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