What is the “learning” step in Gaussian Naive Bayes classification?

Question

For conditionally independent features $f_i$, Naive Bayes Classification gives me the classifier

$Classifier(f) := \arg \max_{k} P(C=k) · ∏^n_{i=1} P(f_i|C=k)$

for classes $k$. I understand that for Gaussian Naive Bayes, I can assume normally distributed features, yielding

$Classifier(f) := \arg \max_k P(C=k) · ∏^n_{i=1} \frac{1}{\sqrt{(2πσ_{k,i})}} e^{( -\frac{(f_i - μ_{k,i})^2}{2σ_{k,i})}}$

where $μ_{k,i}$ is the mean of class $k$ and feature $f_i$ (and similiar for variance $σ_{k,i}$).

But where is the "learning step" in this whole procedure?

Answer 1

Naive Bayes uses the class labels from the training set to learn $P(C=k)$ and $P(f_i|C=k)$ directly from the training data.

You learn the mean and variance of each class conditional distribution for each feature just as you would the mean and variance for any other problem. You treat the $f_i$ of each class as if they were their own independent sample.