Coverage of Credible Sets for Regression under Variable Selection
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866914842217021440 |
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| author | Pal, Samhita Ghosal, Subhashis |
| author_facet | Pal, Samhita Ghosal, Subhashis |
| contents | We study the asymptotic frequentist coverage of credible sets based on a novel Bayesian approach for a multiple linear regression model under variable selection. We initially ignore the issue of variable selection, which allows us to put a conjugate normal prior on the coefficient vector. The variable selection step is incorporated directly in the posterior through a sparsity-inducing map and uses the induced prior for making an inference instead of the natural conjugate posterior. The sparsity-inducing map minimizes the sum of the squared l2-distance weighted by the data matrix and a suitably scaled l1-penalty term. We obtain the limiting coverage of various credible regions and demonstrate that a modified credible interval for a component has the exact asymptotic frequentist coverage if the corresponding predictor is asymptotically uncorrelated with other predictors. Through extensive simulation, we provide a guideline for choosing the penalty parameter as a function of the credibility level appropriate for the corresponding coverage. We also show finite-sample numerical results that support the conclusions from the asymptotic theory. We also provide the credInt package that implements the method in R to obtain the credible intervals along with the posterior samples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_13938 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Coverage of Credible Sets for Regression under Variable Selection Pal, Samhita Ghosal, Subhashis Methodology We study the asymptotic frequentist coverage of credible sets based on a novel Bayesian approach for a multiple linear regression model under variable selection. We initially ignore the issue of variable selection, which allows us to put a conjugate normal prior on the coefficient vector. The variable selection step is incorporated directly in the posterior through a sparsity-inducing map and uses the induced prior for making an inference instead of the natural conjugate posterior. The sparsity-inducing map minimizes the sum of the squared l2-distance weighted by the data matrix and a suitably scaled l1-penalty term. We obtain the limiting coverage of various credible regions and demonstrate that a modified credible interval for a component has the exact asymptotic frequentist coverage if the corresponding predictor is asymptotically uncorrelated with other predictors. Through extensive simulation, we provide a guideline for choosing the penalty parameter as a function of the credibility level appropriate for the corresponding coverage. We also show finite-sample numerical results that support the conclusions from the asymptotic theory. We also provide the credInt package that implements the method in R to obtain the credible intervals along with the posterior samples. |
| title | Coverage of Credible Sets for Regression under Variable Selection |
| topic | Methodology |
| url | https://arxiv.org/abs/2406.13938 |