We start by defining an *a priori labeled feature allocation*. Let $\hat{F}_{N,1}$ be the collection of indices in $[N]$ with feature 1, let $\hat{F}_{N,2}$ be the collection of indices in $[N]$ with feature 2, etc. Here, we think of a priori labels as being the ordered, positive natural numbers. This specification is different from (a priori unlabeled) feature allocations as defined above since there is nothing to distinguish the features in a feature allocation other than, potentially, the members of a feature. Consider the following analogy: an a priori labeled feature allocation is to a feature allocation as a classification is to a clustering. Indeed, when each index $n$ belongs to exactly one feature in an a priori feature allocation, feature 1 is just class 1, feature 2 is class 2, and so on.

Another way to think of an a priori labeled feature allocation of $[N]$ is as a matrix of $N$ rows filled with zeros and ones. Each column is associated with a feature. The $(n,k)$ entry in the matrix is one if index $n$ is in feature $k$ and zero otherwise. However, just as—contrary to the classification case—we do not know the ordering of clusters in a 