  We define the $\textit{marginal information}$ of a communication protocol,
and use it to prove XOR lemmas for communication complexity. We show that if
every $C$-bit protocol has bounded advantage for computing a Boolean function
$f$, then every $\tilde \Omega(C \sqrt{n})$-bit protocol has advantage
$\exp(-\Omega(n))$ for computing the $n$-fold xor $f^{\oplus n}$. We prove
exponentially small bounds in the average case setting, and near optimal bounds
for product distributions and for bounded-round protocols.
