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A Bayesian approach to non-intrusive quality assessment of narrow-band speech is presented. The speech features used to assess quality are the sample mean and variance of band-powers evaluated from the temporal envelope in the channels of an auditory filter-bank. Bayesian multivariate adaptive regression splines (BMARS) is used to map features into quality ratings. The proposed combination of features and regression method leads to a high performance quality assessment algorithm that learns efficiently from a small amount of training data and avoids overfitting. Use of the Bayesian approach also allows the derivation of credible intervals on the model predictions, which provide a quantitative measure of model confidence and can be used to identify the need for complementing the training databases.