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We study random families of subsets of $\mathbb{N}$ that are similar toexchangeable random partitions, but do not require constituent sets to bedisjoint: Each element of ${\mathbb{N}}$ may be contained in multiple subsets.One class of such objects, known as Indian buffet processes, has become apopular tool in machine learning. Based on an equivalence between Indian buffetand scale-invariant Poisson processes, we identify a random scaling variablewhose role is similar to that played in exchangeable partition models by thetotal mass of a random measure. Analogous to the construction of exchangeablepartitions from normalized subordinators, random families of sets can beconstructed from randomly scaled subordinators. Coupling to a heavy-tailedscaling variable induces a power law on the number of sets containing the first$n$ elements. Several examples, with properties desirable in applications, arederived explicitly. A relationship to exchangeable partitions is made preciseas a correspondence between scaled subordinators and Poisson-Kingman measures,generalizing a result of Arratia, Barbour and Tavare on scale-invariantprocesses.

Scaled subordinators and generalizations of the Indian buffet process