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.

这篇论文通过基本数学概念推导出了一种限制 Bayesian Knowledge Tracing 参数空间的方法，并引入了一个新的算法来估计参数。

基于参数束缚的贝叶斯知识追踪

Parametric Constraints for Bayesian Knowledge Tracing from First  Principles

In theoretical cognitive science, there is a tension between highly
structured models whose parameters have a direct psychological interpretation
and highly complex, general-purpose models whose parameters and representations
are difficult to interpret. The former typically provide more insight into
cognition but the latter often perform better. This tension has recently
surfaced in the realm of educational data mining, where a deep learning
approach to predicting students' performance as they work through a series of
exercises---termed deep knowledge tracing or DKT---has demonstrated a stunning
performance advantage over the mainstay of the field, Bayesian knowledge
tracing or BKT. In this article, we attempt to understand the basis for DKT's
advantage by considering the sources of statistical regularity in the data that
DKT can leverage but which BKT cannot. We hypothesize four forms of regularity
that BKT fails to exploit: recency effects, the contextualized trial sequence,
inter-skill similarity, and individual variation in ability. We demonstrate
that when BKT is extended to allow it more flexibility in modeling statistical
regularities---using extensions previously proposed in the literature---BKT
achieves a level of performance indistinguishable from that of DKT. We argue
that while DKT is a powerful, useful, general-purpose framework for modeling
student learning, its gains do not come from the discovery of novel
representations---the fundamental advantage of deep learning. To answer the
question posed in our title, knowledge tracing may be a domain that does not
require `depth'; shallow models like BKT can perform just as well and offer us
greater interpretability and explanatory power.

本文考虑深度知识追踪（DKT）在预测学生学习表现时的优越性及其与贝叶斯知识追踪（BKT）模型的比较，提出了 BKT 模型的拓展方案，最终指出虽然 DKT 是一种实用的学生学习模型，但其对数据的优化不在于发现新的表现形式。