A Bayesian solution is derived for digital communication channel equalization with decision feedback. This is an extension to the maximum a posteriori probability symbol-decision equalizer to include decision feedback. A novel scheme of utilizing decision feedback is proposed which not only improves equalization performance but also reduces computational complexity dramatically. It is shown that the Bayesian equalizer has an equivalent structure to the radial basis function network, the latter being a one-hidden-layer artificial neural network widely used in pattern classification and many other areas of signal processing. Two adaptive approaches are developed to realize the Bayesian solution. The maximum likelihood Viterbi algorithm and the conventional decision feedback equalizer are used as two benchmarks to assess the performance of the Bayesian decision feedback equalizer.