  Recently, a number of mostly $\ell_1$-norm regularized least squares type
deterministic algorithms have been proposed to address the problem of
\emph{sparse} adaptive signal estimation and system identification. From a
Bayesian perspective, this task is equivalent to maximum a posteriori
probability estimation under a sparsity promoting heavy-tailed prior for the
parameters of interest. Following a different approach, this paper develops a
unifying framework of sparse \emph{variational Bayes} algorithms that employ
heavy-tailed priors in conjugate hierarchical form to facilitate posterior
inference. The resulting fully automated variational schemes are first
presented in a batch iterative form. Then it is shown that by properly
exploiting the structure of the batch estimation task, new sparse adaptive
variational Bayes algorithms can be derived, which have the ability to impose
and track sparsity during real-time processing in a time-varying environment.
The most important feature of the proposed algorithms is that they completely
eliminate the need for computationally costly parameter fine-tuning, a
necessary ingredient of sparse adaptive deterministic algorithms. Extensive
simulation results are provided to demonstrate the effectiveness of the new
sparse variational Bayes algorithms against state-of-the-art deterministic
techniques for adaptive channel estimation. The results show that the proposed
algorithms are numerically robust and exhibit in general superior estimation
performance compared to their deterministic counterparts.
