Variations in student responses to a multiple exam for latent group discovery
Questions in an exam are detectors of student competency. Students are detectors of the correct answers in a test. What is the variation in the student’s model of the correct exam? The precision error equations can be used to construct a covariance matrix for the students instead of the questions. What makes the difference is what is being averaged. When you want to use a test to tell you something about the questions, you average over the students. The covariance matrix is then indexed by the questions. When you want the test to tell you about the students, you average over the questions. The covariance matrix is indexed by students.
All of this suggests that it would be possible to build a completely parameter-less approach to detecting latent groups in students. This would be a different approach from that involved in topic models which use a specific probability distribution — the Dirichlet distribution. In this approach, you would assume a number of groups and arbitrarily assign students to these groups in a probabilistic fashion (60% group 1, 30% group 2, etc.). One can then see how well this group distribution predicts the observed covariance matrix by use of non-commutative harmonic analysis. Group assignment is thereby completely determined by the data — no parameters are needed.
