PDFs and probability in naive Bayes classification

Question

I have seen a few times the technique of using the Gaussian PDF for continuous features in Naive Bayes. here and here. Illustrated in the first link:

How is this possible? I always learnt that the PDF is not a probability -- as the probability of any exact value of x is zero.

Answer 1

You're right that the statement is wrong. It should be a likelihood:

$$L(c \mid x=v)=\frac{1}{\sqrt{2\pi\sigma_c^2}}e^{-\frac{(v-\mu_c)^2}{2\sigma_c^2}}$$

A likelihood applies here because we are interested in the relative likelihood that a point belongs to each class:

$$P(c=c' \mid x=v) = \frac{L(c=c' \mid x=v)}{\sum_{c_i} L(c=c_i \mid x=v)}.$$