Bayesian Analysis of Generalized Hierarchical Indian Buffet Processes for Within and Across Group Sharing of Latent Features

Bayesian nonparametric hierarchical priors are highly effective in providing flexible models for latent data structures exhibiting sharing of information within and across groups. In this work, we focus on latent feature allocation models, where the data structures correspond to multi-sets or unbounded sparse matrices, which we refer to as generalized hierarchical Indian Buffet processes (HIBP). These are based on hierarchical versions of generalized spike and slab Indian Buffet processes (IBP), where the fundamental development in this regard is the Bernoulli-based HIBP, devised by Thibaux-Jordan (2007), as a hierarchical extension of the IBP devised by Griffiths-Ghahramani (2005). With a focus on Bayesian inference, we provide novel explicit descriptions of the joint, marginal, and posterior distributions of the HIBP, significantly advancing our understanding of these processes. Our results allow for exact sampling for the otherwise complex joint marginal distributions. We provide a general characterization of their posterior distributions as well as highlight bottlenecks for practical implementation. Our main focus then shifts to specific tractable results for the remarkable case of Poisson HIBP, which correspond to generalizations of mixed Poisson random count models arising in genetics, imaging, topic modeling, random occupancy, and species sampling models. We show they also have important relations to Bayesian nonparametric latent class models appearing in the literature. Furthermore, we show that all general HIBP may be coupled to Poisson HIBP, allowing for further analysis of such processes.