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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
(u − mi )2
fi (u) = √
2πσ , so
Λ(u) = f1 (u)
(u − m1 )2 (u − m0 )2
m0 + m1
m1 − m0
σ2 = 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 .
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,
m1 −m0 ln τ +
ln τ + m0 +m1
2 declare H...
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- Spring '08
- The Land