  In this paper, we study the statistical behaviour of the Exponentially
Weighted Aggregate (EWA) in the problem of high-dimensional regression with
fixed design. Under the assumption that the underlying regression vector is
sparse, it is reasonable to use the Laplace distribution as a prior. The
resulting estimator and, specifically, a particular instance of it referred to
as the Bayesian lasso, was already used in the statistical literature because
of its computational convenience, even though no thorough mathematical analysis
of its statistical properties was carried out. The present work fills this gap
by establishing sharp oracle inequalities for the EWA with the Laplace prior.
These inequalities show that if the temperature parameter is small, the EWA
with the Laplace prior satisfies the same type of oracle inequality as the
lasso estimator does, as long as the quality of estimation is measured by the
prediction loss. Extensions of the proposed methodology to the problem of
prediction with low-rank matrices are considered.
