  Maximum likelihood is the most widely used statistical estimation technique.
Recent work by the authors introduced a general methodology for the
construction of estimators for functionals in parametric models, and
demonstrated improvements - both in theory and in practice - over the maximum
likelihood estimator (MLE), particularly in high dimensional scenarios
involving parameter dimension comparable to or larger than the number of
samples. This approach to estimation, building on results from approximation
theory, is shown to yield minimax rate-optimal estimators for a wide class of
functionals, implementable with modest computational requirements. In a
nutshell, a message of this recent work is that, for a wide class of
functionals, the performance of these essentially optimal estimators with $n$
samples is comparable to that of the MLE with $n \ln n$ samples.
  In the present paper, we highlight the applicability of the aforementioned
methodology to statistical problems beyond functional estimation, and show that
it can yield substantial gains. For example, we demonstrate that for learning
tree-structured graphical models, our approach achieves a significant reduction
of the required data size compared with the classical Chow--Liu algorithm,
which is an implementation of the MLE, to achieve the same accuracy. The key
step in improving the Chow--Liu algorithm is to replace the empirical mutual
information with the estimator for mutual information proposed by the authors.
Further, applying the same replacement approach to classical Bayesian network
classification, the resulting classifiers uniformly outperform the previous
classifiers on 26 widely used datasets.
