In the paper Tutorial on Variational Autoencoders, the authors state that
Since $P(X\mid z)$ is an isotropic Gaussian, the negative log probability of $X$ is proportional squared Euclidean distance between $f(z)$ and $X$.
According to the paper, $$P(X) = \int P(X\mid z; \theta)P(z)\,\mathrm dz,$$ and $$P(X\mid z; \theta) = \mathcal N (X\mid f(z; \theta), \sigma^2 ∗ I).$$
I don't see the connection here in proving this.