  Multivariate Item Response Theory (MIRT) is sought-after widely by applied
researchers looking for interpretable (sparse) explanations underlying response
patterns in questionnaire data. There is, however, an unmet demand for such
sparsity discovery tools in practice. Our paper develops a Bayesian platform
for binary and ordinal item MIRT which requires minimal tuning and scales well
on large datasets due to its parallelizable features. Bayesian methodology for
MIRT models has traditionally relied on MCMC simulation, which cannot only be
slow in practice, but also often renders exact sparsity recovery impossible
without additional thresholding. In this work, we develop a scalable Bayesian
EM algorithm to estimate sparse factor loadings from mixed continuous, binary,
and ordinal item responses. We address the seemingly insurmountable problem of
unknown latent factor dimensionality with tools from Bayesian nonparametrics
which enable estimating the number of factors. Rotations to sparsity through
parameter expansion further enhance convergence and interpretability without
identifiability constraints. In our simulation study, we show that our method
reliably recovers both the factor dimensionality as well as the latent
structure on high-dimensional synthetic data even for small samples. We
demonstrate the practical usefulness of our approach on three datasets: an
educational assessment dataset, a quality-of-life measurement dataset, and a
bio-behavioral dataset. All demonstrations show that our tool yields
interpretable estimates, facilitating interesting discoveries that might
otherwise go unnoticed under a pure confirmatory factor analysis setting.
