Deep learning methods for communications over unknown nonlinear channels have attracted considerable interest recently. In this paper, we consider semi-supervised learning methods, which are based on variational inference, for decoding unknown nonlinear channels. These methods, which include Monte Carlo expectation maximization and a variational autoencoder, make efficient use of few pilot symbols and the payload data. The best semi-supervised learning results are achieved with a variational autoencoder. For sufficiently many payload symbols, the variational autoencoder also has lower error rate compared to meta learning that uses the pilot data of the present as well as previous transmission blocks.