Chernoff’s bound binds a tail probability (ie.Pr(X⩾a), wherea⩾EX). Assuming that the distribution ofXisQ, the logarithm of the bound is known to be equal to the value of relative entropy (or minus Kullback‐Leibler distance) forI‐projectionP⁁ofQon a setH≜ {P : EPX = 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 theI‐projection (or generalizedI‐projection). © 2003 American Institute of Physics