Let’s say we are trying to categorize email data into spam/non-spam categories. We will first start by creating a threshold for observations to determine if we should place it into the spam category.

$P(Y=j|X=x0) > 0.9$

This means j will be spam and X will be input an observation of emails. Variables compared include let’s say the inclusion of “Nigerian prince”, the grammar, etc.

If an observation meets this threshold then it will be placed into class 1, spam. If not, then it will be placed in not spam category.

Looking at the image, the spam is in class x1 (blue) and not spam is in class x2 (orange). The line represents where the probability is exactly 0.9. This is called the Bayes Decision Boundary. The points that are found spam will be placed on one side of the decision boundary and nodes found to be not spam will be placed on the other side.