  We show that for any union-closed family $\mathcal{F} \subseteq 2^{[n]},
\mathcal{F} \neq \{\emptyset\}$, there exists an $i \in [n]$ which is contained
in a $0.01$ fraction of the sets in $\mathcal{F}$. This is the first known
constant lower bound, and improves upon the
$\Omega(\log_2(|\mathcal{F}|)^{-1})$ bounds of Knill and W\'{o}jick. Our result
follows from an information theoretic strengthening of the conjecture.
Specifically, we show that if $A, B$ are independent samples from a
distribution over subsets of $[n]$ such that $Pr[i \in A] < 0.01$ for all $i$
and $H(A) > 0$, then $H(A \cup B) > H(A)$.
