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In this paper, Bayesian parameter estimation through the consideration of theMaximum A Posteriori (MAP) criterion is revisited under the prism of theExpectation-Maximization (EM) algorithm. By incorporating a sparsity-promotingpenalty term in the cost function of the estimation problem through the use ofan appropriate prior distribution, we show how the EM algorithm can be used toefficiently solve the corresponding optimization problem. To this end, we relyon variance-mean Gaussian mixtures (VMGM) to describe the prior distribution,while we incorporate many nice features of these mixtures to our estimationproblem. The corresponding MAP estimation problem is completely expressed interms of the EM algorithm, which allows for handling nonlinearities and hiddenvariables that cannot be easily handled with traditional methods. Forcomparison purposes, we also develop a Coordinate Descent algorithm for the$\ell_q$-norm penalized problem and present the performance results viasimulations.

A MAP approach for $\ell_q$-norm regularized sparse parameter estimation using the EM algorithm