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.

True False
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\%.

True False
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.

True False
Solution

False.