 Recognition network

A recognition network is a learned mapping from observed data to parameters of an approximate posterior distribution over latent variables. In a variational autoencoder, it specifies $q_{\varphi }(z\mid x)$, while the generative decoder specifies the opposite conditional direction $p_{\theta }(x\mid z)$. The recognition network supports amortized inference; it need not coincide with the exact posterior of the learned generative model.^[ref]

For a scalar Gaussian recognition law, the neural network can output a mean ${\mu }_{\varphi }(x)$ and log variance ${\ell }_{\varphi }(x)$. Thenwhere $\epsilon \sim N(0,1)$ is parameter-independent noise. Exponentiating the log variance ensures a positive variance, and the factor $1/2$ converts it to the standard deviation used in Gaussian reparameterization.^[ref]

For an original calculation, suppose one input gives $\mu =1$ and $\ell =\log 4$. The recognition variance is4 and its standard deviation is2. A noise draw $\epsilon =1/2$ gives $z=2$, while the posterior mean supplied by the same network remains1. Confusing log variance with log standard deviation would instead produce the wrong scale.

A deterministic network evaluation and a random latent sample are distinct steps. Re-evaluating the same fixed network on the same input returns the same distribution parameters; new noise draws can still yield different $z$ values. Returning only the mean omits the uncertainty represented by that distribution. Its adequacy must be assessed against the intended inference problem, not inferred merely from the fact that it outputs a positive variance.