  We develop new representations for the Levy measures of the beta and gamma
processes. These representations are manifested in terms of an infinite sum of
well-behaved (proper) beta and gamma distributions. Further, we demonstrate how
these infinite sums may be truncated in practice, and explicitly characterize
truncation errors. We also perform an analysis of the characteristics of
posterior distributions, based on the proposed decompositions. The
decompositions provide new insights into the beta and gamma processes (and
their generalizations), and we demonstrate how the proposed representation
unifies some properties of the two. This paper is meant to provide a rigorous
foundation for and new perspectives on Levy processes, as these are of
increasing importance in machine learning.
