Bayes' theorem in Gaussian mixture model for $p(z_i = k | x_i, \mu_k,\Sigma_k)$

Question

Given the $k$th Gaussian distribution $N \sim (\mu_k, \Sigma_k)$, the probability that $x_i$ generated from this Gaussian $k$ can be found via Bayes' rule $$\begin{align}p(z_i = k | x_i,\mu_k, \Sigma_k) &= \frac{p(x_i,z_i =k)}{p(x)} \\ &= \frac{\pi_kN(x_i|\mu_k,\Sigma_k)}{\sum_{k=1}^m\pi_kN(x_k|\mu_k,\Sigma_k)}\end{align}$$ where $p(x,z_i=k)$ is the joint probability density distribution while $p(x)$ is the marginal distribution over the mixture of Gaussians.

Bayes' theorem in machine learning is applied in the following manner, when estimating the posterior of the model parameters $\theta$, $$p(\theta|D) = \frac{p(\theta)p(D|\theta)}{\int p(D|\theta)p(\theta)d\theta}$$ In this case $p(D|\theta)$ is a conditional probability because $\theta$ is a random variable.

1. why is it the case that $N(x_i|\mu_k,\Sigma_k)$ is not a conditional probability but can still be used in Bayes' theorem ?
2. Is the numerator in Bayes' theorem a distribution or a discrete probability? When is it the case where it is a distribution and when is it the case where the numerator is a probability. I know that $p(\theta)p(D|\theta)$ is a distribution over $\theta$ and $\pi_kN(x_i|\mu_k,\Sigma_k)$ is also the joint distribution.

Answer 1

1. why is it the case that $N(x_i|\mu_k,\Sigma_k)$ is not a conditional probability but can still be used in Bayes' theorem ?

$N(x_i|\mu_k,\Sigma_k)$ is a conditional probability density function. It is conditional on cluster assignment $z_i = k$. High-level formula is

$$ p(k | X) \propto p(X|k)\, p(k) $$

2. Is the numerator in Bayes' theorem a distribution or a discrete probability? When is it the case where it is a distribution and when is it the case where the numerator is a probability. I know that $p(\theta)p(D|\theta)$ is a distribution over $\theta$ and $\pi_kN(x_i|\mu_k,\Sigma_k)$ is also the joint distribution.

$p(z_i=k) = \pi_k$ is a Bernoulli distribution (discrete), while $p(x_i|z_i=k) = N(x_i|\mu_k,\Sigma_k)$ is a probability density function, and the result is a probability density.