  The problem of simple $M-$ary hypothesis testing under a generic performance
criterion that depends on arbitrary functions of error probabilities is
considered. Using results from convex analysis, it is proved that an optimal
decision rule can be characterized as a randomization among at most two
deterministic decision rules, of the form reminiscent to Bayes rule, if the
boundary points corresponding to each rule have zero probability under each
hypothesis. Otherwise, a randomization among at most $M(M-1)+1$ deterministic
decision rules is sufficient. The form of the deterministic decision rules are
explicitly 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 based
approaches are all covered under the proposed formulation. A numerical example
is presented for prospect theory based binary hypothesis testing.
