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A Bayesian solution is derived for digital communication channel equalization with decision feedback. This is an extension of the maximum a posteriori probability symbol-decision equalizer to include decision feedback. A novel scheme utilizing decision feedback that not only improves equalization performance but also reduces computational complexity greatly is proposed. It is shown that the Bayesian equalizer has a structure equivalent to that of 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 asses the performance of the Bayesian decision feedback equalizer.< >