  This paper addresses the sparse representation (SR) problem within a general
Bayesian framework. We show that the Lagrangian formulation of the standard SR
problem, i.e., $\mathbf{x}^\star=\arg\min_\mathbf{x} \lbrace \|
\mathbf{y}-\mathbf{D}\mathbf{x} \|_2^2+\lambda\| \mathbf{x}\|_0 \rbrace$, can
be regarded as a limit case of a general maximum a posteriori (MAP) problem
involving Bernoulli-Gaussian variables. We then propose different tractable
implementations of this MAP problem that we refer to as "Bayesian pursuit
algorithms". The Bayesian algorithms are shown to have strong connections with
several well-known pursuit algorithms of the literature (e.g., MP, OMP, StOMP,
CoSaMP, SP) and generalize them in several respects. In particular, i) they
allow for atom deselection; ii) they can include any prior information about
the probability of occurrence of each atom within the selection process; iii)
they can encompass the estimation of unkown model parameters into their
recursions.
