  Given an intractable distribution $p$, the problem of variational inference
(VI) is to find the best approximation from some more tractable family $Q$.
Commonly, one chooses $Q$ to be a family of factorized distributions (i.e., the
mean-field assumption), even though~$p$ itself does not factorize. We show that
this mismatch leads to an impossibility theorem: if $p$ does not factorize,
then any factorized approximation $q\in Q$ can correctly estimate at most one
of the following three measures of uncertainty: (i) the marginal variances,
(ii) the marginal precisions, or (iii) the generalized variance (which can be
related to the entropy). In practice, the best variational approximation in $Q$
is found by minimizing some divergence $D(q,p)$ between distributions, and so
we ask: how does the choice of divergence determine which measure of
uncertainty, if any, is correctly estimated by VI? We consider the classic
Kullback-Leibler divergences, the more general R\'enyi divergences, and a
score-based divergence which compares $\nabla \log p$ and $\nabla \log q$. We
provide a thorough theoretical analysis in the setting where $p$ is a Gaussian
and $q$ is a (factorized) Gaussian. We show that all the considered divergences
can be \textit{ordered} based on the estimates of uncertainty they yield as
objective functions for~VI. Finally, we empirically evaluate the validity of
this ordering when the target distribution $p$ is not Gaussian.
