  Sparse representations have proven their efficiency in solving a wide class
of inverse problems encountered in signal and image processing. Conversely,
enforcing the information to be spread uniformly over representation
coefficients exhibits relevant properties in various applications such as
digital communications. Anti-sparse regularization can be naturally expressed
through an $\ell_{\infty}$-norm penalty. This paper derives a probabilistic
formulation of such representations. A new probability distribution, referred
to as the democratic prior, is first introduced. Its main properties as well as
three random variate generators for this distribution are derived. Then this
probability distribution is used as a prior to promote anti-sparsity in a
Gaussian linear inverse problem, yielding a fully Bayesian formulation of
anti-sparse coding. Two Markov chain Monte Carlo (MCMC) algorithms are proposed
to generate samples according to the posterior distribution. The first one is a
standard Gibbs sampler. The second one uses Metropolis-Hastings moves that
exploit the proximity mapping of the log-posterior distribution. These samples
are used to approximate maximum a posteriori and minimum mean square error
estimators of both parameters and hyperparameters. Simulations on synthetic
data illustrate the performances of the two proposed samplers, for both
complete and over-complete dictionaries. All results are compared to the recent
deterministic variational FITRA algorithm.
