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The problem of simple $M-$ary hypothesis testing under a generic performancecriterion that depends on arbitrary functions of error probabilities isconsidered. Using results from convex analysis, it is proved that an optimaldecision rule can be characterized as a randomization among at most twodeterministic decision rules, of the form reminiscent to Bayes rule, if theboundary points corresponding to each rule have zero probability under eachhypothesis. Otherwise, a randomization among at most $M(M-1)+1$ deterministicdecision rules is sufficient. The form of the deterministic decision rules areexplicitly specified. Likelihood ratios are shown to be sufficient statistics.Classical performance measures including Bayesian, minimax, Neyman-Pearson,generalized Neyman-Pearson, restricted Bayesian, and prospect theory basedapproaches are all covered under the proposed formulation. A numerical exampleis presented for prospect theory based binary hypothesis testing.

Optimal Decision Rules for Simple Hypothesis Testing under General Criterion Involving Error Probabilities