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.

翻译：本研究确立了Rasch模型下贝叶斯自适应测试方法的一致性，填补了其大样本保证理论研究的空白。尽管贝叶斯方法因其有限样本性能及规避冷启动等问题的能力而受到认可，但其在非独立同分布结构中的渐近性质（特别是严格证明）仍存缺失。我们推导出在给定项目序列下潜在特质后验分布收敛至真值的条件，并证明贝叶斯估计量在先验设定错误情况下仍保持稳健性。随后将分析拓展至测试过程中内生选择项目的自适应项目选择方法。此外，我们为项目选择问题建立了贝叶斯决策理论框架，并提出一种新型选择策略，使测试过程与估计量最优性能相匹配。这些理论成果为自适应测试中的贝叶斯方法奠定了理论基础，与其有限样本优势的既有证据形成互补。