  Item response theory (IRT) is a non-linear generative probabilistic paradigm
for using exams to identify, quantify, and compare latent traits of
individuals, relative to their peers, within a population of interest. In
pre-existing multidimensional IRT methods, one requires a factorization of the
test items. For this task, linear exploratory factor analysis is used, making
IRT a posthoc model. We propose skipping the initial factor analysis by using a
sparsity-promoting horseshoe prior to perform factorization directly within the
IRT model so that all training occurs in a single self-consistent step. Being a
hierarchical Bayesian model, we adapt the WAIC to the problem of dimensionality
selection. IRT models are analogous to probabilistic autoencoders. By binding
the generative IRT model to a Bayesian neural network (forming a probabilistic
autoencoder), one obtains a scoring algorithm consistent with the interpretable
Bayesian model. In some IRT applications the black-box nature of a neural
network scoring machine is desirable. In this manuscript, we demonstrate
within-IRT factorization and comment on scoring approaches.
