I know the encoder is variational posterior $q_{\phi}(\mathbf{z} \mid \mathbf{x})$.
I also know that the decoder represents the likelihood: $p_{\theta}(\mathbf{x} \mid \mathbf{z})$.
My question is about the prior $\mathrm{p}(\mathbf{z})$.
I know ELBO can be written as:
$$E_{q_{\phi}(\mathbf{z} \mid \mathbf{x})}[\log (p_{\theta}(\mathbf{x} \mid \mathbf{z}))]-\mathrm{D}_{\mathrm{KL}}( q_{\phi}(\mathbf{z} \mid \mathbf{x}) \| \mathrm{p}(\mathbf{z})) \leq \log (p_{\theta}( \mathbf{x}))$$
And for the VAE, the variational posterior is
$$ q_{\boldsymbol{\phi}}(\mathbf{z} \mid \mathbf{x}^{(i)})= \mathcal{N}( \boldsymbol{\mu}^{(i)}, \boldsymbol{\sigma}^{2(i)} \mathbf{I}),$$
and prior is
$$ \mathrm{p}(\mathbf{z})=\mathcal{N}( \boldsymbol{0}, \mathbf{I}).$$
So
$$\mathrm{D}_{\mathrm{KL}}\left(\mathrm{q}_{\Phi}(\mathbf{z} \mid \mathbf{x}) \| p_{z}(\mathbf{z})\right)=\frac{1}{2} \sum_{j=1}^{J}\left(1+\log \left(\sigma_{j}^{2}\right)-\sigma_{j}^{2}-\mu_{j}^{2}\right)$$
That's one way I know the prior plays a role, in helping determine part of the loss function.
Is there any other role that the prior plays for the VAE?