The information divergence of a probability measure P from an exponential family E over a finite set is defined as infimum of the divergences of P from Q subject to Q∈E. All directional derivatives of the divergence from E are explicitly found. To this end, behaviour of the conjugate of a log-Laplace transform on the boundary of its domain is analysed. The first order conditions for P to be a maximizer of the divergence from E are presented, including new ones when P is not projectable to E
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