Random signed measures

TL;DR

This paper introduces and studies random signed measures, extending point process theory to real-valued measures, providing existence results, representations, and applications in Bayesian nonparametrics for topics, graphs, and regression.

Contribution

It offers the first existence theorem for random signed measures, extends Kingman's representation to the real-valued case, and develops new Bayesian models using these measures.

Findings

Established existence of random signed measures

Extended Kingman's representation to signed measures

Developed Bayesian models for topics, graphs, and regression

Abstract

Point processes and, more generally, random measures are ubiquitous in modern statistics. However, they can only take positive values, which is a severe limitation in many situations. In this work, we introduce and study random signed measures, also known as real-valued random measures, and apply them to constrcut various Bayesian non-parametric models. In particular, we provide an existence result for random signed measures, allowing us to obtain a canonical definition for them and solve a 70-year-old open problem. Further, we provide a representation of completely random signed measures (CRSMs), which extends the celebrated Kingman's representation for completely random measures (CRMs) to the real-valued case. We then introduce specific classes of random signed measures, including the Skellam point process, which plays the role of the Poisson point process in the real-valued case, and the Gaussian random measure. We use the theoretical results to develop two Bayesian nonparametric models -- one for topic modeling and the other for random graphs -- and to investigate mean function estimation in Bayesian nonparametric regression.