Abstract—Nonparametric Bayesian methods offer a convenient paradigm to deal with uncertain model structure. However, priors such as the (hierarchical) Dirichlet process prior on partitions and the Indian buffet process prior on binary matrices are not always flexible enough to express our prior beliefs. We propose a much larger family of nonparametric exchangeable priors by relaxing the concept of consistency. We discuss the consequences of this point of view and propose novel ways to specify and learn these priors. In particular, we introduce new flexible priors and inference procedures to extend the DP, HDP and IBP models. An experiment on text data illustrates how flexible priors can be useful to increase our modeling capabilities. I.