feature allocation can similarly be recovered from the order-of-appearance label collections $(\indfao_{n})$.

We can also recover a new *random ordered feature allocation* $\labrandfa_{N}$ from the sequence $(\indfar_{n})$. In particular, $\labrandfa_{N}$ is the sequence—rather than the collection—of features $\{n: x_{k} \in \indfar_{n}\}$ such that the feature with smallest label $\randlabfa_{k}$ occurs first, and so on. This construction achieves our goal of avoiding the combinatorial factors needed to work with the distribution of $\randfa_{N}$, while retaining exchangeability and consistency.

\[ex:feat\_6\_unif\_rand\_lab\_spec\] Once more, consider the feature allocation $$\detfa_{6} = \{\{2,5,4\},\{3,4\},\{6,4\},\{3\},\{3\}\}.$$ and the uniform random labeling in . If it happens that $\randlabfa_{4} < \randlabfa_{5} < \randlabfa_{2} < \randlabfa_{1} < \randlabfa_{3}$, then the random ordered feature allocation is $$\labdetfa_{6} = (\{3\},\{6,4\},\{3\},\{2,5,4\},\{3,4\}).$$

Recall that we were motivated by to produce such a random ordering scheme to avoid obfuscating combinatorial factors in the probability of a feature allocation. From another perspective, these factors arise because the random labeling is in some sense more natural than alternative labelings; again, consider random labels as iid parameters for each feature. While order-of-appearance labeling is common due to its pleasant aesthetic representation in matrix form (compare and ), one must be careful to remember that the resulting label sets $(\indfao_{n})$ 