this sequence. We let $\randfasp_{\infty}$ denote the space of consistent feature allocation sequences, of which each random feature allocation is a random element. The sigma field associated with this space is generated by the finite-dimensional sigma fields of the restricted random feature allocations $\randfa_{n}$.

We say that $\randfa_{\infty}$ is exchangeable if $\randfa_{\infty} \eqd \sigma(\randfa_{\infty})$ for every finite permutation $\perm$. That is, for every permutation $\perm$ that changes no indices above $N$ for some $N < \infty$, we require $\randfa_{N} \eqd \sigma(\randfa_{N})$, where $\randfa_{N}$ is the restriction of $\randfa_{\infty}$ to $[N]$.

Labeling features {#sec:label}
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Now that we have defined consistent, exchangeable random feature allocations, we want to characterize the class of all distributions on these allocations. We begin by considering some alternative representations of the feature allocation that are not merely useful, but indeed key to some of our later results.

A number of authors have made use of matrices as a way of representing feature allocations [@griffiths:2006:infinite; @thibaux:2007:hierarchical; @doshi:2009:variational]. This representation, while a boon for intuition in some regards, requires care because a matrix presupposes an order on the features, which is not a part of the feature allocation a priori. We cover this distinction in some detail next.

