# In the Bayes test, applied to a binary hypothesis testing problem where we have to choose one

## Question:

In the Bayes test, applied to a binary hypothesis testing problem where we have to choose one of two possible hypotheses H0 and H1, we minimize the risk R defined by R = C00 p0 P (say H0 | H0 is true) + C10 p0 P (say H1 | H0 is true) + C11 p1 P (say H1 | H1 is true) + C01 p1 P (say H0 | H1 is true). The terms C00, C10, C11, and C01 denote the costs assigned to the four possible outcomes of the experiment: The first subscript indicates the hypothesis chosen and the second the hypothesis that is true. Assume that C10 > C00 and C01 > C11, The p0 and p1 denote the a priori probabilities of hypotheses H0 and H1, respectively.

(a) Given the observation vector x, show that the partitioning of the observation space so as to minimize the risk R leads to the likelihood ratio test: say H0 if A (x) < λ, say H1 if A (x) > λ, where A (x) is the likelihood ratio A (x) ƒx (x | H1)/ ƒx (x | H0) and λ is the threshold of the test defined by Λ = p0 (C10 – C00)/p1 (C01 – C11)

(b) What are the cost values for which the Bayes’ criterion reduces to the minimum probability of error criterion?

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