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.

翻译：变分自编码器和赫尔姆霍茨机使用识别网络（编码器）来近似生成模型（解码器）的后验分布。本文研究了识别网络能够精确建模真实后验分布的充分必要条件。这些结果是在概率图模型/贝叶斯网络的通用框架下推导得出的，其中网络表示一组条件独立性陈述。我们推导了识别网络具备所需性质的全局条件（基于d-分离）和局部条件。研究表明，对于局部条件而言，完美性（每个节点的所有父节点均相互连接）这一性质起着关键作用。