Question 1: Covariance Matrix Properties

The covariance matrix is always:

a) Square
b) Positive Semidefinite
c) Positive Definite
d) Symmetric
e) None of the above

For each of the properties you selected, describe what would happen if the covariance matrix does not satisfy that property.

Question 2: Decision Boundaries

Describe the possible set of decision boundaries that can be generated using Gaussian density functions in a two-class problem.

Question 3: Bayesian Classification Components

Consider a two-class classification problem using Bayesian approach:

a) Explain how the prior probabilities affect the decision boundary when:

• Both classes have equal priors
• One class has a higher prior than the other

b) How does the likelihood function influence the classification decision when:

• The classes have similar covariance matrices
• The classes have different covariance matrices

Question 4: Practical Application

Given a dataset with two classes, where:

• Class 1: Mean = (2, 3), Covariance = [[1, 0], [0, 1]]
• Class 2: Mean = (4, 5), Covariance = [[2, 0], [0, 2]]
• Prior probabilities: P(Class 1) = 0.6, P(Class 2) = 0.4

a) Calculate the posterior probability for a test point (3, 4) b) Determine the class assignment for this point c) Explain how the decision would change if the prior probabilities were equal