Prediction-Powered Statistical Inference

• Prediction-powered statistical inference is a framework that combines abundant ML predictions with limited high-quality labels to perform valid statistical estimation.
• It corrects bias using a 'rectifier' approach, ensuring confidence intervals remain reliable even when ML predictions are systematically biased.
• By enhancing sample efficiency, these methods can significantly reduce the number of labels needed, benefiting applications in genomics, astronomy, and more.

Prediction-powered statistical inference is a class of methodologies that enable valid, efficient statistical inference in the presence of both a small, high-quality labeled (“gold-standard”) dataset and a potentially much larger dataset with ML predictions. These frameworks systematically harness the information in the cheap, abundant predictions, while using a smaller labeled set to estimate and correct any predictive bias, thereby yielding provably valid confidence intervals and often achieving greater sample efficiency than classical inference. Most importantly, these procedures are designed to guarantee validity even when the ML predictions are systematically biased or uninformative.

1. Foundations and General Structure

The central goal of prediction-powered inference (PPI) is to perform valid statistical inference—such as mean, quantile, linear regression, or logistic regression estimation—when only a small fraction of the dataset is labeled, and the rest is available only as input features. A machine learning system trained elsewhere provides predictions across the large unlabeled set, but no assumptions are made about the accuracy, calibration, or distributional correctness of this ML model.

Let $(X_i, Y_i)$, $i=1,\dots,n$ be the labeled data and $X_j$, $j=1,\dots,N$ the unlabeled inputs, with point-wise predictions $f(X_j)$ generated by the ML model.

The inference procedure consists of:

1. Computing an estimator that combines labeled and predicted data:
   • For mean estimation:

   $$\hat{\theta}^{PP} = \underbrace{\frac{1}{N} \sum_{i=1}^N f(X_i)}_{\text{Prediction average}} - \underbrace{\frac{1}{n} \sum_{i=1}^n (f(X_i) - Y_i)}_{\text{Rectifier (bias correction)}}$$

2. Constructing a prediction-powered confidence interval:

$$\hat{\theta}^{PP} \pm z_{\alpha/2}\sqrt{\frac{\operatorname{Var}(f(X) - Y)}{n} + \frac{\operatorname{Var}(f(X))}{N}}$$

where $z_{\alpha/2}$ is the appropriate quantile from the standard normal distribution.

For general convex estimation tasks (e.g., regression), PPI operates in terms of risk minimization or M-estimation. The procedure leverages the labeled sample to form a rectifier—an empirical estimate of the bias in the ML predictions—then combines it with predictions on the large unlabeled set via a correction formula applicable to means, quantiles, or regression coefficients.

2. Statistical Guarantees and Assumptions

A defining feature of prediction-powered inference is that no assumptions are made about the ML model’s correctness or calibration. The only requirements are:

• The predictor $f$ is trained independently of both labeled and unlabeled data used for inference.
• Labeled and unlabeled data are sampled i.i.d. from the same distribution (with some extensions to limited forms of data shift).
• Standard regularity conditions (e.g., finite variance) allowing use of the central limit theorem or other standard inferential tools.

Validity guarantee:

• The produced confidence intervals cover the true parameter (mean, regression coefficient, quantile) at the nominal rate, regardless of ML model accuracy.
• If the ML predictions are useless, the estimator reduces to classical inference; if they are informative, confidence intervals shrink accordingly.

The general M-estimation framework is: $$\theta^* = \arg\min_\theta \mathbb{E}[\ell_\theta(X, Y)]$$ and the PPI confidence set for $\theta^*$ is

$$C_\alpha = \{ \theta ~|~ 0 \in R_\delta(\theta) + T_{\alpha-\delta}(\theta) \}$$

where $R_\delta(\theta)$ is the rectifier CI from labels, and $T_{\alpha-\delta}(\theta)$ is a CI from the predictions on unlabeled data; "+" is the Minkowski sum.

3. Efficiency and Data Requirements

The width of prediction-powered confidence intervals is primarily determined by the variance in the prediction error $f(X) - Y$, rather than the variance of $Y$ itself. Explicitly:

• Classical confidence interval width: $\propto 1/\sqrt{n}$
• PPI confidence interval width:

$$\propto \sqrt{ \frac{1}{n} \operatorname{Var}(f(X) - Y) + \frac{1}{N} \operatorname{Var}(f(X)) }$$

Thus, if predictions are highly accurate ($f(X) \approx Y$), the intervals can be much narrower, especially when $N \gg n$.

Empirical results reported demonstrate that PPI can halve or better the number of labeled samples required to achieve a given statistical power compared to classical methods. In all settings, CIs constructed by naively imputing ML predictions failed to achieve valid coverage.

4. Applications Across Domains

Prediction-powered inference has demonstrated substantial benefits in a broad range of scientific domains, including:

Domain	Statistical Task	Labeled N (Classical)	Labeled N (PPI)	Key Benefit
Proteomics	Odds ratio (PTM/IDR in proteins)	799	316	Tighter CI
Astronomy	Fraction of galaxies (e.g., spiral)	449	189	Reduced labeling
Genomics	Quantile estimation (expression)	900	764	Precise quantile CIs
Remote Sensing	Amazon deforestation proportion	35	21	Saves sampling cost
Census	Linear/logistic regression	6653	5569	Fewer labels
Ecology	Counting/plankton (label shift)	---	---	Shift-robust

Use cases include:

• Leveraging AlphaFold’s protein predictions with a few experimental labels for robust inferential tasks.
• Substantially reducing required manual galaxy labeling in astronomy.
• Calibrating and shrinking inference in environmental and census data where model predictions can be reliably debiased.

5. Methodological Extensions and Integrations

The original PPI approach has prompted multiple methodological refinements to improve computational tractability and statistical adaptability:

• PPI++: Introduces an adaptive tuning parameter $\lambda$, which is automatically estimated to interpolate between classical and PPI inference depending on the accuracy of predictions, ensuring confidence intervals are never wider than classical ones, even if predictions are poor.
• Cross-prediction-powered inference: Uses all labeled data efficiently for both model training and bias correction, improving interval stability and avoiding wasteful sample-splitting.
• Assumption-lean/data-adaptive approaches (e.g., PSPA/POP-Inf): Guarantee validity for arbitrary estimands and provide automatic variance-minimizing weighting of prediction vs. label information.
• Bootstrap-based PPBoot: Enables prediction-powered inference for arbitrary estimation problems, including those without explicit asymptotic variance formulas, by using simple resampling strategies.
• Bayesian PPI: Allows credible intervals and new estimands via explicit modeling of the uncertainty in both ML predictions and debiasing adjustments.

Contemporary extensions accommodate generalized loss functions, allow for stratified and federated settings, integrate with empirical Bayes shrinkage, and are applicable to non-standard sampling designs and partially missing covariates.

6. Practical Implementation Considerations

PPI and its descendants are robust to ML misspecification, requiring only independent, i.i.d. data splits and mild regularity for inference. Implementation typically involves:

1. Training a predictive model on external or independent data.
2. Quantifying and correcting bias (the rectifier) using labeled examples.
3. Computing estimates and confidence intervals using both labeled data and model predictions, via specified analytic formulas or resampling.
4. Adapting the weighting parameter or “power-tuning” ($\lambda$) as appropriate for the observed data, often via plugin variance estimates.
5. Ensuring computational scalability by leveraging convex optimization (for generalized linear models, etc.), plug-in bootstrap resampling for arbitrary estimands, or empirical risk approaches for data-adaptive weighting.

Potential limitations include decreased efficiency when predictions are uninformative, less stable results if models are unstable or data splits are too small, and computational cost for massive datasets or complex modeling tasks.

7. Historical Context and Future Directions

Prediction-powered inference unifies and extends classical surrogate outcome methods from biostatistics and economics, control variate methods, and modern semi-supervised learning paradigms. Its foundational properties—robustness, modularity, sample efficiency—have prompted broad interest in both theory and practice.

Emerging research directions include:

• Federated PPI for privacy-preserving, decentralized inference.
• Stratified and post-stratification for domains with heterogeneity across subpopulations.
• Data-adaptive, assumption-lean frameworks for general estimands and complex dependence structures.
• Extensions to risk control, sequential/anytime-valid inference, and settings where decisions inform (and alter) future data distributions (performativity).

As the framework matures, the potential expands for automated, scientifically rigorous analysis pipelines that optimally blend limited ground-truth annotation with the ubiquity of machine learning predictions.

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Prediction-powered inference thus represents a fundamental advance in semi-supervised statistics, providing a rigorous, flexible, and practical methodology for leveraging predictions in scientific inference without sacrificing validity or efficiency.