• Corpus ID: 246431091

On the proof of posterior contraction for sparse generalized linear models with multivariate responses

@inproceedings{Wang2022OnTP,
  title={On the proof of posterior contraction for sparse generalized linear models with multivariate responses},
  author={Shao-Hsuan Wang and Ray Bai and Hsin-Hsiung Huang},
  year={2022}
}

• Shao-Hsuan Wang, Ray Bai, Hsin-Hsiung Huang
• Published 30 January 2022
• Computer Science

In recent years, the literature on Bayesian high-dimensional variable selection has rapidly grown. It is increasingly important to understand whether these Bayesian methods can consistently estimate the model parameters. To this end, shrinkage priors are useful for identifying relevant signals in high-dimensional data. For multivariate linear regression models with Gaussian response variables, Bai and Ghosh (2018) [5] proposed a multivariate Bayesian model with shrinkage priors (MBSP) for estimation and variable selection in high-dimensional settings. However, the proofs of posterior consistency for the MBSP method (Theorems 3 and 4 of [5]) were incorrect. In this paper, we provide a corrected proof of Theorems 3 and 4 of Bai and Ghosh (2018) [5]. We leverage these new proofs to extend the MBSP model to multivariate generalized linear models (GLMs). Under our proposed model (MBSP-GLM), multiple responses belonging to the exponential family are simultaneously modeled and mixed-type responses are allowed. We show that the MBSP-GLM model achieves strong posterior consistency when p grows at a subexponential rate with n. Furthermore, we quantify the posterior contraction rate at which the posterior shrinks around the true regression coefficients and allow the dimension of the responses q to grow as n grows. Thus, we strengthen the previous results on posterior consistency, which did not provide rate results. This greatly expands the scope of the MBSP model to include response variables of many data types, including binary and count data. To the best of our knowledge, this is the first posterior contraction result for multivariate Bayesian GLMs.