Bayesian Prediction under Moment Conditioning

TL;DR

This paper introduces a Bayesian framework for predictive inference under partial information expressed as moment constraints, unifying several existing methods and providing explicit uncertainty quantification.

Contribution

It develops a finite Bayesian approach using curvature-adaptive exchangeable updating under moment constraints, connecting empirical likelihood and GMM within a unified predictive geometry.

Findings

Provides explicit discrete-Gaussian mixture for predictive uncertainty.

Derives finite-sample bounds based on the information-geometric Hessian.

Recovers classical results like de Finetti's coherence and asymptotic normality as special cases.

Abstract

Prediction is a central task of statistics and machine learning, yet many inferential settings provide only partial information, typically in the form of moment constraints or estimating equations. We develop a finite, fully Bayesian framework for propagating such partial information through predictive distributions. Building on de Finetti's representation theorem, we construct a curvature-adaptive version of exchangeable updating that operates directly under finite constraints, yielding an explicit discrete-Gaussian mixture that quantifies predictive uncertainty. The resulting finite-sample bounds depend on the smallest eigenvalue of the information-geometric Hessian, which measures the curvature and identification strength of the constraint manifold. This approach unifies empirical likelihood, Bayesian empirical likelihood, and generalized method-of-moments estimation within a common predictive geometry. On the operational side, it provides computable curvature-sensitive uncertainty bounds for constrained prediction; on the theoretical side, it recovers de Finetti's coherence, Doob's martingale convergence and local asymptotic normality as limiting cases of the same finite mechanism. Our framework thus offers a constructive bridge between partial information and full Bayesian prediction.