Abstract Knowledge Space Theory (KST) aims at modeling the hierarchical relations between items or skills in a learning process. For example, when studying mathematics in school, students first need to master the rules of summation before being able to learn multiplication. In KST, the knowledge states of individuals are represented by means of partially ordered latent classes. In probabilistic KST models, conditional probability parameters are introduced to model transitions from latent knowledge states to observed response patterns. Since these models account for discrete data by assuming a finite number of latent states, they can be represented by Multinomial Processing Tree (MPT) models (i.e., binary decision trees with parameters referring to the conditional probabilities of entering different states). We prove that standard probabilistic models of KST such as the Basic Local Independence Model (BLIM) and the Simple Learning Model (SLM) can be represented as specific instances of MPT models. Given this close link, MPT methods may be applied to address theoretical and practical issues in KST. By highlighting the MPT–KST link and its implications for modeling violations of local stochastic independence in Item Response Theory (IRT), we hope to facilitate an exchange of theoretical results, statistical methods, and software across these different domains of mathematical psychology and psychometrics.

中文翻译：

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用多项式处理树模型表示知识空间理论的概率模型

摘要知识空间理论（KST）旨在建模学习过程中项目或技能之间的层次关系。例如，在学校学习数学时，学生首先需要掌握求和的规则，然后才能学习乘法。在 KST 中，个人的知识状态通过部分有序的潜在类来表示。在概率 KST 模型中，条件概率参数被引入到从潜在知识状态到观察到的响应模式的模型转换中。由于这些模型通过假设有限数量的潜在状态来考虑离散数据，因此它们可以由多项处理树 (MPT) 模型表示（即，二元决策树的参数是指进入不同状态的条件概率）。我们证明了 KST 的标准概率模型，例如基本局部独立模型 (BLIM) 和简单学习模型 (SLM)，可以表示为 MPT 模型的特定实例。鉴于这种密切联系，MPT 方法可用于解决 KST 中的理论和实践问题。通过强调 MPT-KST 链接及其对项目响应理论 (IRT) 中局部随机独立性的建模违规的影响，我们希望促进跨数学心理学和心理测量学的这些不同领域的理论结果、统计方法和软件的交流。