Reparameterization Trick
Definition
The reparameterization trick expresses a sample from $q_\phi(z)$ as a deterministic transformation of parameter-free noise:
$$z = T(\epsilon,\phi),\qquad \epsilon\sim p_0(\epsilon).$$For a Gaussian,
$$z=\mu+\sigma\epsilon,\qquad \epsilon\sim\mathcal N(0,1).$$Why it matters
It makes the randomness independent of $\phi$, allowing gradients of the ELBO to pass through the sampling step.