$7

For conditional inference, the same paper states that the M-estimation, influence-function, and efficiency results are directly reusable with localized or stratified weighting, although rigorous optimality with bandwidth selection and function-valued targets is left open (Lee et al., 7 Jun 2026).

Local Prediction-Powered Inference proves that, for local linear regression, the leading bias is $\mu(x)=E[Y\mid X=x]$8, the expected error is $\mu(x)=E[Y\mid X=x]$9, and

$\eta(x_0;\theta)=E[\ell(Y;\theta)\mid X=x_0]$0

It further derives coverage expansions showing that bias correction improves coverage error from $\eta(x_0;\theta)=E[\ell(Y;\theta)\mid X=x_0]$1 to $\eta(x_0;\theta)=E[\ell(Y;\theta)\mid X=x_0]$2 for $\eta(x_0;\theta)=E[\ell(Y;\theta)\mid X=x_0]$3, and from $\eta(x_0;\theta)=E[\ell(Y;\theta)\mid X=x_0]$4 to $\eta(x_0;\theta)=E[\ell(Y;\theta)\mid X=x_0]$5 for $\eta(x_0;\theta)=E[\ell(Y;\theta)\mid X=x_0]$6 (Gu et al., 2024).

StratPPI proves asymptotic normality for a stratified prediction-powered M-estimator and, in the mean case with natural allocations $\eta(x_0;\theta)=E[\ell(Y;\theta)\mid X=x_0]$7, establishes

$\eta(x_0;\theta)=E[\ell(Y;\theta)\mid X=x_0]$8

with strict improvement unless both $\eta(x_0;\theta)=E[\ell(Y;\theta)\mid X=x_0]$9 and $\theta_0(x_0)$0 are identical across strata. This is a direct conditional efficiency statement because the strata themselves encode the conditioning structure (Fisch et al., 2024).

PPI++ proves asymptotic normality with covariance

$\theta_0(x_0)$1

and shows that, with $\theta_0(x_0)$2 chosen to minimize $\theta_0(x_0)$3, the asymptotic variance is no larger than that of the labeled-only estimator. The CC estimator strengthens this guarantee for linear regression by showing

$\theta_0(x_0)$4

so prediction-based augmentation cannot be less efficient than labels-only (Angelopoulos et al., 2023, Gronsbell et al., 2024).

PPCI proves a nonasymptotic error decomposition, pointwise asymptotic normality, consistent variance estimation, and minimax-optimal convergence rates up to logarithmic factors. Its pointwise confidence interval is

$\theta_0(x_0)$5

and its minimax lower bound scales as

$\theta_0(x_0)$6

under Sobolev-RKHS assumptions (Sui et al., 5 Mar 2026).

EPI shows that

$\theta_0(x_0)$7

with $\theta_0(x_0)$8 no larger than the fully supervised asymptotic variance, and that the semiparametric efficiency bound is attained when the centered auxiliary functions span the predictable component of the supervised score. For hypothesis testing, the empirical likelihood ratio has a weighted chi-squared limit, yielding confidence sets by ELR inversion in addition to Wald-type intervals (Wang et al., 18 Dec 2025).

5. Computation, cross-fitting, and diagnostics

Implementation differs by framework, but several recurring devices are standard. Cross-fitting or strict sample splitting is repeatedly recommended when the predictor or auxiliary function is learned from the inference data, because reusing labeled outcomes for both training and inference leads to leakage or “double-dipping.” In mean estimation, cross-fitted PPI requires only $\theta_0(x_0)$9-consistency of the out-of-fold predictor for asymptotic normality, while single-fit procedures need explicit variance correction in special cases such as kernel ridge regression with unpenalized intercept (Lee et al., 7 Jun 2026, Song et al., 28 Jan 2026).

Local methods require kernel choice, bandwidth selection, and stability of the local design matrix. Local PP regression recommends a symmetric positive kernel with sufficient moments, local linear order $\eta(x_0;\theta_0(x_0))=0$0 as a robust default, and cross-validation on labeled residuals or plug-in bandwidth selectors. When the local matrix is near-singular, a stabilized rectifier

$\eta(x_0;\theta_0(x_0))=0$1

is proposed (Gu et al., 2024).

RKHS-based PPCI requires solving $\eta(x_0;\theta_0(x_0))=0$2 on unlabeled folds, selecting $\eta(x_0;\theta_0(x_0))=0$3 by an L-curve balancing approximation error and a variance proxy, and then solving the localized empirical moment equation for $\eta(x_0;\theta_0(x_0))=0$4. The method emphasizes that the weight function must be learned solely from covariates, never from labels (Sui et al., 5 Mar 2026).

EPI places the computational burden on the empirical likelihood dual. With stacked supervised and centered auxiliary constraints, the labeled weights take the form

$\eta(x_0;\theta_0(x_0))=0$5

where $\eta(x_0;\theta_0(x_0))=0$6 solve the dual moment equations. Practical implementations use basis expansions or cross-fitted learned auxiliaries, Newton or quasi-Newton optimization for the profile empirical likelihood, and plug-in estimation of the weighted chi-squared limit for ELR confidence sets (Wang et al., 18 Dec 2025).

Diagnostics focus on the assumptions that make conditional prediction-powered inference credible. Recommended checks include comparing covariate distributions across labeled and unlabeled samples, auditing model provenance to verify independence between training and inference, monitoring dual feasibility and weight stability in empirical likelihood, examining localized or subgroup residual variance $\eta(x_0;\theta_0(x_0))=0$7, and checking that labeled support is adequate in each subgroup or neighborhood. Several papers also recommend sensitivity analyses with alternative bases, kernels, learners, or stratifications (Wang et al., 18 Dec 2025, Song et al., 28 Jan 2026).

6. Limitations, variants, and open directions

The strongest efficiency results require calibration conditions. In semiparametric optimality theory, efficiency requires score calibration; in EPI, efficiency requires the auxiliary span to match the predictable component of the supervised score; in CC-style augmentation, gains depend on residual information in $\eta(x_0;\theta_0(x_0))=0$8 beyond the regression on $\eta(x_0;\theta_0(x_0))=0$9. When predictions are weak, methods such as PPI++ and CC shrink or project the augmentation so that performance reverts toward labels-only inference, but the conditional efficiency gain itself can vanish (Lee et al., 7 Jun 2026, Wang et al., 18 Dec 2025, Gronsbell et al., 2024).

Conditional inference also introduces structural difficulties absent from global PPI. Local polynomial and RKHS methods face the curse of dimensionality through the effective local sample size or the leverage term $\theta_w=\frac{E[g(Y)w(Z)]}{E[w(Z)]},$0. Stratified procedures can become unstable when the number of strata is large and some $\theta_w=\frac{E[g(Y)w(Z)]}{E[w(Z)]},$1 are small. Kernel-based targets require bandwidth selection and undersmoothing for valid inference. Several papers explicitly note that rigorous optimality theory for bandwidth choice, multi-parameter function estimation, and practical conditional diagnostics remains incomplete (Gu et al., 2024, Fisch et al., 2024, Sui et al., 5 Mar 2026, Lee et al., 7 Jun 2026).

Assumption violations can be more damaging in conditional settings. Reusing training data for inference yields anti-conservative confidence intervals, and missing-not-at-random labeling can bias both classical and prediction-powered procedures. The practical guidance literature therefore treats no double-dipping, comparability of labeled and unlabeled samples, complete covariates, and positivity within the conditioning set as central design requirements rather than optional refinements (Song et al., 28 Jan 2026).

Recent extensions broaden the inferential toolkit. A bootstrap-based approach treats the observed covariates as fixed and performs conditional inference through calibration of $\theta_w=\frac{E[g(Y)w(Z)]}{E[w(Z)]},$2 followed by two-level resampling, with especially strong gains for regression and correlation-type targets; it also reports the “surprise” that, for purely marginal means, unlabeled covariates may provide no standard-deviation gain after calibration under the working models (Efron, 26 Jun 2026). A conformal extension replaces point imputation by calibrated set-valued imputation and yields valid marginal inference for means, Z-estimation, M-estimation, and e-values while naturally accommodating privacy, robustness, and time-series calibration through the conformal miscoverage term $\theta_w=\frac{E[g(Y)w(Z)]}{E[w(Z)]},$3. That work emphasizes, however, that its guarantees are primarily marginal rather than fully conditional (Csillag et al., 17 Oct 2025).

Taken together, the literature presents prediction-powered conditional inference as a collection of closely related strategies: weighted residual correction, local polynomial rectification, stratified aggregation, tuned control variates, RKHS localization, empirical likelihood calibration, bootstrap conditioning, and conformal set-based imputation. Their unifying principle is that conditional targets are made inferentially tractable by expressing them as weighted or localized moments and then using predictions to remove variance rather than to redefine the target.