Regularized Bayesian methods in psychometric modeling

Abstract

Psychometric modeling aims to uncover the underlying information between an individual’s responses to items, providing valuable insights into their cognitive abilities, personality traits, and learning progress. One major challenge lies in the model estimation, where the likelihood function can be multidimensional and intractable. While Markov Chain Monte Carlo (MCMC) simulations have been widely used for parameter estimation in Bayesian frameworks, the computational burden associated with MCMC methods, particularly for high-dimensional datasets, limits their scalability and practicality in many real-world applications. This thesis introduces regularized Bayesian methods in psychometric modeling, which operates by shrinking irrelevant or redundant parameters, thus improving model interpretability and reducing overfitting. In addition, the partially confirmatory framework is incorporated to connect different levels of substantive knowledge availability within a unified exploratory-confirmatory continuum, enabling relations to be made partially from substantive knowledge and data. Four studies were conducted within psychometric modeling: Chapter 2: Two regularized Bayesian algorithms for Q-matrix inference based on saturated cognitive diagnosis modeling were developed to address the dual needs of accuracy and efficiency. One is a Gibbs sampler, an MCMC-based method for its precision, and the other is an efficient regularized variational Bayesian expectation maximization (VBEM) algorithm. This dual-channel approach extends the model’s applicability across a variety of settings. Chapter 3: A comparative study on various Bayesian regularization methods to Q-matrix inference in CDMs was investigated, where models were formulated under different link functions, using different penalized forms with different strategies for Q-matrix recovery. Chapter 4: A regularized variational approximation method was introduced for factor analysis, being an efficient alternative for its counterpart - the recently developed partially confirmatory factor analysis (PCFA) modeling based on MCMC estimation. Chapter 5: Building on the regularized variational approximation method for PCFA modeling, a variational Bayesian expectation-maximization algorithm (VBEM) for variable selection was developed. This extends the multiple-indicators multiple-causes (MIMIC) model within the broader context of structural equation modeling (SEM), incorporating variable selection not only in the measurement model but also in the structural model.