Using Bayes's theorem, we derive a unit-wise recurrence as well as a backward recursion similar to the forward-backward algorithm. The resulting Bayesian recurrent units can be integrated as recurrent neural networks within deep learning frameworks, while retaining a probabilistic interpretation from the direct correspondence with hidden Markov models. Whilst the contribution is mainly theoretical, experiments on speech recognition indicate that adding the derived units at the end of state-of-the-art recurrent architectures can improve the performance at a very low cost in terms of trainable parameters.