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The posterior pθ(z|x) is the probability distribution of a latent representation given an observed data point; it is derived using Bayes' rule from the prior and likelihood but is usually expensive or intractable to compute directly.
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.
The posterior pθ(z|x) is the probability distribution of a latent representation given an observed data point; it is derived using Bayes' rule from the prior and likelihood but is usually expensive or intractable to compute directly.
VAEs approximate the true posterior with an encoder-defined distribution qϕ(z|x), enabling efficient amortized inference and training by maximizing the evidence lower bound (ELBO).
The prior pθ(z) is the probability distribution assigned to latent variables before observing the data.
The likelihood pθ(x|z) is the probability of observed data x conditional on a latent variable z, typically computed by the decoder.
The posterior pθ(z|x) is the updated probability distribution of latent variables after observing x.
The distribution qϕ(z|x) is an encoder-based approximation to the usually intractable true posterior pθ(z|x).
Bayes' rule relates the posterior to the likelihood and prior through pθ(z|x) = pθ(x|z)pθ(z) / pθ(x).
The evidence pθ(x) is the probability of the observed data after integrating the joint distribution over all possible latent variables.
“Can you explain what "The posterior pθ(z|x) is the probability distribution of a latent representation given an observed data point; it is derived using Bayes' rule from the prior and likelihood but is usually expensive or intractable to compute directly." means in simple terms?”