# 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 = C_{00} p_{0} P (say H_{0} | H_{0} is true) + C_{10} p_{0} P (say H_{1} | H_{0} is true) + C_{11} p_{1} P (say H_{1} | H_{1} is true) + C_{01} p_{1} P (say H_{0} | H_{1} is true). The terms C_{00}, C_{10}, C_{11}, and C_{01} 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 C_{10} > C_{00} and C_{01} > C_{11}, The p_{0} and p_{1} denote the a priori probabilities of hypotheses H_{0} and H_{1}, 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 H_{0} if A (x) < λ, say H_{1} if A (x) > λ, where A (x) is the likelihood ratio A (x) ƒx (x | H_{1})/ ƒx (x | H_{0}) and λ is the threshold of the test defined by Λ = p_{0} (C_{10} – C_{00})/p_{1} (C_{01} – C_{11})

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

## Step by Step Answer:

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