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We describe the combinatorial stochastic process underlying a sequence ofconditionally independent Bernoulli processes with a shared beta process hazardmeasure. As shown by Thibaux and Jordan [TJ07], in the special case when theunderlying beta process has a constant concentration function and a finite andnonatomic mean, the combinatorial structure is that of the Indian buffetprocess (IBP) introduced by Griffiths and Ghahramani [GG05]. By reinterpretingthe beta process introduced by Hjort [Hjo90] as a measurable family ofDirichlet processes, we obtain a simple predictive rule for the general case,which can be thought of as a continuum of Blackwell-MacQueen urn schemes (orequivalently, one-parameter Hoppe urn schemes). The corresponding measurablefamily of Perman-Pitman-Yor processes leads to a continuum of two-parameterHoppe urn schemes, whose ordinary component is the three-parameter IBPintroduced by Teh and G\"or\"ur [TG09], which exhibits power-law behavior, asfurther studied by Broderick, Jordan, and Pitman [BJP12]. The idea extends toarbitrary measurable families of exchangeable partition probability functionsand gives rise to generalizations of the beta process with matching buffetprocesses. Finally, in the same way that hierarchies of Dirichlet processeswere given Chinese restaurant franchise representations by Teh, Jordan, Beal,and Blei [Teh+06], one can construct representations of sequences of Bernoulliprocesses directed by hierarchies of beta processes (and their generalizations)using the stochastic process we uncover.

The continuum-of-urns scheme, generalized beta and Indian buffet processes, and hierarchies thereof