Maximizing the Kullback-Leibler distance

Nihat Ay proposed the following problem [1], motivated from statistical learning theory: Let $\mathcal{E}$ be an exponential family. Find the maximizer of the Kullback-Leibler distance $D(P\Vert \mathcal{E})$ from $\mathcal{E}$. A maximizing probability measure $P$ has a lot of interesting properties. For example, the restriction of $\hat{P}$ to the support of $P$ will be equal to $P$, i.e. $\hat{P}(x)=P(x)\hat{P}(Z)$ if $x\in \text{ensuremath}\mathrm{supp}(P)$ (for the proof in the most general case see [2]). This simple property can be used to transform the problem into another form. The first observation is that probability measures having this "projection property" always come in pairs $P_1,P_2$, such that $P_1$ and $P_2$ have the same sufficient statistics $A$ and disjoint supports. Therefore we can solve the original problem by investigating the kernel of the sufficient statistics $\ker A$. If we find all local maximizers of $$\bar{D}(M):=\sum _{x}M(x)\log |M(x)|,M\in \ker A,$$ subject to $\Vert M{\Vert }_{{\ell }_1}\le 2$, then we know all maximizers of the original problem. The talk will present the transformed problem and its relation to the original problem. In the end I will give some consequences for the solutions of the original problem.
[1] N. Ay: An Information-Geometric Approach to a Theory of Pragmatic Structuring. The Annals of Probability 30 (2002) 416-436.
[2] F. Matúš: Optimality conditions for maximizers of the information divergence from an exponential family. Kybernetika 43, 731-746.