使用变分近似的动态非补偿性多维项目反应理论模型

中文摘要 AI 辅助

多维项目反应理论（MIRT）是一种统计测试理论，能够根据测试中的作答精确估计学习者的多个潜在技能。针对MIRT，已提出了补偿性和非补偿性两类模型：前者假设各技能可以相互补充，而后者假设它们不能相互补充。在许多测量多种技能的测试中，非补偿性假设是令人信服的；因此，将非补偿性模型应用于此类数据对于实现无偏且准确的估计至关重要。与测试不同，在日常学习中，潜在技能会随时间变化。为了监测技能的成长，研究者们已经探索了MIRT模型的动态扩展。然而，其中大多数假设了补偿性模型，迄今为止，尚未提出能够在非补偿性假设下再现技能连续潜在状态的模型。为了在非补偿性假设下实现准确的技能追踪，我们通过结合线性动态系统和非补偿性模型，提出了一种非补偿性MIRT模型的动态扩展。这导致了技能的后验分布变得复杂，我们通过最小化近似后验与真实后验之间的Kullback-Leibler散度，用高斯分布来近似该后验。模型参数的学习算法通过蒙特卡洛期望最大化推导得出。模拟研究验证了所提方法能够准确再现潜在技能，而动态补偿性模型则遭受显著的估计不足误差。此外，在实际数据集上的实验表明，我们的动态非补偿性模型能够推断出实用的技能追踪，并阐明非补偿性与补偿性模型在技能追踪上的差异。

英文摘要

Multidimensional item response theory (MIRT) is a statistical test theory that precisely estimates multiple latent skills of learners from the responses in a test. Both compensatory and non-compensatory models have been proposed for MIRT: the former assumes that each skill can complement other skills, whereas the latter assumes they cannot. This non-compensatory assumption is convincing in many tests that measure multiple skills; therefore, applying non-compensatory models to such data is crucial for achieving unbiased and accurate estimation. In contrast to tests, latent skills will change over time in daily learning. To monitor the growth of skills, dynamical extensions of MIRT models have been investigated. However, most of them assumed compensatory models, and a model that can reproduce continuous latent states of skills under the non-compensatory assumption has not been proposed thus far. To enable accurate skill tracing under the non-compensatory assumption, we propose a dynamical extension of non-compensatory MIRT models by combining a linear dynamical system and a non-compensatory model. This results in a complicated posterior of skills, which we approximate with a Gaussian distribution by minimizing the Kullback-Leibler divergence between the approximated posterior and the true posterior. The learning algorithm for the model parameters is derived through Monte Carlo expectation maximization. Simulation studies verify that the proposed method is able to reproduce latent skills accurately, whereas the dynamical compensatory model suffers from significant underestimation errors. Furthermore, experiments on an actual data set demonstrate that our dynamical non-compensatory model can infer practical skill tracing and clarify differences in skill tracing between non-compensatory and compensatory models.

发表机构

• the graduate university for advanced studies, sokendai（综合研究大学院大学（Sokendai））
• the institute of statistical mathematics（统计数学研究所）

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