  The Jensen inequality is a widely used tool in a multitude of fields, such as
for example information theory and machine learning. It can be also used to
derive other standard inequalities such as the inequality of arithmetic and
geometric means or the H\"older inequality. In a probabilistic setting, the
Jensen inequality describes the relationship between a convex function and the
expected value. In this work, we want to look at the probabilistic setting from
the reverse direction of the inequality. We show that under minimal constraints
and with a proper scaling, the Jensen inequality can be reversed. We believe
that the resulting tool can be helpful for many applications and provide a
variational estimation of mutual information, where the reverse inequality
leads to a new estimator with superior training behavior compared to current
estimators.
