MIT6_262S11_lec21

ym of rvs called the observation we usually assume

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Unformatted text preview: y many other rv’s, there is a sequence ￿ Y (m) = (Y1, Y2, . . . , Ym) of rv’s called the observation. We usually assume that Y1, Y2, . . . , are I ID conditional on H = 0 and I ID conditional on H = 1. Thus, if the Yn are continuous, fY (m)|H (￿ | ￿) = y ￿ m ￿ n=1 fY |H (yn | ￿). 3 Assume that, on the basis of observing a sample value ￿ ￿ of Y , we must make a decision about H , i.e., choose y H = 0 or H = 1, i.e., detect whether or not H is 1. Decisions in probability theory, as in real life, are not necessarily correct, so we need a criterion for making a choice. We might maximize the probability of choosing correctly, for example, or, given a cost for the wrong choice, might minimize the expected cost. Note that the probability experiment here includes not only the experiment of gathering data (i.e., measuring ￿ the sample value ￿ of Y ) but also the sample value of y the hypothesis. 4 From Bayes’, recognizing that f (￿ ) = p0f (￿ |0) + p1f (￿ |1) y y y Pr {H =￿ | ￿ } = y y p￿fY |H (￿ | ￿) ￿ p0fY |H (￿ | 0...
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