We present techniques for constructing classifiers that combine statistical information from training data with tangent approximations to known transformations, and we demonstrate the techniques by applying them to a face recognition task. Our approach is to build Bayes classifiers with approximate class-conditional probability densities for measured data. The high dimension of the measurements in modern classification problems such as speech or image recognition makes inferring probability densities from feasibly sized training data sets difficult. We address the difficulty by imposing severely simplifying assumptions and exploiting a priori information about transformations to which classification should be invariant. For the face recognition task, we used a five parameter group of such transformations consisting of rotation, shifts, and scalings. On the face recognition task, a classifier based on our techniques has an error rate that is 20% lower than that of the best algorithm in a reference software distribution.