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Unformatted text preview: f π1 = π0 , the prior is said to be uniform, because it means the hypotheses are equally likely.
For the uniform prior the threshold for the MAP rule is one, and the MAP rule is the same as the
ML rule. Does it make sense that if π0 > π1 , then the threshold for the MAP rule (in LRT form)
is greater than one? Indeed it does, because a larger threshold value in the LRT means there are
fewer observations leading to deciding H1 is true, which is appropriate behavior if π0 > π1 .
The MAP rule has a remarkable optimality property, as we now explain. The average error
probability, which we call pe , for any decision rule can be written as pe = π0 pfalse alarm + π1 pmiss .
A decision rule is speciﬁed by underlining one number from each column of the joint probability
matrix. The corresponding pe is the sum of all numbers in the joint probability matrix that are
not underlined. From this observation it easily follows that, among all decision rules, the MAP 58 CHAPTER 2. DISCRETE-TYPE RANDOM VARIABLES decisi...
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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.
- Spring '08
- The Land