# PPI++: Efficient Prediction-Powered Inference

arXiv: https://arxiv.org/abs/2311.01453 · HTML: https://mlsanity.com/paper/2311.01453/

## Key facts

- Cited by 117 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 33 references that are in the map.
- First posted on arXiv 2 Nov 2023.

## Builds on (most cited first, 9 of 9)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [XGBoost: A Scalable Tree Boosting System](https://mlsanity.com/paper/1603.02754/) (2016)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Retiring Adult: New Datasets for Fair Machine Learning](https://mlsanity.com/paper/2108.04884/) (2021)
- [Prediction-Powered Inference](https://mlsanity.com/paper/2301.09633/) (2023)
- [Veridical Data Science](https://mlsanity.com/paper/1901.08152/) (2019)
- [Cross-Prediction-Powered Inference](https://mlsanity.com/paper/2309.16598/) (2023)
- [Semi-supervised Inference: General Theory and Estimation of Means](https://mlsanity.com/paper/1606.07268/) (2016)
- [High-dimensional semi-supervised learning: in search for optimal inference of the mean](https://mlsanity.com/paper/1902.00772/) (2019)

## Built on by (most cited first, 30 of 117)

- [AutoEval Done Right: Using Synthetic Data for Model Evaluation](https://mlsanity.com/paper/2403.07008/) (2024)
- [Limits to scalable evaluation at the frontier: LLM as Judge won't beat twice the data](https://mlsanity.com/paper/2410.13341/) (2024)
- [Active Statistical Inference](https://mlsanity.com/paper/2403.03208/) (2024)
- [Predictions as Surrogates: Revisiting Surrogate Outcomes in the Age of AI](https://mlsanity.com/paper/2501.09731/) (2025)
- [How to Correctly Report LLM-as-a-Judge Evaluations](https://mlsanity.com/paper/2511.21140/) (2025)
- [Prediction-Powered Ranking of Large Language Models](https://mlsanity.com/paper/2402.17826/) (2024)
- [How Benchmark Prediction from Fewer Data Misses the Mark](https://mlsanity.com/paper/2506.07673/) (2025)
- [Assumption-Lean and Data-Adaptive Post-Prediction Inference](https://mlsanity.com/paper/2311.14220/) (2023)
- [Stratified Prediction-Powered Inference for Hybrid Language Model Evaluation](https://mlsanity.com/paper/2406.04291/) (2024)
- [Noisy but Valid: Robust Statistical Evaluation of LLMs with Imperfect Judges](https://mlsanity.com/paper/2601.20913/) (2026)
- [Cost-Optimal Active AI Model Evaluation](https://mlsanity.com/paper/2506.07949/) (2025)
- [Prediction-Powered E-Values](https://mlsanity.com/paper/2502.04294/) (2025)
- [Prediction De-Correlated Inference: A safe approach for post-prediction inference](https://mlsanity.com/paper/2312.06478/) (2023)
- [A Framework for Efficient Model Evaluation through Stratification, Sampling, and Estimation](https://mlsanity.com/paper/2406.07320/) (2024)
- [Multi-Armed Bandits With Machine Learning-Generated Surrogate Rewards](https://mlsanity.com/paper/2506.16658/) (2025)
- [Reliable and Scalable Robot Policy Evaluation with Imperfect Simulators](https://mlsanity.com/paper/2510.04354/) (2025)
- [Semi-Supervised Risk Control via Prediction-Powered Inference](https://mlsanity.com/paper/2412.11174/) (2024)
- [Adaptive Prediction-Powered AutoEval with Reliability and Efficiency Guarantees](https://mlsanity.com/paper/2505.18659/) (2025)
- [A Note on the Prediction-Powered Bootstrap](https://mlsanity.com/paper/2405.18379/) (2024)
- [Task-Agnostic Machine-Learning-Assisted Inference](https://mlsanity.com/paper/2405.20039/) (2024)
- [Efficient Randomized Experiments Using Foundation Models](https://mlsanity.com/paper/2502.04262/) (2025)
- [Prediction-Powered Causal Inferences](https://mlsanity.com/paper/2502.06343/) (2025)
- [Demystifying Prediction Powered Inference](https://mlsanity.com/paper/2601.20819/) (2026)
- [Multiple-Prediction-Powered Inference](https://mlsanity.com/paper/2603.27414/) (2026)
- [Local Prediction-Powered Inference](https://mlsanity.com/paper/2409.18321/) (2024)
- [AI-Powered Bayesian Inference](https://mlsanity.com/paper/2502.19231/) (2025)
- [No Free Lunch: Non-Asymptotic Analysis of Prediction-Powered Inference](https://mlsanity.com/paper/2505.20178/) (2025)
- [Probably Approximately Correct Labels](https://mlsanity.com/paper/2506.10908/) (2025)
- [Doubly-Robust LLM-as-a-Judge: Externally Valid Estimation with Imperfect Personas](https://mlsanity.com/paper/2509.22957/) (2025)
- [Regression for the Mean: Auto-Evaluation and Inference with Few Labels through Post-hoc Regression](https://mlsanity.com/paper/2411.12665/) (2024)

## Abstract (arXiv)

We present PPI++: a computationally lightweight methodology for estimation and inference based on a small labeled dataset and a typically much larger dataset of machine-learning predictions. The methods automatically adapt to the quality of available predictions, yielding easy-to-compute confidence sets -- for parameters of any dimensionality -- that always improve on classical intervals using only the labeled data. PPI++ builds on prediction-powered inference (PPI), which targets the same problem setting, improving its computational and statistical efficiency. Real and synthetic experiments demonstrate the benefits of the proposed adaptations.