The Stick-Breaking Construction of the Beta Process as a Poisson Process

We show that the stick-breaking construction of the beta process due to Paisley et al. (2010) can be obtained from the characterization of the beta process as a Poisson process. This is achieved by showing that the mean measure of the underlying Poisson process is equal to that of the beta process. We then present a number of consequences of this derivation; in particular, we show how it can be used to derive error bounds on truncated beta processes that are significantly tighter than those in the literature.