Random signed measures
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866929597878108160 |
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| author | Passeggeri, Riccardo |
| author_facet | Passeggeri, Riccardo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_12623 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Random signed measures Passeggeri, Riccardo Probability Statistics Theory Methodology 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. |
| title | Random signed measures |
| topic | Probability Statistics Theory Methodology |
| url | https://arxiv.org/abs/2411.12623 |