  Chernoff's bound binds a tail probability (ie. $Pr(X \ge a)$, where $a \ge
EX$). Assuming that the distribution of $X$ is $Q$, the logarithm of the bound
is known to be equal to the value of relative entropy (or minus
Kullback-Leibler distance) for $I$-projection $\hat P$ of $Q$ on a set
$\mathcal{H} \triangleq \{P: E_PX = a\}$. Here, Chernoff's bound is related to
Maximum Likelihood on exponential form and consequently implications for the
notion of complementarity are discussed. Moreover, a novel form of the bound is
proposed, which expresses the value of the Chernoff's bound directly in terms
of the $I$-projection (or generalized $I$-projection).
