## Bayesian MAP estimation using Gaussian and diffused‐gamma prior

### Abstract

For sparse and high‐dimensional data analysis, a valid approximation of ${\ell }_{0}$‐norm has played a key role. However, there is not much study on the ${\ell }_{0}$‐norm approximation in the Bayesian literature. In this article, we introduce a new prior, called Gaussian and diffused‐gamma prior, which leads to a nice ${\ell }_{0}$‐norm approximation under the maximum a posteriori estimation. To develop a general likelihood function, we utilize a general class of divergence measures, called Bregman divergence. Due to the generality of Bregman divergence, our method can handle various types of data such as count, binary, continuous, etc. In addition, our Bayesian approach provides many theoretical and computational advantages. To demonstrate the validity and reliability, we conduct simulation studies and real data analysis. The Canadian Journal of Statistics 46: 399–415; 2018 © 2018 Statistical Society of Canada

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