I'm learning about VAEs, and need to go this deep to understand them. However, the question is for Bayes theorem with probability distributions. I learnt about Bayes theorem from this video. Excellent explanation, with a simple example. There, are 4 events, a person is librarian ($P(H)$), or not ($P(\lnot H)$), and a person matches the description ($P(E)$), or not ($P(\lnot E)$). We want to know the probability of a person is librarian, given it matches the description. And we're calculating it in a nice visual way. It helps a lot that the probabilities are single numbers.
It is a lot harder to understand this with probability distributions, and I'm having a hard time trying to make up a real world example.
Or is it right to think about $P(H)$, and $P(\lnot H)$ together as a discrete probability distribution where on the x axis 0 is $P(\lnot H)$, and 1 is $P(H)$, and same with $E$?
In VAEs $p(z)$, and $q(z|x)$ are both normal probability distributions, and I can't really imagine a real world example similar.
Say example $p(z)$ is the probability of outcomes if two fair dices are rolled as in the picture:
What could $p(x|z)$ could be as a probability distribution? Say $z$ (the value of outcome) is 6, and we're checking the probability of one of the cubes is $x$ (6 possibilities).
However, in the VAE example, we know $x$ (in fact, that's the only thing we know for sure, and $p(z)$. Say, following for the above example, what's the probability of one of the cubes has the value of 6 ($x$), given ($z$).
Are those correct examples? If not, what could be an understandable, real world example, representing a VAE kind of situation with probabilities?