  We study the problem of model selection type aggregation with respect to the
Kullback-Leibler divergence for various probabilistic models. Rather than
considering a convex combination of the initial estimators $f_1, \ldots, f_N$,
our aggregation procedures rely on the convex combination of the logarithms of
these functions. The first method is designed for probability density
estimation as it gives an aggregate estimator that is also a proper density
function, whereas the second method concerns spectral density estimation and
has no such mass-conserving feature. We select the aggregation weights based on
a penalized maximum likelihood criterion. We give sharp oracle inequalities
that hold with high probability, with a remainder term that is decomposed into
a bias and a variance part. We also show the optimality of the remainder terms
by providing the corresponding lower bound results.
