Levy Measure Decompositions for the Beta and Gamma Processes
This provides a rigorous mathematical foundation for Levy processes, which are increasingly important in machine learning applications.
The authors developed new mathematical representations for the Levy measures of beta and gamma processes using infinite sums of proper beta and gamma distributions, and demonstrated practical truncation methods with explicit error characterization.
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.