Confused about the use of Probability Density Functions in machine learning I am a bit confused about the usage of probability density functions in machine learning. Lets say I have two classes $C_1$ and $C_2$ which are lets say gaussian . Then I can classify a point $x$ by taking the class for which the posterior is bigger. So I just calculate the following two expressions: $$P(x|C_1)P(C_1) \;and\; P(x|C_2)P(C_2)$$
When doing such calculations in our lecture we always talked about probabilities. As far as I know we always "acted" as if these are probabilities but this isn't technically right? I mean plugging in $x$ into the functions doesn't give you a "probability" but I am assuming that it gives you something proportional to the probability.

The formula gives you joint probabilities
$$
P(x, C_i) = P(x|C_i) \, P(C_i)
$$
but you are presumably using Bayes theorem here, so this should be
$$
P(C_i|x) = \frac{P(x|C_i) \, P(C_i)}{\sum_i \, P(x|C_i) \, P(C_i) }
$$
Where you don't need the denominator if you want to do things like optimization, sampling, or comparing the probabilities. This lets us ignore the denominator. If you were interested in the conditional probabilities themselves, you'd need the denominator.
