Elements of Hypothesis Testing
Most signal detection problems can be cast in the framework of M-ary hypothesis testing, in which we have an observation (possibly a vector or function) on the basis of which we wish to decide among M possible statistical situations describing the observations. For example, in an M-ary communications receiver we observe an electrical waveform that consists of one of M possible signals corrupted by random channel or receiver noise, and we wish to decide which of the M possible signals is present. Obviously, for any given decision problem, there are a number of possible decision strategies or rules that could be applied; however, we would like to choose a decision rule that is optimum in some sense. There are several useful definitions of optimality for such problems, and in this chapter we consider the three most common formulations — Bayes, minimax, and Neyman-Pearson — and derive the corresponding optimum solutions. In general, we consider the particular problem of binary (M=2) hypothesis testing, although the extension of many of the results of this chapter to the general M-ary case is straightforward and will be developed in the exercises.
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