are not exchangeable. We will use random labeling extensively below since, among other nice properties, it preserves exchangeability of the sets of feature labels associated with the indices.

Exchangeable feature probability function {#sec:efpf}
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In general, given a probability of a random feature allocation, $\mbp(\randfa_{N} = \detfa_{N})$, we can find the probability of a random ordered feature allocation $\mbp(\labrandfa_{N} = \labdetfa_{N})$ as follows. Let $H$ be the number of distinct features of $\randfa_{N}$, and let $(\ok_{1},\ldots,\ok_{H})$ be the multiplicities of these distinct features in decreasing order. Then $$\label{eq:order_mult}
    \mbp(\labrandfa_{N} = \labdetfa_{N})
        = \binom{ K }{ \ok_{1}, \ldots, \ok_{H} }^{-1} \mbp(\randfa_{N} = \detfa_{N}),$$ where $$\binom{ K }{ \ok_{1}, \ldots, \ok_{H} } := \frac{ K! }{ \ok_{1}! \cdots \ok_{H}!}.$$

For partitions, the effect of this multiplicative factor is the same across all partitions with the same number of clusters; for some number of clusters $K$, it is just $1 / K!$. In the general feature case, the multiplicative factor may be different for different feature configurations with the same number of features.

Consider $\randfa_{N}$ constructed as in . Denote the sizes of the two features by $M_{N,1}$ and $M_{N,2}$. Then $$\begin{aligned}
    \nonumber
    \mbp(\labrandfa_{N} = \labdetfa_{N})
        &= \frac{1}{2} q_{A}^{M_{N,1}} (1-q_{A})^{N - M_{N,1}} q_{B}^{M_{N,2}} 