Written Problems on Learning

1. (Problem 3.2 From Pattern Classification by Duda, Hart and Stork)
   
   Let x have a uniform density:
   
   
   
   (a) Suppose that n samples D={x_1, ..., x_n} are drawn independently according to p(x|theta). Show that the maximum likelihood estimate for theta is max[D]--that is, the value of the maximum element in D.
   
   (b) Suppose that n=5 points are drawn from the distribution and the maximum value of which is 0.6.  Plot the likelihood p(D|theta) in the range 0<=theta<=1.  Explain in words why you do not need to know the values of the other four points.
2. AIMA 20.10a
3. Give one advantage of using MAP learning over ML learning, and one advantage of ML learning over MAP learning. Describe the conditions under which you would use MAP learning over ML and vice versa.