Compound Sequential Decisions Against the Well-Informed Antagonist

We consider causally estimating (filtering) the components of a noise-corrupted sequence relative to a reference class of filters. The noiseless sequence to be filtered is designed by a "well-informed antagonist", meaning it may evolve according to an arbitrary law, unknown to the filter, based, among other things, on past noisy sequence components. We show that this formulation is more challenging than that of an individual noiseless sequence (aka the "semi-stochastic" setting) in the sense that any deterministic filter, even one guaranteed to do well on every noiseless individual sequence, fails under some well-informed antagonist. On the other hand, we constructively establish the existence of a randomized filter which successfully competes with an arbitrary given finite reference class of filters, under every antagonist. Our noise model allows for channels whose noisy output depends on the l past channel outputs (in addition to the noiseless input symbol). Memoryless channels are obtained as a special case of our model by taking l = 0. In this case, our scheme coincides with one that was recently shown to compete with an arbitrary reference class in the semi-stochastic setting. Hence, our results show that the latter scheme is universal also under the well-informed antagonist.