Weak Factor Recovery in Sparse Bayesian Factor Models under Exchangeable Shrinkage Process Prior

Sparse Bayesian factor models provide a flexible framework for learning low-dimensional latent structure in multivariate data, but inference on the number of active factors remains challenging when some factors are weak or highly sparse. We study weak factor recovery under a variance based exchangeable shrinkage process (ESP) prior. We derive the posterior factor activation mechanism and show that, conditional on the global and local shrinkage parameters, activation is governed by the standardised loading energy together with the ESP prior activation odds. This yields an explicit activation boundary and clarifies how signal strength, sparsity, the number of observed variables, and the truncation level affect recovery. Simulation experiments show a clear transition between shrinkage and recovery around this boundary and demonstrate that sparsity can substantially affect activation even when raw loading energy is held fixed. They further show that factor activation and recovery of the relative loading pattern across variables can behave differently. These results clarify when weak latent factors become difficult to distinguish from redundant components under ESP shrinkage.