Supercharging Bayesian Inference with Reliable AI-Informed Priors
Authors: Jongwoo Choi, Sean O'Hagan
Organizations: Department of Statistics, University of Connecticut · Department of Statistics, University of Chicago
Abstract
Modern predictive systems encode beliefs that can act as useful prior information for statistical inference in data-limited settings. Using them for prior construction introduces a tradeoff: an informative prior built from a predictive model can sharpen inference from limited data, but also risks propagating error from the model into the posterior. We propose a framework for AI-informed prior elicitation that mitigates this tension by rectifying the AI-induced law that generates synthetic data before using it to inform a prior. The rectified law can be embedded into synthetic data-driven prior elicitation techniques, including as a base measure in a Dirichlet process (DP) prior on the data-generating process. We refer to the resulting prior and corresponding posterior as the rectified AI prior and rectified AI posterior. We establish Gaussian asymptotics for the rectified AI posterior under non-vanishing prior strength and derive a first-order expression for its centering bias. Our rectified AI priors substantially reduce bias compared to standard approaches, improve the coverage of credible intervals, and make AI-powered prior information more reliable. We additionally apply the rectified AI prior to a real skin disease classification task and show that it can meaningfully boost predictive performance.
We establish a formal connection between the decades-old surrogate outcome model in biostatistics and economics and the emerging field of prediction-powered inference (PPI). The connection treats predictions from pre-trained models, prevalent in the age of AI, as cost-effective surrogates for expensive outcomes. Building on the surrogate outcomes literature, we develop recalibrated prediction-powered inference, a more efficient approach to statistical inference than existing PPI proposals. Our method departs from the existing proposals by using flexible machine learning techniques to learn the optimal ``imputed loss'' through a step we call recalibration. Importantly, the method always improves upon the estimator that relies solely on the data with available true outcomes, even when the optimal imputed loss is estimated imperfectly, and it achieves the smallest asymptotic variance among PPI estimators if the estimate is consistent. Computationally, our optimization objective is convex whenever the loss function that defines the target parameter is convex. We further analyze the benefits of recalibration, both theoretically and numerically, in several common scenarios where machine learning predictions systematically deviate from the outcome of interest. We demonstrate significant gains in effective sample size over existing PPI proposals via three applications leveraging state-of-the-art machine learning/AI models.
Many applications require statistically valid inference across many related tasks, while using only a handful of high-quality labels per hypothesis. In AI evaluation, these tasks may correspond to model behaviors across prompts, subgroups, or hypotheses; in social science surveys, they may correspond to related questions, populations, or measurement conditions. Prediction-powered inference (PPI) uses abundant but inexpensive proxy measurements to improve inference from limited, ground-truth labels, but commonly used methods treat tasks independently and therefore fail to exploit shared structure across related tasks. This limitation is especially important in settings where only a small number of labels are available per task. To address this issue, we introduce a multi-task prediction-powered inference framework that uses labeled data from related tasks to improve power while preserving task-specific inference. Our methods exploit the shared structure in the proxy-ground-truth relationship through cross-task recalibration, while retaining within-task rectification and power tuning to construct accurate point estimates and confidence intervals. We prove that efficiency gains beyond power-tuned PPI are only possible when the proxy-ground-truth relationship contains nonlinear structure; affine cross-task recalibrations are asymptotically equivalent to using the original proxy. We complement our theoretical findings with experiments on synthetic and semi-synthetic datasets, as well as a case study auditing language models on election-related information during the 2024 U.S. presidential election. Using a large human-annotation study, we show that cross-task recalibration can substantially reduce confidence interval widths when labels are scarce.
Nicolas Emmenegger, Ellery Stahler, Chara Podimata
Large language models enable inexpensive AI-generated annotations, but using them reliably for causal inference remains challenging. Naively pooling AI and human data induces bias, while existing methods such as Prediction-Powered Inference (PPI; Angelopoulos et al., 2023a) treat AI outputs as proxies of true labels -- an assumption often violated for generative model outputs in practice. We propose Generative Augmented Inference (GAI), a framework that treats AI outputs as general, potentially high-dimensional informative features for learning human labels rather than as surrogates. GAI flexibly models this relationship using nonparametric methods, enabling consistent estimation and valid inference from combined human and AI data. We establish asymptotic normality and show that, under random labeling, GAI strictly improves asymptotic efficiency over human-data-only estimation whenever AI outputs are informative for true labels. Empirical studies on real-world datasets demonstrate that GAI significantly reduces estimation error and improves confidence interval quality across diverse generative data sources relative to human-only and PPI-based estimation.