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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.01358 |
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| _version_ | 1866915530498113536 |
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| author | Marusic, Juraj Medina, Marco Avella Rush, Cynthia |
| author_facet | Marusic, Juraj Medina, Marco Avella Rush, Cynthia |
| contents | We provide a theoretical framework for a wide class of generalized posteriors that can be viewed as the natural Bayesian posterior counterpart of the class of M-estimators in the frequentist world. We call the members of this class M-posteriors and show that they are asymptotically normally distributed under mild conditions on the M-estimation loss and the prior. In particular, an M-posterior contracts in probability around a normal distribution centered at an M-estimator, showing frequentist consistency and suggesting some degree of robustness depending on the reference M-estimator. We formalize the robustness properties of the M-posteriors by a new characterization of the posterior influence function and a novel definition of breakdown point adapted for posterior distributions. We illustrate the wide applicability of our theory in various popular models and illustrate their empirical relevance in some numerical examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_01358 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A theoretical framework for M-posteriors: frequentist guarantees and robustness properties Marusic, Juraj Medina, Marco Avella Rush, Cynthia Statistics Theory Machine Learning We provide a theoretical framework for a wide class of generalized posteriors that can be viewed as the natural Bayesian posterior counterpart of the class of M-estimators in the frequentist world. We call the members of this class M-posteriors and show that they are asymptotically normally distributed under mild conditions on the M-estimation loss and the prior. In particular, an M-posterior contracts in probability around a normal distribution centered at an M-estimator, showing frequentist consistency and suggesting some degree of robustness depending on the reference M-estimator. We formalize the robustness properties of the M-posteriors by a new characterization of the posterior influence function and a novel definition of breakdown point adapted for posterior distributions. We illustrate the wide applicability of our theory in various popular models and illustrate their empirical relevance in some numerical examples. |
| title | A theoretical framework for M-posteriors: frequentist guarantees and robustness properties |
| topic | Statistics Theory Machine Learning |
| url | https://arxiv.org/abs/2510.01358 |