Poisson Hierarchical Indian Buffet Processes for Within and Across Group Sharing of Latent Features-With Indications for Microbiome Species Sampling Models

In this work, we present a comprehensive Bayesian posterior analysis of what we term Poisson Hierarchical Indian Buffet Processes, designed for complex random sparse count species sampling models that allow for the sharing of information across and within groups. This analysis covers a potentially infinite number of species and unknown parameters, which, within a Bayesian machine learning context, we are able to learn from as more information is sampled. To achieve our refined results, we employ a range of methodologies drawn from Bayesian latent feature models, random occupancy models, and excursion theory. Despite this complexity, our goal is to make our findings accessible to practitioners, including those who may not be familiar with these areas. To facilitate understanding, we adopt a pseudo-expository style that emphasizes clarity and practical utility. We aim to express our findings in a language that resonates with experts in microbiome and ecological studies, addressing gaps in modeling capabilities while acknowledging that we are not experts ourselves in these fields. This approach encourages the use of our models as basic components of more sophisticated frameworks employed by domain experts, embodying the spirit of the seminal work on the Dirichlet Process. Ultimately, our refined posterior analysis not only yields tractable computational procedures but also enables practical statistical implementation and provides a clear mapping to relevant quantities in microbiome analysis.