Bayesian MAP estimation using Gaussian and diffused-gamma prior

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  • Author: Gyuhyeong Goh and Dipak K. Dey
  • Date: 05 April 2019
  • Copyright: Image copyright of Patrick Rhodes

For sparse and high‐dimensional data analysis, a valid approximation of l^0-norm has played a key role. However, there is not much study on the l^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 l^0-norm approximation under the maximum a posteriori estimation. To develop a general likelihood function, authors in a paper published in The Canadian Journal of Statistics utilize a general class of divergence measures, called Bregman divergence.

The paper is available via this link and the authors explain their findings in further detail below:

Bayesian MAP estimation using Gaussian and diffused‐gamma prior

Gyuhyeong Goh and Dipak K. Dey

The Canadian Journal of Statistics, Volume 46, Issue 3, September 2018, pages 399-415

DOI: https://doi.org/10.1002/cjs.11458

thumbnail image: Bayesian MAP estimation using Gaussian and diffused-gamma prior

In the era of Big Data, when a statistical model is applied to data, the data analyst frequently encounters the situation where the number of candidate variables exceeds the number of observations, often referred to as high-dimensional data problems. In modern high-dimensional data analysis, imposing a constraint on the number of relevant variables, called l0-norm regularization, plays a key role to break the curse of high-dimensionality. While there has been dramatic growth in Bayesian theory and methods in various fields of statistics, there is not much study on the l0-norm regularization method for high-dimensional data analysis in a Bayesian framework.

In this paper, the authors introduce a novel Bayesian method based on a nice l0-norm approximation method for performing Bayesian inference on high-dimensional data. To develop a generalized Bayesian regression model, the authors consider a general class of divergence measures, called Bregman divergence. Due to the generality of Bregman divergence, the proposed method can handle various types of data such as count, binary, continuous, etc. In addition, the new Bayesian method possesses many theoretical and computational advantages. To assess validity and reliability of the newly-developed method, the authors conduct extensive simulation studies. As a real data application, the method is exemplified through a statistical classification for gene expression data obtained from a leukemia microarray study.

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