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.

本篇研究文章探讨了针对各种概率模型使用 Kullback-Leibler 距离的模型选择类型聚合问题。文章提出了两种聚合方法，并使用惩罚极大似然准则选择聚合权重，给出了高概率的锐利的神谕不等式和相应的下界结果。