This is a recurrent problem I'm having difficulties addressing.
Suppose I have some trained and ready to go Gaussian mixture model with $k$ clusters $c_1 c_2 c_3... c_k$ of known means and variances. Now suppose I used this model to generate new data (after all it is a generative model. I learned the probability distribution, so I can sample from it).
I used the GMM to get some new example, new information $v \in \mathbb R^d$. I would like to know how well does it fit the model, or in other words $P(v)$.
But from the model description $P(v) = \sum_{i=1}^{k} P(v| c_i) P(c_i)$ where $P(v|c_i) = \frac{1}{\sqrt{2^d \pi ^d |\Sigma_i|}}e^{-\frac{1}{2}(v - \mu _i)^T\Sigma_i ^{-1}(v - \mu _i)}$ from the definition of the model as a gaussian mixture model.
My problem is that $P(v|c_i)$ is not a probability. It is a density. It can be greater than $1$. And so in turn, $P(v)$ is also not a probability, but a density. And can also be greater than one. So the question "What is the probability that our model will generate the example $v$" is not a logical question.
But I need to answer that question regardless. I need to give some meaning to $P(v)$.
This also occurs in hidden markov models. Same problem. Suppose my machine is in state $1$ and i want to know what's the probability that it will output "Hello".
The question "What is the probability?" is meaningless, since the set of all possible outputs for my machine is infinite. I fall into the same pitfall as with GMM where I have to go with density, or just say "its zero. the probability is always zero. Always."
How do we handle this situation? is it ok to just imagine densities as probabilities?
Firstly, it should be noted that there are many ways to model a process with infinitely many outcomes that have positive probabilities for individual outcomes. So perhaps, the real answer to your question is to model your process or system differently.
Assuming you have an appropriate model, then there is no reason to talk about the probability of any given outcome. But you can still be concerned with the relative probability of being near certain outcomes. In particular, if you have random variable $X$ with density $f$ and two points $x$ and $y$ with $f(x)=9$ and $f(y)=3$ you can say that you are 3 times as likely to get an outcome near $x$ then near $y$. In this case, near can be thought of in the following sense $$\frac{f(x)}{f(y)}=\frac{\lim_{\epsilon \to 0}\mathbb{P}(x-\epsilon \leq X \leq x+\epsilon)}{\lim_{\epsilon \to 0}\mathbb{P}(y-\epsilon \leq X \leq y+\epsilon)}.$$ In this way you can talk about relative probabilities even if you can't talk about actual probabilities.