Bayes-Optimal Sequential Multi-Hypothesis Testing in Exponential Families

Bayesian sequential testing of multiple simple hypotheses is a classical sequential decision problem. But the optimal policy is computationally intractable in general, because the posterior probability space is exponentially increasing in the number of hypotheses (i.e, the curse of dimensionality in state space). We consider a specialized problem in which observations are drawn from the same exponential family. By reconstructing the posterior probability vector using the natural sufficient statistic, it is shown that the intrinsic dimension of the posterior probability space cannot exceed the number of parameters governing the exponential family, or the number of hypotheses, whichever is smaller. For univariate exponential families commonly used in practice, the probability space is of one or two dimension in most cases. Hence, the optimal policy can be attainable with only moderate computation. Geometric interpretation and illustrative examples are presented. Simulation studies suggest that the optimal policy can substantially outperform the existing method. The results are also extended to the sequential sampling control problem.