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Unformatted text preview: clare H1 true |H0 ) pmiss = P (declare H0 true |H1 ) pe = π0 pfalse alarm + π1 pmiss . The MAP rule based on a given prior probability distribution (π1 , π0 ) minimizes the average error probability, pe , computed using the same prior. Example 3.10.1 Suppose under hypothesis Hi , the observation X has the N (mi , σ 2 ) distribution, for i = 0 or i = 1, where the parameters are known and satisfy: σ 2 > 0 and m0 < m1 . Identify the ML and MAP decision rules and their associated error probabilities, pf alse alarm and pmiss . Assume prior probabilities π1 and π0 are given where needed. Solution: The pdfs are given by 1 (u − mi )2 fi (u) = √ exp − 2σ 2 2πσ , so Λ(u) = f1 (u) f0 (u) (u − m1 )2 (u − m0 )2 + 2σ 2 2σ 2 m0 + m1 m1 − m0 u− 2 σ2 = exp − = exp Observe that Λ(X ) > 1 if and only if X ≥ X m0 +m1 , 2 . so the ML rule for this example is: > γM L declare H1 is true < γM L declare H0 is true. where γM L = m0 +m1 . 2 The LRT for a general threshold τ is equivalent to ln Λ(X ) > ln τ < ln τ declare H1 is true declare H0 is true. or equivalently, X > < σ2 m1 −m0 σ2 m1 −m0 ln τ + ln τ + m0 +m1 2 m0 +m1 2 declare H...
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