  Mason's Conjecture asserts that for an $m$--element rank $r$ matroid $\M$ the
sequence $(I_k/\binom{m}{k}: 0\leq k\leq r)$ is logarithmically concave, in
which $I_k$ is the number of independent $k$--sets of $\M$. A related
conjecture in probability theory implies these inequalities provided that the
set of independent sets of $\M$ satisfies a strong negative correlation
property we call the \emph{Rayleigh condition}. This condition is known to hold
for the set of bases of a regular matroid. We show that if $\omega$ is a weight
function on a set system $\Q$ that satisfies the Rayleigh condition then $\Q$
is a convex delta--matroid and $\omega$ is logarithmically submodular. Thus,
the hypothesis of the probabilistic conjecture leads inevitably to matroid
theory. We also show that two--sums of matroids preserve the Rayleigh condition
in four distinct senses, and hence that the Potts model of an iterated two--sum
of uniform matroids satisfies the Rayleigh condition. Numerous conjectures and
auxiliary results are included.
