  We derive a construction of the beta process that allows for the atoms with
significant measure to be drawn first. Our representation is based on an
extension of the Sethuraman (1994) construction of the Dirichlet process, and
therefore we refer to it as a stick-breaking construction. Our first proof uses
a limiting case argument of finite arrays. To this end, we present a finite
sieve approximation to the beta process that parallels that of Ishwaran &
Zarepour (2002) and prove its convergence to the beta process. We give a second
proof of the construction using Poisson process machinery. We use the Poisson
process to derive almost sure truncation bounds for the construction. We
conclude the paper by presenting an efficient sampling algorithm for
beta-Bernoulli and beta-negative binomial process models.
