Understanding conditional probability formulas in the context of class-conditionals in generative models I am trying to understand the theory behind probabilistic generative models a bit better.
If I model the class-conditionals as Gaussians, the formula is this:
$$
   \frac{1}{2\pi^{\frac{D}{2}}|\Sigma|^{\frac{1}{2}}}\exp\left(-\frac{1}{2}({x}-{\mu_k)^T{\Sigma}({x}-{\mu_k})}\right)
$$
My understanding is this is a conditional probability because $ \mu_k $ is used in the formula, a separate mean for each class, which is a random variable. So the outcome is now conditional on that random value.
And the $ \mu_k $ is the only notational difference between the above formula and the one for the non-conditional multivariate Gaussian for $ p(x) $.
Something similar is explained here:
https://towardsdatascience.com/3-conditionals-every-data-scientist-should-know-1916d48b078a
for the poisson, where it says that the non-conditional is $ p(y) = \frac{e^{-\lambda }\lambda ^y}{y!} $,  but if $ \lambda $ itself depends on another random variable $X$, it now becomes a conditional of the form $ p(y|x) = \frac{e^{-f(X=x) }\lambda ^y}{y!} $
I understand that, when a parameter of a probability distribution depends on another random variable, we now have a conditional distribution.
However, I don't know how to relate the above two formulas for the conditional distributions to this one: $ p(x|y) = \frac{p(x,y)}{p(y)} $.
What are $p(x,y)$ and $p(y)$ in these cases?
I don't even know what theory to look at because the explanations for conditional probability are usually simple coin-toss examples. And the machine learning literature I have looked at just assumes that the reader knows this.
Simple, possibly step-by-step help is much appreciated!

In your multivariate normal example write the class-conditional multivariate normal density for $X$ as $f(x; \mu_k, \Sigma)$  or shorter as $f_k(x)$. Let the class $K$ be a random variable with distribution $\DeclareMathOperator{\P}{\mathbb{P}}  \P(K=k)=\pi_k$.  The unconditional distribution of $X$ is then a mixture distribution with the density of $X$ given by
$$ f(x)=\sum_k \pi_k f(x; \mu_k, \Sigma)  $$
Relating this to the formula you cite (with $y$ replaced by $k$ to conform to this example), that is,
$$ p(x \mid k)= \frac{p(x,k)}{p(k)} $$ and further, for the example, replace $p$ with $f$).
Then we have:
\begin{align}
 p(k)      &= \pi_k \\
 p(x,k)    &= \pi_k \cdot f(x; \mu_k,\Sigma) \\
 p(x\mid k)&= f(x; \mu_k,\Sigma)
\end{align}
Maybe one problem relating to this is that most text-book examples of jiunt distributions are either joint discrete or joint continuous, while this is an example of a mixed distribution, $X$ is continuous while $K$ is discrete.
