p(x) in Gaussian mixture model I've recently learned about Gaussian mixture models (GMM) in school. The formula for a GMM goes something like this:
$$p(x) = \sum_{k=1}^{K} \pi_i\mathcal{N}(x|\mu_i,\,\sigma^{2}_i)$$
Now, I know from probability theory that given a certain sample $x$, the probability for it should sum to 1.
For example, if I roll a six-sided die, the probability that I either get or don't get the number $5$ is equal to 1.
So what exactly does $p(x)$ here represent? The probability that a certain $x$ falls in the model? I feel like I'm missing something fundamental here.

The first this is that the $p(x)$ represents a density and not a probability in the classical way that an event $x$ is likely to happen with probability $p(x)$.
Then the correct way to think of $p(x)$ is as the weighted sum of densities on a given point $x$. With that said think about the following example:
You have a Normal distribution centred around $-1$ and a Normal distribution centred around $1$. Now someone tells you that a point $x$ is likely to belong to the first normal distribution with probability $\pi =0.2$ then the probability for the point to belong to the second is $1-\pi =0.8$. With that in mind then someone ask you what is the density on the point $x=0$, this naturally will be the sum of both normal distributions on the point $x=0$. But is it only that? You have the information that the second is 4 times more likely than the first one. So, you would expect the density of the second normal distribution to contribute 4 times more than the first one. Hence, you will have that the density on a particular point $x=0$ will be $p(0) = 0.2 * Norm(0;-1,\sigma) + 0.8*Norm(0;1,\sigma)$
And again is not a probability is a density, in discrete case this changes. Hope that helps
