of EFPFs is characterized in a different way; we will see that the class of feature allocations with EFPFs is equivalent to the class of FAs obtained from feature frequency models together with singletons of a certain distribution. Indeed, we will find that the class of clusterings with EPPFs is, in this way, analogous to the class of feature allocations with EFPFs when both are considered as subclasses of the general class of feature allocations. The diagram also shows several examples that we use to illustrate and develop our theory.

Feature allocations {#sec:feature}
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We consider data sets with $N$ points and let the points be indexed by the integers $[N] := \{1,2,\ldots,N\}$. We also explicitly allow $N = \infty$, in which case the index set is $\mathbb{N} = \{1,2,3,\ldots\}$. For our discussion of feature allocations and partitioning it is sufficient to focus on the indices rather than the data points; thus, we will be discussing models for collections of subsets of $[N]$ and $\mathbb{N}$.

Our introduction to feature allocations follows @broderick:2012:clusters. We define a *feature allocation* $\detfa_{N}$ of $[N]$ to be a multiset of non-empty subsets of $[N]$ called *features*, such that no index $n$ belongs to infinitely many 