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Matrix factorization (MF) has been widely applied to collaborative filtering in recommendation systems. Its Bayesian variants can derive posterior distributions of user and item embeddings, and are more robust to sparse ratings. However, the Bayesian methods are restricted by their update rules for the posterior parameters due to the conjugacy of the priors and the likelihood. Variational autoencoders (VAE) can address this issue by capturing complex mappings between the posterior parameters and the data. However, current research on VAEs for collaborative filtering only considers the mappings based on the explicit data information while the implicit embedding information is overlooked. In this paper, we first derive evidence lower bounds (ELBO) for Bayesian MF models from two viewpoints: user-oriented and item-oriented. Based on the ELBOs, we propose a VAE-based Bayesian MF framework. It leverages not only the data but also the embedding information to approximate the user-item joint distribution. As suggested by the ELBOs, the approximation is iterative with cross feedback of user and item embeddings into each other's encoders. More specifically, user embeddings sampled at the previous iteration are fed to the item-side encoders to estimate the posterior parameters for the item embeddings at the current iteration, and vice versa. The estimation also attends to the cross-fed embeddings to further exploit useful information. The decoder then reconstructs the data via the matrix factorization over the currently re-sampled user and item embeddings.