Random signed measures

Point processes and, more generally, random measures are ubiquitous in statistics. However, they can only take positive values, which is a severe limitation in many situations. In this presentation, we introduce random signed measures, also known as real-valued random measures, and apply them to construct various Bayesian non-parametric models.
In particular, we provide an existence result for random signed measures, allowing us to obtain a canonical definition for random signed measures and solve a long-standing 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 sentiment topic modeling and the other for random graphs -- and to investigate mean function estimation in Bayesian nonparametric regression.