Properties of the Bayesian Knowledge Tracing Model

Bayesian knowledge tracing has been used widely to model student learning. However, the name \Bayesian knowledge tracing" has been applied to two related, but distinct, models: The first is the Bayesian knowledge tracing Markov chain which predicts the student-averaged probability of a correct application of a skill. We present an analytical solution to this model and show that it is a function of three parameters and has the functional form of an exponential. The second form is the Bayesian knowledge tracing hidden Markov model which can use the individual student's performance at each opportunity to apply a skill to update the conditional probability that the student has learned that skill. We use a fixed point analysis to study solutions of this model and find a range of parameters where it has the desired behavior.