Consistent Group selection using Global-local prior in High dimensional setup
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915626161799168 |
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| author | Paul, Sayantan Ghosh, Prasenjit Chakrabarti, Arijit |
| author_facet | Paul, Sayantan Ghosh, Prasenjit Chakrabarti, Arijit |
| contents | We consider the problem of model selection when grouping structure is inherent within the regressors. Using a Bayesian approach, we model the mean vector by a one-group global-local shrinkage prior belonging to a broad class of such priors that includes the horseshoe prior. In the context of variable selection, this class of priors was studied by Tang et al. (2018). A modified form of the usual class of global-local shrinkage priors with polynomial tail on the group regression coefficients is proposed. The resulting threshold rule selects the active group if within a group, the ratio of the $L_2$ norm of the posterior mean of its group coefficient to that of the corresponding ordinary least square group estimate is greater than a half. In the theoretical part of this article, we have used the global shrinkage parameter either as a tuning one or an empirical Bayes estimate of it depending on the knowledge regarding the underlying sparsity of the model. When the proportion of active groups is known, using $τ$ as a tuning parameter, we have proved that our method is oracle. In case this proportion is unknown, we propose an empirical Bayes estimate of $τ$. Even if this empirical Bayes estimate is used, then also our half-thresholding rule captures the truly important groups and obtains optimal estimation rate of the group coefficients simultaneously. Though our theoretical works rely on a special form of the design matrix, for general design matrices also, our simulation results show that the half-thresholding rule yields results similar to that of Yang and Narisetty (2020). As a consequence of this, in a high dimensional sparse group selection problem, instead of using the so-called `gold standard' spike and slab prior, one can use the one-group global-local shrinkage priors with polynomial tail to obtain similar results. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2302_04715 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Consistent Group selection using Global-local prior in High dimensional setup Paul, Sayantan Ghosh, Prasenjit Chakrabarti, Arijit Statistics Theory We consider the problem of model selection when grouping structure is inherent within the regressors. Using a Bayesian approach, we model the mean vector by a one-group global-local shrinkage prior belonging to a broad class of such priors that includes the horseshoe prior. In the context of variable selection, this class of priors was studied by Tang et al. (2018). A modified form of the usual class of global-local shrinkage priors with polynomial tail on the group regression coefficients is proposed. The resulting threshold rule selects the active group if within a group, the ratio of the $L_2$ norm of the posterior mean of its group coefficient to that of the corresponding ordinary least square group estimate is greater than a half. In the theoretical part of this article, we have used the global shrinkage parameter either as a tuning one or an empirical Bayes estimate of it depending on the knowledge regarding the underlying sparsity of the model. When the proportion of active groups is known, using $τ$ as a tuning parameter, we have proved that our method is oracle. In case this proportion is unknown, we propose an empirical Bayes estimate of $τ$. Even if this empirical Bayes estimate is used, then also our half-thresholding rule captures the truly important groups and obtains optimal estimation rate of the group coefficients simultaneously. Though our theoretical works rely on a special form of the design matrix, for general design matrices also, our simulation results show that the half-thresholding rule yields results similar to that of Yang and Narisetty (2020). As a consequence of this, in a high dimensional sparse group selection problem, instead of using the so-called `gold standard' spike and slab prior, one can use the one-group global-local shrinkage priors with polynomial tail to obtain similar results. |
| title | Consistent Group selection using Global-local prior in High dimensional setup |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2302.04715 |