Posterior Robustness with Milder Conditions: Contamination Models Revisited
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866915503467921408 |
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| author | Hamura, Yasuyuki Irie, Kaoru Sugasawa, Shonosuke |
| author_facet | Hamura, Yasuyuki Irie, Kaoru Sugasawa, Shonosuke |
| contents | Robust Bayesian linear regression is a classical but essential statistical tool. Although novel robustness properties of posterior distributions have been proved recently under a certain class of error distributions, their sufficient conditions are restrictive and exclude several important situations. In this work, we revisit a classical two-component mixture model for response variables, also known as contamination model, where one component is a light-tailed regression model and the other component is heavy-tailed. The latter component is independent of the regression parameters, which is crucial in proving the posterior robustness. We obtain new sufficient conditions for posterior (non-)robustness and reveal non-trivial robustness results by using those conditions. In particular, we find that even the Student-$t$ error distribution can achieve the posterior robustness in our framework. A numerical study is performed to check the Kullback-Leibler divergence between the posterior distribution based on full data and that based on data obtained by removing outliers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_00281 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Posterior Robustness with Milder Conditions: Contamination Models Revisited Hamura, Yasuyuki Irie, Kaoru Sugasawa, Shonosuke Methodology Statistics Theory Robust Bayesian linear regression is a classical but essential statistical tool. Although novel robustness properties of posterior distributions have been proved recently under a certain class of error distributions, their sufficient conditions are restrictive and exclude several important situations. In this work, we revisit a classical two-component mixture model for response variables, also known as contamination model, where one component is a light-tailed regression model and the other component is heavy-tailed. The latter component is independent of the regression parameters, which is crucial in proving the posterior robustness. We obtain new sufficient conditions for posterior (non-)robustness and reveal non-trivial robustness results by using those conditions. In particular, we find that even the Student-$t$ error distribution can achieve the posterior robustness in our framework. A numerical study is performed to check the Kullback-Leibler divergence between the posterior distribution based on full data and that based on data obtained by removing outliers. |
| title | Posterior Robustness with Milder Conditions: Contamination Models Revisited |
| topic | Methodology Statistics Theory |
| url | https://arxiv.org/abs/2303.00281 |