Compound Bayes Predictors for Sequences with Apparent Markov Structure

Sequential predictors for binary sequences with no assumptions upon the existence of an underlying process are discussed. The rule offered here induces an expected proportion of errors which differs by 0(n-½) from the Bayes envelope with respect to the observed kth order Markov structure. This extends the compound sequential Bayes work of Robbins, Hannan and Blackwell from sequences with perceived 0th order structure to sequences with perceived kth order structure. The proof follows immediately from applying the 0th order theory to 2k separate subsequences. These results show the essential robustness of procedures which play Bayes with respect to (a perhaps randomized) version of an estimate of the distribution of the past. Such procedures still have asymptotically good properties even when the underlying assumptions for which they were originally developed no longer hold.