2015/10/25 by Lancelot F. James, James, Lancelot F., Peter Orbanz +3
Economics, Econometrics and Finance · #Economics of Agriculture and Food Markets #FOS: Mathematics #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.1510.07309
openalex publication_date 2015/10/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study random families of subsets of ℕ that are similar to exchangeable random partitions, but do not require constituent sets to be disjoint: Each element of ℕ may be contained in multiple subsets. One class of such objects, known as Indian buffet processes, has become a popular tool in machine learning. Based on an equivalence between Indian buffet and scale-invariant Poisson processes, we identify a random scaling variable whose role is similar to that played in exchangeable partition models by the total mass of a random measure. Analogous to the construction of exchangeable partitions from normalized subordinators, random families of sets can be constructed from randomly scaled subordinators. Coupling to a heavy-tailed scaling variable induces a power law on the number of sets containing the first n elements. Several examples, with properties desirable in applications, are derived explicitly. A relationship to exchangeable partitions is made precise as a correspondence between scaled subordinators and Poisson-Kingman measures, generalizing a result of Arratia, Barbour and Tavare on scale-invariant processes.