  Bayesian nonparametric hierarchical priors are highly effective in providing
flexible models for latent data structures exhibiting sharing of information
between and across groups. Most prominent is the Hierarchical Dirichlet Process
(HDP), and its subsequent variants, which model latent clustering between and
across groups. The HDP, may be viewed as a more flexible extension of Latent
Dirichlet Allocation models (LDA), and has been applied to, for example, topic
modelling, natural language processing, and datasets arising in health-care. We
focus on analogous latent feature allocation models, where the data structures
correspond to multisets or unbounded sparse matrices. The fundamental
development in this regard is the Hierarchical Indian Buffet process (HIBP),
which utilizes a hierarchy of Beta processes over J groups, where each group
generates binary random matrices, reflecting within group sharing of features,
according to beta-Bernoulli IBP priors. To encompass HIBP versions of
non-Bernoulli extensions of the IBP, we introduce hierarchical versions of
general spike and slab IBP. We provide explicit novel descriptions of the
marginal, posterior and predictive distributions of the HIBP and its
generalizations which allow for exact sampling and simpler practical
implementation. We highlight common structural properties of these processes
and establish relationships to existing IBP type and related models arising in
the literature. Examples of potential applications may involve topic models,
Poisson factorization models, random count matrix priors and neural network
models
