Chernoff's bound forms

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).