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Bayesian nonparametric hierarchical priors provide flexible models forsharing of information within and across groups. We focus on latent featureallocation models, where the data structures correspond to multisets orunbounded sparse matrices. The fundamental development in this regard is theHierarchical Indian Buffet process (HIBP), devised by Thibaux and Jordan(2007). However, little is known in terms of explicit tractable descriptions ofthe joint, marginal, posterior and predictive distributions of the HIBP. Weprovide explicit novel descriptions of these quantities, in the Bernoulli HIBPand general spike and slab HIBP settings, which allows for exact sampling andsimpler practical implementation. We then extend these results to the morecomplex setting of hierarchies of general HIBP (HHIBP). The generality of ourframework allows one to recognize important structure that may otherwise bemasked in the Bernoulli setting, and involves characterizations via dynamicmixed Poisson random count matrices. Our analysis shows that the standardchoice of hierarchical Beta processes for modeling across group sharing is notideal in the classic Bernoulli HIBP setting proposed by Thibaux and Jordan(2007), or other spike and slab HIBP settings, and we thus indicate tractablealternative priors.

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