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In a variational autoencoder (VAE), the encoder and decoder form a probabilistic autoencoding pipeline around a latent space. The encoder takes an input data point x (e.g., an image) and maps it into a probability distribution over latent variables z, rather than a single latent vector. This latent distribution is typically parameterized (commonly as a Gaussian) by neural networks that output parameters such as the mean and (optionally) variance. The latent space is the probabilistic bottleneck: sampling from the encoder’s distribution produces a latent sample z. The decoder then maps from this latent space back to the input space, producing a reconstructed output x′ that is probabilistically modeled to be as similar as possible to the original x. Training is done jointly by optimizing an objective that balances (1) reconstruction quality (how close x′ is to x) and (2) regularization that aligns the encoder’s approximate posterior with a chosen prior over latents. This is commonly handled via the evidence lower bound (ELBO), and gradients through sampling are enabled using the reparameterization trick.
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In a variational autoencoder (VAE), probabilistic modeling describes how observed data x is generated from a latent variable z. The prior pθ(z) represents assumptions about latent variables before observing data, commonly a standard multivariate Gaussian. The likelihood pθ(x|z) describes the probability of observing x given z and is modeled by the probabilistic decoder, often as a Gaussian distribution centered on the decoder output. Together, they define the joint distribution pθ(x,z) = pθ(x|z)pθ(z), and the data likelihood is obtained by marginalizing over the latent variables: pθ(x) = ∫ pθ(x|z)pθ(z) dz.
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