Sufficiency of pairwise KL dominance for finite-state experiments

Determine whether pairwise dominance of all ordered Kullback–Leibler divergences characterizes stopping dominance for finite-state statistical experiments; specifically, establish whether, for experiments with finite state space and all distinct states i,j, the inequalities KL(F_i\|F_j)\ge KL(G_i\|G_j) imply F\succeq_s G, and hence decision dominance.

Background

The paper completely characterizes stopping dominance and decision dominance for binary experiments using the two directed Kullback–Leibler divergences. In the proposed extension to a finite state space, the authors show that pairwise KL inequalities are necessary: for every ordered pair of states i\ne j, KL(F_i|F_j)\ge KL(G_i|G_j). They also note that the equivalence between stopping and decision dominance does not rely on having only two states.

The unresolved issue is whether these pairwise information inequalities are sufficient when there are more than two states. The binary proof relies on representing an experiment by a scalar likelihood ratio and embedding target values by stopping a continuous one-dimensional process at two boundaries. With multiple states, likelihood ratios form a vector, and a continuous vector-valued process may not hit prescribed target likelihood-ratio vectors. Consequently, the paper does not establish sufficiency of pairwise KL dominance in the general finite-state case.