Problems tagged with "bayes classifier"
Problem #32
Tags: bayes error, bayes classifier
Shown below are two conditional densities, \(p_1(x \,|\, Y = 1)\) and \(p_0(x \given Y = 0)\), describing the distribution of a continuous random variable \(X\) for two classes: \(Y = 0\)(the solid line) and \(Y = 1\)(the dashed line). You may assume that both densities are piecewise constant.
Part 1)
Suppose \(\pr(Y = 1) = 0.5\) and \(\pr(Y = 0) = 0.5\). What is the prediction of the Bayes classifier at \(x = 1.5\)?
Solution
Class 0
Part 2)
Suppose \(\pr(Y = 1) = 0.5\) and \(\pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution?
Part 3)
Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the prediction of the Bayes classifier at \(x = 1.5\)?
Solution
Class 1
Part 4)
Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution?
Problem #33
Tags: bayes classifier
Suppose the Bayes classifier achieves an error rate of 15\% on a particular data distribution. True or False: It is impossible for any classifier trained on data drawn from this distribution to achieve better than 85\% accuracy on a finite test set that is drawn from this distribution.
Solution
False.
Problem #47
Tags: bayes error, bayes classifier
Part 1)
Suppose a particular probability distribution has the property that, whenever data are sampled from the distribution, the sampled data are guaranteed to be linearly separable. True or False: the Bayes error with respect to this distribution is 0\%.
Solution
True.
Part 2)
Now consider a different probability distribution. Suppose the Bayes classifier achieves an error rate of 0\% on this distribution. True or False: given a finite data set sampled from this distribution, the data must be linearly separable.
Solution
False.