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**Feature allocations, probability functions, and paintboxes.**
*(English)*
Zbl 1329.62278

Summary: The problem of inferring a clustering of a data set has been the subject of much research in Bayesian analysis, and there currently exists a solid mathematical foundation for Bayesian approaches to clustering. In particular, the class of probability distributions over partitions of a data set has been characterized in a number of ways, including via exchangeable partition probability functions (EPPFs) and the Kingman paintbox. Here, we develop a generalization of the clustering problem, called feature allocation, where we allow each data point to belong to an arbitrary, non-negative integer number of groups, now called features or topics. We define and study an “exchangeable feature probability function” (EFPF) – analogous to the EPPF in the clustering setting – for certain types of feature models. Moreover, we introduce a “feature paintbox” characterization – analogous to the Kingman paintbox for clustering – of the class of exchangeable feature models. We provide a further characterization of the subclass of feature allocations that have EFPF representations.

### MSC:

62H30 | Classification and discrimination; cluster analysis (statistical aspects) |

62F15 | Bayesian inference |

60G09 | Exchangeability for stochastic processes |

60G57 | Random measures |