Loss Function

reconstruction loss

KL Divergence

Compares the Posterior ($q(z|x)$; $z$ latent variables given $x$ inputs) and Prior distributions

Prior

The prior distribution typically does not depend directly on the input x. Instead, the prior is usually chosen to be a standard distribution that is fixed before training and does not change with the input. The most common choice for the prior is a standard normal or gaussian distribution, $N(0,I)$, where $\mu$= 0is a zero mean vector and $I$ is the identity matrix as the covariance, indicating that the latent variables are independent and identically distributed.

1. Why a Gaussian Distribution?
2. Training and inference computation are more tractable.
3. Facilitates application of the reparametrization trick and calculation of KL divergence

4. Regularization - comes from the KL Divergence loss term which penalizes deviations from the prior distribution. Thus the organizes the latent space that effectively prevent overfitting.

5. Why fixed prior that does not adapt to specific inputs?

6. forces the model to learn the encoded distributions to resemble the prior ($N(0,I)$; covariance $I$ is an identity matrix, ensuring the latent variables are independent and identically distributed) and reconstruct the input variables accurately.

7. Advanced Models have conditional priors - how do they work?

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### Math
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