  Variational autoencoders and Helmholtz machines use a recognition network
(encoder) to approximate the posterior distribution of a generative model
(decoder). In this paper we study the necessary and sufficient properties of a
recognition network so that it can model the true posterior distribution
exactly. These results are derived in the general context of probabilistic
graphical modelling / Bayesian networks, for which the network represents a set
of conditional independence statements. We derive both global conditions, in
terms of d-separation, and local conditions for the recognition network to have
the desired qualities. It turns out that for the local conditions the property
perfectness (for every node, all parents are joined) plays an important role.
