  There are two major routes to address the ubiquitous family of inverse
problems appearing in signal and image processing, such as denoising or
deblurring. A first route relies on Bayesian modeling, where prior
probabilities are used to embody models of both the distribution of the unknown
variables and their statistical dependence with the observed data. The
estimation process typically relies on the minimization of an expected loss
(e.g. minimum mean squared error, or MMSE). The second route has received much
attention in the context of sparse regularization and compressive sensing: it
consists in designing (often convex) optimization problems involving the sum of
a data fidelity term and a penalty term promoting certain types of unknowns
(e.g., sparsity, promoted through an 1 norm). Well known relations between
these two approaches have lead to some widely spread misconceptions. In
particular, while the so-called Maximum A Posterori (MAP) estimate with a
Gaussian noise model does lead to an optimization problem with a quadratic
data-fidelity term, we disprove through explicit examples the common belief
that the converse would be true. It has already been shown [7, 9] that for
denoising in the presence of additive Gaussian noise, for any prior probability
on the unknowns, MMSE estimation can be expressed as a penalized least squares
problem, with the apparent characteristics of a MAP estimation problem with
Gaussian noise and a (generally) different prior on the unknowns. In other
words, the variational approach is rich enough to build all possible MMSE
estimators associated to additive Gaussian noise via a well chosen penalty. We
generalize these results beyond Gaussian denoising and characterize noise
models for which the same phenomenon occurs. In particular, we prove that with
(a variant of) Poisson noise and any prior probability on the unknowns, MMSE
estimation can again be expressed as the solution of a penalized least squares
optimization problem. For additive scalar denois-ing the phenomenon holds if
and only if the noise distribution is log-concave. In particular, Laplacian
denoising can (perhaps surprisingly) be expressed as the solution of a
penalized least squares problem. In the multivariate case, the same phenomenon
occurs when the noise model belongs to a particular subset of the exponential
family. For multivariate additive denoising, the phenomenon holds if and only
if the noise is white and Gaussian.
