Bayesian nonparametric hierarchical priors provide flexible models for sharing of information within and across groups. We focus on 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), devised by Thibaux and Jordan (2007). However, little is known in terms of explicit tractable descriptions of the joint, marginal, posterior and predictive distributions of the HIBP. We provide explicit novel descriptions of these quantities, in the Bernoulli HIBP and general spike and slab HIBP settings, which allows for exact sampling and simpler practical implementation. We then extend these results to the more complex setting of hierarchies of general HIBP (HHIBP). The generality of our framework allows one to recognize important structure that may otherwise be masked in the Bernoulli setting, and involves characterizations via dynamic mixed Poisson random count matrices. Our analysis shows that the standard choice of hierarchical Beta processes for modeling across group sharing is not ideal in the classic Bernoulli HIBP setting proposed by Thibaux and Jordan (2007), or other spike and slab HIBP settings, and we thus indicate tractable alternative priors.

翻译：贝叶斯非参数分层先验为组内与组间信息共享提供了灵活的建模框架。我们聚焦于潜在特征分配模型，其中数据结构对应于多重集或无界稀疏矩阵。此类研究的基础是Thibaux与Jordan（2007）提出的分层印度自助餐过程（HIBP）。然而，关于HIBP的联合分布、边缘分布、后验分布及预测分布的显式可解描述仍知之甚少。我们针对伯努利HIBP及广义spike-and-slab HIBP设定，提供了这些量的显式新描述，从而实现了精确采样与更简便的实际应用。随后，我们将这些结果推广至更复杂的广义HIBP分层结构（HHIBP）设定中。我们框架的通用性有助于识别在伯努利设定中可能被掩盖的重要结构，并涉及通过动态混合泊松随机计数矩阵的刻画。分析表明，Thibaux与Jordan（2007）提出的经典伯努利HIBP设定或其他spike-and-slab HIBP设定中，用于建模组间共享的标准分层贝塔过程并非理想选择，因此我们指出了可行的替代先验形式。