#reparameterization-trick

Question 1: Single choice

A VAE encoder outputs $\mu_\phi(x)$ and $\sigma_\phi(x)$ , and the loss contains $\mathbb{E}_{z \sim q_\phi(z \mid x)}[\log p_\theta(x \mid z)]$ , estimated with a sampled $z$ . Why is the sample written as $z = \mu_\phi(x) + \sigma_\phi(x) \odot \epsilon$ with $\epsilon \sim \mathcal{N}(0, I)$ ?

Without it the decoder parameters $\theta$ would get no gradient from the reconstruction term1
Sampling $z$ this way is what makes the KL term to the prior $\mathcal{N}(0, I)$ computable in closed form2
The noise moves into $\epsilon$ , so $z$ is differentiable in $\phi$ and gradients reach the encoder3
It removes sampling from training, since $\epsilon$ is drawn once and then kept fixed4

#reparameterization-trick

Timed test