Bayesian factor models provide a flexible framework for the analysis of high-dimensional data, combining dimensionality reduction with probabilistic quantification of uncertainty and, importantly, allowing the latent dimension itself to be learned from the data. In recent years, sparse formulations have further expanded their scope. Imposed factor loadings sparsity, indeed, can improve interpretability and identifiability.

Motivated by applications involving multiple related studies, I will first discuss APAFA (Bortolato and Canale, 2026), a Bayesian multi-study factor model designed to flexibly learn which latent components are shared across studies and which are study-specific, while avoiding overly restrictive assumptions on their dependence structure.

Building on this, I will then turn to a complementary perspective in which sparsity is imposed on the factor scores. This leads to a new class of identifiable sparse factor models in which interpretable latent structure can be recovered without relying on conventional constraints on the loadings.

Together, these developments illustrate how different notions of sparsity can be exploited to address some of the central inferential challenges in modern Bayesian factor modelling.

Bortolato, E., & Canale, A. (2026). Adaptive partition factor analysis. Journal of the American Statistical Association, http://doi.org/10.1080/01621459.2026.2621517