Prediction-Powered Conditional Inference

TL;DR

This work introduces a reproducing kernel-based localization method that learns a data-adaptive weight function from covariates and reformulates the target conditional moment at the test point as a weighted unconditional moment, yielding a prediction-powered estimator and confidence interval that reduce variance when the predictor is informative while preserving validity regardless of predictor accuracy.

Abstract

We study prediction-powered conditional inference in the setting where labeled data are scarce, unlabeled covariates are abundant, and a black-box machine-learning predictor is available. The goal is to perform statistical inference on conditional functionals evaluated at a fixed test point, such as conditional means, without imposing a parametric model for the conditional relationship. Our approach combines localization with prediction-based variance reduction. First, we introduce a reproducing kernel-based localization method that learns a data-adaptive weight function from covariates and reformulates the target conditional moment at the test point as a weighted unconditional moment. Second, we incorporate machine-learning predictions through a correction-based decomposition of this localized moment, yielding a prediction-powered estimator and confidence interval that reduce variance when the predictor is informative while preserving validity regardless of predictor accuracy. We establish nonasymptotic error bounds and minimax-optimal convergence rates for the resulting estimator, prove pointwise asymptotic normality with consistent variance estimation, and provide an explicit variance decomposition that characterizes how machine-learning predictions and unlabeled covariates improve statistical efficiency. Numerical experiments on simulated and real datasets demonstrate valid conditional coverage and substantially sharper confidence intervals than alternative methods.

Figures (10)

• Figure 1: Protocol for prediction-powered conditional inference. The procedure takes labeled data, unlabeled covariates, an ML predictor, and a test point $x_0$ as inputs. A localization step uses the covariate distribution to learn weights that capture the local structure around $x_0$. The upper block estimates a bias correction from labeled data, while the lower block computes a plug-in term using predictions from the unlabeled data. These components are combined to produce a valid confidence interval $\mathcal{C}(x_0)$ for the conditional target $\theta_0(x_0)$.
• Figure 2: Empirical $\sigma^2_{Y-f}$, $\sigma^2_Y$, and $\sigma^2_{f}$ for the conditional mean at different (age, sex) test points in the census income data. Results are based on $1000$ replications.
• Figure 3: Empirical coverage rate and average width of nominal $95\%$ confidence intervals for the conditional mean at the test point ($\text{age}=70, \text{sex}=1$) in the census income data. The red horizontal line indicates the nominal $95\%$ coverage level. Results are based on $1000$ replications.
• Figure 4: Diabetes progression data illustration. Left: split conformal predictive intervals for the outcome $Y$ at test covariates $X$. Right: a bootstrap uncertainty band for the conditional mean $\theta_0(x)=\mathbb{E}[Y| X=x]$.
• Figure 5: Empirical RMSE, coverage rate, and average width of nominal 95% confidence intervals for the conditional mean evaluated at different test points in the simulated data. Results are based on 1000 replications per test point.

Theorems & Definitions (44)

• proposition 1
• theorem 1
• theorem 2
• theorem 3
• corollary 1
• proposition 2
• theorem 4
• theorem 5
• corollary 2
• lemma 1: Theorem 3.8 of adams2003sobolev