PE1-3.2 Statistical Decision Theory
You can apply the Bayes criterion to binary and M-ary hypothesis testing, and compare it with the minimax and Neyman-Pearson criteria.
The three criteria differ only in what you are allowed to assume: Bayes needs prior probabilities and costs, minimax protects against the worst-case prior when you have none, Neyman-Pearson fixes false alarm rate and maximises detection. Radar uses Neyman-Pearson for exactly that reason — you can specify a tolerable false alarm rate but not a sensible prior on aircraft. Every one of them ends in the same shape of answer, a likelihood ratio compared against a threshold, and only the threshold changes.
Work through these
Introduction to statistical decision theory
The framework for deciding between hypotheses when the evidence is noisy. Everything in the rest of this subject is a special case of this item.
Bayes criterion: binary hypothesis testing
The criterion that minimises average cost, given prior probabilities and a cost for each kind of mistake. It is the most general of the three and needs the most assumptions.
Bayes criterion: M-ary hypothesis testing
The same idea with more than two possibilities. The structure generalises cleanly, which is worth seeing once.
The likelihood ratio test
The quantity every one of these criteria reduces to. If you understand this ratio, the three criteria differ only in where the threshold is set.
Minimax criterion
What to do when you do not know the prior probabilities: assume the worst case. It is the conservative choice and it costs performance when the worst case does not happen.
Neyman-Pearson criterion
What to do when the costs are not comparable: fix one error rate and minimise the other. It is the criterion radar and detection actually use.
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