Bivariate Markov processes (BMPs) described by Ephraim and Mark (2012) consist of a pair of stochastic processes in the continuous time, one observable and the other latent, that are jointly Markov. In the present article, the navigation behavior and the learning process of a user of a web-based tutoring system are jointly modeled as BMPs constrained by assumptions that are coherent with the concepts of competence-based knowledge space theory. Such constraints are expressed as formal assumptions about the web-based system and about the nature of the learning process. Scenarios are considered where the observed process is the navigation of an individual through the pages of an intelligent tutoring system, whereas the latent learning process consists of transitions among states in a competence structure. The approach seems to be rather general and flexible in modeling learning scenarios with different assumptions. As an example, BMP models are developed for some exemplary scenarios. Maximum likelihood parameter estimation via expectation–maximization algorithm is presented. The results of a simulation study showed that the parameter values are well-recovered by the estimation algorithm. The results of the application of a bivariate Markov model to the real data of students navigating the intelligent tutoring system Stat-Knowlab showed that the proposed approach provides useful insight into students’ learning processes.

Ephraim和Mark（2012）描述的双变量马尔可夫过程（BMP）由连续时间中的一对随机过程组成，一个可观察到，另一个潜在，这两个过程共同称为马尔可夫。在本文中，基于Web的补习系统的导航行为和用户的学习过程被联合建模为BMP，这些BMP受与基于能力的知识空间理论的概念相一致的假设约束。这些约束表示为关于基于Web的系统以及学习过程的性质的形式上的假设。考虑的场景是，所观察到的过程是个人通过智能补习系统页面的导航，而潜伏学习过程则是由能力结构中各州之间的过渡组成。在用不同的假设对学习情景进行建模时，该方法似乎相当通用且灵活。作为示例，针对一些示例性场景开发了BMP模型。提出了通过期望最大化算法的最大似然参数估计。仿真研究的结果表明，估计算法可以很好地恢复参数值。将二元马尔可夫模型应用于在智能辅导系统Stat-Knowlab中导航的学生的真实数据的结果表明，该方法为学生的学习过程提供了有用的见识。仿真研究的结果表明，估计算法可以很好地恢复参数值。将二元马尔可夫模型应用于在智能辅导系统Stat-Knowlab中导航的学生的真实数据的结果表明，该方法为学生的学习过程提供了有用的见识。仿真研究的结果表明，估计算法可以很好地恢复参数值。将二元马尔可夫模型应用于在智能辅导系统Stat-Knowlab中导航的学生的真实数据的结果表明，该方法为学生的学习过程提供了有用的见识。