The problem is to find a test that is in some way optimal. There are several approaches to finding such a test. The subject is covered in detail in many books on statistics, for example, see [72, 51, 80, 83].
In the Bayesian approach we assign costs to our decisions; in particular we introduce positive numbers , , where is the cost incurred by choosing hypothesis when hypothesis is true. We define the conditional risk of a decision rule for each hypothesis asa priori probabilities or priors. We define the Bayes risk as the overall average cost incurred by the decision rule : Bayes rule as the rule that minimizes the Bayes risk .
Very often in practice we do not have control over or access to the mechanism generating the state of nature and we are not able to assign priors to various hypotheses. In such a case one criterion is to seek a decision rule that minimizes, over all , the maximum of the conditional risks, and . A decision rule that fulfills that criterion is called a minimax rule.
In many problems of practical interest the imposition of a specific cost structure on the decisions made is not possible or desirable. The Neyman–Pearson approach involves a trade-off between the two types of errors that one can make in choosing a particular hypothesis. The Neyman–Pearson design criterion is to maximize the power of the test (probability of detection) subject to a chosen significance of the test (false alarm probability).
It is remarkable that all three very different approaches – Bayesian, minimax, and Neyman–Pearson – lead to the same test called the likelihood ratio test . The likelihood ratio is the ratio of the pdf when the signal is present to the pdf when it is absent:
Living Rev. Relativity 15, (2012), 4
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