What is the expected loss of the bayes optimal classifier

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Machine Learning

Problem 1. The feature space consists of three possible points (events) A, B, C, which occur with probability 0.2, 0.3, 0.5, respectively. For each event there are two possible labels +1 or -1, which happen with probability 0.9, 0.3 and 0.8 respectively (that is, P(1|A) = 0.9, P(1|B) = 0.3, P(1|C) = 0.8). Determine the Bayes optimal classifier. What is the expected loss of the Bayes optimal classifier?

Problem 2. A probability distribution on the real line is a mixture of two classes +1 and -1 with density N(1, 2) (normal distribution with mean 1 and variance 2) and N(4, 1), with prior probabilities 0.3 and 0.7 respectively. What is the Bayes decision rule? Give an estimate for the Bayes risk.

Problem 3. Consider a k-NN classifier for a 2-class problem. What is its expected (classification) loss and how does it compare to the Bayes
optimal, when k = 3, assuming you have sufficiently many data points? How does the empirical loss of 3-NN compare to the Bayes optimal? (Recall that the empirical loss of 1-NN is zero).

Problem 4. Generate 2000 points from two equally weighted spherical Gaussians N(0, I), N((3, 0, . . . , 0), I) in R^p , p = 1, 11, 21, . . . 101 (note, you have to first flip a coin to decide from which Gaussian to sample), where I is the identity matrix and the centers of Gaussians are distance 3 apart. Implement 1-NN and 3-NN classifiers. Test the resulting classifier on a separately generated dataset with 1000 pts. Plot the error rate as a function of p. Observations?

Problem 5. What is the VC-dimension of the set of indicator functions of disks in R² (i.e., functions which are 1 inside a circle -1 outside (but not the other way around!))? What about the indicator functions of rectangular boxes with sides parallel to the axes? You need to explain why.

Reference no: EM132390045

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