  In this paper, Bayesian parameter estimation through the consideration of the
Maximum A Posteriori (MAP) criterion is revisited under the prism of the
Expectation-Maximization (EM) algorithm. By incorporating a sparsity-promoting
penalty term in the cost function of the estimation problem through the use of
an appropriate prior distribution, we show how the EM algorithm can be used to
efficiently solve the corresponding optimization problem. To this end, we rely
on variance-mean Gaussian mixtures (VMGM) to describe the prior distribution,
while we incorporate many nice features of these mixtures to our estimation
problem. The corresponding MAP estimation problem is completely expressed in
terms of the EM algorithm, which allows for handling nonlinearities and hidden
variables that cannot be easily handled with traditional methods. For
comparison purposes, we also develop a Coordinate Descent algorithm for the
$\ell_q$-norm penalized problem and present the performance results via
simulations.
