On the Consistency of Bayesian Adaptive Testing under the Rasch Model

TL;DR

This paper proves the asymptotic consistency of Bayesian adaptive testing under the Rasch model, providing theoretical guarantees for large-sample behavior and robust estimation even with prior mis-specification.

Contribution

It establishes the first rigorous asymptotic consistency results for Bayesian adaptive testing under the Rasch model, including adaptive item selection and decision-theoretic frameworks.

Findings

Posterior distributions converge to true latent traits.

Bayesian estimators are robust to prior mis-specification.

Proposed item selection aligns with optimal estimator performance.

Abstract

This study establishes the consistency of Bayesian adaptive testing methods under the Rasch model, addressing a gap in the literature on their large-sample guarantees. Although Bayesian approaches are recognized for their finite-sample performance and capability to circumvent issues such as the cold-start problem; however, rigorous proofs of their asymptotic properties, particularly in non-i.i.d. structures, remain lacking. We derive conditions under which the posterior distributions of latent traits converge to the true values for a sequence of given items, and demonstrate that Bayesian estimators remain robust under the mis-specification of the prior. Our analysis then extends to adaptive item selection methods in which items are chosen endogenously during the test. Additionally, we develop a Bayesian decision-theoretical framework for the item selection problem and propose a novel selection that aligns the test process with optimal estimator performance. These theoretical results provide a foundation for Bayesian methods in adaptive testing, complementing prior evidence of their finite-sample advantages.