  In submodular optimization we often deal with the expected value of a
submodular function $f$ on a distribution $\mathcal{D}$ over sets of elements.
In this work we study such submodular expectations for negatively dependent
distributions. We introduce a natural notion of negative dependence, which we
call Weak Negative Regression (WNR), that generalizes both Negative Association
and Negative Regression. We observe that WNR distributions satisfy Submodular
Dominance, whereby the expected value of $f$ under $\mathcal{D}$ is at least
the expected value of $f$ under a product distribution with the same
element-marginals.
  Next, we give several applications of Submodular Dominance to submodular
optimization. In particular, we improve the best known submodular prophet
inequalities, we develop new rounding techniques for polytopes of set systems
that admit negatively dependent distributions, and we prove existence of
contention resolution schemes for WNR distributions.
