Consider a training set of multivariate input/output process data. Given a new observation we ask the following questions: is the new observation normal or abnormal? is one of the inputs or outputs abnormal (faulty) and which? For a linear Gaussian model of the process, the problem is solved by Bayesian hypothesis testing. The formulation differs from existing multivariate statistical monitoring methods by taking variance (uncertainty) of the linear regression model into account. In the limit case of zero model variance, the proposed method matches the established methods for anomaly detection and fault isolation. The proposed method might yield an order of magnitude reduction in fault isolation errors compared to the established approaches when regression models have large variance. This is the case for ill-conditioned multivariate regression models even with large training data sets. The paper also shows that isolating faults to a small ambiguity group works much better than trying to isolate a single fault. The proposed method is verified in a Monte Carlo study and in application to jet engine fault isolation.