feature allocation* $\randfa_{N}$ of $[N]$ is a random element of $\randfasp_{N}$. Let $\perm: \mathbb{N} \rightarrow \mathbb{N}$ be a finite permutation. That is, for some finite value $N_{\perm}$, we have $\perm(n) = n$ for all $n > N_{\sigma}$. Further, for any feature $\blockfa \subset \mathbb{N}$, denote the permutation applied to the feature as follows: $\perm(\blockfa) := \{\perm(n): n \in \blockfa\}$. For any feature allocation $F_{N}$, denote the permutation applied to the feature allocation as follows: $\perm(\randfa_{N}) := \{\perm(\blockfa): \blockfa \in \randfa_{N}\}$. Finally, let $\randfa_{N}$ be a random feature allocation of $[N]$. Then we say that a random feature allocation $\randfa_{N}$ is *exchangeable* if $\randfa_{N} \eqd \sigma(\randfa_{N})$ for every permutation of $[N]$.

In addition to exchangeability, we also require our distributions on feature allocations to exhibit a notion of coherence across different ranges of the index. Intuitively, we often imagine the indices as denoting time, and it is natural to suppose that the randomness at time $n$ is coherent with the randomness at time $n+1$. More formally, we say that a feature allocation $\detfa_{M}$ of $[M]$ is a *restriction* of a feature allocation $\detfa_{N}$ of $[N]$ for $M < N$ if $$\detfa_{M} = \{\blockfa \cap [M]: \blockfa \in \detfa_{N}, \blockfa \cap [M] 