A version of the multiple hypotheses testing problem is studied in which the decision procedure is based only on the current observation and the previous decision. Conditions for inconsistency and consistency of the stepwise Bayes rule, which are related to the boundedness of the likelihood ratios, are given. The (typically slow) rate of convergence of the error probabilities of consistent procedures is determined, and a sharp lower bound for the Bayes risk in terms of bounds on the likelihood ratios is derived. A modification of the recursive Sakrison's procedure for a continuous estimation problem is obtained in this setting by embedding the discrete family of original probability distributions into an exponential family.
"Recursive Testing of Multiple Hypotheses: Consistency and Efficiency of the Bayes Rule." Ann. Statist. 22 (2) 616 - 633, June, 1994. https://doi.org/10.1214/aos/1176325487