Empirical Likelihood with Generative AI

2026-05-29

AI Summary

This study addresses the challenge of parameter identification and inference when the likelihood function is intractable, particularly in settings where external information—such as synthetic data from generative AI—can supplement moment conditions. The authors propose a nonparametric Bayesian framework based on exponentially tilted empirical likelihood, which enables efficient and parallelizable inference by projecting posterior samples from a Dirichlet process onto the subspace satisfying the given moment constraints. Innovatively treating generative AI–synthesized data as prior information on observable variables, the work establishes posterior consistency and a Bernstein–von Mises theorem for the projected posterior under both vanishing and persistent prior regimes. In an application to predicting stock returns using overnight news headlines, the method effectively leverages AI-generated auxiliary data for implicit regularization—even without explicit parametric priors—yielding substantially improved predictive performance.

Abstract

Moment conditions are widely used to identify parameters in models where the full likelihood is either unknown or intentionally left unspecified. Empirical likelihood methods address this problem by assigning probability weights to the observed data so that the sample moment conditions hold exactly. Building on this idea, we propose a nonparametric Bayesian framework based on exponentially tilted empirical likelihood. This Bayesian formulation is particularly appealing in settings where prior information is more naturally specified on the observables rather than on the underlying parameters. Such settings arise in the presence of auxiliary data sources or synthetic data generated by modern generative AI models.Inference proceeds by projecting posterior draws from a Dirichlet process onto the moment-restricted model, yielding a computationally efficient procedure that is naturally amenable to parallelization. We establish new Bernstein--von Mises and consistency theorems for the resulting projection posterior under both vanishing-prior and persistent-prior regimes. In an application to return prediction using overnight news headlines, we show that AI-generated auxiliary data can provide a useful source of indirect regularization when informative priors on the parameter itself are unavailable.

Problem

• empirical likelihood
• moment conditions
• generative AI
• nonparametric Bayesian
• auxiliary data

Innovation

• empirical likelihood
• generative AI
• nonparametric Bayesian
• moment conditions
• posterior projection