This situation is illustrated in figure 35 in this

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Unformatted text preview: recall that if is small, then fi (u) is the approximate probability that the observation is within a length interval centered at u, if Hi is the true hypothesis. For this reason, we still call fi (u) the likelihood of X = u if Hi is the true hypothesis. We also define the likelihood ratio, Λ(u), for u in the support of f1 or f0 , by f1 (u) Λ(u) = . f0 (u) A likelihood ratio test (LRT) with threshold τ is defined by: Λ(X ) >τ <τ declare H1 is true declare H0 is true. Just as for the case of discrete-type observations, the maximum likelihood (ML) test is the LRT with threshold τ = 1, and the maximum a posteriori probability (MAP) decision rule is the LRT 114 CHAPTER 3. CONTINUOUS-TYPE RANDOM VARIABLES with threshold τ = π0 , where (π1 , π0 ) is the prior probability distribution. In particular, the ML π1 rule is the special case of the MAP rule for the uniform prior: π1 = π0 . The following definitions are the same as in the case of discrete-type observations: pfalse alarm = P (de...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.

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