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?