Prediction-Powered Inference

• Prediction-powered inference is a statistical framework that integrates abundant ML predictions with a small set of gold-standard data to achieve valid and efficient inference.
• It combines imputed estimates from large unlabeled datasets with bias correction from labeled data to accurately estimate means, quantiles, and regression coefficients.
• The approach is widely applied in fields like proteomics, astronomy, and genomics, enhancing data efficiency and rigor in statistical conclusions.

Prediction-powered inference (PPI) is a statistical framework that enables valid inference when a small set of gold-standard (labeled) data is supplemented with a much larger set of ML predictions on unlabeled examples. It is designed to yield provably valid confidence intervals for quantities such as means, quantiles, and regression coefficients, regardless of the accuracy or methodology of the ML predictor. Prediction-powered inference leverages the abundance and low cost of ML predictions, correcting for their possible biases, to produce more data-efficient and statistically rigorous conclusions in scientific, industrial, and applied domains.

1. Conceptual Foundations and Overview

Prediction-powered inference addresses the common scenario in modern data analysis where high-quality labeled data are costly or scarce but large amounts of unlabeled data can be annotated with predictions from a ML model. The naive use of these predictions for inference introduces bias due to inevitable model imperfections. PPI overcomes this by "rectifying" the bias via a problem-specific correction term—referred to as the "rectifier"—quantified using a small trusted set of labeled data.

Given a parameter of interest $\theta^*$ (e.g., the mean of $Y$), PPI constructs an estimator that combines imputed estimates from the predictions $f(X_i)$ over the large unlabeled dataset and a bias correction term from the gold-standard data. For mean estimation, the canonical formula is:

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

Here, the first term is a precise estimate based on the massive unlabeled set, while the second term debiases the estimator using the labeled set. For more general statistical quantities (e.g., quantiles or regression coefficients), the approach uses an estimating equation or risk minimizer, with the rectifier defined as the expectation of the difference between gradients evaluated at the true outcome and the prediction.

2. Methodological Details and Algorithms

PPI instantiates its methodology for a broad range of classical estimands:

• Mean estimation: Uses the above formula. The variance of $\hat{\theta}_{\text{PP}}$ is decomposed into the variance due to predictions and the variance of the error correction, leading to confidence intervals:

$$\text{CI} = \hat{\theta}_{\text{PP}} \pm 1.96 \cdot \sqrt{ \frac{\sigma_f^2}{N} + \frac{\sigma^2_{f-Y}}{n} }.$$

• Quantile estimation: Expresses the $q$-quantile as a pinball loss minimizer. The correction term becomes:

$$\Delta_\theta = E[ \mathbb{I}\{Y \leq \theta\} - \mathbb{I}\{f(X) \leq \theta\} ],$$

and confidence intervals are constructed by "rectifying" the imputed empirical distribution function.

• Regression coefficients (linear/logistic regression): The target parameter is the minimizer of an empirical risk. The imputed prediction is treated as an outcome; corrections for each coordinate are calculated using gradients on labeled versus predicted data. Confidence intervals are then formed using a normal approximation.

For all cases, Theorem 1 establishes that as long as one constructs confidence intervals for the rectifier (labeled data) and the imputed component (unlabeled predictions), their Minkowski sum is a valid confidence set for $\theta^*$.

3. Assumptions and Statistical Validity

PPI imposes only mild requirements: standard regularity for moment existence and a nondegeneracy condition on the estimation problem (e.g., that the population subgradient equation $E[g_{\theta^*}(X, Y)] = 0$ holds for convex problems). No assumptions are made about the ML predictor's form, consistency, or unbiasedness—a marked contrast to most inference after prediction methods.

The bias due to predictions is always estimated and corrected for on the labeled data, ensuring finite-sample validity. The coverage guarantees are both nonasymptotic and asymptotic, and the estimator is always valid for the pre-specified confidence level, regardless of the predictive quality of $f$.

4. Efficiency Gains and Trade-offs

A haLLMark of PPI is that estimation efficiency is directly linked to the accuracy of the ML predictions and the size of the unlabeled dataset:

• If $f(X)$ is highly accurate, then $\text{Var}(f(X) - Y)$ is small, resulting in confidence intervals substantially narrower than those derived from labeled data alone.
• As $N$ grows, the predictive variance can become negligible compared to the error correction variance estimated over $n$ examples.
• In the extreme of noisy predictions or small $N$, the benefits diminish, but validity is retained.

This means PPI can yield dramatic improvements in effective sample size, sometimes achieving the same inferential power with significantly fewer labeled points. However, in cases where predictions add little information, PPI's intervals may align with, but not outperform, classical methods.

5. Case Studies and Applications

PPI has demonstrated its practical benefits in a spectrum of scientific and applied settings:

• Proteomics: Used AlphaFold structure predictions to estimate odds ratios between protein disorder regions and post-translational modifications. Rectifying with experimental labels produced valid, substantially tighter confidence intervals.
• Astronomy (Galaxy Zoo): Human-labeled spiral fraction estimation was enhanced by using ML predictions, reducing interval width beyond classical inference.
• Genomics: Transformer-based gene expression predictions were combined with sparse labeled data to estimate expression quantiles with tight uncertainty bounds.
• Remote sensing: Accurate forest cover fraction and deforestation rates were inferred using satellite image predictions rectified by costly human-verified labels.
• Census and social sciences: Regression analyses benefited from "cheap" ML-predicted outcomes, reducing the labeled sample size needed for valid inference.
• Ecology (plankton counting): ML-based abundance predictions were adjusted with human-labeled subsets to efficiently estimate population sizes.

In all these cases, PPI intervals are compared to standard (label-only) and naive (ML-only) approaches, consistently showing superior or equivalent inferential efficiency with statistical rigor.

6. Broader Implications and Future Directions

PPI represents a general methodology for unlocking the potential of "weakly-labeled" or "prediction-rich" regimes, where the primary limiting factor has traditionally been the cost of high-quality annotation. Its agnosticism to the ML method ensures broad applicability, including in contexts involving black-box or ensemble predictors.

The framework's ability to yield valid $p$-values and confidence intervals makes it an appealing normalization protocol across scientific disciplines. PPI substantially lowers the barrier to data-efficient discovery in fields ranging from natural sciences to social research and policy analysis. Its algorithms extend to risk minimization, hypothesis testing, and more general estimating equation-based models.

Potential directions for expansion include adaptation to stratification, cross-fitting, integration with Bayesian inference, extensions to federated and local inference, and further optimizations to computational and statistical efficiency. As data modalities and ML predictors continue to evolve, PPI provides a foundation for rigorous inference in ever more complex and data-rich environments.