Problems tagged with "bayes error"

Problem #032

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 #047

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.

Problem #093

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)

What is \(\pr(1 \leq X \leq 3 \given Y = 0)\)?

Part 2)

Suppose \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the prediction of the Bayes classifier at \(x = 2.5\)?

Solution

Class 0.

Part 3)

Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution?

Part 4)

Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the prediction of the Bayes classifier at \(x = 2.5\)?

Solution

Class 1.

Part 5)

Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution?

Solution

Video explanation: https://youtu.be/mczKmVUJauI

Problem #110

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 that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)?

Part 2)

Suppose \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the prediction of the Bayes classifier at \(x = 1.5\)?

Solution

Class 0.

Part 3)

Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution?

Part 4)

Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)?

Part 5)

Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the prediction of the Bayes classifier at \(x = 1.5\)?

Solution

Class 1.

Part 6)

Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution?

Problem #116

Tags: covariance, Gaussians, bayes error

Suppose that, in a binary classification setting, the true underlying class-conditional densities \(p(\vec x \given Y=0)\) and \(p(\vec x \given Y=1)\) are known to each be multivariate Gaussians with full covariance matrices. Suppose, also, that \(\pr(Y = 1) = \pr(Y = 0) = \frac{1}{2}\).

True or False: it is possible that the Bayes error in this case is exactly zero.

True False
Solution

False.