Variational Bayesian inference for sparse item response theory models

Item response theory (IRT) model is a widely appreciated statistical method in exploring the relationship between individual latent traits and item responses. In this paper, a sparse IRT model is established to address the sparsity of factor loadings. A global and local shrinkage prior is imposed to penalize the factor loadings: the global parameter controls the amount of shrinkage at the column levels, while the local parameter adjusts the penalty of factor loadings within each column. We develop a variational Bayesian procedure to conduct posterior inference. By exploiting a stochastic representation for logistic function, we frame sparse IRT model as a mixture model mixing with Pólya-Gamma distribution. Such a strategy admits a conjugate posterior for the latent quantity, thus leading to a straightforward posterior computation. We assess the performance of the proposed method via a simulation study. A real example related to personality assessment is analysed to illustrate the usefulness of methodology.