clustering a priori, we do not a priori know the ordering of features in a feature allocation. To make use of a matrix representation for a feature allocation, we will need to introduce or find such an order.

The reasoning above suggests that introducing an order for features in a feature allocations would be useful. The next example illustrates that the probability $\mbp(\randfa_{N} = \detfa_{N})$ in some sense undercounts features when they contain exactly the same indices: e.g., $\blockfa_{j} = \blockfa_{k}$ for some $j \ne k$. This fact will suggest to us that it is not merely useful, but indeed a key point of our theoretical development, to introduce an ordering on features.

\[ex:two\_bern\]

Given $q_{A}, q_{B} \in (0,1)$, draw $\indpart_{n,A} \iid \bern(q_{A})$ and $\indpart_{n,B} \iid \bern(q_{B})$, independently, and construct the random feature allocation by collecting those indices with successful draws: $$\randfa_{N} := \{\{n: n \le N, \indpart_{n,A} = 1\}, \{n: n \le N,
\indpart_{n,B} = 1\}\}.$$ One caveat here is that if either of the two sets in the multiset $\randfa_{N}$ is empty, we do not include it in the allocation. Note that calling the features $A$ and $B$ was merely for the purposes of construction, and in defining 