A probabilistic framework for the polytomous extension of knowledge space theory (KST) is proposed. It consists in a probabilistic model, called polytomous local independence model, that is developed as a generalization of the basic local independence model. The algorithms for computing “maximum likelihood” (ML) and “minimum discrepancy” (MD) estimates of the model parameters have been derived and tested in a simulation study. Results show that the algorithms differ in their capability of recovering the true parameter values. The ML algorithm correctly recovers the true values, regardless of the manipulated variables. This is not totally true for the MD algorithm. Finally, the model has been applied to a real polytomous data set collected in the area of psychological assessment. Results show that it can be successfully applied in practice, paving the way to a number of applications of KST outside the area of knowledge and learning assessment. Electronic supplementary material The online version of this article (10.1007/s11336-020-09722-5) contains supplementary material, which is available to authorized users.

中文翻译：

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将基本局部独立模型扩展到多分数据

提出了知识空间理论（KST）的多态扩展的概率框架。它包含一个概率模型，称为多分局部独立模型，它是作为基本局部独立模型的推广而开发的。计算模型参数的“最大似然”(ML) 和“最小差异”(MD) 估计的算法已在模拟研究中推导出和测试。结果表明，这些算法在恢复真实参数值的能力上有所不同。无论操纵变量如何，ML 算法都能正确恢复真实值。对于 MD 算法，这并不完全正确。最后，该模型已应用于在心理评估领域收集的真实多分数据集。结果表明它可以成功地应用于实践，为知识和学习评估领域之外的许多 KST 应用铺平了道路。电子补充材料本文的在线版本（10.1007/s11336-020-09722-5）包含补充材料，可供授权用户使用。