CVAE Problem Setup

This note isolates the basic factorization target and the role of each model component.

Problem Setup

We assume an input-conditioned generation task with:

input $x$ ,
latent strategy $z$ ,
output sequence $y = (y_1, \dots, y_T)$ .

The frozen entangled base model is:

p_\theta(y \mid x).

The target factorization is:

discrete $z$ :

p(y \mid x) = \sum_{z=1}^{K} p_\psi(y \mid x, z)\, p_\phi(z \mid x),

continuous $z$ :

p(y \mid x) = \int p_\psi(y \mid x, z)\, p_\phi(z \mid x)\, dz.

Interpretation:

$p_\phi(z \mid x)$ is a strategy router,
$p_\psi(y \mid x, z)$ is a strategy-conditioned decoder,
$q_\xi(z \mid x, y)$ is a variational inference model used during training.

Training data for post-training is drawn from the frozen base distribution:

(x, y) \sim \mathcal{D}, \qquad \mathcal{D}(x, y) = \mathcal{D}_x(x)\, p_\theta(y \mid x).

Why this setup matters

The methodology is not about learning $p(y \mid x)$ from scratch. It is about refactoring an already strong entangled model so that the latent variable can act as a reusable high-level strategy handle.

Next: CVAE model components