Statistical Learning: 4.6 Gaussian Discriminant Analysis (One Variable)  Summary and Q&A
TL;DR
Gaussian Discriminant Analysis is a technique used to classify data based on Gaussian density functions, with decision boundaries determined by mean and variance values.
Questions & Answers
Q: What is the mathematical form of the Gaussian density function used in discriminant analysis?
The mathematical form of the Gaussian density function includes constants and an exponential term that depends on the variable. It also involves the mean and variance for the observations in each class.
Q: Why is it assumed that the variance is the same in each class?
Assuming equal variances across classes is a convenience that determines whether the resulting discriminant function is linear or quadratic. It simplifies the calculations and provides easier decision boundaries.
Q: How can we simplify the complicated expression obtained from applying Bayes' formula?
By taking logarithms and discarding terms that do not depend on the class, the expression can be simplified. This leads to a linear function involving the variable, mean, variance, and prior probability for each class.
Q: What is the decision boundary when the probabilities of two classes are equal?
When the probabilities of two classes are equal, the decision boundary is exactly at the average mean of the two classes. Intuitively, this is the point at which it is equally likely to classify an observation into either class.
Summary & Key Takeaways

Gaussian Discriminant Analysis uses Gaussian density functions to classify data based on mean and variance values of each class.

Decision boundaries are determined by comparing the discriminant scores of each class, obtained using the Gaussian density functions.

When dealing with data instead of populations, mean and variance estimates are used to calculate the decision boundaries.