EP-GIG Priors and Applications in Bayesian Sparse Learning
This work addresses the need for flexible priors in Bayesian sparse learning, offering a novel method that is incremental in building upon existing generalized hyperbolic distributions.
The paper tackles the problem of constructing sparsity-inducing priors for Bayesian sparse learning by proposing the EP-GIG framework, a mixture of exponential power distributions with a generalized inverse Gaussian density, which enables EM algorithms for efficient inference and extends to grouped variable selection and logistic regression.
In this paper we propose a novel framework for the construction of sparsity-inducing priors. In particular, we define such priors as a mixture of exponential power distributions with a generalized inverse Gaussian density (EP-GIG). EP-GIG is a variant of generalized hyperbolic distributions, and the special cases include Gaussian scale mixtures and Laplace scale mixtures. Furthermore, Laplace scale mixtures can subserve a Bayesian framework for sparse learning with nonconvex penalization. The densities of EP-GIG can be explicitly expressed. Moreover, the corresponding posterior distribution also follows a generalized inverse Gaussian distribution. These properties lead us to EM algorithms for Bayesian sparse learning. We show that these algorithms bear an interesting resemblance to iteratively re-weighted $\ell_2$ or $\ell_1$ methods. In addition, we present two extensions for grouped variable selection and logistic regression.