Truncated random measures
© 2019 ISI/BS. Completely random measures (CRMs) and their normalizations are a rich source of Bayesian nonparametric priors. Examples include the beta, gamma, and Dirichlet processes. In this paper, we detail two major classes of sequential CRM representations—series representations and superpositi...
| Main Authors: | , , , |
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| Other Authors: | |
| Format: | Article |
| Language: | English |
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Bernoulli Society for Mathematical Statistics and Probability
2021
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| Online Access: | https://hdl.handle.net/1721.1/134549 |
| _version_ | 1826208479616958464 |
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| author | Campbell, Trevor Huggins, Jonathan H How, Jonathan P Broderick, Tamara |
| author2 | Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory |
| author_facet | Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory Campbell, Trevor Huggins, Jonathan H How, Jonathan P Broderick, Tamara |
| author_sort | Campbell, Trevor |
| collection | MIT |
| description | © 2019 ISI/BS. Completely random measures (CRMs) and their normalizations are a rich source of Bayesian nonparametric priors. Examples include the beta, gamma, and Dirichlet processes. In this paper, we detail two major classes of sequential CRM representations—series representations and superposition representations—within which we organize both novel and existing sequential representations that can be used for simulation and posterior inference. These two classes and their constituent representations subsume existing ones that have previously been developed in an ad hoc manner for specific processes. Since a complete infinite-dimensional CRM cannot be used explicitly for computation, sequential representations are often truncated for tractability. We provide truncation error analyses for each type of sequential representation, as well as their normalized versions, thereby generalizing and improving upon existing truncation error bounds in the literature. We analyze the computational complexity of the sequential representations, which in conjunction with our error bounds allows us to directly compare representations and discuss their relative efficiency. We include numerous applications of our theoretical results to commonly-used (normalized) CRMs, demonstrating that our results enable a straightforward representation and analysis of CRMs that has not previously been available in a Bayesian nonparametric context. |
| first_indexed | 2024-09-23T14:05:59Z |
| format | Article |
| id | mit-1721.1/134549 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T14:05:59Z |
| publishDate | 2021 |
| publisher | Bernoulli Society for Mathematical Statistics and Probability |
| record_format | dspace |
| spelling | mit-1721.1/1345492024-01-02T18:40:04Z Truncated random measures Campbell, Trevor Huggins, Jonathan H How, Jonathan P Broderick, Tamara Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory Massachusetts Institute of Technology. Laboratory for Information and Decision Systems © 2019 ISI/BS. Completely random measures (CRMs) and their normalizations are a rich source of Bayesian nonparametric priors. Examples include the beta, gamma, and Dirichlet processes. In this paper, we detail two major classes of sequential CRM representations—series representations and superposition representations—within which we organize both novel and existing sequential representations that can be used for simulation and posterior inference. These two classes and their constituent representations subsume existing ones that have previously been developed in an ad hoc manner for specific processes. Since a complete infinite-dimensional CRM cannot be used explicitly for computation, sequential representations are often truncated for tractability. We provide truncation error analyses for each type of sequential representation, as well as their normalized versions, thereby generalizing and improving upon existing truncation error bounds in the literature. We analyze the computational complexity of the sequential representations, which in conjunction with our error bounds allows us to directly compare representations and discuss their relative efficiency. We include numerous applications of our theoretical results to commonly-used (normalized) CRMs, demonstrating that our results enable a straightforward representation and analysis of CRMs that has not previously been available in a Bayesian nonparametric context. 2021-10-27T20:05:31Z 2021-10-27T20:05:31Z 2019 2019-10-28T17:28:40Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/134549 en 10.3150/18-BEJ1020 Bernoulli Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Bernoulli Society for Mathematical Statistics and Probability arXiv |
| spellingShingle | Campbell, Trevor Huggins, Jonathan H How, Jonathan P Broderick, Tamara Truncated random measures |
| title | Truncated random measures |
| title_full | Truncated random measures |
| title_fullStr | Truncated random measures |
| title_full_unstemmed | Truncated random measures |
| title_short | Truncated random measures |
| title_sort | truncated random measures |
| url | https://hdl.handle.net/1721.1/134549 |
| work_keys_str_mv | AT campbelltrevor truncatedrandommeasures AT hugginsjonathanh truncatedrandommeasures AT howjonathanp truncatedrandommeasures AT brodericktamara truncatedrandommeasures |