  Bayesian Knowledge Tracing (BKT) is a probabilistic model of a learner's
state of mastery corresponding to a knowledge component. It considers the
learner's state of mastery as a "hidden" or latent binary variable and updates
this state based on the observed correctness of the learner's response using
parameters that represent transition probabilities between states. BKT is often
represented as a Hidden Markov Model and the Expectation-Maximization (EM)
algorithm is used to infer these parameters. However, this algorithm can suffer
from several issues including producing multiple viable sets of parameters,
settling into a local minima, producing degenerate parameter values, and a high
computational cost during fitting. This paper takes a "from first principles"
approach to deriving constraints that can be imposed on the BKT parameter
space. Starting from the basic mathematical truths of probability and building
up to the behaviors expected of the BKT parameters in real systems, this paper
presents a mathematical derivation that results in succinct constraints that
can be imposed on the BKT parameter space. Since these constraints are
necessary conditions, they can be applied prior to fitting in order to reduce
computational cost and the likelihood of issues that can emerge from the EM
procedure. In order to see that promise through, the paper further introduces a
novel algorithm for estimating BKT parameters subject to the newly defined
constraints. While the issue of degenerate parameter values has been reported
previously, this paper is the first, to our best knowledge, to derive the
constrains from first principles while also presenting an algorithm that
respects those constraints.
