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Exercise 1.11
Thank you Prof. Yaser. Your book is really easy to follow. I have just started it for a week and I am trying to finish every exercises in the book.
About exercise 1.11, I don't know where to check the answer so I post it here. Could you please tell me whether my answers are right or wrong? Is there any place that I can check my answer on exercise by myself? Ex 1.11: Dataset D of 25 training examples. X = R, Y = {1, +1} H = {h1, h2} where h1 = +1, h2 = 1 Learning algorithms: S  choose the hypothesis that agrees the most with D C  choose the hypothesis deliberately P[f(x) = +1] = p (a) Can S produce a hypothesis that is guaranteed to perform better than random on any point outside D? Answer: No In case that all examples in D have yn = +1 (b) Is it possible that the hypothesis that C produces turns out to be better than the hypothesis that S produces? Answer: Yes (c) If p = 0.9, what is the probability that S will produce a better hypothesis than C? Answer: P[P(Sy = f) > P(Cy = f)] where Sy is the output hypothesis of S, Cy is the output hypothesis of C + Since yn = +1, Sy = +1. Moreover, P[f(x) = +1] = 0.9 > P(Sy = f) = 0.9 + We have, P(Cy = +1) = 0.5, P(Cy = 1) = 0.5, P[f(x) = +1] = 0.9, P[f(x) = 1] = 0.1 > P[Cy = f] = 0.5*0.9 + 0.5*0.1 = 0.5 Since 0.9 > 0.5, P[P(Sy = f) > P(Cy = f)] = 1 (d) Is there any value of p for which it is more likely than not that C will produce a better hypothesis than S? Answer: p < 0.5 I am not sure that my answer of (a) and for (c) is not conflict. Thank You and Best Regards, 
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