{"ID":2862274,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.01358","arxiv_id":"2510.01358","title":"A theoretical framework for M-posteriors: frequentist guarantees and robustness properties","abstract":"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.","short_abstract":"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 condi...","url_abs":"https://arxiv.org/abs/2510.01358","url_pdf":"https://arxiv.org/pdf/2510.01358v1","authors":"[\"Juraj Marusic\",\"Marco Avella Medina\",\"Cynthia Rush\"]","published":"2025-10-01T18:37:05Z","proceeding":"math.ST","tasks":"[\"math.ST\",\"stat.ML\"]","methods":"[]","has_code":false}