Optimality conditions for maximizers of the information divergence from an exponential family

The information divergence of a probability measure P from an exponential family E over a nite set is de ned as in mum 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 rst 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 .