I am relatively new to higher-level statistics and I just cannot seem to wrap my head around this. I am trying to derive ELBO for a Variational Autoencoder. Here is the formula for the PDF of a Multivariate Normal Distribution I was given:
$$p(a) = (det(2 \pi \Sigma))^{\frac{-1}{2}}exp(-\frac{1}{2}(a-\mu)^T\Sigma^{-1}(a-\mu))$$
I'm then told to derive $$p(z,x) = p(z)p(x|z)$$ where $$p(z) = N(z;0,I)$$ and $$p(x|z) = Bernoulli(x; f(x))$$
In this model, x is a fixed datapoint in a dataset I am trying to maximize the log-likelihood of. Z is the set of latent space variable for that datapoint.
If the mean is $0$, then the result of the left-hand determinant section is undefined, because $0^{-\frac{1}{2}}$ is undefined. Am I missing something?