Allowing Growing Dimensional Binary Outcomes via the Multivariate Probit Indian Buffet Process

There is a rich literature on infinite latent feature models, with much of the focus being on the Indian Buffet Process (IBP). The current literature focuses on the case in which latent features are generated independently, so that there is no within-sample dependence in which features are selected. Motivated by ecology applications in which latent features correspond to which species are discovered in a sample, we propose a new class of dependent infinite latent feature models. Our construction starts with a probit Indian buffet process, and then incorporates dependence among features through a correlation matrix in a latent multivariate Gaussian random variable. We show that the proposed approach preserves many appealing theoretical properties of the IBP. To address the challenge of modeling and inference for a growing-dimensional correlation matrix, we propose Bayesian modeling approaches to reduce effective dimensionality. We additionally describe extensions to incorporate covariates and hierarchical structure to enable borrowing of information. For posterior inference, we describe Markov chain Monte Carlo sampling algorithms and efficient approximations. Simulation studies and applications to fungal biodiversity data provide compelling support for the new modeling class relative to competitors.