Homework5

Homework5 - a. Assume that the doctor is also informed...

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Machine Learning Winter 2010 Homework #5 Due: Friday February 19 2010 (5:00pm) Deadline: TBA 1. Exercise 6.1 2. Exercise 6.2 3. Exercise 6.6 4. A group of diseases occurs in three dierent variants h 1 , h 2 , and h 3 . Each can cause one of three characteristic syndromes D 1 , D 2 , and D 3 . The probabilities P ( D i | h j ) are well-known from statistical material, they are reliable and will not change. Specically, the following conditional probabilities are given: h 1 h 2 h 3 D 1 0.6 0.1 0.5 D 2 0.3 0.1 0.1 D 3 0.1 0.8 0.4 From time to time, patients come with one of the syndromes D 1 , D 2 , D 3 . In each case, the doctor wants to make a preliminary diagnosis. Then he would verify or falsify his hypothesis (one of h 1 , h 2 , h 3 ) by a closer examination. In case of a wrong guess, more examinations must be done in order to check dierent hypotheses. Clearly, doctor and patients have an interest to maximize the probability f of a correct preliminary diagnosis.
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Unformatted text preview: a. Assume that the doctor is also informed about the frequencies of h 1 , h 2 , h 3 . In other words, he knows the prior probabilities q j := P ( h j ), j = 1, 2, 3. As we know, the MAP hypothesis has the best chance to hit the target hypothesis. How large is f in case q 1 = q 2 = q 3 = 1/3? b. Due to an epidemy of h 3 , the frequencies (priors) suddenly changed into q 1 = q 2 = 0.1 and q 3 = 0.8. What is f now, if the MAP hypothesis is used as preliminary diagnosis? c. Can you make a correct preliminary diagnosis with probability 0.45, even if you are completely ignorant of prior probabilities? (Hint: A purely random guess would succeed with probability 0.33. Using the table of the P ( D i | h j ), it should be possible to improve upon this trivial strategy.) Compare the result to (a)....
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