# If the bit is zero the number of photons x arriving

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Unformatted text preview: he numbers in the column for X = 2. Similarly the conditional probability P (H0 |X = 2) is the bottom number in the column for X = 2 divided by the sum of the numbers in the column for X = 2. Since the denominators are the same (both denominators are equal to P {X = 2}) it follows that whether P (H1 |X = 2) &gt; P (H1 |X = 2) is equivalent to whether the top entry in the column for X = 2 is greater than the bottom entry in the column for X = 2. Thus, the MAP decision rule can be speciﬁed by underlining the larger entry in each column of the joint probability matrix. For our original example, the MAP rule is given by H1 H0 X=0 X=1 X=2 X=3 0.00 0.02 0.06 0.12 0.32 0.24 0.16 0.08 ← underlines indicate the MAP decision rule . Thus, if the observation is X = k , the MAP rule declares hypothesis H1 is true if π1 p1 (k ) &gt; π0 p0 (k ), π0 or equivalently if Λ(k ) &gt; π1 , where Λ is the likelihood ratio deﬁned above. Therefore, the MAP rule is equivalent to the LRT with threshold τ = π0 . π1 I...
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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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