Variational autoencoders and Helmholtz machines use a recognition network (encoder) to approximate the posterior distribution of a generative model (decoder). In this paper we establish some necessary and some 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 perfectness property (for every node, all parents are joined) plays an important role.

In 1986, Tomaszewski made the following conjecture. Given $n$ real numbers $a_{1},\ldots,a_{n}$ with $\sum_{i=1}^{n}a_{i}^{2}=1$, then of the $2^{n}$ signed sums $\pm a_{1} \pm \cdots \pm a_{n}$, at least half have absolute value at most 1. Hendriks and van Zuijlen [ An Improvement of the Boppana-Holzman Bound for Rademacher Random Variables}, arXiv:2003.02588, 2020] and Boppana, Hendriks, and van Zuijlen [ Tomaszewski's Problem on Randomly Signed Sums, Revisited, arXiv:2003.06433, 2020] independently proved that a proportion of at least 0.4276 of these sums has absolute value at most 1. Using different techniques, we improve this bound to 0.46.