Foundation Priors

2025-11-30

AI Summary

This paper addresses the misuse of synthetic data generated by foundation models (e.g., large language models) as if it were genuine empirical observations. We formalize such synthetic data as a “foundation prior”—a structured, subjective prior encoding user-specific beliefs and trust in the model. Methodologically, we introduce the first formulation of synthetic outputs as an exponentially tilted generalized prior, seamlessly embeddable within standard Bayesian inference, explicitly integrating prompt engineering, user epistemic priors, and model reliability. Our contributions are threefold: (1) establishing a rigorous theoretical foundation for synthetic data as a structured subjective prior; (2) developing a computationally tractable and interpretable joint inference framework; and (3) enabling safe, compliant deployment in latent-variable modeling, experiment design guidance, and complex model optimization—thereby mitigating empirical misinterpretation and ensuring methodological integrity.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Other Foundations of Machine Learning
• Humans and AI: Other Foundations of Human Computation & AI

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Economics, Online Markets and Human Computation: Trust and reliance of crowd workers and data experts on GenAI

Abstract

Foundation models, and in particular large language models, can generate highly informative responses, prompting growing interest in using these ''synthetic'' outputs as data in empirical research and decision-making. This paper introduces the idea of a foundation prior, which shows that model-generated outputs are not as real observations, but draws from the foundation prior induced prior predictive distribution. As such synthetic data reflects both the model's learned patterns and the user's subjective priors, expectations, and biases. We model the subjectivity of the generative process by making explicit the dependence of synthetic outputs on the user's anticipated data distribution, the prompt-engineering process, and the trust placed in the foundation model. We derive the foundation prior as an exponential-tilted, generalized Bayesian update of the user's primitive prior, where a trust parameter governs the weight assigned to synthetic data. We then show how synthetic data and the associated foundation prior can be incorporated into standard statistical and econometric workflows, and discuss their use in applications such as refining complex models, informing latent constructs, guiding experimental design, and augmenting random-coefficient and partially linear specifications. By treating generative outputs as structured, explicitly subjective priors rather than as empirical observations, the framework offers a principled way to harness foundation models in empirical work while avoiding the conflation of synthetic ''facts'' with real data.

Problem

• Model-generated synthetic data reflects user biases and model patterns
• The framework treats generative outputs as structured subjective priors
• It integrates synthetic data into statistical workflows without conflating with real observations

Innovation

• Foundation prior as structured subjective priors for synthetic data
• Exponential-tilted Bayesian update with trust parameter weighting
• Incorporating synthetic data into statistical and econometric workflows