  We consider the entropy of sums of independent discrete random variables, in
analogy with Shannon's Entropy Power Inequality, where equality holds for
normals. In our case, infinite divisibility suggests that equality should hold
for Poisson variables. We show that some natural analogues of the Entropy Power
Inequality do not in fact hold, but propose an alternative formulation which
does always hold. The key to many proofs of Shannon's Entropy Power Inequality
is the behaviour of entropy on scaling of continuous random variables. We
believe that R\'{e}nyi's operation of thinning discrete random variables plays
a similar role to scaling, and give a sharp bound on how the entropy of ultra
log-concave random variables behaves on thinning. In the spirit of the
monotonicity results established by Artstein, Ball, Barthe and Naor, we prove a
stronger version of concavity of entropy, which implies a strengthened form of
our discrete Entropy Power Inequality.
